ontransportpropertiesof majoranafermionsin superconductors ... · topological superconductors are...

On transport properties ofMajorana fermions in

superconductors:free & interacting

Proefschrift

ter verkrijging vande graad van Doctor aan de Universiteit Leiden,

op gezag van Rector Magnificus prof. mr. C.J.J.M. Stolker,volgens besluit van het College voor Promoties

te verdedigen op woensdag 12 Juni 2019klokke 15.00 uur

door

Nikolay Vladislavovich Gnezdilov

geboren te Novosibirsk, Rusland in 1991

Promotores: Prof. dr. C.W. J. BeenakkerProf. dr. S. I. Mukhin (National University of Science

and Technology, Moscow, Russia)

Promotiecommissie: Dr. A.R. Akhmerov (TU Delft)Prof. dr. A. Brinkman (Universiteit Twente)Dr. A. BoyarskiProf. dr. E. R. ElielProf. dr. H. Schiessel

Casimir PhD Series Delft-Leiden 2019-19ISBN 978-90-8593-405-9An electronic version of this thesis can be foundat https://openaccess.leidenuniv.nl

Front cover: Spectral function of the SYK impurity lattice with the linesof zeros. Chiral Majorana mode propagating along the edge of a two-dimensional topological superconductor, with a graphene fragment in thelower left corner. Back cover: Tunneling spectroscopy of a black hole chip(SYK quantum dot), on the background of a graph of the low-high voltageduality derived in this thesis. The author’s name is tilted representing the“rotating” Holstein-Primakoff approach proposed in this thesis.

https://openaccess.leidenuniv.nl

To my parents.Ìîèì ðîäèòåëÿì.

Contents

1 Introduction 1

1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Electrical signatures of Majorana surface states . . . . . . . 41.3 The Sachdev-Ye-Kitaev model in solid state systems . . . . 61.4 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . 101.4.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 101.4.3 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . 111.4.4 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . 111.4.5 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . 121.4.6 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . 121.4.7 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Wiedemann-Franz-type relation between shot noise and

thermal conduction of Majorana surface states in a three-

dimensional topological superconductor 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Surface-sensitive thermal and electrical conduction . . . . . 17

2.2.1 Description of the geometry . . . . . . . . . . . . . . 172.2.2 Scattering formulas . . . . . . . . . . . . . . . . . . . 182.2.3 Electron-hole symmetry enforced upper bound on

the shot noise power . . . . . . . . . . . . . . . . . . 182.3 Combined effects of electron-hole and time-reversal symme-

tries on the shot noise power . . . . . . . . . . . . . . . . . 192.3.1 Surface Hamiltonian with tunnel coupling to metal

contacts . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.2 Condition on the coupling matrix for equality of

shot noise and thermal conductance . . . . . . . . . 21

vi CONTENTS

2.4 Formulation and solution of the surface scattering problem 222.4.1 Reduction to an effectively 2D geometry . . . . . . . 222.4.2 Single-surface transmission matrix . . . . . . . . . . 232.4.3 Transmission matrix for coupled top and bottom

surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 242.5 Thermal conductance and corresponding shot noise power . 26

2.5.1 Single surface . . . . . . . . . . . . . . . . . . . . . . 262.5.2 Coupled surfaces . . . . . . . . . . . . . . . . . . . . 272.5.3 Finite aspect ratio . . . . . . . . . . . . . . . . . . . 282.5.4 Locality condition . . . . . . . . . . . . . . . . . . . 29

2.6 Numerical solution of the full 3D scattering problem . . . . 302.6.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . 302.6.2 Translationally invariant system . . . . . . . . . . . 312.6.3 Disorder effects . . . . . . . . . . . . . . . . . . . . . 33

2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.8 Appendix: Matrix Green’s function of the surface Hamil-

tonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.8.1 Single surface . . . . . . . . . . . . . . . . . . . . . . 362.8.2 Coupled top and bottom surfaces . . . . . . . . . . . 37

3 Topologically protected charge transfer along the edge of

a chiral p-wave superconductor 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Full counting statistics of Majorana edge states . . . . . . . 41

3.2.1 Finite temperature . . . . . . . . . . . . . . . . . . . 433.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4 Appendix: Effect of a tunnel barrier at the NS contact . . . 47

4 Valley switch in a graphene superlattice due to pseudo-

Andreev reection 49

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Graphene superlattice with anti-unitary symmetry . . . . . 504.3 Topological phase transitions . . . . . . . . . . . . . . . . . 524.4 Valley switch . . . . . . . . . . . . . . . . . . . . . . . . . . 544.5 Robustness of the valley switch . . . . . . . . . . . . . . . . 554.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.7 Appendix: Scattering matrix of the graphene superlattice . 57

CONTENTS vii

5 Low-high voltage duality in tunneling spectroscopy of the

Sachdev-Ye-Kitaev model 59

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Tunneling Hamiltonian . . . . . . . . . . . . . . . . . . . . . 615.3 Tunneling current . . . . . . . . . . . . . . . . . . . . . . . . 625.4 Low–high voltage duality . . . . . . . . . . . . . . . . . . . 635.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.6 Appendix: Outline of the calculation . . . . . . . . . . . . . 65

5.6.1 Generating function of counting statistics . . . . . . 655.6.2 Saddle point solution . . . . . . . . . . . . . . . . . . 665.6.3 Average current and shot noise power . . . . . . . . 67

6 Isolated zeros in the spectral function as signature of a

quantum continuum 69

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.2 Isolated zeros in the spectral function . . . . . . . . . . . . 716.3 SYK model . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.4 Cluster of SYK nodes . . . . . . . . . . . . . . . . . . . . . 756.5 Holographic fermions . . . . . . . . . . . . . . . . . . . . . . 776.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.7 Appendix: Outline of the calculation . . . . . . . . . . . . . 80

6.7.1 Inclusion of general coupling between discrete levels 806.7.2 Mean-field treatment of the SYK model with two

component impurity . . . . . . . . . . . . . . . . . . 816.7.3 Multiple states coupled to the SYK continuum . . . 836.7.4 Effective momentum coupling by the SYK chain . . 85

7 Gapless odd-frequency superconductivity induced by the

Sachdev-Ye-Kitaev model 87

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.3 SYK proximity effect . . . . . . . . . . . . . . . . . . . . . . 897.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.5 Appendix: Gor’kov Green’s function for the QD variables . 94

8 First-order dipolar phase transition in the Dicke model

with innitely coordinated frustrating interaction 99

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.2 Dicke Hamiltonian for a Cooper pair boxes array . . . . . . 101

viii CONTENTS

8.3 Diagonalization of the frustrated Dicke model . . . . . . . . 1048.3.1 Tunneling regime . . . . . . . . . . . . . . . . . . . . 1048.3.2 Rotating Holstein-Primakoff representation . . . . . 1068.3.3 Superradiant dipolar regime . . . . . . . . . . . . . . 108

8.4 First order dipolar phase transition . . . . . . . . . . . . . . 1118.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.6 Appendix: Bogoliubov’s transformation for the frustrated

Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.7 Appendix: Quantum phase transition in the Dicke model . 119

Bibliography 123

Summary 139

Samenvatting 143

Curriculum Vitæ 147

List of Publications 149

Chapter 1

Introduction

1.1 Preface

Majorana fermions are charge neutral fermionic modes that equal theirown antiparticles [1] and originate from a real solution of the complexDirac equation [2]. Those recently attracted a lot of attention in thecontext of subgap states in superconductors [3–11]. In superconductingsystems, oppositely charged electron and hole quasiparticle excitationscan be naturally thought as particle-antiparticle partners. The particle-hole symmetry relates the creation operator of the quasiparticle excitationγ†E at energy E to its annihilation operator γ−E at the opposite energy.The p-wave superconducting pairing allows for the subgap solutions ofthe Bogoliubov-De Gennes (BdG) Hamiltonian at E = 0 on vortices intwo dimensions [3–5, 8] or as the bound states at the ends of a one-dimensional nanowire [6, 9], which makes the superconductor topologicallynon-trivial [10, 12]. These solutions realize Majorana zero modes γ†0 = γ0(often referred to as Majoranas) that obeys the anticommutation relationγi, γj = 2δij . It follows from the reality of Majorana zero modes (γ†i =γi) that two of those are needed to encode a single complex fermion, sothat each Majorana enters as an equal-weight superposition of the electronand hole excitations. This results in 2N degeneracy of the ground statedescribed by 2N Majoranas, non-Abelian exchange statistic of the latterones [4, 5, 13], and makes them especially attractive in the perspective oftopological quantum computation [6, 14, 15].

Once Majorana zero modes are stacked together with sufficient overlap,they form a subgap chiral edge Majorana mode. The left panel in Fig.

2 Chapter 1. Introduction

Figure 1.1. Left panel: A stack of superconducting nanowires [6] that form adispersive (see Fig. 1.2) Majorana subgap edge state. The figure is taken fromM. Diez, I. C. Fulga, D. I. Pikulin, J. Tworzydło, and C. W. J. Beenakker, NewJournal of Physics 16, 063049 (2014). Right panel: 3D topological insulator inproximity to superconductor and ferromagnet hosts Majorana edge mode. Thefigure is reprinted with permission from A. R. Akhmerov, Johan Nilsson, and C.W. J. Beenakker, Phys. Rev. Lett. 102, 216404 (2009). Copyright 2009 by theAmerican Physical Society.

Figure 1.2. Dispersion of a chiral p-wave superconductor drawn from the tight-binding model defined on the infinite stripe geometry (lattice constant a). Thebulk of the system is gapped, while the Majorana edge states have the lineardispersion with a slope proportional to the bulk gap ∆0.

1.1 schematically shows an array of p-wave nanowires [6] that hosts aMajorana edge state. Also, a surface of a three-dimensional topologicalinsulator [10, 16–18] in proximity to superconductor and ferromagnet ispredicted to form a Majorana edge mode along the boundary between aferromagnet and superconductor [19, 20], as it is shown in the right panel

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1.1 Preface 3

of Fig. 1.1. Moreover, Majorana edge modes at the interface betweenquantum anomalous Hall insulator and an s-wave superconductor [21]have been recently reported to be observed in experiment [22].

Chiral Majorana edge states in p-wave superconductors with a spin-triplet px + ipy pair potential [4, 7], whose dispersion is shown in Fig1.2, can be thought as a superconducting analog of chiral edge modesof the quantum Hall effect. However, because Majorana fermions arecharge neutral, quantized electrical signatures of the Majorana edge modeare lacking. On the other hand, a temperature gradient is predicted todrive a heat current along the edge carried by Majorana fermions, sothat the thermal conductance G is quantized at the electronic quantumG0 = π2k2

BT/3h times one-half. 1 Study of electrical alternatives to theheat conductance measurements is the goal of Chapters 2 and 3.

3D TI

SC

B

Figure 1.3. Left panel: An array of the superconducting nanowires in themagnetic field strongly coupled through a disordered quantum dot. The figure isreprinted with permission from Aaron Chew, Andrew Essin, and Jason Alicea,Phys. Rev. B 96, 121119(R) (2017). Copyright 2017 by the American PhysicalSociety. Right panel: Disordered surface of 3D TI in proximity to a s-wavesuperconductor in the magnetic field. The figure is reprinted with permissionfrom D. I. Pikulin and M. Franz, Phys. Rev. X 7, 031006 (2017). Copyright2017 by the American Physical Society.

The edge states of topological superconductors discussed above aredescribed by the low energy non-interacting theories. In contrast, thesecond part of the thesis is devoted to the strongly interacting Majoranazero-modes and to the Sachdev-Ye-Kitaev (SYK) model in particular [27,28]. The interactions are supposed to be so strong, that the quasiparticledescription of the model is no longer possible. However, the SYK model isexactly solvable in the large N limit and establishes connections between

1 The thermal conductance G = 1/2 × G0 is due to the central charge c = 1/2 ofthe field theory of Majorana edge mode [4, 23–26].

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non-Fermi liquids, black hole horizons, and many-body quantum chaos[27, 29, 30].

The SYK model can be realized as a low energy theory of an arrayof superconducting nanowires in the topological phase [6, 9] once thoseare strongly coupled through a disordered quantum dot in the magneticfield oriented in the plane of the quantum dot [31]. Additionally, a surfaceof a three-dimensional topological insulator with an irregular opening inproximity to a superconductor is proposed [32] to be described by the SYKmodel at low energies (see Fig. 1.3). Chapters 5, 6, and 7 are addressedto direct and indirect transport signatures of the non-quasiparticle natureof the SYK model to characterize these systems.

1.2 Electrical signatures of Majorana surface statesTopological superconductors are analogous to topological insulators [10,16–18]: Both combine an excitation gap in the bulk with gapless statesat the surface, without localization by disorder as long as time-reversalsymmetry is preserved. However, the nature of the surface excitationsis entirely different: In a topological insulator these are Dirac fermions,relativistic electrons or holes of charge ±e, while a topological supercon-ductor has charge-neutral Majorana fermions on its surface. A transportexperiment that aims to detect the Majorana surface states cannot be asroutine as electrical conduction — the direct analog for Majorana fermionsof the electrical conductance of Dirac fermions is the thermal conductanceGthermal. The challenge of low-temperature thermal measurements is onereason why Majorana surface states have not yet been detected in a trans-port experiment on candidate materials for topological superconductivity[33–37].

There exists a purely electrical alternative to thermal detection of Ma-jorana fermions [38]. Particle-hole symmetry enforces that a Majoranafermion at the Fermi level is an equal-weight electron-hole superposition,so while the average charge is zero, the charge fluctuations have a quan-tized variance of

VarQ = 12(+e)2 + 1

2(−e)2 = e2. (1.1)

Quantum fluctuations of the charge can be detected electrically in a shotnoise measurement. For a single fully transmitted Majorana mode par-ticle hole symmetry enforces a one-to-one relationship between the zero

1.2 Electrical signatures of Majorana surface states 5

temperature shot noise power P and the thermal conductance,

P/P0 = G/G0 = 12 Tr tt†, P0 = e3V/h, (1.2)

where t is the rank-one transmission matrix between two contacts alongthe edge (biased at voltage V > 0, see Fig. 1.4). By definition 2 [39, 40],

P =∫ ∞−∞

dt 〈δI(0)δI(t)〉 = τ−1 VarQ, (1.3)

the shot noise power is the correlator of the current fluctuations andgives the variance of the charge transferred between the contacts in atime τ . Eq. (1.2) implies that VarQ has the universal value 1

2P0τ for afully transmitted Majorana mode. These produce a quantized shot noisepower P of 1

2e2/h per eV of applied bias [38]. The factor 1/2 reminds us

that a Majorana fermion is “half a Dirac fermion”.

Figure 1.4. Nonlocal current and voltage measurement to detect the charge-neutral Majorana edge mode in a two-dimensional topological superconductor.A bias voltage V excites the edge mode, producing a fluctuating current δI(t)and voltage δV (t), detected at a remote contact. Because the bulk of the su-perconductor is grounded, these nonlocal signals are evidence for conduction bygapless edge excitations.

In Chapter 3 of this thesis, we extend the shot noise quantization to thehigher moments of charge fluctuations of a single Majorana edge-mode.We explore the relation between the thermal conductance and electrical

2 The dissipated power in a contact resistance R = 1/G, measured in a frequencyband width ∆f , equals Pdiss = 2RP∆f . To avoid the factor of two, an alternativedefinition of P has a factor of two in front of the integral in Eq. (1.3). With ourdefinition, the Johnson-Nyquist formula for thermal noise is Pthermal = 2kBTG, whilePdiss = 4kBT∆f .


shot noise for the surface of a three-dimensional topological superconduc-tor in Chapter 2.

1.3 The Sachdev-Ye-Kitaev model in solid statesystems

The extreme case of strongly-correlated Majorana zero-modes is underscrutiny in the second part of the thesis. We are going to be focused onthe SYK model, which describes N Majoranas with randomized infinite-range interaction in 0 + 1 dimensions. It was proposed by Kitaev [27] asa simplified version of the disordered quantum Heisenberg model studiedearlier by Sachdev and Ye [28], so that the system would not have areplica symmetry breaking [41, 42] in the large N limit [43, 44]. The SYKHamiltonian [27, 29, 30] reads

H = 14!

N∑i,j,k,l=1

Jijklγiγjγkγl , (1.4)

where γi are Majorana zero-modes: γ†i = γi and γi, γj = 2δij . Thecouplings Jijkl can be drawn independently from Gaussian distributionwith zero mean Jijkl = 0 and finite variance J2

ijkl = 3!J2/N3.The SYK model comprises several perculiar properties [27, 29, 30]:

1. it possesses an exact large N solution at strong coupling J lackingquasiparticles;

2. emergent conformal symmetry in the infrared;

3. it saturates the upper bound on quantum chaos [45], which is alsothe case for holographic duals of black hole horizons [46].

Both 1 and 2 can be understood in terms of two-point Green’s functionfound in the long time limit 1 Jτ N :

G(τ, τ ′) = −N−1N∑i=1

⟨T γi(τ)γi(τ ′)

⟩= −

(4πJ2

)−1/4 sgn(τ − τ ′)√|τ − τ ′|

. (1.5)

In frequency representation, the Green’s function (1.5) scales as a power-law 1/

√ω, that has a branch cut rather then quasiparticle like pole struc-

1.3 The Sachdev-Ye-Kitaev model in solid state systems 7

ture. The result (1.5) originates from the solution of a saddle-point equa-tion:

J2∫dτ ′G(τ, τ ′)G(τ ′, τ ′′)3 = −δ(τ − τ ′′) . (1.6)

Equation (1.6) remains invariant under time reparametrization τ 7→ f(τ):

G(τ, τ ′) 7→[∂τf(τ)∂τ ′f(τ ′)

]∆G(f(τ), f(τ ′)

), (1.7)

where f(τ) is an arbitrary monotonic differentiable function due to theconformal invariance and ∆ = 1/4 is an anomalous fermionic scaling di-mension [27, 29, 30, 47]. The absence of quasiparticles in the SYK modelqualitatively describes the strange metal phase, that exists above the crit-ical temperature in high-Tc superconductors [48, 49], including linear intemperature resistivity in the case of one-dimensional extension of theSYK model [50].

The property of maximal chaos 3 in the SYK model is usually formu-lated in terms of the so-called Out-Of-Time-Order-Correlation function(OTOC), first introduced by Larkin and Ovchinnikov [51]. It shows howfast the perturbation at time t = 0 given by the operator V (0) spreadsthrough the system to influence the later measurement W (t) [45, 52]:

F (t) = Tr(e−βH [W (t), V (0)]2

). (1.8)

It was recently shown by Maldacena, Shenker, and Stanford [45], thatfor a many-body quantum system OTOC can not grow faster then expo-nentially with a characteristic time-scale tL ≥ ~/ (2πkBT ) known as theLyapunov time. The SYK model (1.4) is very nonlocal: All degrees of free-dom are strongly coupled among each other. This leads to the exponentialgrowth of the OTOC function [27, 29, 30]:∑

i,j

〈γi(0)γj(t)γi(0)γj(t)〉 ∝ eλLt , (1.9)

that precisely saturates the upper bound on the Lyapunov time λL =1/tL = 2πkBT/~ [45]. This is why the SYK model is often refereed as amaximal scrambler.

A possibility to study all these intriguing properties in physical observ-ables inspired a few proposals of realizing the SYK model in a condensedmatter platform as a low-energy effective description of an interacting


quantum system [31, 32, 53–55]. The SYK model with Majorana (real)zero-modes is claimed to be a low-energy theory of the Fu-Kane super-conductor [8] in a magnetic field with a disordered opening [32], whereasRef. [31] suggests to use N Majorana nanowires [9] coupled through adisordered quantum dot.

In Ref. [53] Chen et al suggested to use a graphene flake to realizethe SYK model with the conventional (complex) fermionic zero-modes 3

(cSYK model) [47]:

H =N∑

i,j,k,l=1Jij;kl c

†ic†jckcl , (1.10)

where the couplings are hermitian and antisymmetric Jij;kl = J∗kl;ij =−Jji;kl = −Jij;lk. Graphene in a perpendicular magnetic field B is knownto have Landau levels [57], which are quantized as En ' ~v

√2n eB/~c

with integer n. The 0th Landau level is degenerate 4 and robust [58] underdisorder unless the chiral symmetry is broken. Thus, one gets fermioniczero-modes separated from the higher bands by the gap controlled by themagnetic field. As those are robust under disorder, the authors propose tomake the boundary of the flake sufficiently irregular. This would make thewave functions of 0th Landau levels Φi(r) random in the real space. Inclu-sion of the Coulomb interaction V (r− r′) would enable one to project theinteraction term on the basis of 0th Landau levels, so that the projectionis governed by the following overlap:

Jij;kl =∫dr∫dr′Φ∗i (r)Φ∗j (r′)V (r− r′)Φk(r)Φl(r′) , (1.11)

that makes the low-energy theory zero dimensional 5 with the cSYK ef-fective Hamiltonian (1.10). The comparison of the generated overlap to

3 The features of the SYK model mentioned in Section 1.3 are valid for bothmodels with real (Majorana) and complex fermions. The difference is that for the cSYKmodel the two-point function (1.5) contains the asymmetry parameter of the non-Fermiliquid [56] if the system is away from the charge neutrality point (chemical potential6= 0). We briefly address this issue in the Appendix of Chapter 5. Another distinctionbetween Dirac and Majorana cases is that the right hand side of the reparametrisationprescription (1.7) for complex fermions is multiplied by g(τ)/g(τ ′), where the functiong(τ) appears because of U(1) gauge invariance [47].

4 Degeneracy of 0th Landau level is proportional to the amount of magnetic fluxthat flows through the system.

5 The system can be thought as a strongly interacting disordered quantum dot.

1.3 The Sachdev-Ye-Kitaev model in solid state systems 9

the expected Gaussian distribution is shown in Fig 1.5. It is stated thatthe cSYK model well describes the low-energy theory of the disorderedgraphene flake of the characteristic size l ' 0.15 µm that hosts N ' 100zero-modes at field strength B ∼ 20 T [53]. Another proposal for realiza-tion of the cSYK model (1.10) with ultracold gases was made by Danshitaet al [54]. However, all the proposals require for a large number of sta-ble fermionic zero modes separated from the higher bands with a notablegap, sufficient amount of disorder in the system to randomize the wavefunctions of the zero-modes in the real space, and dominating two-bodyinteraction to be projected on the basis of zero-modes.

a

Figure 1.5. Left panel: Holography in a lab. Strongly interacting disorderedgraphene quantum dot described by 0 + 1-dimensional cSYK model (1.10) atthe boundary of 1 + 1-dimensional anti-de Sitter space with a black hole [55].The figure is reprinted by permission from Springer Nature: Marcel Franz andMoshe Rozali, Nature Reviews Materials 3, 491–501 (2018). Copyright 2018 bySpringer Nature. Right panel: Statistics of the effective coupling Jij;jk (1.11)for the disordered graphene flake. The figure is reprinted with permission fromAnffany Chen, R. Ilan, F. de Juan, D. I. Pikulin, and M. Franz, Phys. Rev. Lett.121, 036403 (2018). Copyright 2018 by the American Physical Society.

To characterize the "black hole on a chip" discussed in Refs. [31, 32,53] propose to measure the local density of states of the SYK quantumdot in a tunneling spectroscopy. The differential tunneling conductanceat weak coupling

G = dI

dV∝ ImGR(V ) ∝ V −1/2 (1.12)

reproduces the scaling of the SYK saddle-point solution (1.5) for the volt-ages in the range of J/N V J . The scaling of the differential

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conductance (1.5) reveals the emergent conformal symmetry of the SYKmodel.

1.4 This thesis

Bellow, I briefly highlight the main results presented in the thesis.

1.4.1 Chapter 2

In the first chapter we compare the thermal conductance Gthermal (attemperature T ) and the electrical shot noise power Pshot (at bias volt-age V kBT/e) of Majorana fermions on the two-dimensional surfaceof a three-dimensional topological superconductor. We present analyticaland numerical calculations to demonstrate that, for a local coupling be-tween the superconductor and metal contacts, Gthermal/Pshot = LT/eV(with L the Lorenz number). This relation is ensured by the combi-nation of electron-hole and time-reversal symmetries, irrespective of themicroscopics of the surface Hamiltonian, and provides for a purely electri-cal way to detect the charge-neutral Majorana surface states. A surfaceof aspect ratio W/L 1 has the universal shot-noise power Pshot =(W/L)× (e2/h)× (eV/2π).

1.4.2 Chapter 3

It was found in Ref. [38], that the Majorana fermions propagating alongthe edge of a topological superconductor with px + ipy pairing deliver ashot noise power of 1

2 × e2/h per eV of voltage bias. In this chapter we

calculate the full counting statistics of the transferred charge and find thatit becomes trinomial in the low-temperature limit, distinct from the bino-mial statistics of charge-e transfer in a single-mode nanowire or charge-2etransfer through a normal-superconductor interface. All even-order corre-lators of current fluctuations have a universal quantized value, insensitiveto disorder and decoherence. These electrical signatures are experimen-tally accessible, because they persist for temperatures and voltages largecompared to the Thouless energy.

1.4 This thesis 11

1.4.3 Chapter 4

There exists an analogy of topological superconductors reveals while con-sidering graphene superlattice shown in Fig. 1.6. Dirac electrons in

Figure 1.6. Graphene superlattice with a periodic potential modulation. Dif-ferent colors distinguish the carbon atoms on the A and B sublattice, each ofwhich has an ionic potential VAn

, VBn, n = 1, 2, 3, induced by the substrate.

graphene have a valley degree of freedom that is being explored as a carrierof information. In that context of “valleytronics” one seeks to coherentlymanipulate the valley index. In this chapter we show that reflection from asuperlattice potential can provide a valley switch: Electrons approachinga pristine-graphene–superlattice-graphene interface near normal incidenceare reflected in the opposite valley. We identify the topological origin ofthis valley switch, by mapping the problem onto that of Andreev reflec-tion from a topological superconductor, with the electron-hole degree offreedom playing the role of the valley index. The valley switch is ideal ata symmetry point of the superlattice potential, but remains close to 100%in a broad parameter range.

1.4.4 Chapter 5

In this chapter we study how the non-Fermi liquid behavior of the closedsystem in equilibrium manifests itself in an open system out of equilib-


rium. We calculate the current-voltage characteristic of a quantum dot,described by the complex-valued SYK model, coupled to a voltage sourcevia a single-channel metallic lead (coupling strength Γ). A one-parameterscaling law appears in the large-N conformal regime, where the differen-tial conductance G = dI/dV depends on the applied voltage only throughthe dimensionless combination ξ = eV J/Γ2. Low and high voltages arerelated by the duality G(ξ) = G(π/ξ). This provides for an unambiguoussignature of the conformal symmetry in tunneling spectroscopy.

1.4.5 Chapter 6

We study the observable properties of quantum systems which involve aquantum continuum as a subpart. We show in a very general way thatin any system, which consists of at least two isolated states coupled to acontinuum, the spectral function of one of the states exhibits an isolatedzero at the energy of the other state. Several examples of quantum systemsexhibiting such isolated zeros are discussed. Although very general, thisphenomenon can be particularly useful as an indirect detection tool for thecontinuum spectrum in the lab realizations of quantum critical behavior.

1.4.6 Chapter 7

In this chapter we demonstrate that a single fermion quantum dot acquiresodd-frequency Gor’kov anomalous averages in proximity to strongly cor-related Majorana zero-modes, described by the SYK model. Despite thepresence of finite anomalous pairing, superconducting gap vanishes for theintermediate coupling strength between the quantum dot and Majoranas.The increase of the coupling leads to smooth suppression of the originalquasiparticles.

1.4.7 Chapter 8

In the last chapter we consider another model with infinity ranged inter-action. We found analytically a first-order quantum phase transition in aCooper pair box array of N low-capacitance Josephson junctions capaci-tively coupled to resonant photons in a microwave cavity. The Hamilto-nian of the system maps on the extended Dicke Hamiltonian of N spins1/2 with infinitely coordinated frustrating interaction. This interactionarises from the gauge-invariant coupling of the Josephson-junction phases

1.4 This thesis 13

to the vector potential of the resonant photons field. In the N 1 semi-classical limit, we found a critical coupling at which the ground state ofthe system switches to one with a net collective electric dipole momentof the Cooper pair boxes coupled to a super-radiant equilibrium photoniccondensate. This phase transition changes from the first to second orderif the frustrating interaction is switched off. A self-consistently “rotating”Holstein-Primakoff representation for the Cartesian components of thetotal superspin is proposed, that enables one to trace both the first- andthe second-order quantum phase transitions in the extended and standardDicke models, respectively.

Chapter 2

Wiedemann-Franz-typerelation between shot noiseand thermal conduction ofMajorana surface states in athree-dimensionaltopological superconductor

2.1 IntroductionIn a two-dimensional (2D) topological superconductor, studied in Ref.[59], there is only a single Majorana edge mode, but a three-dimensional(3D) topological superconductor has a large number N of Majorana sur-face modes connecting a pair of metal contacts (see Fig. 2.1). In theabsence of inter-mode coupling this would give a shot noise power of

Pshot = 12NT P0, P0 = eV

e2

h, (2.1)

The contents of this chapter have been published and reprinted with permissionfrom N. V. Gnezdilov, M. Diez, M. J. Pacholski, and C. W. J. Beenakker, Phys. Rev.B 94, 115415 (2016). Copyright 2016 by the American Physical Society.



16 Chapter 2. Thermal conduction of surface states in a 3D TSC

Figure 2.1. Schematic of a superconductor between a pair of normal-metalcontacts. The thermal conductance Gthermal = J/δT is obtained by applying atemperature difference T , T +δT between the contacts and measuring the result-ing heat current J . For the shot noise measurement one would bias contact N1at voltage V , while keeping the superconductor and N2 grounded. The electricalcurrent I2 into contact N2 fluctuates with noise power Pshot. Both Gthermal ∝ Tand Pshot ∝ V are governed by the Majorana surface states of the topologicalsuperconductor.

for a mode-averaged transmission probability T . Because the thermalconductance equals [7]

Gthermal = 12NT G0, G0 = LT e

2

h, (2.2)

with L = 13(πkB/e)2 the Lorenz number, uncoupled Majorana modes have

a one-to-one relationship between shot noise and thermal conduction.We would expect this relationship to break down as a result of inter-

mode scattering: A pair of coupled Majorana modes is equivalent to asingle Dirac fermion mode, which can be in an eigenstate of charge atVarQ = 0. The thermal conductance would not be affected, as long as Tremains the same, but Pshot would be reduced. Much to our surprise, wediscovered in numerical simulations of a 3D topological superconductorthat Pshot/P0 = Gthermal/G0 with high accuracy. This is remarkable evenin the absence of any disorder, since the modes at top and bottom surfacesare coupled when they reach the metal contact.

We have found that it is the combination of electron-hole and time-reversal symmetry that preserves the relationship between electrical andthermal conduction in a 3D topological superconductor, provided the con-version from Majorana to Dirac fermions at the metal electrode is local

2.2 Surface-sensitive thermal and electrical conduction 17

in space. The general argument is presented in Secs. 2.2 and 2.3. Theimplication is that the shot noise power has a universal limit

Pshot = 12π

W

LP0, (2.3)

for a surface of aspect ratioW/L 1, with corrections from poor couplingto the metal contacts that we calculate in Secs. 2.4 and 2.5. A numer-ical test of our analytical predictions for a model Hamiltonian of a 3Dtopological superconductor is given in Sec. 2.6. We conclude in Sec. 2.7with a discussion in the context of the Wiedemann-Franz relation betweenelectrical and thermal conduction [60].

2.2 Surface-sensitive thermal and electrical con-duction

2.2.1 Description of the geometry

We consider the geometry of Fig. 2.1, a superconductor S connecting twonormal-metal contacts N1 and N2. The superconductor is topologicallynontrivial, with a gapped bulk and a gapless surface. We compare twotransport properties, one thermal and one electrical, both sensitive to thesurface states.

For thermal transport we take the two-terminal thermal conductance

Gthermal = limδT→0

J

δT, (2.4)

giving the heat current J flowing from contact N1 at temperature T + δTto contact N2 at temperature T , in linear response for δT T . Thesuperconductor is a thermal insulator in the bulk, because of the excitationgap, but a thermal conductor on the surface, so Gthermal measures heatconduction along the surface.

For electrical transport both contacts are kept at the same temperatureT . Contact N1 is biased at voltage V relative to ground, while contact N2as well as the superconductor are grounded. Most of the charge currentI1 injected into the superconductor at N1 is short-circuited to ground viathe bulk, which is an ideal electrical conductor. At the remote contact N2a fluctuating current I2(t) remains due to surface conduction from N1 to


N2. Even if the time average 〈I2〉 vanishes, there will be time-dependentfluctuations δI2(t) with low-frequency noise power

Pshot =∫ ∞−∞

dt 〈δI2(0)δI2(t)〉. (2.5)

At low temperatures kBT eV this is predominantly shot noise ∝ V .

2.2.2 Scattering formulas

In a scattering formulation the thermal and electrical transport propertiescan be expressed in terms of the matrix t(E) of transmission amplitudesfrom N1 to N2, at energy E relative to the Fermi level. The transmissionmatrix has a block structure in the electron-hole degree of freedom,

t =(tee tehthe thh

). (2.6)

The submatrix tee describes transmission of an electron as an electron,while the describes transmission of an electron as a hole.

At sufficiently small temperature and voltage the transmission matrixmay be evaluated at the Fermi level (E = 0) and we have the Landauer-type formulas [12, 61, 62]

Gthermal = 12G0 Tr t†t, (2.7)

Pshot = P0 Tr (τ+ − τ2−), (2.8)

τ± = t†eetee ± t†hethe. (2.9)

Eq. (2.8) may equivalently be written in terms of the full transmissionmatrix,

Pshot/P0 = 12 Tr (1 + τz)t†t− 1

4 Tr[(1 + τz)t†τzt

]2, (2.10)

with the help of the Pauli matrix τz =( 1 0

0 −1)acting on the electron-hole

degree of freedom.

2.2.3 Electron-hole symmetry enforced upper bound onthe shot noise power

Electron-hole symmetry at the Fermi level equates

t = τxt∗τx, (2.11)

2.3 Combined effects of electron-hole and time-reversal symmetries onthe shot noise power 19

with τx =(0 11 0)the Pauli matrix that exchanges electrons and holes. It

follows that

Tr τzt†t = 12 Tr (tτzt† + t∗τzt

T)= 1

2 Tr (tτzt† + τxtτxτzτxt†τx) = 0. (2.12)

Eqs. (2.7) and (2.10) can therefore be combined into

Pshot/P0 = Gthermal/G0 − δp,

δp = 14 Tr

[(1 + τz)t†τzt

]2.

(2.13)

Since (1 + τz)2 = 2(1 + τz) the term δp can be written as the trace of apositive definite matrix,

δp = TrX2 ≥ 0, X = 14(1 + τz)t†τzt(1 + τz) = X†, (2.14)

so the dimensionless shot noise power Pshot/P0 is bounded from above bythe dimensionless thermal conductance Gthermal/G0.

In Ref. [38] it was demonstrated that this inequality becomes a strictequality for a rank-one transmission matrix t, as in 1D transmission viathe unpaired Majorana edge mode of a 2D topological superconductor[59, 63]. Only particle-hole symmetry is needed in that case. In the nextsection we will show that the combination of particle-hole symmetry andtime-reversal symmetry achieves approximate equality on the 2D surfaceof a 3D topological superconductor, irrespective of the rank of t.

2.3 Combined effects of electron-hole and time-reversal symmetries on the shot noise power

2.3.1 Surface Hamiltonian with tunnel coupling to metalcontacts

The surface Hamiltonian of Majorana fermions in the x–z plane has theform [64]

H = vpxσx + vpzσz, (2.15)

with velocity v, momentum operators pα = −i∂/∂xα, and Pauli matricesσα acting on the spin degree of freedom. (The 2×2 unit matrix is σ0 and wehave set ~ to unity.) We assume that there is no valley degeneracy of the


surface states, so the surface spectrum consists of a single nondegeneratecone with dispersion relation E2 = v2p2.

A disorder potential V (x, z)σα is forbidden by the combination ofelectron-hole symmetry and time-reversal symmetry,

H = −H∗, H = σyH∗σy. (2.16)

This insensitivity to impurity scattering is a unique property of a topo-logical superconductor in symmetry class DIII 1 [65, 66]. A spatialmodulation of the Fermi velocity v(x, z) is allowed by symmetry [67], andthis is the only source of scattering on the surface.

The normal metal has propagating modes labeled by a spin degreeof freedom σ, electron-hole degree of freedom τ , and orbital degree offreedom ν. The superconducting surface has only evanescent modes atthe Fermi level, because of the vanishing density of states. A Majoranafermion with spin σ′ at point r is coupled to the metal by the couplingmatrix element 〈σ′, r|Ξ|σ, τ, ν〉. The scattering matrix is given by [68]

S(E) = 1 + iΞ†(H − 12 iΞΞ† − E)−1Ξ, (2.17)

near the Fermi level where the energy dependence of the coupling matrixΞ can be neglected.

The scattering matrix is unitary, SS† = S†S = 1, with electron-holeand time-reversal symmetries 2

S(E) = τxS∗(−E)τx, S(E) = σyS

T(E)σy. (2.18)

The corresponding symmetry relations for the coupling matrix are

Ξ = Ξ∗τx, Ξ = σyΞ∗σy. (2.19)

1 Symmetry class DIII refers to a superconductor with broken spin-rotation sym-metry. Class-CI superconductors with preserved spin-rotation symmetry may also hostgapless surface states, but these are Dirac fermions rather than Majorana fermions anddo not have the same protection against disorder scattering.

2 The symmetry operations (2.18) on the scattering matrix are taken in a time-reversally symmetric basis of the propagating modes in the normal metal. If transversemomentum q is used to label the modes, these operations should be accompanied byq 7→ −q.

2.3 Combined effects of electron-hole and time-reversal symmetries onthe shot noise power 21

2.3.2 Condition on the coupling matrix for equality of shotnoise and thermal conductance

We now restrict ourselves to the Fermi level, E = 0, and determine acondition on the coupling matrix Ξ that ensures the equality Pshot/P0 =Gthermal/G0 of the dimensionless shot noise power and thermal conduc-tance. According to Eqs. (2.13) and (2.14), a necessary and sufficientcondition is that the electron-electron block X of the transmission matrixproduct t†τzt vanishes identically. Here we establish a sufficient conditioninvolving only the coupling matrix:

ΞτzΞ† ≡ 0⇒ t†τzt ≡ 0⇒ X ≡ 0⇔ Pshot/P0 = Gthermal/G0.

(2.20)

No requirements are made on the rank of t, which may involve strongmode-mixing by the surface Hamiltonian.

The combination of the electron-hole and time-reversal symmetries(2.19) implies that

Ξτx = σyΞσy ⇒ σyΞτzΞ† = −ΞτzΞ†σy, (2.21a)Ξτx = Ξ∗ ⇒ ΞτzΞ† = −(ΞτzΞ†)T, (2.21b)

so the matrix ΞτzΞ† is antisymmetric and it anticommutes with σy. Be-cause the only 2× 2 matrix with both these properties is identically zero,we conclude that ΞτzΞ† vanishes if it is block-diagonal in 2× 2 matrices:

ΞτzΞ† = 0 if 〈σ′, r′|ΞτzΞ†|σ, r〉 = Γσσ′(r)δ(r − r′). (2.22)

The 2 × 2 matrix Γ(r) acts on the spin degree of freedom of a Majoranafermion at position r on the superconducting surface. We will refer to thecondition (2.22) as a locality condition on the coupling matrix productΞτzΞ†.

In the next section we will explicitly solve a simple model where thelocality condition holds, but we argue that it is a natural assumption fora weakly disordered NS interface between the normal metal (N) and thetopological superconductor (S). If the disorder mean free path is largecompared to the superconducting coherence length, a Majorana fermiontransmitted through the NS interface is locally converted into an electron-hole superposition via a 2× 4 matrix K and then scattered nonlocally inthe normal metal via a unitary matrix U without mixing electrons andholes — so U commutes with τz and UτzU † = τz. Substitution of Ξ = KUgives the desired locality to ΞτzΞ† = KτzK

†.


Figure 2.2. Superconducting surface layer in the x–z plane with tunnel couplingto a normal-metal in the region |z| > L/2 (shaded, tunnel rate Γ). This 2Dscattering geometry effectively describes the transmission from N1 to N2 via thetop or bottom surface of the 3D topological superconductor in Fig. 2.1. Theextension d of the contact region is assumed to be sufficiently large that topand bottom surfaces can be treated independently. (Coupling of top and bottomsurfaces is described in Fig. 2.3.)

2.4 Formulation and solution of the surface scat-tering problem

2.4.1 Reduction to an effectively 2D geometry

The transmission matrix t refers to a 3D scattering problem, and this ishow we will calculate it numerically later on. For an analytical treatmenta reduction to an effectively 2D geometry is desirable. Referring to Fig.2.1, in a typical thin-film geometry one has d W , so the contributionsfrom the top and bottom surfaces in the x–z plane dominate over thecontributions from the lateral surfaces in the y–z plane.

The normal-metal contact region in the x–y plane is connected tothe superconducting surface in the x–z plane at z = ±L/2. We ignorethe curvature of the surface at this connection and replace the contactregion by the region |z| > L/2 in the x–z plane. Tunneling into the metalelectrode in the contact region is described by the effective Hamiltonian

H = H − 12 iΓ(z)σ0, Γ(z) = Γθ(|z| − L/2), (2.23)

2.4 Formulation and solution of the surface scattering problem 23

with θ(s) the unit step function. If the tunnel barrier is sufficiently trans-parent (tunnel rate Γ v/d), a particle approaching z = ±L/2 via thetop surface (y = d/2) will enter the metal contact before reaching thebottom surface (y = −d/2), so that we can treat top and bottom surfacesseparately. This produces the effectively 2D geometry of Fig. 2.2, whichwe will analyze in the next subsection. The regime Γ . v/d, when topand bottom surfaces cannot be treated separately, is considered in Sec.2.4.3.

2.4.2 Single-surface transmission matrix

The normal metal has a nonzero density of states at the Fermi level, witha set of M transverse momenta q (in the x-direction). At each q thereare four propagating modes, including the spin and electron-hole degreeof freedom. We collect the total number of 4M mode indices in the labelα, with Pauli matrices σi, τi acting, respectively, on the spin and electron-hole degree of freedom. The scattering matrix element Sαα′(z, z′;E) atenergy E relates an outgoing mode α at z to an incoming mode α′ at z′.

The full 4M × 4M scattering matrix S(z, z′;E) describes both trans-mission (when z > L/2 and z′ < −L/2 or the other way around) andreflection (when z, z′ > L/2 or z, z′ < −L/2). In accord with Eq. (2.17)it is given by

S(z, z′;E) = 11δ(z − z′) + iΓW †(z)G(z, z′;E)W (z′), (2.24)

in terms of a 2M × 2M matrix Green’s function G(z, z′;E),

(H− E)G(z, z′;E) = 11δ(z − z′), (2.25)

and a 2M × 4M coupling matrix W (z). The rank of the matrix G is onlyhalf the rank of S, because the Majorana fermions on the superconductingsurface lack the electron-hole degree of freedom of the Dirac fermions inthe normal metal. 3

Particle conservation (unitarity) requires that∫|z′′|>L/2

dz′′ S(z, z′′;E)S†(z′, z′′;E) = 11δ(z − z′), (2.26)

3 The factor of two difference between the number of independent modes in thenormal metal and the superconductor is the reason that the mode-matching approachof Ref. [69] cannot be applied here.


for |z|, |z′| > L/2, which is satisfied if

W (z)W †(z) = 11. (2.27)

Although in the general treatment of the previous section we allowedfor mode-mixing on the superconducting surface, for a tractable analyt-ical calculation we now simplify to the Hamiltonian (2.15) with a uni-form velocity v. Because other sources of scattering are excluded by thecombination of electron-hole and time-reversal symmetry, the transversemomentum q is not coupled by the effective Hamiltonian (2.23) and Gdecomposes into 2× 2 q-dependent blocks G(z, z′; q, E).

This matrix Green’s function is calculated in Appendix 2.8.1. To ob-tain the transmission matrix from N1 to N2 we must take z > L/2 andz′ < −L/2. At the Fermi level the result is

G(z, z′; q, 0) = 12iv

exp[−ξ(z − z′ − L)]ξ coshLq + q sinhLq

(−ξ − κ iqiq ξ − κ

), (2.28a)

ξ =√q2 + κ2, κ = 1

2Γ/v, z > L/2, z′ < −L/2. (2.28b)

The 4M × 4M transmission matrix t(z, z′) at the Fermi level followsfrom Eq. (2.24),

t(z, z′) =∑q

t(z, z′; q),

t(z, z′, q) = iΓw†(z, q)G(z, z′; q, 0)w(z′, q),(2.29)

with a q-dependent 2 × 4M coupling matrix w(z, q). The unitarity con-straint reads

w(z, q)w†(z, q′) = δqq′σ0. (2.30)

2.4.3 Transmission matrix for coupled top and bottom sur-faces

The approach outlined above for the case of uncoupled top and bottomsurfaces can be readily generalized to allow for a coupling of the twosurfaces via the contact region. In the effective 2D representation theregion of nonzero Γ now extends over a finite interval,

Γ(z) = Γ [θ(|z| − L/2)− θ(|z| − L/2− d)] . (2.31)

2.4 Formulation and solution of the surface scattering problem 25

Figure 2.3. Panel a: Same as Fig. 2.2, but now with antiperiodic boundaryconditions at z = ±(L+ d) to include the effect of a coupling of top and bottomsurface via the contact region of finite length d. Panel b shows the correspondencebetween trajectories in the 2D and 3D representation. Coupling of the red andgreen trajectories, at the same transverse momentum q, elevates the rank of theq-dependent transmission matrix from one to two.

A particle on the top surface crossing the contact region without beingabsorbed continues on the bottom surface. The corresponding 2D geom-etry is shown in Fig. 2.3. It has a finite extent 2L+ 2d in the z-direction,with antiperiodic boundary conditions at z = ±(L + d) to account for aπ Berry phase.

The calculation of the Green’s function is given in Appendix 2.8.2.Instead of Eq. (2.28) we now have

G(z, z′; q, 0) = 12iv

cosh ξsξ coshLq cosh ξd+ q sinhLq sinh ξd

×(−ξ − κ tanh ξs iq tanh ξsiq tanh ξs ξ − κ tanh ξs

), (2.32a)

s = L+ d− z + z′ ∈ (−d, d),− L/2− d < z′ < −L/2, L/2 < z < L/2 + d. (2.32b)


The single-layer result (2.28) is recovered in the limit d→∞.The key distinction between the single-surface Green’s function (2.28)

and the coupled-surface result (2.32) is that — while both are 2× 2 ma-trices — the latter is of rank two but the former is only of rank one (oneof the two eigenvalues of the matrix (2.28) vanishes).

2.5 Thermal conductance and corresponding shotnoise power

We use the 2D surface theory of the previous section to calculate thethermal conductance Gthermal, including the effects of poor coupling to themetal contacts, strong coupling of top and bottom surfaces, and effects ofa finite aspect ratio. Subject to the locality condition (2.22) these resultsapply as well to the shot noise power Pshot = Gthermal × P0/G0.

2.5.1 Single surface

The thermal conductance (2.7) follows from the transmission matrix (2.29)upon integration,

Gthermal = 12G0

∫ ∞L/2

dz

∫ −L/2−∞

dz′Tr t†(z, z′)t(z, z′). (2.33)

Because of the unitarity condition (2.30) the coupling matrix drops outand only the 2 × 2 matrix Green’s function enters. Substitution of theresult (2.28) for a single surface gives

Gthermal/G0 = 12Γ2∑

q

∫ ∞L/2

dz

∫ −L/2−∞

dz′TrG†(z, z′; q, 0)G(z, z′; q, 0)

= 12κ

2∑q

(ξ coshLq + q sinhLq)−2. (2.34)

For W L the sum over transverse momenta may be replaced by anintegration,

∑q → (W/2π)

∫∞−∞ dq, resulting in

Gthermal = G0W

2π

∫ ∞0

dqκ2

(ξ cosh qL+ q sinh qL)2 . (2.35)

The coupling strength of the superconductor to the metal contactsis quantified by the product κL = ΓL/2v. In the strong-coupling limit

2.5 Thermal conductance and corresponding shot noise power 27

Figure 2.4. Thermal conductance of the surface of a 3D topological supercon-ductor (aspect ratio W/L 1) as a function of the coupling strength to thenormal-metal contacts. The solid curve is calculated from Eq. (2.35), the dashedcurve is the effective-length approximation (2.37). In the strong-coupling limitGthermal → (W/L)(G0/2π).

κL→∞ the thermal conductivity approaches the universal valueL

WGthermal →

12πG0, for κL→∞, (2.36)

This is the Majorana fermion analogue [67, 70] of the Dirac fermion con-ductivity in graphene [69, 71]. The effect of a finite coupling strength canbe understood in terms of an effective length Leff = L+ 1/κ, so that

Gthermal ≈1

2πG0 ×W

Leff= G0W

2πLκL

1 + κL, (2.37)

see Fig. 2.4.

2.5.2 Coupled surfaces

If we allow for coupling of the top and bottom surfaces via the metalcontact, we would use the Green’s function (2.32) instead of Eq. (2.28),to arrive at

Gthermal = G0W

2π

∫ ∞0

dq2κ2 sinh2 ξd

(ξ cosh qL cosh ξd+ q sinh qL sinh ξd)2 . (2.38)


Figure 2.5. Thermal conductance for coupled top and bottom surfaces. Ford/L 1 we recover twice the single-surface plot in Fig. 2.4. The solid curves forfinite d are calculated from Eq. (2.38), the dashed curves are the approximation(2.39).

There are now contributions from two surfaces in parallel, so (L/W )Gthermal →2×G0/2π in the large-κ limit. As shown in Fig. 2.5, the finite-d effect isaccurately described by a reduction factor tanh2 κd,

Gthermal ≈G0W

2πL2κL tanh2 κd

1 + κL. (2.39)

2.5.3 Finite aspect ratio

Deviations from the universal limit (2.36) of the thermal conductivityappear even for strong coupling to the metal contacts, if the aspect ratioof the surface area is not large enough. The relevant variable is the ratior = P/L of the perimeter P = 2(W + d) of the metal contacts relative totheir separation L.

Transverse momenta are quantized with antiperiodic boundary condi-tions, because of the π Berry phase,

qn = (2n+ 1)π/P, n = 0,±1,±2, . . . . (2.40)

For strong coupling to the contacts (κ 1/L, 1/d) we find from Eq. (2.35)

2.5 Thermal conductance and corresponding shot noise power 29

Figure 2.6. Thermal conductivity in the limit κ→∞ of strong coupling to themetal contacts, as a function of the aspect ratio r = 2(W + d)/L. The curve iscalculated from Eq. (2.41). A cube has r = 4 and σthermal = 0.95×G0/2π.

the thermal conductivity σthermal = (L/P)Gthermal as the sum

σthermal = G0

∞∑n=−∞

12r cosh2[(2n+ 1)π/r]

. (2.41)

As shown in Fig. 2.6, the universal limit is reached rather quickly; for acube geometry, L = W = d⇒ r = 4, we are only 5% below the universallimit.

2.5.4 Locality condition

In this model calculation the general locality condition (2.22) on the cou-pling matrix reads

w(z, q)τzw†(z, q′) = δqq′R(z), (2.42)

since δqq′ 7→ δ(x− x′) upon Fourier transformation. The 2× 2 matrix R,acting on the spin degree of freedom, may depend on z but it should notdepend on q. We only need to impose locality in x, because in Eq. (2.24)we have already taken a local coupling in z.

The electron-hole and time-reversal symmetry constraints

w(z, q) = w∗(z,−q)τx, w(z, q) = σyw∗(z,−q)σy (2.43)


require thatR(z) = −RT(z) = −σyR(z)σy, (2.44)

and this is only possible for a 2×2 matrix 4 if R(z) ≡ 0. Then Eq. (2.29)gives t†(z, z′)τzt(z, z′) ≡ 0 and thus Eq. (2.13) implies the equality

Pshot/P0 = Gthermal/G0. (2.45)

All the results presented above for the thermal conductance then applyalso to the shot noise power.

Within the 2D model calculation of this section we cannot ascertainthat the locality condition (2.42) holds. For that purpose we need toperform a fully 3D calculation, as we will do in the next section.

2.6 Numerical solution of the full 3D scatteringproblem

2.6.1 Model Hamiltonian

Our numerical simulation is based on the Bogoliubov-De Gennes Hamil-tonian [10]

H(k) =(

k2

2m + V (r)− EF

)σ0 ⊗ τz

+ ∆ (kzσz ⊗ τx − kyσ0 ⊗ τy − kxσx ⊗ τx) , (2.46)

discretized on a cubic lattice (lattice constant a0, hopping energy t0).The disorder potential is V and the p-wave pair potential is ∆. This is ageneric model of a 3D topological superconductor in symmetry class DIII,without spin-rotation symmetry but with electron-hole and time-reversalsymmetries

H(k) = −τxH∗(−k)τx, H(k) = σyH∗(−k)σy. (2.47)

The geometry is that of Fig. 2.1, in the normal-metal regions we set∆ ≡ 0. A tunnel barrier of height Ubarrier (two lattice sites wide) isintroduced at the NS interfaces z = ±L/2. The scattering matrix iscalculated using the Kwant toolbox [72] . We fixed the Fermi energyat EF = 2.5 t0 and took a relatively large pair potential ∆ = 0.4EF toeliminate bulk conduction without requiring a large L.

4 Any 2× 2 antisymmetric matrix R is a scalar multiple of σy, so if we insist thatR anticommutes with σy we must have R ≡ 0.

2.6 Numerical solution of the full 3D scattering problem 31

2.6.2 Translationally invariant system

Figure 2.7. Comparison of the analytical result (2.38) for the thermal con-ductance with a numerical simulation of the Hamiltonian (2.46). The couplingstrength κ at the NS interface is the single fit parameter for the comparison.These are calculations in the simplest case V0 = 0, B0 = 0, αso = 0, W/L→∞.

For a direct test of the analytical calculation from the previous sectionwe first consider a translationally invariant system along the x-direction(W L, d ' 100a0, no disorder). Results are shown in Fig. 2.7. Theanalytical result Eq. (2.38) describes the numerics very well, with thecoupling strength κ as the single fit parameter. This demonstrates thevalidity of the 2D representation of the 3D scattering problem, includingthe effect of coupling between top and bottom surfaces.

We next investigate the extent to which the relation (2.45) holds,still in the translationally invariant system, by adding to the Hamilto-nian (2.46) the spin-orbit coupling

Vso = αsokxσy ⊗ τz, (2.48)

in order to mix the modes from top and bottom surface. (The same Vso isadded to superconducting and normal regions.) Note that Vso preservesthe electron-hole and time-reversal symmetries (2.47). We break time-reversal symmetry by imposing on the normal metal the magnetic fieldB = B0θ(|z| − L/2)x, in the gauge A = B0θ(|z| − L/2) yz.

As shown in Fig. 2.8, both a nonzero αso and a nonzero B0 are neededfor a difference between dimensionless shot noise power and thermal con-ductance. The nonzero αso is needed to couple the modes from top and


Figure 2.8. Numerical results for the shot noise power (red solid curves) andthermal conductance (blue dashed curves), in a superconductor with L = d =100 a0, W/L → ∞, V0 = 0, Ubarrier = 0. The magnetic field and spin-orbitcoupling are varied in, respectively, the bottom and top panel. This simulationdemonstrates the inequality Pshot/P0 ≤ Gthermal/G0, with equality if either B0or αso vanishes.

bottom surface — otherwise the transmission matrix would be of rank oneand the equality (2.45) would hold irrespective of whether time-reversalsymmetry is broken or not [38]. The nonzero B0 is needed because ofthe argument from Sec. 2.3.2 that mode coupling in the presence of time-reversal symmetry is not effective at violating the relation between shotnoise power and thermal conductance.

2.7 Discussion 33

Figure 2.9. Numerical results for the shot noise power (red solid curve) andthermal conductance (blue dashed curve) as a function of the disorder potentialstrength V0, in a superconductor with L = d = W = 20 a0 (B0 = 0, αso = 0,Ubarrier = 0). Shot noise and thermal conductance differ by less than 10−3, evenin the presence of significant disorder.

2.6.3 Disorder effects

We now break translational invariance by adding a disorder potential V ,uniformly distributed in the interval (−V0,+V0), randomly fluctuatingfrom site to site throughout the superconductor. We also added disorderon the normal side of the NS interface (in a sheet of width 10a0). Becausethe calculations are now computationally more expensive we took a smallersuperconductor, a cube of size L = W = d = 20a0. Results are shown inFig. 2.9.

Without disorder the thermal conductivity is close to the limit ex-pected from Fig. 2.5 for a cube aspect ratio: σthermal/G0 = 0.95/2π ⇒Gthermal/G0 = 0.605. Disorder has a significant effect, but the dimension-less shot noise and thermal conductance remain nearly indistinguishable.

2.7 Discussion

As a particle that is its own antiparticle, a Majorana fermion must becharge-neutral — but it need not be in an eigenstate of charge. This isa key distinction between a Majorana fermion as a fundamental particlesuch as a neutrino, or as a composite quasiparticle in a superconductor.


For the latter only the expectation value of the charge must vanish, sothere may be quantum fluctuations of the charge. Here we have shownhow we can exploit this property for a purely electrical detection of theMajorana surface states in a 3D topological superconductor, obviating thedifficulty of thermal measurements.

We like to think of the relation

Gthermal/Pshot = LT/eV, L ≡ 13(πkB/e)2, (2.49)

between thermal conductance and electrical shot noise power as the Ma-jorana counterpart of the electronic Wiedemann-Franz relation

Gthermal/Gelectrical = LT (2.50)

between thermal and electrical conductance [60]. The analogy is quitedirect: Eq. (2.50) expresses the fact that a nonequilibrium electron trans-ports energy and charge in a fixed ratio. The same holds for Eq. (2.49),with the electron charge Q = e replaced by the Majorana charge varianceVarQ = e2.

There exists an altogether different “Wiedemann-Franz type relation”for Majorana fermions, relating heat and particle currents rather thanheat and charge currents:

Gthermal/Gparticle = 12G0, G0 = 1

3(πkB)2 T. (2.51)

The “particle conductance” Gparticle is not directly measurable (since Ma-jorana fermions do not couple to the chemical potential), but it can beformally defined in terms of the Landauer formula Gparticle = NT /h or interms of an equivalent Kubo formula [60]. The factor 1/2 is the “topo-logical” or “central” charge C = 1/2 of a Majorana fermion [4]. No suchfactor appears in Eq. (2.49), because both the thermal conductance andthe shot noise power are proportional to C, so it drops out of the ratio.

One direction for future research is to generalize the relation (2.49) totopological superconductors with more than a single species of Majoranafermions on their surface. This is a key difference with 3D topological in-sulators, which have a Z2 topological quantum number, so at most a singleDirac cone on the surface. In contrast, 3D topological superconductorshave a Z topological quantum number, allowing for multiple Majoranacones [10, 16, 73].

Another direction to explore is how the class-DIII topological super-conductors with Majorana surface states considered here compare with

2.8 Appendix: Matrix Green’s function of the surface Hamiltonian 35

the class-CI topological superconductors with Dirac surface states. ForEq. (2.51) the difference is simply a factor of two, to account for a centralcharge C = 1 of Dirac fermions [70]. We do not expect such a simplecorrespondence for the relation (2.49).

From the experimental point of view, the usefulness of Eq. (2.49) isthat it provides a purely electrical way to access the transport proper-ties of Majorana surface states. The shot noise measurements shouldbe performed at energies eV well below the superconducting gap ∆. InCuxBi2Se3 this is about 0.6 meV [34]. Shot noise dominates over ther-mal noise if eV & 3kBT [63], so if one would perform the experiment atV = 0.1 meV, a readily accessible temperature range T . 0.3 K would do.

2.8 Appendix: Matrix Green’s function of thesurface Hamiltonian

We solve the differential equation[−iσx

∂

∂z+ qσz − 1

2 iΓ(z)σ0 − E]G(z, z′; q, E) = 11δ(z − z′) (2.52)

to obtain the 2× 2 matrix Green’s function that determines the transmis-sion matrix of the Majorana fermions between the normal-metal contacts.(To simplify the notation we have set v ≡ 1.) For a similar calculation ingraphene, see Ref. [74].

We first consider in Sec. 2.8.1 the case of uncoupled top and bottomsurfaces, when the z-coordinate ranges over the entire real axis and thetunnel coupling in the contact region is given by

Γ(z) = Γθ(|z| − L/2), −∞ < z <∞. (2.53)

In Sec. 2.8.2 we incorporate the finite extension d of the contact region,by setting

Γ(z) = Γ [θ(|z| − L/2)− θ(|z| − L/2− d)] , |z| < L+ d. (2.54)

Antiperiodic boundary conditions at z = ±(L + d) then couple the topand bottom surfaces.


2.8.1 Single surface

We defineε(z) = E + 1

2 iΓ(z), ε0 = E + 12 iΓ, (2.55)

with Γ(z) given by Eq. (2.53). The Green’s function that decays at infinityis

G(z, z′; q, E) = Pz↔0 exp[∫ z

0dz1 (iσzε(z1) + qσy)

][M− i

2σz+iσzθ(z − z′)]

· P0↔z′ exp[∫ z′

0dz2 (−iσzε(z2) + qσy)

], (2.56)

where Pz1↔z2 indicates a monotically increasing or decreasing ordering ofthe z-dependent non-commuting operators, from z1 leftmost to z2 right-most. The matrix M is determined by the requirement thatlimz→±∞ G(z, z′; q, E) = 0:(−ε0 ± i

√q2 − ε2

0, q

)· exp[±1

2L(iEσz + qσy)](M ± 12 iσz) = 0. (2.57)

The row-spinor on the left-hand-side is orthogonal (without taking com-plex conjugates) to the column-spinor

|±〉 =(ε0 ∓ i

√q2 − ε2

0q

), (2.58)

which is an eigenstate of

(iε0σz + qσy)|±〉 = ±√q2 − ε2

0|±〉. (2.59)

The square root should be taken such that Re√q2 − ε2 > 0.

The result is

M =(M1 M2M2 M1

),

M1 =E(ξ2

0 − 12 iΓE) cosh(Lξ0) + Eξξ0 sinh(Lξ0) + 1

2 iΓq2

2ξξ20 cosh(Lξ0) + 2(ξ2

0 − 12 iΓE)ξ0 sinh(Lξ0)

,

M2 =q(ξ2

0 − 12 iΓE) cosh(Lξ0) + qξξ0 sinh(Lξ0) + 1

2 iΓqE2ξξ2

0 cosh(Lξ0) + 2(ξ20 − 1

2 iΓE)ξ0 sinh(Lξ0),

(2.60)

2.8 Appendix: Matrix Green’s function of the surface Hamiltonian 37

with the definitions

ξ0 =√q2 − E2, ξ =

√q2 − (E + 1

2 iΓ)2. (2.61)

As a check, we take the limit E → 0, q → 0, whenM → 12 iσ0, as it should.

Note the symmetry relations

σyMT(−q, E)σy = M(q, E), M∗(−q,−E) = −M(q, E), (2.62)

which ensure that the Green’s function (2.56) satisfies the required time-reversal and electron-hole symmetries:

σyGT(z′, z;−q, E)σy = G(z, z′; q, E), (2.63)G∗(z, z′;−q,−E) = −G(z, z′; q, E). (2.64)

To obtain the transmission matrix we set z > L/2 and z′ < −L/2:

G(z, z′; q, E) = exp[(z − L/2)

(iEσz − 1

2Γσz + qσy)]

·exp [(L/2)(iEσz + qσy)]

·(M + 1

2 iσz)

exp[(L/2)(iEσz − qσy)]

· exp[−(z′ + L/2)

(iEσz − 1

2Γσz − qσy)]. (2.65)

For E = 0 this simplifies to Eq. (2.28) in the main text.

2.8.2 Coupled top and bottom surfaces

The Green’s function for coupled top and bottom surfaces is still of theform (2.56), with Γ(z) now given by Eq. (2.54). Instead of a decay atinfinity we now have the antiperiodic boundary conditions

G(L+ d, z′; q, E

)= −G

(−L− d, z′; q, E

). (2.66)

The condition (2.57) on the matrix M is replaced by

exp[

12L (iEσz + qσy)

]exp [d (iε0σz + qσy)]

· exp[

12L (iEσz + qσy)

] (M + 1

2 iσz)

=

= − exp[−1

2L (iEσz + qσy)]

exp [−d (iε0σz + qσy)]

· exp[−1

2L (iEσz + qσy)] (M − 1

2 iσz), (2.67)


with solution

M =(M1 M2M2 M1

), (2.68)

M1 =E(ξ2

0 − 12 iΓE) cosh(Lξ0) tanh(ξd) + Eξξ0 sinh(Lξ0) + 1

2 iΓq2 tanh(ξd)

2ξξ20 cosh(Lξ0) + 2(ξ2

0 − 12 iΓE)ξ0 sinh(Lξ0) tanh(ξd)

,

M2 =q(ξ2

0 − 12 iΓE) cosh(Lξ0) tanh(ξd) + qξξ0 sinh(Lξ0) + 1

2 iΓqE tanh(ξd)2ξξ2

0 cosh(Lξ0) + 2(ξ20 − 1

2 iΓE)ξ0 sinh(Lξ0) tanh(ξd).

For the transmission matrix we set −L/2− d < z′ < −L/2 and L/2 <z < L/2 +d. The energy-dependent Green’s function is then given by Eq.(2.65) with M from Eq. (2.68). At zero energy this produces the result(2.32) from the main text.

Chapter 3

Topologically protectedcharge transfer along theedge of a chiral p-wavesuperconductor

3.1 Introduction

The second moment of the transferred charge in a Majorana mode is di-rectly determined by the quantized thermal conductance. Higher momentsare not so constrained, and one might ask whether they are quantizedas well. Here we calculate the full probability distribution P (q) of thetransferred charge, including also the effects of finite temperature. In thezero-temperature limit we find that the characteristic (moment generat-ing) function F (χ) = 〈eiχq〉, related to P (q) by a Fourier transform, hasthe form

F (χ) =[1 + 1

4(eieχ − 1) + 14(e−ieχ − 1)

]N. (3.1)

This describes a trinomial counting statistics where N = eV τ/h attemptstransfer either −e, 0, or +e charge, with probabilities 1/4, 1/2, and 1/4,

The contents of this chapter have been published and reprinted with permissionfrom N. V. Gnezdilov, B. van Heck, M. Diez, J. A. Hutasoit, and C. W. J. Beenakker,Phys. Rev. B 92, 121406(R) (2015). Copyright 2015 by the American Physical Society.



40 Chapter 3. Edge transport of a chiral p-wave superconductor

respectively.This result for the statistics of charge transported by a chiral Majorana

edge mode can be contrasted with the characteristic function for the chargetransmitted by an electronic mode in a nanowire [75],

Felectron(χ) =[1 + T (eieχ − 1)

]N, (3.2)

where T ∈ [0, 1] is the transmission probability. The two key distinctionswith Eq. (3.1) are that the counting statistics is binomial, rather thantrinomial, and that the transfer probability is not quantized. A chiralquantum Hall edge mode would have a quantized T = 1, but then therewould be no charge fluctuations at all.

To see that these distinctions are not merely a consequence of thepresence of a superconductor, we compare with the corresponding resultfor the charge transferred through a normal-metal-superconducting (NS)point contact [76],

FNS(χ) =[1 +RA(e2ieχ − 1)

]N. (3.3)

The transmission probability is replaced by the probability RA for An-dreev reflection [77] and the unit of transferred charge is doubled, but thestatistics remains binomial and not quantized.

Resonant tunneling through a Majorana zero-mode, bound to a vortexor to the end of a nanowire, provides another point of comparison [78–80]. For two contacts biased at voltage ±V/2 and coupled to the zero-mode with tunnel probabilities T1, T2, the charge entering contact 1 hascharacteristic function [80]

Fzero-mode =[1 + T1(T1 + T2)−1(eieχ − 1)

]N. (3.4)

The statistics is binomial and dependent on the tunnel probabilities, ex-cept for a symmetric junction (when T1 = T2 it drops out). 1

1 For a symmetric coupling to the Majorana zero-mode (T1 = T2), the trinomialand binomial characteristic functions (3.1) and (3.4) are related by lnFzero-mode(χ) =12 ieNχ+ 1

2 lnF (χ), so cumulants of order two and higher differ only by a factor of two.This explains the correspondence between the Bernoulli numbers in Eq. (3.18) and theEuler polynomial of Ref. [80]. For general Tn there is no such correspondence betweentrinomial and binomial distributions.

3.2 Full counting statistics of Majorana edge states 41

3.2 Full counting statistics of Majorana edge statesOur analysis follows the scattering theory of counting statistics pioneeredby Levitov and Lesovik [75], in the convenient formulation of Klich [81].The characteristic function is given by

F (χ) = Tr ρ0 exp[ieχ

∑E>0

c†(E)Pc(E)]

× exp[−ieχ

∑E>0

c†(E)M(E)c(E)], (3.5)

M(E) = S†(E)PS(E), S =(r′ t′

t r

), P =

(0 00 σz

). (3.6)

The sum over energies is understood as∑E → (τ/h)

∫dE in the limit

τ → ∞. The trace gives the expectation value with density matrix ρ0 ofthe fermion operators c, c†, representing the quasiparticles injected into theedge from the two contacts, contact 1 at voltage V and contact 2 grounded.The scattering matrix S relates incoming and outgoing quasiparticles,with reflection and transmission subblocks. The matrix P selects thequasiparticles at contact 2, where the current is measured. The Paulimatrix σz appearing in P acts on the electron-hole degree of freedom, toaccount for the fact that electron and hole quasiparticles contribute withopposite sign to the electrical current.

Because different energies are uncoupled, we may perform the trace ateach energy separately, so that we may write Eq. (3.5) in the form

lnF (χ) =∑E>0

ln Tr(e−βc

†(E)E(E)c(E)

× eieχc†(E)Pc(E)e−ieχc†(E)M(E)c(E)

)− lnZ, (3.7)

E(E) = E − eV(σz 00 0

). (3.8)

Here β = 1/kBT and Z = Tr e−βc†Ec is the partition function at temper-ature T (the same for both contacts) and chemical potential µ (equal to±eV for electrons and holes at contact 1, and equal to 0 at contact 2).

With the help of the formula [81, 82]

Tr∏nec†Anc = Det

(1 +

∏neAn), (3.9)


the expression (3.7) reduces to

lnF (χ) = τ

h

∫ ∞0

dE ln Det(1−F + FeieχPe−ieχM

), (3.10)

F(E) = [1 + eβE(E)]−1. (3.11)

The matrix exponentials simplify because P2n = P2, P2n+1 = P,M2n =M2,M2n+1 =M (in view of unitarity, SS† = 1), hence

eieχP = 1 + P2(cos eχ− 1) + iP sin eχ,e−ieχM = 1 +M2(cos eχ− 1)− iM sin eχ.

(3.12)

In the zero-temperature limit F(E)P → 0 for E > 0, so the factor eieχPin Eq. (3.10) may be replaced by unity. To first order in V we then have

lnF0(χ) = (eV τ/h) ln Det[1+

+ 12(1 + σz)t†

(cos eχ− 1− iσz sin eχ

)t], (3.13)

with transmission matrix t evaluated at the Fermi energy E = 0.These formulas hold for any channel connecting two metal contacts

via a superconductor. We now use that the connection is via an unpairedMajorana edge mode, which implies that the 2N×2N transmission matrixhas rank one, irrespective of the number 2N of electron-hole modes inthe metal contact: t = T 1/2 uvT with unit vectors u, v and transmissionprobability T . Particle-hole symmetry at the Fermi level requires thatt = σxt

∗σx, hence the matrix t†σzt vanishes identically:

t†σzt = −iσxtTσyt = −iT (uTσyu)σxvvT = 0. (3.14)

Similarly, tσzt† = 0 while tt† has a single nonzero eigenvalue equal to T .We thus arrive at

lnF0(χ) = N ln[1 + 1

2T (cos eχ− 1)], N = eV τ/h, (3.15)

which for T = 1 is the result (3.1) announced in the introduction.The corresponding trinomial probability distribution P (q) of the trans-

ferred charge q = 0,±1,±2, . . . ±N (in units of the electron charge e) isgiven by

P (q) = (2− T )N−|q|T |q|N !2N+|q|(N − |q|)!|q|!× 2F1

[|q|−N

2 , |q|+1−N2 , |q|+ 1, T 2

(2−T )2

], (3.16)

3.2 Full counting statistics of Majorana edge states 43

Figure 3.1. Noise power P of the Majorana edge mode at temperature T , asa function of bias voltage V . The solid curve is calculated from Eq. (3.22), thedashed line is the low-temperature asymptote (3.23). These are results for asingle-channel contact (N = 1) to a fully transmitted edge mode (T = 1). Herethe voltage is assumed to be small on the scale of the superconducting gap ∆0.See Fig. 3.2 for the voltage dependence in the regime kBT eV . ∆0.

with 2F1 the hypergeometric function. For T = 1 this simplifies to

P (q) = 2−2N(

2NN − q

), (3.17)

which looks like a displaced binomial distribution. Cumulants are coeffi-cients in the Taylor series lnF (χ) =

∑p〈〈qp〉〉(iχ)p/p!, giving for T = 1

the result

〈〈qp〉〉 =

2N (2p+1 − 1) p!(p+1)!Bp+1 for p even,

0 for p odd,(3.18)

with Bp+1 the Bernoulli number. The first few values are

N−1〈〈qp〉〉 = 12 ,−

14 ,

12 ,−

178 ,

312 for p = 2, 4, 6, 8, 10. (3.19)

All of this is at zero-temperature.

3.2.1 Finite temperature

The general finite-temperature formulas are complicated, for a compactexpression we take the case N = 1, T = 1 of a single-channel contact


to a fully transmitted edge mode. The determinant in Eq. (3.10) thenevaluates to

lnF (χ) = τ

h

∫ ∞0

dE ln[1 + 1

2f(1− f)(cos 2eχ− 1)

+ (f + 12fV − ffV )(cos eχ− 1)

], (3.20)

f(E) = (1 + eβE)−1, fV = f(E − eV ) + f(E + eV ). (3.21)

This is the multinomial statistics of transferred charge 0,±e,±2e, withthe interpretation that charge ±e is transferred via the Majorana edgemode and charge ±2e is transferred via Andreev reflection into the bulksuperconductor. The corresponding noise power is

P = −1τ

limχ→0

d2

dχ2 lnF (χ) =

= e2

h

∫ ∞0

dE[1

2fV (1− 2f) + 3f − 2f2] (3.22)

= e2

h×

(kBT + 12eV ) for kBT . eV,

2kBT for eV kBT.(3.23)

As a check, we note that the thermal noise power is related to the contactconductance 2 G = e2/h by

Pthermal ≡ limV→0

P = 2kBTG, (3.24)

in accordance with the Johnson-Nyquist relation.From Fig. 3.1 we see that the slope dP/dV is within 10% of the quan-

tized value e2/2h for voltages eV & 3kBT . This lower limit on the voltageis to be combined with an upper limit set by the superconducting energygap. From Fig. 3.2 we estimate that the quantization is preserved even inthe presence of strong disorder when eV . ∆0/2. For a realistic gap [83,84] of 0.2 meV the quantized shot-noise regime would then be accessibleat temperatures below 0.4 K, which is a quite feasible requirement.

2 The contact conductance G in the geometry of Fig. 1.4 is given in terms of the2N×2N reflection matrix r at the contact by G = 1

2 (e2/h) Tr (1−r†τzrτz). For N = 1,T = 1 we have r = abT, r† = σxba

Tσx, with unit vectors a, b, hence G = e2/h.3 The Hamiltonian of the chiral p-wave superconductor on a two-dimensional square

lattice is H = ε(k)σz + ∆0(σx sin kxa+ σy sin kya), ε(k) = −2t0(cos kxa+ cos kya)− µ.

3.3 Discussion 45

Figure 3.2. Voltage dependence of the zero-temperature shot noise power,calculated numerically 3 for the tight-binding Hamiltonian of a chiral p-wave su-perconductor (black solid and dashed curves in the band structure are the coun-terpropagating Majorana modes at opposite edges). The superconducting region(W = L = 100 in units of the lattice constant a) is connected to metallic leads(superconductor and lead 2 grounded, lead 1 at voltage V ). The electron-to-electron and electron-to-hole transmission matrices tee(E) and the(E) from lead1 to 2 are calculated as a function of the energy E = eV . The differential shotnoise power in lead 2 then follows from [12, 62] dP/dV = (e3/h)Tr (T+ − T 2

−),with T± = t†eetee ± t

†hethe. The black solid curve is the result for a clean system,

the red dashed curve is obtained in the presence of a random on-site potentialUn ∈ [−∆0/2,∆0/2]. The red arrow indicates the Thouless energy ET = ~v/L.This simulation demonstrates that the quantized shot noise dP/dV = e3/2h isinsensitive to disorder for voltages |V | . ∆0/e — even if eV is large comparedto ET.

3.3 Discussion

We discuss three further issues regarding the robustness of the quantizedshot noise of the Majorana edge mode.

For the simulations in Fig. 3.2 we took parameters t0 = ∆0, µ = −2∆0 + U , for whichthe velocity of the edge mode equals v = a∆0/~. (In the normal metal leads we set∆0 to zero, keeping the same t0 and µ.) In the clean system U ≡ 0, in the disorderedsystem U is varied randomly from site to site. The scattering matrix was calculatedusing the kwant code from [72].


1) Impurity scattering along the edge has no effect because of thechirality of the edge mode, prohibiting backscattering. The contact re-sistance may in principle reduce T below unity, but this effect can beminimized by using an extended contact: If each of the two contactscontains 2N electron-hole modes with tunnel probability Γ to the edgemode, then T ' [min(1, NΓ)]2. Hence contact resistances have no ef-fect on the quantized shot noise if NΓ & 1. We need to avoid a largethermal noise due to Andreev reflection from such an extended contact,which is of order kBTNΓ2. Both conditions, maximal coupling (T = 1)with minimal thermal noise (Pthermal ' kBTe

2/h), are satisfied if we take1/N Γ 1/

√N , so that NΓ 1, NΓ2 1.

2) Loss of phase coherence has no effect on the quantized shot noise.The coherent electron-hole superposition of a Majorana fermion is fragileindeed, coupling to the electromagnetic environment will project it ontoan electron or a hole, effectively measuring the charge of the quasiparticle.The trinomial statistics, however, remains unaffected, because each of theN current pulses still transfers the same amount of charge ±e with equalprobability 1/4.

3) The shot noise quantization is a macroscopic effect, preserved onscales large compared to the Thouless energy ET = ~v/L. This is theenergy scale on which electrons and holes dephase after traveling withvelocity v over a distance L and which governs transport experimentsin an interferometer [85, 86]. As demonstrated in Fig. 3.2, raising thevoltage to eV ≈ ET has a negligible effect on the shot noise. The reasonfor this unusual insensitivity is that the electron and hole component of theMajorana mode acquire the same phase factor at finite energy [87], so nodephasing can occur. The fact that the energy scale for the quantizationis set by ∆0 rather than by ET is crucial for the observability of the effect.

In conclusion, we have identified unique electrical signatures of a charge-neutral Majorana mode propagating along the edge of a topological su-perconductor: A trinomial statistics of transferred charge, with quantizedcumulants persisting in a macroscopic system, since they are insensitiveto impurity scattering or loss of phase coherence. A promising physicalsystem to search experimentally for the shot noise quantization could bean array of parallel nanowires [59, 88–91] or parallel chains of magneticatoms [92, 93], all on a superconducting substrate.

As a direction for further theoretical research we point to the effectof interactions among the Majorana fermions. Two recent studies [94, 95]

3.4 Appendix: Effect of a tunnel barrier at the NS contact 47

have found an interaction-driven quantum phase transition from centralcharge c = 1/2 to c = 3/2. Because the coefficient 1/2 in the shot noisepower (1.2) originates from the central charge of the Majorana mode, itwould be interesting to see what is the effect of this phase transition onthe charge transfer statistics. A related extension of our results would beto topological superconductors with a higher Chern number, supportingmultiple Majorana modes at each edge, for which physical realizationshave been recently predicted [96, 97].

3.4 Appendix: Effect of a tunnel barrier at theNS contact

A tunnel barrier at the normal-metal–superconductor (NS) interface isneeded to suppress the thermal noise from Andreev reflection. At zerotemperature this is of no concern, but it turns out that a tunnel bar-rier is still advantageous because it reduces the voltage sensitivity of theshot noise power. We show this in Fig. 3.3, where we compare the zero-temperature dP/dV for the two cases.

Figure 3.3. Voltage dependence of the zero-temperature shot noise power in themodel of Fig. 3.2. The black curves are with a tunnel barrier at the NS interfaces(transmission probability Γ = 0.32 in each of N = 100 modes), the blue curveis without any barrier. For V . ET there is no difference, for larger V a tunnelbarrier helps to preserve the quantization all the way up to the superconductinggap.

Chapter 4

Valley switch in a graphenesuperlattice due topseudo-Andreev reflection

4.1 IntroductionThe precession of a spin in a magnetic field has analogues for the pseu-dospin degrees of freedom that characterize quasiparticles in condensedmatter. The K,K′ valley index of Dirac electrons in graphene is a sucha pseudospin — it is actively studied because it might play a role as acarrier of information in “valleytronics”, the valley-based counterpart ofspintronics [98]. The analogue of the magnetic field for valley precessioncan be provided by a superlattice potential: [99–102] When graphene isdeposited on a substrate with a commensurate honeycomb lattice, thevalleys are coupled by the periodic modulation of the potential on thecarbon atoms [103–105]. The coupling can be represented by an artifi-cial magnetic field [106] that rotates the valley pseudospin as the electronpropagates through the superlattice [99], analogously to the spin-rotationby an exchange field in spintronics [107].

The contents of this chapter have been published and reprinted with permissionfrom C. W. J. Beenakker, N. V. Gnezdilov, E. Dresselhaus, V. P. Ostroukh, Y. Herasy-menko, İ. Adagideli, and J. Tworzydło, Phys. Rev. B 97, 241403(R) (2018). Copyright2018 by the American Physical Society.



50 Chapter 4. Pseudo-Andreev reflection in a graphene superlattice

Here we present a valley precession effect without a counterpart inspintronics: A quantized 180 precession angle upon reflection, such thatthe electron switches valleys. This valley switch is analogous to Andreevreflection at the interface between a normal metal and a topological super-conductor [11]. Andreev reflection is the reflection as a hole of an electronincident on the superconductor, which happens with unit probability ifthe normal-superconductor interface contains a Majorana zero-mode. Theanalogous effect happens in the superlattice because of an anti-unitarysymmetry that is formally equivalent to the charge-conjugation symmetryin a superconductor.

4.2 Graphene superlattice with anti-unitary sym-metry

We consider the Dirac Hamiltonian [103–105]

H =

V0 + µ vFp− 0 αvFp+ V0 − µ −β∗ 0

0 −β V0 − µ vFp−α∗ 0 vFp+ V0 + µ

. (4.1)

It acts on the spinor Ψ = (ψKA, ψKB,−ψK′B, ψK′A) that contains the sub-lattice (A,B) and valley (K,K′) degrees of freedom of a conduction electronmoving in the x–y plane of a carbon monolayer (graphene), with velocityvF and momentum p = (px, py), p± ≡ px± ipy. In terms of Pauli matricesσi and τi acting, respectively, on the sublattice and valley indices, we maywrite

H = vF(pxσx + pyσy) + µτz ⊗ σz+ 1

2(τx ⊗ σx)Re (α− β)− 12(τy ⊗ σy)Re (α+ β)

− 12(τx ⊗ σy)Im (α− β)− 1

2(τy ⊗ σx)Im (α+ β). (4.2)

For simplicity, we have shifted the zero of energy such that V0 = 0.An epitaxial substrate induces a periodic potential modulation, which

triples the size of the unit cell: it is enlarged by a factor√

3 ×√

3 andcontains six rather than two carbon atoms. The parameters V0, µ (real)and α, β (complex) are determined by the substrate potentials on these six

4.2 Graphene superlattice with anti-unitary symmetry 51

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Figure 4.1. Left panel: Hexagonal lattice of graphene, decorated by a periodicpotential modulation. Different colors distinguish the carbon atoms on the A andB sublattice, each of which has an ionic potential VAn , VBn , n = 1, 2, 3, inducedby the substrate. The lattice constant a0 of the original hexagonal lattice isincreased by a factor

√3 in the superlattice. Right panel: The Dirac points at

the K and K′ points of the original Brillouin zone of graphene (dashed hexagon)are folded onto the Γ point at the center of the superlattice Brillouin zone (yellowhexagon).

atoms. 1 The Brillouin zone remains hexagonal, but the two Dirac conesat opposite corners K, K′ of the original Brillouin zone of graphene arefolded onto the center Γ of the superlattice Brillouin zone. Depending onthe relative magnitude of α, β, µ a gap may open or a linear or quadraticband crossing may appear [103–105].

If there is translational invariance in the y-direction, 2 the momentumcomponent py ≡ q is a good quantum number and we may consider the

1As derived in Ref. [105], the relationship between the parameters V0, µ, α, β in theDirac Hamiltonian (4.1) and the ionic potentials VAn , VBn , n = 1, 2, 3, in Fig. 4.1 is:6V0 =

∑n(VAn +VBn ), 6µ =

∑n(VAn−VBn ), 6α = 2VA1−VA2−VA3 +i

√3(VA2−VA3 ),

6β = 2VB1 − VB2 − VB3 + i√

3(VB2 − VB3 ). When comparing with Ref. [105], note thatwe are using a basis in which the single-valley Dirac Hamiltonian vF(p · σ) is the samein both valleys.

2 For definiteness, we are considering the orientation of the graphene lattice shownin Fig. 4.1, with the x-direction parallel to the zigzag edge. A rotation of the lattice byan angle φ is equivalent to multiplication of α and β by eiφ.


Hamiltonian H(q) at a fixed q. Time-reversal symmetry is expressed by

(τy ⊗ σy)H∗(q)(τy ⊗ σy) = H(−q), (4.3)

where the complex conjugation should also be applied to the momentumoperator px = −i~∂/∂x ≡ −i~∂x. Note the sign change of q. An addi-tional anti-unitary symmetry without inversion of q exists if β∗ = α,

H(q) = vF(−i~σx∂x + qσy) + µτz ⊗ σz (4.4)− (τy ⊗ σy)Reα− (τx ⊗ σy)Imα, if β∗ = α,

⇒ τxH∗(q)τx = −H(q). (4.5)

4.3 Topological phase transitionsThe symmetry (4.5) is formally identical to charge-conjugation symmetryin a superconductor, where τx switches electron and hole degrees of free-dom. Because the symmetry operation C = τxK (with K = complex con-jugation) squares to +1, it is symmetry class D in the Altland-Zirnbauerclassification of topological states of matter 3 [108]. This correspondenceopens up the possibility of a phase transition into a phase that is analo-gous to a topological superconductor [10, 16] — with the K and K′ valleysplaying the role of electron and hole.

The q-dependent topological quantum number Q(q) of the Hamilto-nian (4.4) follows from Kitaev’s Pfaffian formula [6],

Q(q) = sign Pf[τxH(q)]px=0. (4.6)

(The multiplication by τx ensures that the Pfaffian Pf is calculated of anantisymmetric matrix.) We find

Q(q) = sign (v2Fq

2 + µ2 − |α|2). (4.7)

The graphene superlattice is always topologically trivial (Q = +1) forlarge |q|. However, provided that |µ| < |α|, it is topologically nontrivial(Q = −1) in an interval near q = 0. There is a pair of topological phasetransitions at

q = ±qc, vFqc =√|α|2 − µ2. (4.8)

3 For µ = 0 the Hamiltonian (4.4) also has chiral symmetry (it anticommutes withσz), so then the symmetry class would be BDI rather than D. The difference is notessential for our analysis.

4.3 Topological phase transitions 53

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Figure 4.2. Top panel: Energy spectrum of superlattice graphene with Hamil-tonian (4.4) at µ = 0.5, α = 1 = β (in units of ~vF/a0). Middle panel: Cor-responding reflection probabilities from a superlattice-graphene strip of lengthL = 10 a0 in the x-direction, connected at x = 0 and x = L to heavily dopedpristine-graphene electrodes. In the topologically nontrivial regime near normalincidence (|q| < qc =

√|α|2 − µ2) the reflection is completely into the opposite

valley. Bottom panel: Topological order parameter OL(q) = Det r, computedfrom Eq. (4.10). The switch from intra-valley to inter-valley reflection lines upwith the switch from OL(q) = +1 to −1.


To probe the topological phase transition, we contact the graphenesuperlattice at x = 0 and x = L by pristine-graphene electrodes, heavilydoped so that the Fermi energy in the pristine graphene is high above theDirac point. By analogy with the conductance of a Kitaev wire [11], the2 × 2 reflection matrix r in the large-L limit should be fully diagonal inthe topologically trivial phase and fully off-diagonal in the topologicallynontrivial phase. In the superconducting problem this means that thereis complete Andreev reflection from the topological superconductor. Herethe analogue is a complete valley switch: An incident electron in valley Kis reflected in the other valley K′ with unit probability when |q| < qc.

4.4 Valley switchLet us see how this expectation is borne out by an explicit calculation(detailed in the appendix). The energy spectrum of the Hamiltonian (4.4)is shown in Fig. 4.2. The intervalley coupling splits the Dirac cone atthe Γ point into a pair of cones at k = (0,±qc) which are gapped out if|µ| > |α|. The same figure shows the E = 0 reflection probabilities |rK′K|2and |rKK|2 with and without a valley switch. We clearly see the transitionfrom complete inter-valley to complete intra-valley reflection at q = ±qc.

The analytical formulas that govern this transition are simplest for thespecial case µ = 0, when qc = |α|/vF. We find the reflection matrix

r =(rKK rKK′

rK′K rK′K′

), rKK = rK′K′ = −Z−1 sinh 2qL,

rKK′ = r∗K′K = −iZ−1(α/|α|) sinh 2qcL, (4.9)Z = cosh 2qcL+ cosh 2qL.

(We have set ~ ≡ 1.) In the topologically trivial regime |q| > qc theoff-diagonal elements of r vanish ∝ exp[−2(|q| − qc)L], while in the topo-logically nontrivial regime |q| < qc it is the diagonal elements that vanish∝ exp[−2(qc − |q|)L].

To confirm that the valley switch at |q| = qc is due to a topologicalphase transition in a finite system, we calculate the L-dependent topolog-ical order parameter [38]

OL(q) = Det r = − tanh(ξ−L) tanh(ξ+L),

ξ± = |α| ±√v2

Fq2 + µ2.

(4.10)

4.5 Robustness of the valley switch 55

Unlike the Pfaffian invariant (4.6), the determinant (4.10) crosses oversmoothly from +1 to −1 in an interval around |q| = qc. This intervalbecomes narrower and narrower with increasing L, approaching the dis-continuous topological phase transition in the infinite-L limit. At |q| = qcthe determinant of r vanishes, signifying the opening of a reflectionlessmode, a mode that is transmitted with unit probability through the su-perlattice.

4.5 Robustness of the valley switch

The anti-unitary symmetry (4.5) is broken if we move away from thesymmetry point β∗ = α. Let us find out how sensitive the valley switcheffect is to the symmetry breaking. The simplest formulas appear forµ = 0. To quantify the magnitude of the valley switch we calculate theratio ρ = |rKK/rK′K|2 at q = 0. We find

ρ = (|α|2 − |β|2)2

4|α∗Ξ + βΞ∗|2 , Ξ =√αβ coth(

√αβL), (4.11)

→ 14(|α| − |β|)2 |αβ|−1 for L→∞ if arg (αβ) 6= π,

see Fig. 4.3 for a plot. Only the absolute value of the complex amplitudesα, β enters in the large-L limit, provided that αβ is not on the negative realaxis (when reflection is quenched by the opening of a propagating mode). 4

The valley switch happens with 100% probability if |α| = |β|, but theratio |α/β| may differ from unity by as much as a factor of two and still90% of the reflected intensity at normal incidence happens with a valleyswitch. Fig. 4.4 shows that this robustness to variation of parameterspersists for µ 6= 0 – as long as µ < |α| ≈ |β|.

Concerning the robustness of the valley switch to disorder, we firstlynote that forward scattering events which only couple the transverse mo-menta within the topologically nontrivial interval |q| < qc do not spoil thetopological protection. Secondly, large-angle scattering with a mean freepath longer than the penetration depth ξ = 1/qc into the superlattice willalso not affect the valley switch.

4 In terms of the atomic potentials, the condition argαβ = π is equivalent toVA1 (VB3 − VB2 ) + VA2 (VB2 − VB1 ) + VA3 (VB1 − VB3 ) = 0, VA1 (VB2 + VB3 − 2VB1 ) +VA2 (VB1 + VB2 − 2VB3 ) + VA3 (VB1 + VB3 − 2VB2 ) > 0.


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Figure 4.3. Plot of the fraction 1/(1 + ρ) = |rK′K|2/(|rKK|2 + |rK′K|2) of thereflected intensity at normal incidence (q = 0) that is reflected in the oppositevalley, calculated from Eq. (4.11). The reflection with valley switch happensat close to unit probability provided that the product αβ stays away from thenegative real axis.

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Figure 4.4. Reflection probabilities from the superlattice-graphene strip withand without a valley switch, for four values of µ. The parameters α, β are quitefar from the ideal symmetry point β∗ = α, but still the reflection near normalincidence happens predominantly in the opposite valley (|rK′K|2 ≈ 1) providedµ < |α| ≈ |β|.

4.6 Conclusion 57

4.6 Conclusion

We have presented a topological mechanism that switches the K,K′ valleyindex of Dirac fermions in graphene. Unlike scattering processes thatrequire control on the atomic scale, such as intervalley reflection froman armchair edge, our valley switch relies on the long-range effect of asuperlattice potential.

The valley switch is protected by a topological invariant, the Pfaffian(4.6), originally introduced by Kitaev to describe a topologically nontriv-ial superconductor [6]. Because of this topological protection the switchhappens with 100% probability even in the presence of a large Fermi en-ergy mismatch at the interface with the superlattice. It is analogous tothe 100% Andreev reflection from a Majorana zero-mode, which is alsounaffected by a Fermi energy mismatch at the interface with the super-conductor [11].

We have identified the symmetry point of the superlattice Hamiltonianat which an anti-unitary symmetry appears that is analogous to charge-conjugation symmetry in a superconductor, and we have checked that thevalley switch remains close to 100% in a broad parameter range aroundthat symmetry point. We expect that the analogy between intervalleyreflection and Andreev reflection revealed in this work can provide furtheruseful additions to the valleytronics toolbox [98].

4.7 Appendix: Scattering matrix of the graphenesuperlattice

The calculation of the scattering matrix of the graphene superlattice re-gion 0 < x < L, sandwiched between heavily doped pristine-graphenecontacts, proceeds as follows. (See Refs. [69, 109] for similar calculationsin graphene.)

We start from the Dirac HamiltonianH(px, py), given by Eq. (4.2). Weconsider solutions of the Dirac equation HΨ = EΨ at energy E that areplane waves in the y-direction, Ψ(x, y) = Ψq(x)eiqy. The four-componentspinor Ψq(x) in the region 0 < x < L is a solution of

∂

∂xΨq(x) = Ξ(q)Ψq(x), Ξ(q) = iv−1

F σx[E −H(0, q)], (4.12)


resulting in the transfer matrix

Ψq(L) = T (q)Ψq(0), T (q) = eΞ(q). (4.13)

The next step is to transform to a basis of right-moving and left-movingmodes in the contact regions x < 0, x > L. The Dirac Hamiltonian inthose regions is

Hcontact = vF(pxσx + pyσy)− Vdoping. (4.14)

[We use the same valley-isotropic basis Ψ = (ψKA, ψKB,−ψK′B, ψK′A) asin Eq. (4.2).] In the limit Vdoping → ∞ of infinitely doped contacts theright-moving modes Ψ+(x, y) and left-moving modes Ψ−(x, y) are givenfor x < 0 by

Ψ+ = c+Ke

ikx+iqy

1100

+ c+K′e

ikx+iqy

0011

,

Ψ− = c−Ke−ikx+iqy

1−100

+ c−K′e−ikx+iqy

001−1

,(4.15)

with vFk = Vdoping → ∞. The same expression with x 7→ x − L appliesfor x > L.

The transfer matrix in the basis (c+K, c−K, c

+K′ , c

−K′) is

T (q) = HT (q)H, H = 1√2

1 1 0 01 −1 0 00 0 1 10 0 1 −1

. (4.16)

After this “Hadamard transform” [109] we can directly read off the ele-ments of the reflection matrix r from the x = 0 interface,

r =(rKK rKK′

rK′K rK′K′

)= −(T−−)−1 · T−+, (4.17a)

T−− =(T22 T24T42 T44

), T−+ =

(T21 T23T41 T43

). (4.17b)

The final results are lengthy and not recorded here, but they are easilyderived using a computer algebra system.

Chapter 5

Low-high voltage duality intunneling spectroscopy ofthe Sachdev-Ye-Kitaevmodel

5.1 Introduction

The Sachdev-Ye-Kitaev model, a fermionic version [27] of a disorderedquantum Heisenberg magnet [28, 110], describes how N fermionic zero-energy modes are broadened into a band of width J by random infinite-range interactions. The phase diagram of the SYK Hamiltonian can besolved exactly in the large-N limit [29, 30, 47], when a conformal sym-metry emerges at low energies that forms a holographic description of thehorizon of an extremal black hole in a (1+1)-dimensional anti-de Sitterspac [27, 47, 110, 111].

To be able to probe this holographic behaviour in the laboratory, it isof interest to create a “black hole on a chip" [31, 32, 53], that is, to realizethe SYK model in the solid state. Ref. [32] proposed to use a quan-

The contents of this chapter have been published and reprinted with permissionfrom N. V. Gnezdilov, J. A. Hutasoit, and C. W. J. Beenakker, Phys. Rev. B 98,081413(R) (2018). Copyright 2018 by the American Physical Society.



60 Chapter 5. Current voltage characteristic of the SYK quantum dot

Figure 5.1. Tunneling spectroscopy of a graphene flake, in order to probe thecomplex-valued SYK model [53]. We calculate the current I driven by a voltageV through a single-channel point contact (coupling strength Γ) into a grapheneflake on a grounded conducting substrate. At the charge neutrality point a chiralsymmetry ensures that the zeroth Landau level (degeneracy N = eΦ/h for anenclosed flux Φ) is only broadened by electron-electron interactions (strength J).For a sufficiently random boundary the quantum dot can be described by theSYK Hamiltonian (5.1).

tum dot formed by an opening in a superconducting sheet on the surfaceof a topological insulator. In a perpendicular magnetic field the quan-tum dot can trap vortices, each of which contains a Majorana zero-mode[8]. Chiral symmetry ensures that the band only broadens as a result offour-Majorana-fermion terms in the Hamiltonian, a prerequisite for thereal-valued SYK model. A similar construction uses an array of Majoranananowires coupled to a quantum dot [31]. Since it might be easier to startfrom conventional electrons rather than Majorana fermions, Ref. [53] sug-gested to work with the complex-valued SYK model of interacting Diracfermions in the zeroth Landau level of a graphene quantum dot. Chiralsymmetry at the charge-neutrality point again suppresses broadening ofthe band by two-fermion terms.

The natural way to study a quantum dot is via transport proper-ties. Electrical conduction through chains of SYK quantum dots has beenstudied in Refs. [50, 112–118]. For a single quantum dot coupled to atunnel contact, as in Fig. 5.1, Refs. [31, 32, 53] studied the limit of neg-ligibly small coupling strength Γ, in which the differential conductanceG = dI/dV equals the density of states of the quantum dot. Conformalsymmetry in the large-N limit gives a low-voltage divergence ∝ 1/

√V ,

until eV drops below the single-particle level spacing δ ' J/N [119–121].Here we investigate how a finite Γ affects the tunneling spectroscopy.

5.2 Tunneling Hamiltonian 61

We focus on the complex-valued SYK model for Dirac fermions, as in thegraphene quantum dot of Ref. [53]. Our key result is that in the large-N conformal symmetry regime J/N eV J the zero-temperaturedifferential conductance of the quantum dot depends on Γ, J , and V onlyvia the dimensionless combination ξ = eV J/Γ2. Low and high voltagesare related by the duality G(ξ) = G(π/ξ), providing an experimentalsignature of the conformal symmetry.

5.2 Tunneling HamiltonianWe describe the geometry of Fig. 5.1 by the Hamiltonian

H = HSYK +∑p

εpψ†pψp +

∑i,p

(λic†iψp + λ∗iψ†pci),

HSYK = (2N)−3/2∑ijklJij;kl c

†ic†jckcl, (5.1)

Jij;kl = J∗kl;ij = −Jji;kl = −Jij;lk.

The annihilation operators ci, i = 1, 2, . . . represent the N = hΦ/e in-teracting Dirac fermions in the spin-polarized zeroth Landau level of thegraphene quantum dot (enclosing a flux Φ). Two-fermion terms c†icj aresuppressed by chiral symmetry when the Fermi level µ = 0 is at the charge-neutrality point (Dirac point) [53]. The operators ψp represent electronsat momentum p in the single-channel lead (dispersion εp = p2/2m, lin-earized near the Fermi level), coupled to mode i in the quantum dot withcomplex amplitude λi. The tunneling current depends only on the sum of|λi|2, via the coupling strength

Γ = πρlead∑i|λi|2, ρlead = (2π~vF)−1. (5.2)

If T ∈ (0, 1) is the transmission probability into the quantum dot, one hasΓ ' T Nδ ' T J .

The Hamiltonian HSYK is the complex-valued SYK model [47] if wetake random couplings Jij;kl that are independently distributed Gaussianswith zero mean 〈Jij;kl〉 = 0 and variance 〈|Jij;kl|2〉 = J2. The zerothLandau level then broadens into a band of width J , corresponding toa single-particle level spacing δ ' J/N (more precisely, δ ' J/N lnN)[119]. In the energy range δ ε J the retarded Green’s functions canbe evaluated in saddle-point approximation [47],

GR(ε) = −iπ1/4

√β

2πJΓ (1/4− iβε/2π)Γ (3/4− iβε/2π) , (5.3)


where β = 1/kBT and Γ(x) is the Gamma function. At zero temperaturethis simplifies to

GR(ε) = −iπ1/4 exp[

14 iπ sgn(ε)

]|Jε|−1/2 . (5.4)

Quantum fluctuations around the saddle point cut off the low-ε divergencefor |ε| < δ [119–121].

5.3 Tunneling current

The graphene flake lies on a grounded substrate, 1 so the current is entirelydetermined by the transmission of electrons through the point contact.The current operator I is given by the commutator

I = ie~

[H,∑pψ†pψp

]= i e

~∑n,p

(λnc†nψp − λ∗nψ†pcn

). (5.5)

We calculate the time-averaged expectation value of I using the Keldyshpath integral technique [123–125], which has previously been applied tothe SYK model in Refs. [112, 113, 117, 126]. The expectation value I ofthe tunneling current is given by the first derivative of cumulant generatingfunction: [124]

I = −i limχ→0

∂

∂χlnZ(χ) , (5.6)

Z(χ) =⟨TC exp

(−i∫Cdt

[H + 1

2χ(t)I])⟩. (5.7)

Here TC indicates time-ordering along the Keldysh contour [123] of thecounting field χ(t), equal to +χ on the forward branch of the contour(from t = 0 to t = ∞) and equal to −χ on the backward branch (fromt =∞ to t = 0). The calculation is worked out in the Appendix.

The result for the differential conductance is

G = dI

dV= e2

h

∫ +∞

−∞dε f ′(ε− eV ) 4Γ ImGR(ε)

|1 + iΓGR(ε)|2 , (5.8)

1 We assume that the grounded substrate does not spoil the non-Fermi-liquid stateof the quantum dot. This might happen if the coupling becomes too strong, accordingto S. Banerjee and E. Altman, [122].

5.4 Low–high voltage duality 63

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Figure 5.2. Differential conductance G = dI/dV calculated from Eq. (5.8), asa function of dimensionless voltage ξ = eV J/Γ2 for three different temperatures.On the semi-logarithmic scale the duality between low and high voltages showsup as a reflection symmetry along the dotted line (where ξ =

√π).

where f(ε) = (1+eβε)−1 is the Fermi function. Substitution of the confor-mal Green’s function (5.3) gives upon integration the finite temperaturecurves plotted in Fig. 5.2.

At zero temperature f ′(ε− eV )→ −δ(ε− eV ) and substitution of Eq.(5.4) produces a single-parameter function of ξ = eV J/Γ2,

G(ξ) = e2

h

2√

2√2 + π1/4ξ−1/2 + π−1/4ξ1/2 . (5.9)

5.4 Low–high voltage dualityThe T = 0 differential conductance (5.9) in the conformal regime J/N eV J satisfies the duality relation

G(ξ) = G(π/ξ), if N−1(J/Γ)2 ξ, 1/ξ (J/Γ)2. (5.10)

The V -to-1/V duality is visible in the semi-logarithmic Fig. 5.2 by a reflec-tion symmetry of the differential conductance along the ξ =

√π axis. The

symmetry is precise at T = 0, and is broken in the tails with increasingtemperature.


The voltage range in which V and 1/V are related by Eq. (5.10) coversthe full conformal regime for N ' (J/Γ)4. In this voltage range the 1/

√V

tail at high voltages crosses over to a√V decay at low voltages. The high-

voltage tail reproduces the 1/√V differential conductance that follows [31,

32, 53] from the density of states in the limit Γ → 0 (since ξ → ∞ forΓ → 0). The density of states gives [119–121] a crossover to a

√V decay

when eV drops below the single-particle level spacing δ ' J/N . Ourfinite-Γ result (5.9) implies that this crossover already sets in at largervoltages eV ' Γ2/J , well above δ for N (J/Γ)2.

The symmetrically peaked profile of Fig. 5.2 is a signature of conformalsymmetry in as much as this produces a power-law singularity in theretarded propagator at low energies. It is not specific for the square-rootsingularity (5.4), other exponents would give a qualitatively similar low-high voltage duality. For example, the generalized SYK2p model with2p ≥ 4 interacting Majorana fermion terms has a ε(1−p)/p singularity [29,112], corresponding to the duality G(ξp) = G(Cp/ξp) with Cp a numericalcoefficient and ξp = (eV )2(p−1)/pJ2/pΓ−2. In contrast, a disordered Fermiliquid such as the non-interacting SYK2 model, with Hamiltonian H =∑ij Jijc

†icj , has a constant propagator at low energies and hence a constant

dI/dV in the range J/N eV J .

5.5 Conclusion

We have shown that tunneling spectroscopy can reveal a low–high voltageduality in the conformal regime of the Sachdev-Ye-Kitaev model of Ninteracting Dirac fermions. A physical system in which one might searchfor this duality is the graphene quantum dot in the lowest Landau level,proposed by Chen, Ilan, De Juan, Pikulin, and Franz [53].

As argued by those authors, one should be able to reach N of order 102

for laboratory magnetic field strengths in a sub-micrometer-size quantumdot. This leaves two decades in the conformal regime J/N eV J .If we tune the tunnel coupling strength near the ballistic limit Γ . J , itshould be possible even for these moderately large values of N to achieveN ' (J/Γ)4 and access the duality over two decades of voltage variation.For such large Γ the condition on temperature, kBT Γ2/J would thenalso be within experimental reach (J ' 34 meV from Ref. [53] and Γ '10 meV has kBT = 10−2 Γ2/J at T = 300 mK).

5.6 Appendix: Outline of the calculation 65

5.6 Appendix: Outline of the calculation

We describe the calculation leading to Eq. (5.8) for the current-voltagecharacteristics, generalizing it to nonzero chemical potential µ and in-cluding also the shot noise power. We set ~ and e to unity, except for thefinal formulas.

5.6.1 Generating function of counting statistics

Arbitrary cumulants of the current operator (5.5) can be obtained fromthe generating function (5.7). A gauge transformation allows us to writeequivalently

Z(χ) =⟨TC exp

(−i∫CH(t) dt

)⟩, (5.11)

H(t) = HSYK +∑pεpψ

†pψp − µ

∑nc†ncn

+∑n,p

(eiχ(t)/2λnc

†nψp + e−iχ(t)/2λ∗nψ

†pcn

). (5.12)

For generality we have added a chemical potential term ∝ µ. (In the maintext we take µ = 0, corresponding to a quantum dot at charge neutrality.)

We need the advanced and retarded Green’s functionsGA(ε) =(GR(ε)

)∗and the Keldysh Green’s function

GK(ε) = F(ε)(GR(ε)−GA(ε)

), F(ε) = tanh(βε/2). (5.13)

These are collected in the matrix Green’s function G, which on the Keldyshcontour has the representation [125, 127]

G =(GR GK

0 GA

)= Lσ3

(G++ G+−

G−+ G−−

)L†, (5.14)

L = 1√2

(1 −11 1

), σ3 =

(1 00 −1

), (5.15)

in terms of the Green’s functions on the forward and backward branches


of the contour:

G++(t, t′) = −iN−1∑n

⟨T cn(t)c†n(t′)

⟩, (5.16a)

G+−(t, t′) = iN−1∑n

⟨c†n(t′)cn(t)

⟩, (5.16b)

G−+(t, t′) = −iN−1∑n

⟨cn(t)c†n(t′)

⟩, (5.16c)

G−−(t, t′) = −iN−1∑n

⟨T −1 cn(t)c†n(t′)

⟩. (5.16d)

The operators T and T −1 order the times in increasing and decreasingorder, respectively.

5.6.2 Saddle point solution

In the regime J/N ε J the Green’s function of the SYK model isgiven by the saddle point solution [47]

GR(ε) = −iCe−iθ

√β

2πJΓ(

14 − iβε2π + iE

)Γ(

34 − iβε2π + iE

) , (5.17)

with the definitions

e2πE =sin(π4 + θ

)sin(π4 − θ

) , C = (π/ cos 2θ)1/4 . (5.18)

The angle θ ∈ (−π/4, π/4) is a spectral asymmetry angle [128], determinedby the charge per site Q ∈ (−1/2, 1/2) on the quantum dot according to[56]

Q = N−1∑i〈c†ici〉 − 1

2 = −θ/π − 14sin 2θ. (5.19)

For µ = 0, when Q = 0, one has θ = 0, C = π1/4. In good approximation(accurate within 15%),

θ ≈ −12πQ ⇒ C ≈ (π/ cosπQ)1/4. (5.20)

In the mean-field approach the quartic SYK interaction (5.1) is re-placed by a quadratic one with the kernel G−1 from Eq. (5.14). A Gaussian


integration over the Grassmann fields gives the generating function

lnZ =∫ ∞−∞

dε2π ln

det[1− ΓΞ(ε)Λ†G(ε)Λ

]det [1− ΓΞ(ε)G(ε)]

, (5.21)

Λ =(

cos (χ/2) i sin (χ/2)i sin (χ/2) cos (χ/2)

), Ξ(ε) = −i

(1 2F(ε− V )0 −1

). (5.22)

The matrix Ξ(ε) is the Keldysh Green’s function of the lead, integratedover the momenta. This evaluates further to

lnZ =∫ ∞−∞

dε

2π ln[1 +

iΓ(GR −GA

)(1 + iΓGR) (1− iΓGA)

·(

[1−F(ε)F(ε− V )] (cosχ− 1) + i [F(ε)−F(ε− V )] sinχ)].

(5.23)

At zero temperature the distribution function simplifies to F(ε) 7→sgn (ε), hence

lnZ =∫ V

0

dε

2π ln

1 +2iΓ

(GR −GA

) (eiχ − 1

)(1 + iΓGR) (1− iΓGA)

. (5.24)

5.6.3 Average current and shot noise power

A p-fold differentation of Z(χ) with respect to χ gives the p-th cumu-lant of the current. In this way the full counting statistics of the chargetransmitted through the quantum dot can be calculated [124]. The firstcumulant, the time-averaged current I from Eq. (5.6), is given by

I = e

h

∫ +∞

−∞dε

iΓ [F(ε)−F(ε− V )](GR −GA

)(1 + iΓGR) (1− iΓGA) , (5.25)

which is Eq. (5.8) from the main text.At zero temperature the differential conductance G = dI/dV is

G(ξ) = 2e2

h

[1 + 1

2 sin (π/4 + θ)

(√ξ

C+ C√

ξ

)]−1

, (5.26)


with ξ = eV J/Γ2. The duality relation

G(ξ) = G(C4/ξ) (5.27)

reduces to the one from the main text, G(ξ) = G(π/ξ), when we setµ = 0⇒ θ = 0⇒ C = π1/4.

The second cumulant, the shot noise power P , follows similarly from

P = − limχ→0

∂2

∂χ2 lnZ(χ). (5.28)

The Fano factor F , being the ratio of the shot noise power and the currentat zero temperature, is simply given by

F = dP/dV

dI/dV= e

(1− h

e2G

). (5.29)

It has the same one-parameter scaling and duality as G. The fact thathigher order cumulants of the current have the same scaling as the differ-ential conductance is a consequence of the single-point-contact geometry,with a single counting field χ(t). This does not carry over to a two-point-contact geometry.

Chapter 6

Isolated zeros in the spectralfunction as signature of aquantum continuum

6.1 IntroductionThe energy levels of a generic quantum system can be organized either indiscrete or continuum spectra. The discrete spectrum is associated withthe existence of stable bound states, corresponding to localized long-livedquasiparticles with well defined energy, while the continuum spectrumreflects either multiparticle states with finite phase volume – e.g., particle-hole continuum, a thermodynamically large number of interacting degreesof freedom – a thermal bath, or the absence of quasiparticles as such – aquantum critical continuum. The qualitative difference between discretestates and continua is manifest in the spectral function: It either exhibitssharp peaks in the former case or a (peakless) smooth profile in the latterone. In more complicated quantum systems which have both discrete andcontinuum subparts, the spectral function can take a distinct shape. Thewell-known example is the Fano resonance [129–131], which arises whenone probes the continuum in the presence of the isolated states, whose

The contents of this chapter have been published and reprinted with permissionfrom N. Gnezdilov, A. Krikun, K. Schalm, and J. Zaanen, Phys. Rev. B 99, 165149(2019). Copyright 2019 by the American Physical Society.



70 Chapter 6. Isolated zeros in the spectral function

interference leads to a characteristic line shape with a neighboring peakand zero.

The continued strong interest in quantum criticality and various non-Fermi liquid models inherently concerns the study of a quantum systemwith a continuum rather than a discrete spectrum. A defining feature ofthe non-Fermi liquid is the absence of the stable quasiparticles. No dis-crete peaks are seen in the spectral function; rather it has the appearanceof a power law [132]. The same type of spectra are characteristic of quan-tum critical systems [133], which are described by conformal field theoryand a lack of the quasiparticles. Conformal field theories also genericallyexhibit power laws, controlled by the anomalous dimensions of the oper-ators. These non-Fermi liquid and quantum critical ideas appear to bevery relevant to the unconventional states of strongly correlated quan-tum matter, most notoriously the strange metal phase observed in hightemperature superconductors and heavy fermion systems [134]. This ex-perimental relevance in turn triggered the active theoretical effort in thelast decades, aimed at building controlled models of non-Fermi liquidsand/or quantum critical systems. Among the latest developments is theSachdev-Ye-Kitaev model [27, 28], which has a power law spectral func-tion due to the strong entanglement between the constituent fermions. Afew proposals have arisen recently on the experimental realization of theSYK model [31, 32, 53], which makes the question about the observableproperties of the quantum critical continuum especially important. Thiscase of the 0 + 1 dimensional “quantum dot” systems is special since themultiparticle phase space shrinks to zero and any observed continuumspectrum can not be of quasiparticle nature. Therefore the detection ofcontinuum in a quantum dot signals the interesting physics, whether it isrelated to the particular SYK model or not.

In this paper we point out a very basic and therefore very generalfeature of quantum systems containing the continuum as a subpart. Weshow that the spectral function exhibits a distinct line shape characterizedby an isolated zero arising when one probes a discrete subpart of thesystem that consists both of discrete states and a continuum. It is the“mirror” of the Fano resonance. A similar effect has been pointed outby one of us in the form of interference effects between different decaychannels of the core holes created in high energy photoemission processes[130]. This distinct spectral function zero is in principle observable inexperiment and it can therefore be used as a signature of the presence of

6.2 Isolated zeros in the spectral function 71

a continuum subpart in quantum systems, in particular in the laboratoryrealizations of quantum criticality. The important aspect is that this probeis indirect – it does not interact with the continuum system. This can bea significant advantage since the quantum critical systems are notoriouslyfragile and the direct measurement could easily destroy them.

We shall first discuss the generic mechanism of the phenomenon andthen demonstrate how it works in several examples with continuum sub-systems: (1) two single fermion quantum dots coupled to an SYK quantumdot, (2) a one-dimensional wire coupled to a chain of the SYK nodes, and(3) a holographic model of a local quantum critical system with a periodiclattice.

6.2 Isolated zeros in the spectral function

Consider two fermions χA, χB with discrete quasiparticle energies ΩA, ΩB,respectively, coupled to a fermion ψ with a continuum spectrum character-ized by a Green’s function G(ω). The Euclidean action for the full systemreads:

S = Sχ + Sψ + Sint, (6.1)

Sχ =∫dt

∑σ=A,B

χσ (∂t + Ωσ)χσ,

Sψ = −∫dtdt′ ψ(t)G

(t− t′

)−1ψ(t′),

Sint =∫dt

∑σ=A,B

(λσ ψ χσ + λ∗σχσψ

).

As shown in Appendix 6.7.1 our effect is present for any tunneling cou-plings λσ, however for brevity here we consider the case λA = λB ≡ λ.Integrating out the fermion ψ in the continuum gives the Green’s functionfor the fermions χA, χB:

Gσσ′(ω)−1=(ω−ΩA−|λ|2G(ω) −|λ|2G(ω)−|λ|2G(ω) ω−ΩB−|λ|2G(ω)

). (6.2)

We are interested in the spectral function of a single fermion χA: AA(ω) =− 1π ImGRAA(ω), where the retarded Green’s function is obtained from the

inverse of the matrix (6.2) after analytic continuation ω → ω + iδ sign(ω)


with δ = 0+. The result is

AA(ω) = −|λ|2

π

ImGR(ω)|D(ω)|2

(ω − ΩB)2 , (6.3)

D(ω)=(ω − ΩA)(ω − ΩB)−|λ|2(2ω − ΩA − ΩB)GR(ω).

In (6.3) GR is the retarded Green’s function of the fermion ψ. Here we’vetaken the limit δ → 0+ assuming the imaginary part of GR stays finite. InEqs. (6.14) and (6.30) in Appendices 6.7.1 and 6.7.2 we give the resultsfor finite δ, i.e., a universal finite lifetime for the fermions χA,B.

Our main observation is that the spectral function of fermion χAhas a double zero exactly at the energy level of fermion χB: ω = ΩB. Thephysics of the zero in (6.3) can be better understood by considering thematrix structure of the imaginary part of the Green’s function

ImGRσσ′∼(

(ω − ΩB)2 (ω − ΩB)(ω − ΩA)(ω − ΩB)(ω − ΩA) (ω − ΩA)2

). (6.4)

This matrix is degenerate and has a single zero eigenvalue with eigenvector

|χ0〉 = (ω − ΩA)|χA〉 − (ω − ΩB)|χB〉 . (6.5)

In other words this linear combination of probes is completely obliviousto the continuum. One immediately sees that for ω = ΩB the probefermion χA aligns with the “oblivious” combination and thus does notget absorbed/reflected. This explains the double zero at the resonancefrequency. In the degenerate case Ω ≡ ΩA = ΩB the double zero atω = Ω is canceled by the same multiplying factor in the denominator D(ω)and it’s easy to understand the absence of the effect: For any energy thecombination (6.5) is now fixed to |χ0〉 ∼ |χA〉−|χB〉 and it never coincideswith one of the probes.

Though similar, this occurrence of the double zero for the discreteprobe is the obverse of the Fano resonance, as the latter follows whenone integrates out the discrete fermions χA,B. The Green’s function forthe continuum fermion ψ then becomes (setting the χB coupling to zero)G−1ψψ = G−1 − |λ|2/(ω − ΩA), with the spectral function

Aψ(ω)=− 1π

ImGRψψ(ω)=− 1π

(ω − ΩA)2ImGR(ω)|ω − ΩA − λ2GR(ω)|2 . (6.6)

The double zero at the energy ω = ΩA is again obvious but the polestructure is different. This Fano response has a pole at ω = ωp near ΩA

6.3 SYK model 73

shifted by the interaction with continuum. For small λ we can approximateits location as ωp = ΩA + λ2G(ΩA). In this case the spectrum functionnear the pole ω ∼ ωp takes the familiar Fano form [131]

Aψ ∝(ε+ qΓ)2

ε2 + Γ2 , (6.7)

with the parameters ε = (ω − ΩA − λ2ReG(ΩA)), Γ = λ2ImG(ΩA), andq = ReG(ΩA)/ImG(ΩA).

One obvious extension of the action is to include a direct couplingbetween fermions S′χ =

∫dt (κχAχB + κ∗χBχA). We analyze this in Ap-

pendix 6.7.1. The complex zeros of the spectral function AA(ω) are givenin this case by (ΩB−Reκ± iImκ). The real part of the coupling shifts theposition of the zero, while the imaginary one moves it off the real axis. Inthe latter case the exact real zero in the spectral function gets supersededby the localized depression with finite minimum value, but the overall lineshape stays intact. This is reminiscent of the Fano resonance, where theexact zero is never observed in practice, but the full line shape points outthe characteristic physics [130, 131].

6.3 SYK model

To give an explicit example of how this zero arises we first focus onthe 0 + 1 dimensional model featuring quantum critical continuum – theSachdev-Ye-Kitaev (SYK) model [27, 28] with quartic interaction amongcomplex fermions [47]. We couple the continuum SYK model at chargeneutrality point (µSY K = 0) to two discrete states with combined Hamil-

Figure 6.1. Cheburashkian geometry [135]. Two quantum dots with de-screte energy levels ΩA and ΩB are symmetrically coupled to an SYK quantumdot.


Figure 6.2. Isolated zero for the SYK model. For a given coupling to acontinuum λ = 0.135, F = 10, the dependence of the spectral function of state Aon the energy level of the state B is shown. The isolated zero is always present atΩB . The peak at ΩA gets sharpened due to the proximity of zero, but is destroyedif ΩA = ΩB [not shown, see Eq. (6.3)]. Green arrows show the positions of thepoles of the Green’s function defined as zeros of the determinant D(ω) in thespectral function in Eq. (6.3). Note that they do not coincide exactly with themaxima of the spectral function due to the proximity of zeros.

Figure 6.3. Asymptotic shape of the peak. Depending on the value of thecoupling λ the shape of the peak at ΩA may differ considerably in the asymptoticcases when ΩB ΩA (red lines) and ΩB ΩA (black lines). When λ 1 (leftpanel) the two cases are identical, for intermediate values of λ (middle panel) thewidth of the peak changes, for strong coupling (right panel) the position of thepeak is affected as well.

tonianH =

∑σ=A,B

Ωσχ†σχσ +HSY K +Hint, (6.8)

HSY K = 1(2N)3/2

N∑i,j,k,l=1

Jij;kl ψ†iψ†jψkψl,

Hint = 1√N

N∑i=1

∑σ=A,B

(λiψ

†iχσ + λ∗iχ

†σψi

).

6.4 Cluster of SYK nodes 75

The fermions χA, χB can be thought of as a pair of single state tunablequantum dots symmetrically coupled to the SYK system, the latter mightbe experimentally realized in a graphene flake based device [53]. We call itCheburashkian geometry [135], Fig. 6.1. The random couplings Jik;kl andλi are independently distributed as Gaussian with zero mean Jij;kl = 0 =λi and finite variance |Jij;kl|2 = J2 (Jij;kl = −Jji;kl = −Jij;lk = (Jkl;ij)∗),|λi|2 = λ2. After disorder averaging, one finds that in the large N , longtime limit 1 Jτ N the spectral function AA(ω) for the fermionχA coincides with Eq. (6.3), where the coupling strength is given bythe variance |λi|2 = λ2 and the continuum GR originates from the SYKGreen’s function

GR(ω) = −iπ1/4 eiπsgn(ω)/4√J |ω|

(6.9)

at zero temperature, see the details in Appendix 6.7.2.The line shape of the spectral function of χA depends on the frequen-

cies ΩA and ΩB in a characteristic manner. We give examples in Figs. 6.2and 6.3. By tuning those one gets a distinct identifier of existence of theSYK continuum spectrum.

In case of multiple M states, a spectral function of a single stateAA(ω) ∝ ImGR(ω)

∏M−1m=1 (ω − ΩBm)2 contains M −1 isolated zeros. This

is described in Appendix 6.7.3, where we assume M N to avoid atransition in the SYK model to a Fermi liquid phase [122].

6.4 Cluster of SYK nodes

One can generalize the previous case by clustering evenly separated quan-tum SYK dots, coupled to a 1D wire [116, 136]. Here we can consider theitinerant fermion χp either in Galilean continuum or in an independent

Figure 6.4. SYK chain. An evenly spaced chain of SYK impurities is intro-duced to the one-dimensional system with propagating fermion χ.


crystal with arbitrary periodicity. It has a given dispersion relation ξp andinteracts locally with an SYK quantum dot at point xn, see Fig. 6.4. Thecorresponding Hamiltonian is similar to Eq. (6.8) with

H =∑p

ξpχ†pχp +

∑xn

HSY K(xn) +Hint(xn), (6.10)

HSY K(xn)= 1(2N)3/2

N∑i,j,k,l=1

Jnij;kl ψ†i,nψ

†j,nψk,nψl,n,

Hint(xn)= 1√N

N∑i=1

(λi,nψ

†i,nχ(xn) + λ∗i,nχ

†(xn)ψi,n).

It is clear that momentum plays now the role of the quantum numberA,B in our earlier model (6.1). Therefore we are dealing with the caseof many coupled states M > 2. Importantly however, the momentummodes are not all cross coupled through the interaction mediated by manySYK continua. As usual for the periodic structures, see Appendix 6.7.4,the SYK chain introduces an Umklapp-like effect: After integrating outall the SYK fermions the effective interaction only couples the momentaseparated by the integer number of the reciprocal lattice units ∆p = 2π/a.Note that only a discrete set of momentum modes is coupled, thereforethe SYK fermions stay in the quantum critical phase. The inverse Green’sfunction is labeled by the Bloch momentum p

G(p, ω)−1 =

=

. . . · · · · · ·... ω− ξp− λ2G(ω) −λ2G(ω)... −λ2G(ω) ω− ξp−∆p− λ2G(ω)

. (6.11)

Following the same logic as before we find that the energy distribution ofthe spectral density at given momentum p has a discrete set of zeros atfrequencies ω = . . . , ξp−∆p, ξp+∆p, ξp+2∆p, . . . . Varying p this introducescontinuous lines of zeros separated by ∆p in the momentum resolved spec-tral function A(p, ω), as seen in Fig. 6.6, top panel. One can considerdisordering the periodic structure by randomly shifting the nodes. Thesharp lines in the Brillouin zone will smear out, since the Bloch momen-tum will not be well defined anymore. Therefore we expect the lines ofzeros to smear out as well, but for the weak disorder the characteristicline shape will remain visible.

6.5 Holographic fermions 77

Figure 6.5. Holographic fermions. The spectral function of the fermioncorresponds to the propagator in the holographic model. It is affected by the nearhorizon geometry, which induces locally critical continuum contribution GAdS .

6.5 Holographic fermions

This isolated zero phenomenon in the case of a finite number of spatial di-mensions can be illustrated in a holographic model of a strange metal [137].In this framework one describes strongly entangled quantum systems interms of a classical gravitational problem with one extra dimension, seeFig. 6.5. At finite chemical potential these systems can flow to a novel“strange metal” low energy sector that exhibits local quantum criticality,i.e. it contains a continuum sector. The fermionic spectral function canbe computed from the propagator of a dual holographic fermion X in thecurved space with a charged black hole horizon. We refer the reader tothe seminal works [138–140]. Importantly, the dynamics of a holographicfermion can be understood as a momentum dependent probe of the quan-tum critical continuum sector. In holographic models this sector is local,i.e. the system falls apart into local domains, each of which is a contin-uum theory in its own right [141]. The SYK model is an explicit proposalfor the microscopic theory of one such domain. This explains why at lowenergy this holographic theory is similar to (6.10) with the SYK Hamil-tonian replaced by the more abstract local quantum critical Hamiltonian[140]. The Green’s function of the critical fermions ψ can be shown tohave the scaling form G(ω) ∼ ω2νp , where νp depends on the momentumof the itinerant probe χp.

Unlike the case with an SYK chain, the holographic continuum comesfrom translationally invariant horizon, therefore it does not introduce theBrillouin zone by itself. The weak interaction between the localized contin-


Figure 6.6. Lines of isolated zeros. Spectral density of a fermion is shownin color. The Brillouin zone boundaries are shown with red grid lines. Thebright bands correspond to the dispersion relation of itinerant fermion ξp, whilethe finite background is due to the critical continuum. The lines of zeros areseen as darker bands and correspond to the dispersion of the Umklapp copies ofthe fermionic dispersion ξp±2π/a. Top panel: In SYK chain. We use free spacedispersion ξp = p2/2m−µ with µ = 0.2,m = 0.5, SYK Green’s function (6.9) hasJ = 10, and the coupling is λ = 0.3. The gray dashed lines show the dispersionsξp±∆p. Bottom panel: In the holographic model for fermions on top ofthe continuum. The model has periodic background with period a.

uum sectors effectively restores translational invariance. The appearanceof isolated zeros in the holographic spectrum becomes apparent once onebreaks translations by introducing a crystal lattice with period a. The

6.6 Conclusion 79

technical details of this construction are discussed in the specialized pa-pers [142–145]. Importantly, in this case it is the periodic potential, whichintroduces the Bloch momentum and makes the dispersion ξp multivalueddue to the appearance of the ξp−2π/a and ξp+2π/a copies. These branchesare coupled by the near horizon continuum G(ω), and this leads to thesimilar pattern in the spectral function as in the SYK chain case, see Fig.6.6, bottom panel. At finite temperature the zeros are smeared and thespectral density never strictly vanishes. However the distinctive line shapeis clearly recognizable.

6.6 Conclusion

We have observed a characteristic phenomenon which arises in quantumsystems having both discrete spectrum and continuum subparts. The in-terference between the discrete parts mediated by the continuum givesrise to the isolated zeros in the spectral function of the one discrete state,which are located at the energy levels of the other states. It is the comple-ment of the known Fano resonance. We derive this effect in full generalityand show that it is present for a very general class of the continuum sub-systems, provided their Green’s functions have finite imaginary part andlack a pole structure.

By studying several examples with and without spatial dimensions,we see that the phenomenon is generally present. It is characterized bythe distinctive “pole-zero” line shape, which is different from that of theFano resonance, since the positions of the pole and zero are in principleindependent of each other. We show that this line shape is not destroyedby the additional couplings or disorder: The exact zero gets smeared, butthe localized depression remains. This leads us to propose that our findingcan serve as a convenient experimental marker for a quantum system toinclude a continuum subpart, such as an SYK continuum in the grapheneflake on a quantum dot or a local quantum critical phase in case of astrange metal.


6.7 Appendix: Outline of the calculation

6.7.1 Inclusion of general coupling between discrete levels

As in the main text, we consider two fermions coupled through continuumbut with an additional direct coupling:

S = Sχ + Sψ + Sint, (6.12)

Sχ =∫ β

0dτ(χA χB

)(∂τ + ΩA κκ∗ ∂τ + ΩB

)(χAχB

),

Sψ = −∫ β

0dτ

∫ β

0dτ ′ ψ(τ)G

(τ − τ ′

)−1ψ(τ ′),

Sint =∫ β

0dτ

∑σ=A,B

(λσ ψ χσ + λ∗σχσψ

),

where κ is the direct coupling strength. After integration over the fermionψ, we derive the inverse Green’s function for the fermions χA, χB:

Gσσ′(iωn)−1 =(

iωn−ΩA−|λA|2G(iωn) −κ−λ∗AλB G(iωn)−κ∗ −λAλ∗B G(iωn) iωn−ΩB−|λB|2G(iωn)

), (6.13)

where ωn = πT (2n+ 1) are Matsubara frequencies. Inversion of the ma-trix (6.13) and analytic continuation iωn → ω + iδ with δ = 0+ gives theretarded Green’s function:

GRσσ′(ω)= 1D(ω)

(ω−ΩB+iδ−|λB|2GR(ω) κ+λ∗AλBGR(ω)

κ∗+λAλ∗BGR(ω) ω−ΩA+iδ−|λA|2GR(ω)

), (6.14)

D(ω)=(ω−ΩA+iδ) (ω−ΩB+iδ)−|κ|2−(|λA|2 (ω−ΩB+iδ)

+ |λB|2 (ω−ΩA+iδ)+κλAλ∗B+κ∗λ∗AλB)GR(ω). (6.15)

The spectral function for fermion χA is defined as the imaginary part ofthe AA block of (6.14):

AA(ω) =− 1π

ImGRAA(ω) = −|λA|2

π

∣∣∣ω − ΩB + κλAλ

∗B

|λA|2∣∣∣2

|D(ω)|2ImGR(ω). (6.16)

The imaginary part of the continuum Green’s function GR is supposedto be finite: ImGR δ = 0+, so that δ can be neglected in (6.16).Zeros of the spectral function (6.16) are given by solutions of the equation


(ω − ΩB)2 + 2Reζ (ω − ΩB) + |ζ|2 = 0, where ζ = κλAλ∗B/|λA|2. In case

of the complex valued ζ = |ζ|eiϕ, the solutions are ω = ΩB − |ζ| cosϕ ±|ζ|√

cos2 ϕ− 1. This results in the appearance of shifted isolated zeros atthe real energy axis only for ϕ = πm: ω = ΩB + (−1)m+1 |ζ|, where m areintegers. So, zeros of the spectral function (6.16) are stable for real valuesof ζ.

In case of negligibility of the direct coupling κ between the isolatedstates on the background of the frequencies of those, we restore the ex-pression for the spectral function from the main text:

AA(ω)=− |λA|2

π

(ω−ΩB)2

|D(ω)|2ImGR(ω), (6.17)

D(ω)= (ω−ΩA)(ω−ΩB)−(|λA|2(ω−ΩB)+|λB|2 (ω−ΩA)

)GR(ω).

The result (6.17) is valid for general couplings λσ.

6.7.2 Mean-field treatment of the SYK model with twocomponent impurity

The Euclidean action for the Hamiltonian (8) in the main text after dis-order averaging is

S =∫ β

0dτ

[ ∑σ=A,B

χσ (∂τ + Ωσ)χσ +N∑i=1

ψi ∂τψi

]

−∫ β

0dτ

∫ β

0dτ ′[λ2

N

N∑i=1

∑σ,σ′=A,B

χσψi(τ) ψiχσ′(τ ′)

+ J2

4N3

N∑i,j,k,l=1

ψiψjψkψl(τ)ψl ψkψjψi(τ ′)]. (6.18)

Introduction of two pairs of nonlocal fields:

1ψ =∫DGψ δ

(Gψ(τ ′, τ)− 1

N

N∑i=1

ψi(τ)ψi(τ ′))

=

=∫DΣψ

∫DGψ exp

[N

∫ β

0dτ

∫ β

0dτ ′Σψ(τ, τ ′)

·(Gψ(τ ′, τ)− 1

N

N∑i=1

ψi(τ)ψi(τ ′))]

(6.19)


and

1χ =∫DGψ δ

Gχ(τ ′, τ)−∑

σ,σ′=A,Bχσ(τ)χσ′(τ ′)

=

=∫DΣχ

∫DGχ exp

[ ∫ β

0dτ

∫ β

0dτ ′Σχ(τ, τ ′)

·

Gχ(τ ′, τ)−∑

σ,σ′=A,Bχσ(τ)χσ′(τ ′)

] (6.20)

allows us to rewrite the action (6.18) as

S =∫ β

0dτ

∫ β

0dτ ′[ ∑σ,σ′=A,B

χσ(τ)(δσσ′δ(τ − τ ′)(∂τ+Ωσ)+Σχ(τ, τ ′)

)χσ′(τ ′)

+N∑i=1ψi(τ)

(δ(τ−τ ′)∂τ+Σψ(τ,τ ′)

)ψi(τ ′)−

(Σχ(τ,τ ′)−λ2Gψ(τ,τ ′)

)Gχ(τ ′,τ)

−N(

Σψ(τ, τ ′)Gψ(τ ′, τ) + J2

4 Gψ(τ, τ ′)2Gψ(τ ′, τ)2)]

. (6.21)

Assuming that all nonlocal fields Gψ, Σψ, Gχ, and Σχ are the functionsof τ − τ ′, we get variational saddle-point equations:

Gψ(τ − τ ′) = − 1N

N∑i=1

⟨Tτψi(τ)ψi(τ ′)

⟩, (6.22)

Σψ(τ − τ ′) = −J2Gψ(τ − τ ′)2Gψ(τ ′ − τ) + λ2

NGχ(τ − τ ′), (6.23)

Gχ(τ − τ ′) = −∑

σ,σ′=A,B

⟨Tτχσ(τ)χσ′(τ ′)

⟩, (6.24)

Σχ(τ − τ ′) = λ2Gψ(τ − τ ′), (6.25)

where Gχ =∑σσ′ Gσσ′ . In the large N and long time limit 1 Jτ N ,

equations (6.22) and (6.23) are simplified to Gψ(iωn)−1 = −Σψ(iωn) andΣψ(τ − τ ′) = −J2Gψ(τ − τ ′)2Gψ(τ ′ − τ) with a known zero temperaturesolution [47]:

Gψ(iωn) = −iπ1/4 sgn(ωn)√J |ωn|

. (6.26)


Thus, we derived an effective action for the impurity fermions:

S =+∞∑

n=−∞

(χn,A χn,B

)G(iωn)−1

(χn,Aχn,B

), (6.27)

so that their Green’s function is

Gσσ′(iωn)= 1D(iωn)

(iωn−ΩB−λ2Gψ(iωn) λ2Gψ(iωn)

λ2Gψ(iωn) iωn−ΩA−λ2Gψ(iωn)

), (6.28)

D(iωn)=(iωn−ΩA) (iωn−ΩB)−λ2(2iωn−ΩA− ΩB)Gψ(iωn). (6.29)

We perform analytic continuation iω → ω + iδ with δ = 0+ to derive theretarded Green’s function:

GRσσ′(ω)= 1D(ω)

(ω−ΩB+ iδ−λ2GRψ (ω) λ2GRψ (ω)

λ2GRψ (ω) ω−ΩA+iδ−λ2GRψ (ω)

), (6.30)

where

GRψ (ω) = −iπ1/4 eiπsgn(ω)/4√J |ω|

, (6.31)

that fulfills ImGRψ δ = 0+. The spectral function of the fermion χA isgiven by the imaginary part of the corresponding matrix element of (6.30)

AA(ω) = − 1π

ImGRAA(ω) = −λ2

π

(ω − ΩB)2

|D(ω)|2ImGRψ (ω), (6.32)

where δ = 0+ is neglected. The result (6.16) coincides with the expression(3) in the main text for GR ≡ GRψ .

6.7.3 Multiple states coupled to the SYK continuum

Here we address the case of M discrete states σ = A,B1, . . . , BM−1 cou-pled to the SYK continuum. To stay in the non-Fermi liquid regime wesuppose M N [122]. Once SYK degrees of freedom are integrated out,


the effective action for χσ fermions is similar to (6.27) in Appendix 6.7.2:

S =+∞∑

n=−∞

∑σ,σ′=A,B1,...,BM−1

χn,σ(δσσ′ (−iωn + Ωσ) + λ2Gψ(iωn)

)χn,σ′ =

=+∞∑

n=−∞

[ ∑σ,σ′=A,B1,...,BM−2

χn,σ(δσσ′ (−iωn + Ωσ) + λ2Gψ(iωn)

)χn,σ′

+ χn,BM−1

(−iωn + ΩBM−1 + λ2Gψ(iωn)

)χn,BM−1

+∑

σ=A,B1,...,BM−2

λ2Gψ(iωn)(χn,BM−1χn,σ + χn,σχn,BM−1

) ]=

=+∞∑

n=−∞

∑σ,σ′=A,B1,...,BM−2

χn,σ

(δσσ′ (−iωn + Ωσ)

+ λ2(−iωn + ΩBM−1

)Gψ(iωn)

−iωn + ΩBM−1 + λ2Gψ(iωn)︸︷︷︸≡GM−1(iωn)

)χn,σ′ =

=+∞∑

n=−∞

∑σ,σ′=A,B1,...,BM−3

χn,σ

(δσσ′ (−iωn + Ωσ)

+ λ2(−iωn + ΩBM−2

)GM−1(iωn)

−iωn + ΩBM−2 + λ2GM−1(iωn)︸︷︷︸≡GM−2(iωn)

)χn,σ′ = . . . =

=+∞∑

n=−∞χn,A

(− iωn+ΩA+λ2 (−iωn + ΩB1)G2(iωn)

−iωn + ΩB1 + λ2G2(iωn)︸︷︷︸≡G1(iωn)

)χn,A , (6.33)

where the derivation of the last line in (6.33) is based on the one-by-oneGaussian integration over all states except χA and Gψ is given by the


expression (6.26). Now we restore G1 back:

G1(iωn)= (−iωn+ΩB1)G2(iωn)−iωn+ ΩB1 +λ2G2(iωn) =

= (−iωn+ΩB1)(−iωn+ΩB2)G3(iωn)(−iωn+ΩB1) (−iωn+ΩB2)+λ2G3(iωn) (−iωn+ΩB1− iωn+ΩB2) =

=∏3m=1 (−iωn+ΩBm)G4(iωn)∏3

m=1 (−iωn+ΩBm)+λ2G4(iωn)∑3m=1

∏3p 6=m

(−iωn+ΩBp

) = . . .

=∏M−1m=1 (−iωn+ΩBm)Gψ(iωn)∏M−1

m=1 (−iωn+ΩBm)+λ2Gψ(iωn)∑M−1m=1

∏M−1p 6=m

(−iωn+ΩBp

). (6.34)Finally,

GAA(iωn)= 1D(iωn)

(M−1∏m=1

(iωn−ΩBm)

−λ2Gψ(iωn)M−1∑m=1

M−1∏p 6=m

(iωn−ΩBp

)), (6.35)

D(iωn)=(iωn−ΩA)M−1∏m=1

(iωn−ΩBm)−λ2Gψ(iωn)

·

(iωn−ΩA)M−1∑m=1

M−1∏p 6=m

(iωn−ΩBp

)+M−1∏m=1

(iωn−ΩBm)

. (6.36)

The spectral function we are interested is

AA(ω) =− 1π

ImGAA (iωn → ω + iδ) =

= −λ2

π

ImGRψ (ω)|D(ω)|2

M−1∏m=1

(ω − ΩBm)2 . (6.37)

Result (6.37) coincides with the two fermion case (6.32) at M = 2.

6.7.4 Effective momentum coupling by the SYK chain

Consider the action corresponding to the Hamiltonian of the SYK chainone with one-dimensional propagating fermions χ, which interact locally


with the SYK nodes. After integrating out the SYK fermions one canwrite down the effective interaction term in the action as

Veff =∫dxdx′λ2χ(x)G(x, x′|ω)χ(x′). (6.38)

Locality of the interaction immediately means that

G(x, x′|ω) = δ(x− x′)ρ(x)G(ω), (6.39)

where we introduce the “position density of nodes” ρ(x) which is a col-lection of periodically distributed delta functions in the case of a chainρ(x) =

∑n δ(x+ na). Moving to momentum space gives

Veff =∫dp d∆p λ2χpχp+∆p ρ(∆p)G(ω), (6.40)

where ρ(∆p) is simply a Fourier transform of ρ(x). For the periodicallydistributed chain it has the form ρ(∆p) =

∑n δ(∆p+2πn/a), which brings

the interaction term to the form used in expression (6.11) of the main text.

Veff =∫dp∑n

λ2χpχp−2πn/aG(ω). (6.41)

Chapter 7

Gapless odd-frequencysuperconductivity inducedby the Sachdev-Ye-Kitaevmodel

7.1 Introduction

In this Chapter we consder the SYK model with Majorana fermions. Wemodify the model via coupling it to a single-state non-interacting quan-tum dot. As we add only a single fermion, this model stays far away fromthe non-Fermi liquid/Fermi liquid transition [122] and it is still exactlysolvable in the large N limit. We demonstrate that the effective theoryfor the fermion in the quantum dot gains the anomalous pairing terms,that make the quantum dot superconducting. Despite the induced super-conductivity, the density of states in the quantum dot has no excitationgap. It has been a while since the phenomenon of gapless superconduc-tivity was found in the superconductors with magnetic impurities, wherefor a specific range of concentration of those, a part of electrons does not

The contents of this chapter have been published and reprinted with permissionfrom N. V. Gnezdilov, Phys. Rev. B 99, 024506 (2019). Copyright 2019 by theAmerican Physical Society.


88 Chapter 7. Superconductivity induced by the SYK model

participate in the condensation process [146, 147]. The anomalous com-ponents of the Gor’kov Green’s function [148, 149] of the quantum dot arecalculated exactly in the large N limit and are odd functions of frequency[150, 151]. Odd-frequency pairing is known to be induced by proximityto an unconventional superconductor [151–154]. Below we obtain inducedodd-frequency gapless superconductivity in zero dimensions as a conse-quence of the proximity to a system described by the SYK model [31, 32].We suggest to use this effect as a way to detect the SYK-like effectivebehavior in a solid-state system.

7.2 The model

Let’s consider the Sachdev-Ye-Kitaev model [27, 29] randomly coupled toa single state quantum dot [155] with the frequency Ωd. The Hamiltonianof the system reads:

H=Ωdd†d+

N∑i=1

λiγi(d†−d

)+ 1

4!

N∑i,j,k,l=1

Jijklγiγjγkγl , (7.1)

where the couplings Jijkl and λi are independently distributed as a Gaus-sian with zero mean Jijkl = 0 = λi and finite variance J2

ijkl = 3!J2/N3,λ2i = λ2/N . The tunneling term in the Hamiltonian (7.1) is similar to one,

that appears for tunneling into Majorana nanowires [15, 80, 152, 156, 157].Once the disorder averaging is done, we decouple the interactions by

introducing four pairs of the non-local fields in the Euclidean action as aresolution of unity [27, 29]:

1 =∫DΣγDGγe

∫dτdτ ′

Σγ (τ,τ ′)2 (NGγ(τ ′,τ)−

∑iγi(τ)γi(τ ′)), (7.2)

1 =∫DΣdDGde

∫dτdτ ′Σd(τ,τ ′)(Gd(τ ′,τ)−d(τ)d(τ ′)), (7.3)

1 =∫DΞdDFde

∫dτdτ ′Ξd(τ,τ ′)(Fd(τ ′,τ)−d(τ)d(τ ′)), (7.4)

1 =∫DΞdDFde

∫dτdτ ′Ξd(τ,τ ′)(Fd(τ ′,τ)−d(τ)d(τ ′)). (7.5)

A variation of the effective action, which is given in Appendix 7.5, with re-spect toGγ , Gd, Fd, Fd and Σγ ,Σd,Ξd, Ξd produces self-consistent Schwinger-Dyson equations [149], that relate those fields to the Green’s functions and

7.3 SYK proximity effect 89

self-energies of Majorana fermions and the fermion in the quantum dot:

Σd(τ) =λ2Gγ(τ), (7.6)

Ξd(τ) =− λ2

2 Gγ(τ), Ξd(τ) = −λ2

2 Gγ(τ), (7.7)

Σγ(τ) =J2Gγ(τ)3 + 2λ2

N

(Gd(τ)−F (τ)

2 − F (τ)2

), (7.8)

Gγ(iωn) =(

iωn − Σγ(iωn))−1

. (7.9)

The Green’s function of Majorana fermions (independent from the sitei) is Gγ(τ) = −N−1∑

i 〈Tτ γi(τ)γi(0)〉 and Gd(τ) = −⟨Tτ d(τ)d(0)

⟩,

Fd(τ) = −〈Tτ d(τ)d(0)〉, Fd(τ) = −⟨Tτ d(τ)d(0)

⟩are normal and anoma-

lous Green’s functions of the quantum dot variables.We are focused on the large N , long time limit 1 Jτ N , where

the conformal symmetry of the SYK model emerges [27, 29]. In thisregime, the backreaction of the quantum dot on Majoranas self-energy(7.8) is suppressed as 1/N . The bare frequency in the equation (7.9) canalso be omitted at low frequencies. Thus, equations (7.8, 7.9) becomeΣγ(τ) = J2Gγ(τ)3 and Gγ(iωn) = −Σγ(iωn)−1, which are the same as inthe case of the isolated SYK model. These equations have a known zerotemperature solution [27, 29] Gγ(iωn) = −iπ1/4sgn(ωn) (J |ωn|)−1/2, whichcontributes to the self-energies (7.6, 7.7) of the quantum dot. The Green’sfunction of Majorana zero-modes has no pole structure, which manifeststhe absence of quasiparticles. Moreover, it behaves as a power-law offrequency, which is the case of quantum criticality [Sac11] and emergenceof the conformal symmetry in the SYK case [27, 29].

7.3 SYK proximity effect

The effective action for the fermion in the quantum dot acquires anoma-lous terms

Seff = −12

+∞∑n=−∞

(dn d−n

)G(iωn)−1

(dnd−n

), (7.10)


Figure 7.1. Top panel: Absolute value of the anomalous averages in thequantum dot coupled to the SYK model as a function of Matsubara frequency.The frequency of the quantum dot is Ωd = 0.1J . Bottom panel: Anomalousaverages in the quantum dot coupled to the SYK/SYK2 model. The frequencyof the dot is Ωd = 0.1J and the coupling strength is λ = 0.2J .

where the Gor’kov Green’s function [148, 149]

G(iωn)−1 =(

iωn−Ωd−λ2Gγ(iωn) λ2Gγ(iωn)λ2Gγ(iωn) iωn+Ωd−λ2Gγ(iωn)

)(7.11)

is found self-consistently in a one loop expansion [149]. Due to negligibilityof the last term in Majoranas self-energy (7.8) mentioned above, the one


loop approximation turns out to be exact in the large N limit. A detailedderivation of the formula (7.11) is presented in Appendix 7.5.

Appearance of the anomalous pairing terms d(τ)Gγ(τ −τ ′)d(τ ′) in theeffective action (7.10) does not require any additional quantum numbers,because those are “glued” by the nonlocality in the imaginary time thatoriginates from the SYK saddle-point solution. The anomalous Green’sfunction which follows from (7.11) is odd in frequency [150, 151]:

F (iωn) =− λ2Gγ(iωn)iωn (iωn − 2λ2Gγ(iωn))− Ω2

d

=

=− F (−iωn). (7.12)

This result (7.12) is well aligned with previously found proximity effectby Majorana zero modes [80, 152] and odd-frequency correlations foundin interacting Majorana fermions [158]. Superconducting pairing growssmoothly while the coupling increases as it is shown in Fig. 7.1, top panel.

It is worthwhile to compare our setting (7.1) to the case when theSYK quantum dot is replaced by a disordered Fermi liquid. The lattercan be described by the SYK2 model: HSYK2 = i

∑ij Jijγiγj . In the

long time limit, the Green’s function of the SYK2 model is GSYK2(iωn) =−i sgn(ωn)/J [32], which is substituted to the result for the anomalouscomponent of the Gor’kov function (7.12). As we show in Fig. 7.1 (bottompanel), the amount of the SYK induced superconductivity is sufficientlyhigher then in the case of the SYK2 model.

In the large N limit, the spectral function of the quantum dot is

A(ω) =− 1π

Im trG (iωn → ω + iδ) =

=− 2π

λ2 (ω2 + Ω2d

)ImGRγ (ω)∣∣∣ω (ω − 2λ2GRγ (ω))− Ω2

d

∣∣∣2 , (7.13)

where δ = 0+ and GRγ (ω) = −iπ1/4eiπsgn(ω)/4 (J |ω|)−1/2. The broadeningδ = 0+ of the fermion in the quantum dot is neglected once the imaginarypart of the SYK Green’s function is finite: λ2ImGRγ (ω) δ = 0+.

In absence of coupling between the single-state quantum dot and theSYK model (λ = 0), there is no particle-hole mixing. Superconductingpairing (Fig. 7.1) appears in the regime of intermediate coupling. Theabsence of the gap in the presence of the anomalous pairing reveals gaplesssuperconductivity [146, 147] in zero dimensions, which can be probed by


Figure 7.2. Density of states in the quantum dot at zero temperatureas a function of frequency. Ωd = 0.1J and δ = 10−3J .

Andreev reflection [77] in the tunneling experiment. The wide broadeningof the peaks in Fig. 7.2 is due to the binding of the fermionic quantum dotwith the SYK quantum critical continuum. Increasing of coupling strengthλ results in grows of the anomalous pairing (7.12) and suppression of theinitial quasiparticle peaks. In strong coupling the system shows divergentbehavior at ω = 0. However, the divergence point might be addressedbeyond the conformal limit [119] ω . J/ (N logN). This changes thescaling of the SYK Green’s function from 1/

√ω to N logN

√ω in the

infrared.In Fig. 7.3 we show, that the behavior of the spectral function of the

quantum dot coupled to the SYK model is qualitatively different fromthe SYK2 case. The SYK2 model, mentioned above, describes disorderedFermi liquid and has a constant density of states ∝ 1/J in the long timelimit.

At finite temperature the saddle-point solution of the SYK model isgiven by [47]:

GRγ (ω) = −iπ1/4

√β

2πJΓ(

14 − iβω2π

)Γ(

34 − iβω2π

) , (7.14)

where β = 1/T is inverse temperature and Γ(x) is the Gamma function.We substitute the finite temperature SYK Green’s function (7.14) in thespectral function of d fermion (7.13). Fig. 7.4 demonstrates that the di-


Figure 7.3. Density of states in the quantum dot coupled to the SYK/SYK2model at zero temperature. The coupling strength is λ = 0.1J and the frequencyof the single state is Ωd = 0.1J .

Figure 7.4. Density of states in the quantum dot at finite temperatureas a function of frequency. The parameters are λ/J = 0.1 = Ωd/J .

vergence around ω ∼ 0 in the quantum dot density of states is regularizedat finite temperature.


7.4 Conclusion

In this paper, we have shown that a single-state spinless quantum dotbecomes superconducting in proximity to a structure whose low-energybehavior can be captured by the Sachdev-Ye-Kitaev model. Anomalousaverages are found exactly in the large N limit and turn out to be oddfunctions of frequency. Appearance of non-zero superconducting pairingdoes not require any additional quantum numbers like spin, because itoriginates from nonlocality of the SYK saddle-point solution. Induced su-perconductivity strikes in the intermediate coupling between the quantumdot and the SYK model. At stronger coupling, the quasiparticle peaks aresmeared out on the background of the SYK quantum critical continuum.We propose to use the peculiar property of the induced gapless supercon-ductivity in zero dimensions to characterize the solid-state systems, thatcan be described by the SYK model as an effective theory in a certainlimit.

7.5 Appendix: Gor’kov Green’s function for theQD variables

The Euclidean action that corresponds to the Hamiltonian (7.1) after dis-order averaging is

S =∫ β

0dτ

[d (∂τ + Ωd) d+ 1

2

N∑i=1

γi∂τγi

]

−∫ β

0dτ

∫ β

0dτ ′[λ2

2N

N∑i=1

γi(d− d

)(τ)γi

(d− d

)(τ ′)

+ J2

8N

N∑i,j,k,l=1

γiγjγkγl(τ)γlγkγjγi(τ ′)]. (7.15)

7.5 Appendix: Gor’kov Green’s function for the QD variables 95

We introduce four pairs of nonlocal fields as a resolution of unity:

1 =∫DGγ δ

(Gγ(τ ′, τ)− 1

N

N∑i=1

γi(τ)γi(τ ′))

=

=∫DΣγ

∫DGγ exp

[N

2

∫ β

0dτ

∫ β

0dτ ′Σγ(τ, τ ′)

·(Gγ(τ ′, τ)− 1

N

N∑i=1

γi(τ)γi(τ ′))]

, (7.16)

1 =∫DGd δ

(Gd(τ ′, τ)− d(τ)d(τ ′)

)=

=∫DΣd

∫DGd exp

[ ∫ β

0dτ

∫ β

0dτ ′Σd(τ, τ ′)

·(Gd(τ ′, τ)− d(τ)d(τ ′)

)], (7.17)

1 =∫DFd δ

(Fd(τ ′, τ)− d(τ)d(τ ′)

)=

=∫DΞd

∫DFd exp

[ ∫ β

0dτ

∫ β

0dτ ′ Ξd(τ, τ ′)

·(Fd(τ ′, τ)− d(τ)d(τ ′)

)], (7.18)

1 =∫DFd δ

(Fd(τ ′, τ)− d(τ ′)d(τ)

)=

=∫DΞd

∫DFd exp

[ ∫ β

0dτ

∫ β

0dτ ′ Ξd(τ, τ ′)

·(Fd(τ ′, τ)− d(τ)d(τ ′)

)]. (7.19)


This allows us to rewrite the action (7.15) as:

S =∫ β

0dτ

∫ β

0dτ ′[d(τ)

(δ(τ − τ ′) (∂τ + Ωd) + Σd(τ, τ ′)

)d(τ ′)

+ d(τ)Ξd(τ, τ ′)d(τ ′) + d(τ)Ξd(τ, τ ′)d(τ ′)

+ 12

N∑i=1

γi(τ)(δ(τ − τ ′)∂τ + Σγ(τ, τ ′)

)γi(τ ′)

− Σd(τ, τ ′)Gd(τ ′, τ)− Ξd(τ, τ ′)Fd(τ ′, τ)− Ξd(τ, τ ′)Fd(τ ′, τ)

− N

2

(Σγ(τ, τ ′)Gγ(τ ′, τ) + J2

4 Gγ(τ, τ ′)4)

− λ2

2 Gγ(τ, τ ′)(Gd(τ, τ ′)−Gd(τ ′, τ) + Fd(τ ′, τ) + Fd(τ ′, τ)

)]. (7.20)

Following Refs. [27–29, 47], we assume that all nonlocal fields are oddfunctions of the time difference τ − τ ′. In the large N , long time limit:1 Jτ N , self-consistent saddle-point equations are

δS

δΣd=0⇒ Gd(τ − τ ′) = −

⟨Tτ d(τ)d(τ ′)

⟩, (7.21)

δS

δΞd=0⇒ Fd(τ − τ ′) = −


⟩(7.22)

δS

δΞd=0⇒ Fd(τ − τ ′) = −


⟩, (7.23)

δS

δΣγ=0⇒ Gγ(τ − τ ′) = − 1

N

N∑i=1

⟨Tτ γi(τ)γi(τ ′)

⟩⇒

⇒ Gγ(iωn)−1 = iωn − Σγ(iωn) ' −Σγ(iωn) (7.24)

7.5 Appendix: Gor’kov Green’s function for the QD variables 97

andδS

δGd=0⇒ Σd(τ − τ ′) = λ2Gγ(τ − τ ′) , (7.25)

δS

δFd=0⇒ Ξd(τ − τ ′) = −λ

2

2 Gγ(τ − τ ′) (7.26)

δS

δFd=0⇒ Ξd(τ − τ ′) = −λ

2

2 Gγ(τ − τ ′) , (7.27)

δS

δGγ=0⇒ Σγ(τ − τ ′) = J2Gγ(τ − τ ′)3

+ λ2

N

(2Gd(τ − τ ′)− Fd(τ − τ ′)− Fd(τ − τ ′)

)' J2Gγ(τ − τ ′)3 . (7.28)

Green’s functions of the fermion in the dot enter the equation for theMajoranas self-energy (7.28) as 1/N , so we neglect them in the largeN limit. Thus, equations (7.24, 7.28) are decoupled from the quan-tum dot and become the standard SYK Schwinger-Dyson equations [27,29] Gγ(iωn)−1 = −Σγ(iωn) and Σγ(τ) = J2Gγ(τ)3 with a known low-frequency solution

Gγ(iωn) = −iπ1/4 sgn(ωn)√J |ωn|

(7.29)

at zero temperature, where ωn = πT (2n + 1) are Matsubara frequencies.Meanwhile, the bare SYK Green’s function (7.29) enters the self-energiesof the quantum dot (7.25, 7.26, 7.27), that, according to the definitions(7.17, 7.18, 7.19), give both normal (dd) and anomalous (dd, dd) compo-nents of the effective action for the d fermion.

The effective action for the fermion in the quantum dot in the basis of~d∗n = 1√

2

(dn d−n

), ~dn = 1√

2

(dn d−n

)Tis given by

S =+∞∑

n=−∞

~d∗n

(−iωn+Ωd+λ2Gγ(iωn) −λ2Gγ(iωn)

−λ2Gγ(iωn) −iωn−Ωd+λ2Gγ(iωn)

)~dn, (7.30)

so that the Gor’kov Green’s function [148] composed from (7.21, 7.22) isfound exactly in the large N limit:

G(iωn)−1 =(

iωn − Ωd − λ2Gγ(iωn) λ2Gγ(iωn)λ2Gγ(iωn) iωn + Ωd − λ2Gγ(iωn)

). (7.31)


The analytic continuation to the real frequencies iωn → ω+iδ with δ = 0+

gives the retarded Green’s function in the particle-hole basis:

GR(ω) = 1(ω + iδ)

(ω + iδ − 2λ2GRγ (ω)

)− Ω2

d

·(ω + iδ + Ωd − λ2GRγ (ω) −λ2GRγ (ω)

−λ2GRγ (ω) ω + iδ − Ωd − λ2GRγ (ω)

), (7.32)

where GRγ (ω) = −iπ1/4eiπsgn(ω)/4 (J |ω|)−1/2.

Chapter 8

First-order dipolar phasetransition in the Dickemodel with infinitelycoordinated frustratinginteraction

8.1 Introduction

Realization of the equilibrium photonic condensates is of great interest forthe fundamental study of a new states of light strongly coupled to quantummetamaterials [159–164]. In particular, cavity quantum electrodynamicsof superconducting qubits is crucial for the quantum computation per-spectives [165–168]. In quantum optics, e.g. in the cavity QED describedby the famous Dicke model [169], the ‘no-go’ theorems made those per-spectives gloomy [170–172], and only dynamically driven condensates areconsidered [173–178]. Nevertheless, it was found that in the equilibriumcircuit QED systems the ‘no-go’ theorems may not hold [162, 163]. In

The contents of this chapter have been published and reprinted with permissionfrom S. I. Mukhin and N. V. Gnezdilov, Phys. Rev. A 97, 053809 (2018). Copyright2018 by the American Physical Society.

https://doi.org/10.1103/PhysRevA.97.053809

100 Chapter 8. Phase transition in the frustrated Dicke model

particular, an array of capacitively coupled Cooper pair boxes to a reso-nant cavity was proven to disobey the ‘no-go’ theorem for an equilibriumsuperradiant quantum phase transition [163]. Nevertheless, another com-plication was found in this case, i.e. it was demonstrated [164, 179, 180],that allowance for the gauge invariance with respect to the electromagneticvector potential of the photon field causes Hamiltonian of the system tomap on the extended Dicke model Hamiltonian of (pseudo)spins one-half,adding to the standard Dicke model a frustrating infinitely coordinatedantiferromagnetic interaction between the spins. Lately, a numerical di-agonalization results for small clusters with N spins were reported [164]to behaved differently depending on the parity of the number of spins N .

Motivated by the above history of the extended Dicke model explo-ration, we present in this paper analytic description of the superradiantequilibrium quantum phase transition in the an array of N 1 Cooperpair boxes strongly coupled to a resonant cavity. The plan of the presentpaper is as follows.

First, we reproduce derivation [164, 179] of the extended Dicke Hamil-tonian with infinitely coordinated antiferromagnetic (frustrating) term.Next, we confirm the absence of the zero modes in the spectrum of thebosonic excitations, as was found in [164]. Then, we introduce a new repre-sentation for the operators of Cartesian components of the total spin (‘su-perspin’) of N spins 1/2: a self-consistently rotating Holstein-Primakoff(RHP) representation. After that, we demonstrate that RHP methodapplied to extended Dicke Hamiltonian reveals the first order quantumphase transition, that sets the system into a double degenerate dipolarordered superradiant state with coherent photonic condensate emergingin the cavity. Besides, in Appendix 8.7 we show that the RHP approachalso reproduces the second order quantum phase transition for the DickeHamiltonian without frustrating interaction term, found earlier by othermethod [181, 182]. We discuss a drastic difference between the criticalvalues of the coupling strength gc in the N →∞ limit for the 1st and 2ndorder phase transitions. In the Summary we present some evaluations ofthe parameters of a possible experimental setup for a validation of ourtheoretical predictions for Cooper pair box arrays in a microwave cavity.

8.2 Dicke Hamiltonian for a Cooper pair boxes array 101

8.2 Dicke Hamiltonian for a Cooper pair boxesarray

In this section we present a derivation of the extended Dicke model Hamil-tonian. We consider a single mode electromagnetic resonant cavity of alinear dimension L coupled to the array of N independent dissipationlessJosephson junctions. It is assumed that the wavelength λ of the cav-ity’s resonant photon is much greater than the inter-junction distance:λ L/N . The vector potential of the electromagnetic field related withthe photon is expressed in the second quantized form:

~A =

√c2h

ωV

(a† + a

)~ε , (8.1)

where h - is Planck’s constant, ω is bare photon frequency, the photoncreation and annihilation bosonic operators are a†, a, ~ε is polarization ofthe electric field, c is velocity of light, and V is the volume of the cavity.

The Hamiltonian of the Cooper pair box array in the cavity then reads:

H = Hph + HJJ , (8.2)

Hph = 12(p2 + ω2q2

), (8.3)

HJJ = EC

N∑i=1

n2i − EJ

N∑i=1

cos(φi −

g

~q

), (8.4)

where the coupling constant is g = 2el√

4π/√V , and l is of the order of

a penetration depth of an electric field into the superconducting islandsforming the Josephson junction, thus, giving the effective thickness acrossof it [180]. For simplicity, we consider all the junctions being identical,with electric field polarization ~ε aligned across a Josephson junction. Herethe two mutually commuting sets of the conjugate variables are intro-duced: [p, q] = −i~ and

[ni, φi

]= −i. The second quantized (harmonic

oscillator) variables of the photonic field are:

p = i

√~ω2(a† − a

)and q =

√~

2ω(a† + a

), (8.5)

where[a, a†

]= 1. An operator 2eni = 2e

(nRi − nLi

)/2 stands for half

of a charge difference at the i-th junction, and equals half the difference


of the number of Cooper pairs populating the left and right islands of aJosephson junction accordingly, multiplied by the elementary charge 2e ofthe Cooper pair. The quantum of the charging energy of a single junctionis EC = (2e)2 /2C.

Following [179] we make a canonical transformation:

φ′i = φi −g

~q and n′i = ni (8.6)

for the JJ’s variables and

p′ = p+ gN∑i=1

ni and q′ = q (8.7)

for photonic variables, so that[p′, φ′i

]= 0 and the other commutation

relations between all the operators remain intact. The Hamiltonian (8.2)becomes

H =12(p2 + ω2q2

)− gp

N∑i=1

ni + g2

2

(N∑i=1

ni

)2

+

+N∑i=1

(EC n

2i − EJ cos φi

), (8.8)

where the primes in the new variables are omitted for brevity. Thus, theinfinitely coordinated interaction term ∝ g2 has appeared in (8.8) afterthe canonical transformation of the Hamiltonian (8.2).

We restrict ourselves to the Cooper pair box limit [183], when thecharging energy EC is large in comparison with Josephson coupling EJ andthe eigenstates of the Hamiltonian (8.4) in the zero order approximationcan be chosen as the eigenstates of the charge difference operators ni. Thelowest bare energy level corresponding to the quantum states

∣∣∣ni = −12

⟩and

∣∣∣ni = 12

⟩is thus twofold degenerate with respect to the direction of

the Cooper pair box dipole moment ~di = 2elni~ε (l - is effective thickness ofthe i-th JJ). This double-degenerate level is separated from the levels withthe greater charge differences by the EC gap. The Josephson tunnelingterm ∼ EJ lifts the degeneracy and opens a gap between the energy levelsof the two states that differ by the wave-function parity ±1 with respect toinversion of the Cooper pair box dipole’s direction. Thus, formed two-levelsystem is naturally described by the Pauli matrices σαi . On the subset of

8.2 Dicke Hamiltonian for a Cooper pair boxes array 103

these lowest energy states the initial Hamiltonian (8.4) of the Cooper pairbox array of N Josephson junctions is represented by a Hamiltonian ofinteracting N spins half:

HJJ =N∑i=1

(EC n

2i − EJ cos φi

)≈ EC

4

N∑i=1

(∣∣∣∣12⟩⟨1

2

∣∣∣∣+ ∣∣∣∣−12

⟩⟨−1

2

∣∣∣∣)i

− EJ2

N∑i=1

(∣∣∣∣12⟩⟨−1

2

∣∣∣∣+ ∣∣∣∣−12

⟩⟨12

∣∣∣∣)i= NEC

4 1− EJ2

N∑i=1

σxi . (8.9)

Here charge and phase difference operators ni and cos φi are projectedon szi and sxi correspondingly, where sαi = 1

2 σαi are spin-1/2 operators

expressed via the Pauli matrices. As a result, initial Hamiltonian (8.8)reduces to the following spin-boson Hamiltonian, modulo energy shiftNEC/4 :

H = 12(p2 + ω2q2

)− gp

N∑i=1

szi − EJN∑i=1

sxi + g2

2

(N∑i=1

szi

)2

. (8.10)

It is important to clarify here the meaning of the spin-boson interactionterm in (8.10), that had emerged when canonical transformation (8.6, 8.7)of the initial gauge-invariant Hamiltonian (8.2) was performed :

−gp szi = −i

√~ω2V

(a† − a

)2el√

4πszi = − ~E ~di (8.11)

and represents the energy of the dipole in the electric field. The electricfield operator in (8.11) is given by

~E = i

√hω

V

(a† − a

)~ε (8.12)

and the dipole moment of the single junction is

~di = 2eszi l~ε . (8.13)

The total dipole moment is then

~d =N∑i=1

~di = 2e l~εN∑i=1

szi . (8.14)


For convenience of the further calculations we perform a unitary trans-

formation U †HU , where U = 1√2

(1 i1 −i

)that interchanges operators of

the Cartesian components of spin half: szi → −syi , s

yi → −sxi and sxi → szi .

Hence, the final Cooper pair box Hamiltonian, that we are going toexplore, becomes:

H = 12(p2 + ω2q2

)+ gp Sy − EJ Sz + g2

2(Sy)2

, (8.15)

where we have introduced operators Sα =∑i sαi of the total spin compo-

nents. The total spin S2 is conserved, because it commutes with (8.15):[S2, H

]= 0. Cooper pairs tunneling is represented by −EJ Sz term, gp Sy

is a dipole coupling strength between Cooper pair box and photonic field,and (g2/2)

(Sy)2

stands for the infinitely coordinated ‘antiferromagnetic’frustrating term.

8.3 Diagonalization of the frustrated Dicke model

8.3.1 Tunneling regime

In this section, we consider the frustrated Dicke Hamiltonian (8.15) andfirst assume that at small coupling strength g the Josepson tunnelingterm −EJ Sz dominates at zero temperature. Then the superspin is inthe large S sector and hence, one is allowed to use the Holstein-Primakofftransformation [184] (HP) in the form:

Sz = S − b†b, (8.16)

Sy = i

√S

2

b†√

1− b†b

2S −

√1− b†b

2S b

' i

√S

2(b† − b

), (8.17)

Sx =

√S

2

b†√

1− b†b

2S +

√1− b†b

2S b

'√S

2(b† + b

), (8.18)

8.3 Diagonalization of the frustrated Dicke model 105

where[b, b†

]= 1. The substitution of (8.16, 8.17) into (8.15) gives Hamil-

tonian of the two linearly coupled harmonic oscillators:

H =ω

(a†a+ 1

2

)− EJ

(S − b†b

)− g√Sω

2(a† − a

) (b† − b

)− g2S

4(b† − b

)2. (8.19)

This model also arises in the case of ultrastrong light-matter couplingregime with polariton dots [185]. Here and in what follows we put ~ =1. With the help of the usual linear Bogoliubov’s transformation of thecreation/annihilation operators (see appendix 8.6) we obtain diagonalizedHamiltonian:

H = −EJ(S + 1

2

)+ 1

2 (ε1 + ε2) + ε1 c†1c1 + ε2 c

†2c2 (8.20)

with the excitations spectrum described by the new oscillator frequencies:

2ε21,2 =EJ

(EJ + g2S

)+ ω2

±√

(EJ (EJ + g2S)− ω2)2 + 4ω2g2SEJ , (8.21)

where the frequencies ε1,2 have to be chosen positive to keep the hermitic-ity of the initial operators p, q, Sy. In contrast with the Dicke modelwithout frustration [181] both energy branches are real in the whole rangeof the coupling constants g, but with a caveat. Namely, the ground stateenergy equals:

E0(S) = −EJ(S + 1

2

)+ 1

2 (ε1 + ε2) . (8.22)

This ground state is stable as long as the ground state energy E0(S) (8.22)has global minimum as a function of the superspin S at the end of theinterval [0, N/2]. One can find value of the coupling strength g = g,at which the minimum becomes double degenerate via solving equationE0 (S = N/2, g = g) = E0 (S = 0, g = g) = ω/2:

g '√

2EJN + (EJ + ω)√

2NEJ

, (8.23)


which can be easily derived from large g asymptotic expression of E0(S):

E0 ' −EJ(S + 1

2

)+ g

2√EJS. (8.24)

For g > g the minimum of E0(S) migrates from S = N/2 to S = 0, seeFigure 8.1. This ‘jump’ of the minimum obviously makes ground stateS = N/2 unstable and leads to an inapplicability of the quasi-classicalHP approximation. Thus, our large S ground state description (8.22) isjustified for g < g.

Figure 8.1. Ground state energy as a function of the superspin S at fixeddimensionless coupling constant λ = g

√N/2EJ . The blue dashed line shows the

double degenerate minimum of the ground state at the coupling strength λ = λ(8.23).

8.3.2 Rotating Holstein-Primakoff representation

In order to make continuation of the theory into the stronger couplingstrength region outside the interval g < g , we substitute in the Hamilto-nian (8.15) the y, z components of the total spin operators with a gener-alised expression of the Holstein-Primakoff representation in a coordinateframe rotated by an angle θ in the z-y plane :

Sz = Jz cos θ − Jy sin θSy = Jz sin θ + Jy cos θ

(8.25)


Here the set of operators of the Cartesian projections of the total spinJx,y,z are

Jz = S − b†b

Jy ' i

√S

2(b† − b

)Jx '

√S2

(b† + b

) . (8.26)

To find θ 6= 0 solution that diagonalises (8.15) we introduce a shift, i√α,

of the photon creation/annihilation operators, similar to [181], in the fol-lowing way:

a† = c† − i

√α

a = c+ i√α

, (8.27)

thus, envisaging formation of a superradiant state. One substitutes (8.25- 8.27) into (8.15) and the Hamiltonian, quadratic in operators c, c† b, b†,becomes:

H =ω(c†c+ 1

2

)− EJ cos θ

(S − b†b

)− g cos θ

√Sω

2(c† − c

) (b† − b

)− g2 cos2 θ S

4(b† − b

)2, (8.28)

where an elimination of the linear in(c† − c

)and

(b† − b

)terms in the

Hamiltonian introduces a system of the two equations:

√2ωα+ g sin θ

(S −

⟨b†b⟩)

= 0 ; (8.29)

EJ sin θ + g cos θ√

2ωα+ g2 cos θ sin θ(S −

⟨b†b⟩− 1

2

)= 0 . (8.30)

We have also made in (8.28) a mean-field decoupling of the products thatare higher than quadratic in b, b†-operators: b†b b†b = 2

⟨b†b⟩b†b−

⟨b†b⟩2

and b†b(b† − b

)+(b† − b

)b†b =

(b† − b

) (1 + 2

⟨b†b⟩)

.A nontrivial solution α 6= 0, θ 6= 0 of the system of equations (8.29)


and (8.30) emerges when g ≥√

2EJ :

cos θ = 2EJg2 , (8.31)

√α = − gS√

2ω

1−

⟨b†b⟩

S

sin θ ' − gS√2ω

sin θ =

= ∓ gS√2ω

√1− 4E2

J

g4 . (8.32)

Under the solutions (8.31), (8.32), the energy of the photonic conden-sate = ωα exactly cancels with the sum of the rest of the c-number termsin the Hamiltonian (8.28):

ωα+ gS sin θ√

2ωα+ g2 sin2 θ

2

(S2 −

⟨b†b⟩2)

=

= g2 sin2 θ

2

(S2 − 2S

⟨b†b⟩

+⟨b†b⟩2− 2S2

+ 2S⟨b†b⟩

+ S2 −⟨b†b⟩2)

= 0 . (8.33)

The c-number terms in the first line of (8.33) have the following meaning:the photonic condensate energy ∼ ωα, the (negative) contribution of thedipole-photon coupling energy ∼ g〈p〉〈Sy〉, and the zero-point oscillationsenergy of the frustration term ∼ g2〈(Sy)2〉/2. The total of these threeterms proves to be zero. This α-independent cancellation, actually, stemsfrom the degeneracy of the energy minima of the diagonal in spin operatorspart of the extended Dicke Hamiltonian (8.15) with respect to 2S + 1different Sy projections and classical part ∼

√α of the photonic operator

p.

8.3.3 Superradiant dipolar regime

The structure of (8.28) is the same as (8.19), though with coefficientsrenormalised with prefactor cos θ due to RHP rotation by an angle θ .Hence, after the Bogoliubov’s transformation similar to the one alreadydescribed in the Appendix 8.6, the diagonalized Hamiltonian expressedvia new second quantized operators e†1,2, e1,2 acquires the form:

H = −g2 cos2 θ

2

(S + 1

2

)+ 1

2 (ε1 + ε2) + ε1 e†1e1 + ε2 e

†2e2 (8.34)


Figure 8.2. Ground state energy E0 as a function of the superspin S at fixeddimensionless coupling constant λ = g

√N/2EJ .

with the positive eigenvalues ε1,2

2ε 21,2 =g4 cos4 θ

2

(S + 1

2

)+ ω2

±

√(g4 cos4 θ

2

(S + 1

2

)− ω2

)2+ 2S ω2g4 cos4 θ (8.35)

and the ground state energy

E0(S) = −g2 cos2 θ

2

(S + 1

2

)+ 1

2 (ε1 + ε2) . (8.36)

The stability of the large S state in this regime is provided by the negativeslope of E0(S) as a function of S (see Figure 8.2) in the strong couplinglimit:

E0(S)∣∣∣g→+∞

' ω

2 −2E2

J

g2

(S + 1

2

). (8.37)

The interval g ≥√

2EJ is characterized with an emergent dipole momentin the Cooper pair box array and the superradint photoinic condensateeither as a metastable state for

√2EJ ≤ g < gc, or as the ground state for

g ≥ gc (the critical strength gc is found below). To see this explicitly, we


express electromagnetic field operators via the new set of Bose-operatorsfound after the Bogoliubov’s transformation:

p =√

2ωα+ iω cos δ√2ε1

(e†1 − e1

)+ iω sin δ√

2ε2

(e†2 − e2

), (8.38)

q = cos δ√2ε1

(e†1 + e1

)+ sin δ√

2ε2

(e†2 + e2

). (8.39)

In turn, the spin operators are expressed via ei, e†i , i = 1, 2 as well:

Jz = S

1−

⟨b†b⟩

S

' S , (8.40)

Jy = −iEJ√S sin δ

g√ε1

(e†1 − e1

)+ iEJ

√S cos δ

g√ε2

(e†2 − e2

), (8.41)

Jx = −EJ√S sin δ

g√ε1

(e†1 + e1

)+ EJ

√S cos δ

g√ε2

(e†2 + e2

). (8.42)

The Bogoliubov’s ‘angle’ δ can be found from the consistency relation:

tan 2δ = 2√

2S ω g2 cos2 θ

g4 cos4 θ(S + 1

2

)− 2ω2

. (8.43)

We find for g ≥√

2EJ the following non-zero expectation values in theground state of Hamiltonian (8.34). For the electric field ~E :⟨

~E · ~ε⟩√

V

4π = 〈p〉 =√

2ωα ' −gS sin θ =

= ∓gS

√1− 4E2

J

g4 ; (8.44)

for the modulus of the Josephson tunneling energy of the Cooper pairs (itdecreases):

−EJ〈Sz〉 = −EJ〈Jz〉 cos θ ' −S 2E2J

g2 ; (8.45)

and a for the emergent finite mean value of the dipole moment:

〈d〉 = 2el〈Sy〉 = 2el〈Jz〉 sin θ ' ±2elS

√1− 4E2

J

g4 . (8.46)

8.4 First order dipolar phase transition 111

Hence, results (8.44) and (8.46) indicate that upon an increase of thecoupling strength g >

√2EJ there is a state with the energy given in

(8.36), which is characterized by an emergent superradiant electromag-netic field 〈p〉 6= 0 in the cavity together with a finite dipole moment ofthe Cooper pair boxes: 〈d〉 6= 0. The latter means that rotation angle θintroduced in (8.25) regulates an extent of a Cooper pair wave functionbetween the superconducting islands forming each Josephson junction inthe Josephson junction array, see Figure 8.3. Namely, when θ progres-sively deviates from zero, the Cooper pairs become localized in one of thetwo superconducting islands constituting a given Josephson junction, andas a result, the latter acquires a dipole moment.

d=0, θ=0 d=el, θ=π/2 d=-el, θ=-π/2

Figure 8.3. Schematic layout of the amplitude distributions of the Cooperpair’s wave function in the adjacent islands of a single JJ and correspondingdipole moment values depending on the rotation angle θ, see text and equations(8.25).

8.4 First order dipolar phase transition

In this section we calculate a critical coupling gc, at which a first orderphase transition between the tunneling and dipolar states described inSections 8.3.1 and 8.3.3 takes place.

In Figure 8.4 we plotted ground state energies calculated for tun-neling and dipolar states as functions of coupling g: E0(S) and E0(S),see (8.22) and (8.36) correspondingly. A dimensionless coupling constantλ = g

√N/2EJ is used. In the strong coupling limit, g

√2EJ , the g

dependence of the both branches of energy is very well approximated by(8.24) and (8.37).

Hence, in the thermodynamic limit N →∞, solution of E0 = E0 gives


the critical value gc of the coupling constant:

gc '√

2EJN + ω

√2

NEJ. (8.47)

Here a crucial difference with respect to [181] is that the critical pointcorresponds to λc ≈ N and not 1 as in the standard Dicke model withoutfrustration. Hence, transition now is size-dependent, where the ‘size’ of thesystem is the total number N of Cooper pair box’s inside the microwavecavity.

Figure 8.4. Ground state energy as a function of dimensionless coupling con-stant λ = g

√N/2EJ . Blue line is for the Josephson tunneling state in the interval

λ < λc ≈ N . Red line is for dipolar ordered state. Red dashed line shows thedipolar state in the metastable region preceding the first order phase transitionat λc.

At λ =√N , i.e. g =

√2EJ , the ground state becomes degenerate and a

dipolar branch E0(S) first appears. For√N < λ < N , i.e.

√2EJ < g <

gc, the dipolar state minimal energy E0(S) is higher then the tunnelingground state energy E0(S). Hence, the system remains in the tunnelingstate (i.e. dipolar disordered). At λ = λc the ground state energy E0(S)crosses the dipole state energy branch E0(S) for the second time and goesabove E0(S). At the critical coupling g = gc (i.e. λ = λc) the first orderphase transition from the tunneling state to dipolar ordered state takesplace. It is, indeed, a first order transition, since at g = gc dipole momentin the dipolar state is already finite: 〈d〉 ≈ ±2elS, see (8.46), while in


Figure 8.5. Photon field 〈p〉 emerging in the cavity as a function of dimensionlesscoupling constant λ = g

√N/2EJ . The first order transition to the state with the

macroscopic photon occupation number⟨a†a⟩6= 0 occurs at the critical coupling

λc ' N +ω/EJ . The blue dotted line shows the metastable solution for 〈p〉, thatappears at λ =

√N .

the tunneling state it equals zero. Namely, the first order phase transitionresults in

〈p〉 = −Sg sin θ =

0 , g < gc

∓Sg√

1− 4E2J/g

4, g ≥ gc(8.48)

see Figure 8.5, and

−EJ⟨Sz⟩

= −SEJ cos θ =−SEJ , g < gc

−E2J2S/g2, g ≥ gc

, (8.49)

⟨d⟩

= 2elS sin θ =

0 , g < gc

±2elS√

1− 4E2J/g

4, g ≥ gc. (8.50)

The collective dipole moment (8.50) is defined by the angle θ, which isshown in the Figure 8.6.

It is important to mention here, that, comparison of (8.47) with (8.23)gives: g − gc =

√2EJ/N > 0. Hence, we have found the first order phase

transition in the region of validity (i.e. g < g) of the large superspin limitS = N/2 1, that justifies the use of the HP approach. In the limit g →


Figure 8.6. The angle θ, that characterizes rotation of HP, as a function ofdimensionless coupling constant λ = g

√N/2EJ . The colour scheme is chosen

the same as for the Figure 8.5.

+∞ the dipolar ordered ground state energy E0 approaches from below theground state energy of a free resonant photon, ω/2. Simultaneously, at g =gc the ground state energy: E0(S) = 0 < ω/2. Hence, our semiclassicaldescription indicates that after the dipole transition the system graduallyapproaches decoupled state E0(S) = ω/2, but with saturated value of thecollective dipole moment ∝ 〈Sy〉 → N/2. It is not possible to decide inthe framework of our semiclassical approach whether a crossover to a state〈Sy〉 = 0 happens in the g → +∞ limit. The latter state was predictednumerically in finite, even N spin-1/2 cluster realization of the extendedDicke model [164].

The excitations branches (8.35) of the diagonalized Hamiltonian areshown on the Figures 8.7, 8.8.The branch ε1, that grows with the increase of the coupling, goes to theinitial photon’s frequency ω after the first order transition. ε2 approacheszero in the strong coupling limit.

Combining together (8.47) and expression g = 2el√

4π/(√V ), one may

formulate the condition for the dipolar quantum phase transition as:

4el√π/V =

√2NEJ , (8.51)

where l is a penetration depth of electric field into Cooper pair box super-conducting island and V is volume of the microwave cavity. Taking into


Figure 8.7. Excitation branches ε1, ε1 (8.21, 8.35) as the functions of dimen-sionless coupling constant λ = g

√N/2EJ . The vertical axis is shown in the

logarithmic scale. At the critical coupling λc ≈ N the frequency ε1 jumps toε1 ≈ ω. The color scheme is the same as in Figure 8.4.

Figure 8.8. Excitation branches ε2, ε2 (8.21, 8.35) as a function of dimension-less coupling constant λ = g

√N/2EJ . The frequencies ε2, ε2 asymptotically

approach zero in the strong coupling limit. The color scheme is the same as inFigure 8.4.

account, that charging energy EC = (2e)2 /2C is of order: EC = 2e2/l,


Figure 8.9. Schematic layout of a Cooper pair box array inside a microwave res-onator of coplanar geometry with one-dimensional superconducting transmissionline (stripline resonator), similar to proposed in [165] for achieving of a strongcoupling g of two-level systems to the resonant photon in our frustrated DickeHamiltonian model.

one may rewrite (8.51) in the following form:

L ≈ l EC l2

EJNΣ , (8.52)

where Σ and L are wave-guide (microwave cavity) cross-section area andlength respectively. Assuming L ≈ Nl we finally find the following condi-tion:

N2 ≈ EC l2

EJΣ . (8.53)

Hence, we come to a similar conclusion (see Figure 8.9) as was alreadymade in [165], that in order to achieve strong coupling limit g ≥ gc fora Cooper pair box array of a ‘thermodynamic size’ N ≈ 102 inside amicrowave resonator, a coplanar geometry with one-dimensional super-conducting transmission line (stripline resonator) should be used, thusproviding inequality S/l2 1, and Cooper pair box should have chargingenergy much greater than Josephson coupling energy: EC EJ .

8.5 Conclusions 117

8.5 Conclusions

In summary, we have demonstrated that strong enough capacitive cou-pling of the Cooper pair box array of low-capacitance Josephson junc-tions to a microwave resonant photon may lead to a first order quantumphase transition. As a result, a dipolar ordered state of Cooper pairs isformed, coupled to emerged coherent photonic condensate. The phasetransition is of the first order due to infinitely coordinated antiferromag-netic (frustrating) interaction, that arises between Cooper pair dipoles ofdifferent Cooper pair boxes. This frustrating interaction is induced by thegauge-invariant coupling of the Josephson junctions to the resonant pho-ton vector-potential in the microwave cavity. The strength of the coherentelectromagnetic radiation field that emerges under the phase transition isproportional to the number N of the Cooper pair boxes in the array and isreminiscent of the superradiant state of Dicke model without frustratingterm found previously [181]. Nevertheless, the phase transition into thelatter state is of second order [181] (see also Figure 8.10 in the Appendix8.7).

The analytical description of the first order quantum phase transi-tion in the Dicke model with infinitely coordinated antiferromagnetic frus-trating interaction has become possible due to a new analytic tool: self-consistently ‘rotating’ Holstein-Primakoff representation for the Cartesiancomponents of the total spin, which is described in this paper. Our ap-proach enables, as a by-product, description of the second order quantumphase transition in the Dicke model without frustrating antiferromagneticinteraction, explored previously by other authors [181]. Nevertheless,‘rotating’ Holstein-Primakoff representation remains to be semiclassical(S →∞). Therefore, the region of ‘spin liquid’ with S ∼ 1 is not attain-able within this method.

8.6 Appendix: Bogoliubov’s transformation forthe frustrated Hamiltonian

Below we show in detail a diagonalization procedure of the Hamiltonian(8.19). Let’s introduce

px = i 1√2ω

(a† − a

)and x =

√ω

2(a† + a

)(8.54)


together with

py = i 1√2EJ

(b† − b

)and y =

√EJ2(b† + b

)(8.55)

and rewrite (8.19) in terms of (8.54, 8.55):

H =− EJ(S + 1

2

)+ 1

2(x2 + ω2 p2

x

)+ 1

2(y2 + E2

J p2y

)+ ωg

√SEJ pxpy + g2SEJ

2 p2y =

=− EJ(S + 1

2

)+ 1

2Kxy+ 12Kpxpy , (8.56)

where

Kxy = x2 + y2 , (8.57)

Kpxpy = ω2p2x + EJ

(EJ + g2S

)p2y + 2ωg

√SEJ pxpy . (8.58)

We diagonalize (8.56) by performing a linear transformation of the quan-tum operators:(

pxpy

)=(

cos γ sin γ− sin γ cos γ

)(p1p2

),

(xy

)=(

cos γ sin γ− sin γ cos γ

)(q1q2

). (8.59)

Then

Kxy = q21 + q2

2 , (8.60)

and

Kpxpy =(ω2 cos2 γ +EJ

(EJ + g2S

)sin2 γ − 2ωg

√SEJ sin γ cos γ

)p2

1

+(ω2 sin2 γ +EJ

(EJ + g2S

)cos2 γ + 2ωg

√SEJ sin γ cos γ

)p2

2

+((ω2 −EJ

(EJ + g2S

))sin 2γ + 2ωg

√SEJ cos 2γ

)p1p2. (8.61)

The diagonalization condition that eliminates the cross-term ∼ p1p2, is:

tan 2γ = 2ωg√SEJ

EJ (EJ + g2S)− ω2 . (8.62)

8.7 Appendix: Quantum phase transition in the Dicke model 119

So, diagonalized operator Kpxpy becomes:

Kpxpy = ε21 p

21 + ε2

2 p22 , (8.63)

where:

2ε21 =EJ

(EJ + g2S

)+ ω2 −

(EJ(EJ + g2S

)− ω2

)cos 2γ

− 2ωg√SEJ sin 2γ , (8.64)

2ε22 =EJ

(EJ + g2S

)+ ω2 +

(EJ(EJ + g2S

)− ω2

)cos 2γ

+ 2ωg√SEJ sin 2γ . (8.65)

Substitution of (8.62) to (8.64) and (8.65) gives the eigenvalues:

2ε21,2 = EJ

(EJ+g2S

)+ω2±

√(EJ(EJ+g2S)−ω2)2+4ω2g2SEJ . (8.66)

The transformation:

p1,2 = i 1√2ε1,2

(c†1,2 − c1,2

)and q1,2 =

√ε1,22(c†1,2 + c1,2

)(8.67)

finally gives the diagonal Hamiltonian (8.20).

8.7 Appendix: Quantum phase transition in theDicke model

We consider the standard Dicke Hamiltonian [169, 181] (modulo our no-tations)

H = 12(p2 + ω2q2

)+ gp Sy − EJ Sz (8.68)

at small coupling g. We apply (8.16, 8.17) to 8.68:

H =− EJ(S + 1

2

)+ ω

(a†a+ 1

2

)+ EJ

(b†b+ 1

2

)− g√Sω

2(a† − a

) (b† − b

). (8.69)

The Bogoliubov’s transformation, similar to those in Appendix 8.6, gives:

H = −EJ(S + 1

2

)+ ε1

(12 + c†1c1

)+ ε2

(12 + c†2c2

), (8.70)


with the excitations spectrum described by the new oscillator frequencies:

2ε21,2 = E2

J + ω2 ±√(

E2J − ω2)2 + 4ω2g2SEJ . (8.71)

The ground state energy equals:

E0(S) = −EJ(S + 1

2

)+ 1

2 (ε1 + ε2) (8.72)

One can check that as a function of S the energy E0(S) has minimumat S = N/2, i.e. at the end of the interval of all possible total spinvalues 0 ≤ S ≤ N/2. This fact justifies the Holstein-Primakoff approach(8.16-8.18) valid in the large spin limit.

However, the lowest branch of excitations becomes imaginary wheng > gc =

√EJ/S:

ε2 =

√√√√E2J + ω2

2 −

√(E2J − ω2)2 + 4ω2g2SEJ

2 (8.73)

Thus, the ground state described above is unstable in the intervalg > gc, compare [181].

The method described in Section 8.3.2 (8.25-8.27) transforms the Hamil-tonian (8.69) into:

H =ω(c†c+ i

√α(c† − c

)+ α+ 1

2

)− EJ cos θ

(S − b†b

)+ EJ sin θ i

√S

2(b† − b

)− g cos θ

√Sω

2(c† − c

) (b† − b

)+ g cos θ

√2ωα i

√S

2(b† − b

)+ g sin θ i

√ω

2(c† − c

) (S −

⟨b†b⟩)

+ g sin θ√

2ωα(S − b†b

). (8.74)

Here we have decoupled cubic in c, b operators terms in a mean-field ap-proximation. Conditions for vanishing of the linear terms ∝

(c† − c

)and(

b† − b)in the Hamiltonian (8.74) are:

√2ωα+ g sin θ

(S −

⟨b†b⟩)

= 0 (8.75)

EJ sin θ + g cos θ√

2ωα = 0 (8.76)

8.7 Appendix: Quantum phase transition in the Dicke model 121

Solving the system of equations (8.75) and (8.76) we find:

cos θ = EJSg2

1−

⟨b†b⟩

S

−1

' EJSg2 ≡

g2cg2 , (8.77)

√α = − gS√

2ω

1−

⟨b†b⟩

S

sin θ ' gS√2ω

√1− g4

cg4 , (8.78)

where both the shift√α and rotation angle θ are non-zero when g > gc.

Thus, using solutions (8.77) and (8.78) we obtain the initial Hamiltonian(8.74) in the form similar to (8.69), but renormalised with cos θ coeffi-cients:

H = EJS

2 cos θ(1− cos2 θ

)− EJ

cos θ

(S + 1

2

)+ ω

(c†c+ 1

2

)+ EJ

cos θ

(b†b+ 1

2

)− g cos θ

√Sω

2(c† − c

) (b† − b

). (8.79)

Next, we perform Bogoliubov’s transformation that diagonalizes (8.79), byperforming a linear transform of Bose-operators c, b into Bose-operatorse1,2, and obtain:

H = EJS

2 cos θ(1− cos2 θ

)− EJ

cos θ

(S + 1

2

)+ ε1

(12 + e†1e1

)+ ε2

(12 + e†2e2

)(8.80)

with the eigenvalues:

2ε21,2 = E2

J

cos2 θ+ ω2 ±

√√√√( E2J

cos2 θ− ω2

)2

+ 4ω2E2J , (8.81)

where both branches are now real for g > gc =√EJ/S due to renormal-

isation of the coefficients with cos θ factors. We have expressed in (8.81)the coupling constant g via cos θ using the self-consistency relation (8.77).As is obvious from (8.77) and (8.78), both the shift

√α and rotation angle

θ progressively deviate from zero with increasing coupling strength g inthe interval g > gc, thus, providing a description of the new stable phaseof the system.


Figure 8.10. The angle θ as a function of dimensionless coupling constantλ = g

√N/2EJ in the Dicke model without frustration term.

The ground state energy of the system is now:

E0(S) = − EJ2 cos θ (S + 1)− EJS

2 cos θ + 12 (ε1 + ε2) , (8.82)

which always has a minimum at the end of the spin interval, at S = N/2,thus justifying the Holstein-Primakoff approximation at finite angles θ.

Thus, we found the second order phase transition that is manifestedby a gradual rotation of the total spin expectation value in the y−z planeby an angle θ:

〈p〉 = −Sg sin θ =

0, g < gc

∓Sg√

1− g4c/g

4, g ≥ gc(8.83)

and

⟨d⟩

= 2eSl sin θ =

0 , g < gc

±2elS√

1− g4c/g

4, g ≥ gc(8.84)

where gc =√

2EJ/N and S = N/2. The angle θ, that describes thetransition is plotted in the Figure 8.10.

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Summary

Majorana fermions in superconductors are the subgap quasiparticle excita-tions that are their own antiparticles. They are an equal weight quantumsuperposition of a positively charged electron and a negatively chargedhole, that makes them charge-neutral but not an eigenstate of charge.Although the expectation value of charge is therefore zero, there are fluc-tuations of charge. This would enable a purely electrical detection ofMajorana states, outdoing low-temperature thermal measurements. Thefluctuations of the charge carried by Majorana fermions in superconduc-tors are studied in the first two chapters of this thesis.

Majorana fermions occur on the boundary of so-called topological su-perconductors, either the one-dimensional edge of a two-dimensional sys-tem or the two-dimensional surface of a three-dimensional system. Inchapter 2 we show that the charge fluctuations (electrical shot noise) areproportional to the thermal conductance, with a coefficient made up onlyfrom fundamental constants and the temperature. This result stems fromthe combination of time reversal and electron-hole symmetries.

The electrical shot noise (the second moment of the charge fluctu-ations) generated by the Majorana fermion in a single edge channel isquantised in units of 1

2×e2/h per eV voltage bias. In chapter 3 we extend

the quantization to all even moments of the charge fluctuations. At lowtemperatures the full probability distribution of the charge fluctuations istrinomial: a Majorana fermion transfers either −e, 0 or +e charge, withthe corresponding probabilities 1/4, 1/2 and 1/4. This result is drasticallydifferent from the usual binomial statistics of electrons.

In chapter 4 we proceed from topological superconductors to graphene,which is a single carbon layer. Although graphene is not superconducting,we demonstrate that there exists an analogy with Andreev reflection ina topological superconductor. In a superlattice of graphene on a suitablesubstrate an anti-unitary symmetry occurs. The latter one is mathemati-

140 Summary

cally equivalent to electron-hole symmetry in a superconductor. The “val-ley index” K,K′ in graphene then plays the role of the electron-hole degreeof freedom. Thus, the electron approaching perpendicularly to the inter-face between pristine graphene and a graphene superlattice is reflected inthe opposite valley, just as Andreev reflection scatters an electron into ahole and vice versa.

The properties of the systems mentioned above can be captured bynon-interacting theories at low energy scale. In the second part of thethesis, we deal with the effects of strong interactions between the parti-cles. In particular we focus on the Sachdev-Ye-Kitaev (SYK) model thatdescribes a metal with strong infinite-range interactions (mean strengthJ) between N fermions (Dirac/Majorana). The model is solvable in thelarge N limit, where a conformal symmetry emerges in the infrared. Thetransport observables resulting from the SYK model are considered in thethree following chapters.

In chapter 5 we predict that the tunneling spectroscopy of the complex-valued SYK model shows a duality between low and high voltage: Thedifferential conductanceG depends on the applied voltage eV only throughthe dimensionless parameter ξ = eV J/Γ2 and obeys the duality relationG(ξ) = G(π/ξ), where Γ characterizes coupling to the probe. This prop-erty of the current-voltage characteristic reveals the underlying conformalsymmetry of the SYK model.

In chapter 6 we show that in a system that contains at least twodiscrete states that are coupled to a quantum continuum, one can seethe isolated zero point in the spectral function of one of the states atthe frequency of the other state. This phenomenon makes it possible totest the continuous behavour of the spectral function of the SYK model.An electron propagating through a lattice of the SYK nodes is scatteredby those in a way that produces a characteristic “holographic” spectralfunction with a line of zeros.

In chapter 7 we point out that a quantum dot becomes superconduct-ing close to Majorana fermions described by the SYK model. Despite theinduced superconducting order parameter, which is an odd function offrequency, there is no excitation gap. The occurrence of “gapless super-conductivity” can be measured spectroscopically.

In the final chapter of this thesis, we consider a system with an in-finitely ranged interaction without disorder, in contrast with the SYKmodel. The interaction arises from the gauge-invariant coupling of N

141

low-capacitance Josephson junctions to a photon in a microwave cavity.The effective theory of the system turns out to be an extended DickeHamiltonian of N spins-1/2 with all-to-all repulsive interaction. We pro-pose a rotating Holstein-Primakoff representation for the total spin of Nparticles, which enables us to uncover the first-order phase transition inthe large N limit.

142 Summary

Samenvatting

Majorana-fermionen in supergeleiders zijn deeltjes die hun eigen antideel-tje zijn. Het zijn deeltjes in een quantumsuperpositie met gelijk gewichtvan positief geladen elektronen en negatief geladen gaten, waardoor zeladingsneutraal zijn maar geen eigentoestand van lading. Hoewel de ver-wachtingswaarde van de lading dus gelijk is aan nul, zijn er fluctuaties vande lading. Dit maakt het mogelijk om Majorana-fermionen langs elektri-sche weg te detecteren, hetgeen bij lage temperaturen veel eenvoudiger isdan detectie via thermische geleiding. De fluctuaties van de lading gedra-gen door Majorana-fermionen in supergeleiders worden bestudeerd in deeerste twee hoofdstukken van dit proefschrift.

Majorana-fermionen treden op aan de buitenkant van zogenaamde to-pologische supergeleiders. Dat kan de één-dimensionale rand zijn van eentwee-dimensionaal systeem of het twee-dimensionale oppervlak van eendrie-dimensionaal systeem. In hoofdstuk 2 leiden we af dat de ladings-fluctuaties (hagelruis) direct gerelateerd zijn aan de thermische geleiding,via een relatie die alleen fundamentele constantes en de temperatuur be-vat. Deze relatie volgt uit de combinatie van tijdomkeersymmetrie enelektron-gat symmetrie.

Het hagelruisvermogen (het tweede moment van de ladingsfluctuaties)dat wordt gegenereerd door de Majorana-fermionen in een enkel rand-kanaal is gequantiseerd in eenheden van 1

2 × e2/h per eV aangelegdespanning. In hoofdstuk 3 breiden we de quantisatie uit naar alle evenmomenten van de ladingsfluctuaties. Bij lage temperaturen is de volle-dige kansverdeling van de ladingsfluctuaties trinominaal: een Majorana-fermion zorgt voor een ladingsoverdracht van hetzij −e, 0 of +e lading,met de bijbehorende kansen 1/4, 1/2 en 1/4. Dit resultaat is fundamenteelanders dan de gangbare binomiale statistiek van gewone fermionen.

In hoofdstuk 4 maken we de overstap van topologische supergeleidersnaar grafeen, wat een enkele koolstoflaag is. Hoewel grafeen niet super-

144 Samenvatting

geleidend is, laten we zien dat er een analogie is met Andreev reflectiein een topologische supergeleider. In een superrooster van grafeen op eengeschikt substraat ontstaat een anti-unitaire symmetrie die wiskundig ge-zien geheel analoog is aan elektron-gat symmetrie in een supergeleider. De“valley index” K,K′ in grafeen speelt dan de rol van de elektron-gat vrij-heidsgraad. Aldus wordt het elektron dat het grensvlak loodrecht nadertgereflecteerd in de tegenovergestelde valley, net zoals bij Andreev reflectievan een elektron als een gat.

Tot zover hebben we vrije deeltjes beschreven. In het tweede deel vanhet proefschrift behandelen we de effecten van wisselwerking tussen dedeeltjes. Een specifiek voorbeeld waar we ons op richten is het Sachdev-Ye-Kitaev (SYK)-model, dat een metaal beschrijft met sterke lange-drachtinteracties (gemiddelde sterkte J) tussen N fermionen (Dirac-fermionen ofMajorana-fermionen). Het model is oplosbaar in de grote-N -limiet, wan-neer een conforme symmetrie optreedt. De experimenteel waarneembareeffecten volgend uit het SYK-model worden in de drie volgende hoofdstuk-ken behandeld.

In hoofdstuk 5 voorspellen we dat de tunneling spectroscopie van hetSYK-model een dualiteit tussen lage en hoge spanning laat zien: het diffe-rentiële geleidingsvermogen G hangt alleen af van de toegepaste spanningV via de dimensieloze parameter ξ = eV J/Γ2 en voldoet aan de dualiteits-relatie G(ξ) = G(π/ξ), waarbij Γ de sterkte van de tunnelkoppeling karak-teriseert. Deze eigenschap van de stroomspanningskarakteristiek onthultde onderliggende conforme symmetrie van het SYK-model.

Hoofdstuk 6 bestudeert de spectrale functie in het SYK-model. Vooreen systeem dat ten minste twee geïsoleerde toestanden bevat die zijngekoppeld aan een kwantumcontinuüm, kan men het geïsoleerde nulpuntin de spectrale functie van één van de toestanden waarnemen met behulpvan de frequentie van de andere toestand. Dit verschijnsel maakt hetmogelijk om de machtwetschaling van de spectrale functie van het SYK-model te testen. Een elektron dat gekoppeld is aan een “quantum dot”in de SYK-toestand wordt door de quantum dot verstrooid op een manierdie een karakteristieke “holografische” spectrale functie voortbrengt.

In hoofdstuk 7 wijzen we erop dat een quantum dot supergeleidendwordt in de nabijheid van Majorana-fermionen die wisselwerken volgenshet SYK-model. Ondanks de geïnduceerde supergeleidende ordeparameteris er geen excitatiekloof. Het optreden van “gapless superconductivity”kan spectroscopisch worden gemeten.

145

In het laatste hoofdstuk van dit proefschrift beschouwen we een geheelander type wisselwerkend systeem zonder wanorde, dit in tegenstelling tothet wanordelijke SYK-model. De wisselwerking komt voort uit de koppe-ling van N Josephson-juncties aan fotonen in een microgolfresonator. Delage energietheorie van het systeem blijkt een gegeneraliseerde Dicke Ha-miltoniaan te zijn van N deeltjes met spin 1/2 die elkaar afstoten. Wegebruiken een zogenaamde Holstein-Primakoff-representatie voor de totalespin van de N deeltjes, waarmee we een eerste orde fase-overgang in degrote-N -limiet kunnen vaststellen.

146 Samenvatting

Curriculum Vitæ

I was born on the 16th of September 1991 in Novosibirsk, Russia. I wentto primary school there, but in 2001 my family moved to Moscow, whereI attended middle and high school.

In 2008 I entered the Faculty of Theoretical and Experimental Physicsof the Moscow Engineering Physics Institute (State University). I wasstudying at the Department of Theoretical Nuclear Physics, where I com-pleted my bachelor (2012) and master (2014) theses under the supervisionof Dr. E. E. Saperstein from the Kurchatov Institute. After my under-graduate studies, I joined the Laboratory of Many-Particle Systems inthe Kurchatov Institute as a part-time researcher. During my master’sI worked on spectral properties of finite Fermi systems combining fieldtheoretical and density functional approaches.

In 2015 I started my Ph.D. studies at the University of Leiden in theTheoretical Nanophysics group led by Prof. Dr. C. W. J. Beenakker. Iinvestigated the transport properties of topological superconductors andgraphene superlattices. Another direction of my research is transport sig-natures of the absence of quasiparticles in low-dimensional systems. I wasalso actively collaborating with Prof. Dr. S. I. Mukhin from the MoscowInstitute of Steel and Alloys (State University) on the light-matter inter-actions with infinitely coordinated potentials. The results of my researchare presented in this thesis.

During my Ph.D. studies, I presented my research in The Netherlands,Germany, Spain, and France. I was a teaching assistant for several courses:Quantum Mechanics I, Superconductivity, and Introduction to Solid StatePhysics.

After finishing the Ph.D., I am moving to the Department of Physicsof Yale University to join the group of Prof. Dr. L. I. Glazman as apostdoc.

148 Curriculum Vitæ

List of Publications

[1] N. V. Gnezdilov and E. E. Saperstein. Calculation of double massdifferences for near-magic nuclei on the basis of a semimicroscopicmodel. JETP Lett. 95, 603 (2012).

[2] N. V. Gnezdilov, I. N. Borzov, E. E. Saperstein and S. V. Tolokon-nikov. Self-consistent description of single-particle levels of magicnuclei. Phys. Rev. C 89, 034304 (2014).

[3] N. V. Gnezdilov, E. E. Saperstein and S. V. Tolokonnikov. Spec-troscopic factors of magic and semimagic nuclei within the self-consistent theory of finite Fermi systems. EPL 107, 62001 (2014).

[4] N. V. Gnezdilov, B. van Heck, M. Diez, J. A. Hutasoit, and C. W.J. Beenakker. Topologically protected charge transfer along the edgeof a chiral p-wave superconductor. Phys. Rev. B 92, 121406(R)(2015) [Chapter 3].

[5] E. E. Saperstein, M. Baldo, N. V. Gnezdilov, and S. V. Tolokon-nikov. Phonon effects on the double mass differences in magic nu-clei. Phys. Rev. C 93, 034302 (2016).

[6] N. V. Gnezdilov, M. Diez, M. J. Pacholski, and C. W. J. Beenakker.Wiedemann-Franz-type relation between shot noise and thermal con-duction of Majorana surface states in a three-dimensional topologicalsuperconductor. Phys. Rev. B 94, 115415 (2016) [Chapter 2].

[7] O. V. Gamayun, V. P. Ostroukh, N. V. Gnezdilov, İ. Adagideli andC. W. J. Beenakker. Valley-momentum locking in a graphene super-lattice with Y-shaped Kekulé bond texture. New Journal of Physics20, 023016 (2018).

https://doi.org/10.1134/S0021364012120053

https://doi.org/10.1103/PhysRevC.89.034304

https://doi.org/10.1209/0295-5075/107/62001



https://doi.org/10.1103/PhysRevC.93.034302


https://doi.org/10.1088/1367-2630/aaa7e5

https://doi.org/10.1088/1367-2630/aaa7e5

[8] S. I. Mukhin and N. V. Gnezdilov. First-order dipolar phase transi-tion in the Dicke model with infinitely coordinated frustrating inter-action. Phys. Rev. A 97, 053809 (2018) [Chapter 8].

[9] C. W. J. Beenakker, N. V. Gnezdilov, E. Dresselhaus, V. P. Os-troukh, Y. Herasymenko, İ. Adagideli, and J. Tworzydło. Valleyswitch in a graphene superlattice due to pseudo-Andreev reflection.Phys. Rev. B 97, 241403(R) (2018), [Editors’ Suggestion] [Chap-ter 4].

[10] N. V. Gnezdilov, J. A. Hutasoit, and C. W. J. Beenakker. Low-highvoltage duality in tunneling spectroscopy of the Sachdev-Ye-Kitaevmodel. Phys. Rev. B 98, 081413(R) (2018) [Chapter 5].

[11] N. Gnezdilov, A. Krikun, K. Schalm, and J. Zaanen. Isolated zerosin the spectral function as signature of a quantum continuum. Phys.Rev. B 99, 165149 (2019) [Chapter 6].

[12] N. V. Gnezdilov. Gapless odd-frequency superconductivity inducedby the Sachdev-Ye-Kitaev model. Phys. Rev. B 99, 024506 (2019)[Chapter 7].

https://doi.org/10.1103/PhysRevA.97.053809






Stellingenbehorende bij het proefschrift

On transport properties of Majorana fermions in superconductors:free & interacting

1. Charge-neutral Majorana surface modes can be detected electrically.Chapter 2

2. The low-to-high voltage duality in the conductance of a quantum dot is anexperimentally accessible signature of conformal symmetry.

Chapter 5

3. The lines of zeros that appear in the energy distribution function of holo-graphic models of strange metals have a quantum-mechanical explanation.

Chapter 6

4. The quantum phase transition in a Cooper pair box array of Josephsonjunctions capacitively coupled to a a microwave cavity is of first order.

Chapter 8

5. The exactly unit transmission probability of an electron through a graphenebilayer at a specific angle of incidence signals a topological phase transitioninduced by a hidden anti-unitary symmetry.

6. A superconductor coupled to a Sachdev-Ye-Kitaev metal has two distinctcritical temperatures, as in Kondo superconductors.

7. To detect the Majorana fusion rule a tri-junction is advantageous over alinear junction.

8. The Lyapunov time of chaotic dynamics in a Sachdev-Ye-Kitaev metal canbe measured via the excitation gap induced by proximity to a supercon-ductor.

Nikolay GnezdilovLeiden, 12 June 2019

ontransportpropertiesof majoranafermionsin superconductors ... · topological superconductors are...

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