学士学位论文 - ustchome.ustc.edu.cn/~wzy7178/docs/my_projects/bc_zhiyuan... · 2017. 7....

学士学位论文

使用矩阵直积态算法定性研究

施温格模型相结构

作者姓名：魏志远

学科专业：物理

导师姓名：Prof. Ignacio Cirac, Dr. Mari Carmen Bañuls

完成时间：二 O一七年五月

University of Science and Technology of ChinaA dissertation for bachelor’s degree

Phase Structure of The Schwinger Model:A Qualitative Study with Matrix Product States

Author’s Name: Zhiyuan Wei

Speciality: Physics

Supervisor: Prof. Ignacio Cirac, Dr. Mari Carmen Bañuls

Finished Time: May, 2017

Contents

Acknowledgements 3

Abstract 4

1 Introduction 5

2 Theoretical Background 72.1 Matrix Product States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 MPS formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Area law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Quantum Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Finite size scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Finite entanglement scaling . . . . . . . . . . . . . . . . . . . . . . 12

2.3 The Schwinger Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Continuum formulation . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Phase structure of the Schwinger model . . . . . . . . . . . . . . . . 142.3.3 Lattice formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.4 Full model and truncated model . . . . . . . . . . . . . . . . . . . . 172.3.5 Lattice spin formulation of the Schwinger model . . . . . . . . . . . 19

3 Numerical Approach 213.1 Results of the Full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 Behavior of the axial fermion density . . . . . . . . . . . . . . . . . 223.1.2 Estimating the critical point for small number of parameters . . . . 233.1.3 Searching for FES in full model . . . . . . . . . . . . . . . . . . . . 25

3.2 Results of the truncated model . . . . . . . . . . . . . . . . . . . . . . . . 303.2.1 Behavior of the axial fermion density . . . . . . . . . . . . . . . . . 303.2.2 Estimating the critical point for small number of parameters . . . . 31

4 Conclusion and Outlook 34

1

CONTENTS 2

5 Appendix: Numerical Techniques 355.1 Searching for the ground state . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Estimating the error of locating pseudo-critical point . . . . . . . . . . . . 375.3 Choosing the best order parameter . . . . . . . . . . . . . . . . . . . . . . 39

Bibliography 41

Acknowledgements

First, I would like to thank Prof. Dr. Ignacio Cirac for letting me work in his researchgroup at MPQ. His great insight and wisdom in physics always stimulate me to pursue thegood science, and I look forward to furthering my science career in his group.

I am very grateful to Dr. Mari Carmen Banuls and Stefan Kuhn for supervising mein my thesis project. Mari Carmen guided us to work on good directions; Stefan led meclearly through all the theory and numerics. Working with them is always inspiring.

I would also like to thank all the people in the theory group at MPQ for the goodexperiences and conversations we shared.

Further, I want to thank Prof. Chuan-Feng Li. Prof. Li introduced me to the world ofquantum science and taught me important methods of doing scientific research, for which Ifeel deeply grateful. Also, thank all our group members in USTC, they create a fascinatingenvironment for academic research. Special thanks to Dr. Jian-Shun Tang and Yi-TaoWang for carefully guiding me in the ROC project.

Thank all my advisors and friends for your supports to me and our joyful moments.The deepest thanks to my family. They paid a great e↵ort to bring me up, and I always

thank their support and love during my life.

3

Abstract

In this thesis, we systematically study the phase structure of Schwinger model using theMatrix Product State (MPS) ansatz. We use a Hamiltonian lattice formulation with Kogut-Susskind staggered fermions and focus on MPS with a small number of variational param-eters. We study two approaches to the Schwinger model, which has previous been usedfor MPS. In the first approach, we integrate out the gauge degree of freedom; while in thesecond approach, we keep gauge degree of freedom but truncate the electric flux to a smalldimension.

By taking the fixed-volume continuum limit in both approaches for the axial fermiondensity, we estimated the critical point for both approaches. We also probed e↵ects ofusing a small number of variational parameters and show the finite-entanglement scalingbehavior in the Schwinger model. Our results show that MPS with small bond dimensiongive a good qualitative estimation for critical point, which is encouraging for exploringhigh-energy models in higher dimensions with tensor networks.

4

Chapter 1

Introduction

Since its invention, the Density Matrix Renormalization Group (DMRG) [1, 2] algorithmhas proven to be very successful for the simulation of one-dimensional spin chains. Later theDMRG algorithm was understood in the framework of the Matrix Product States (MPS) [3–8], which are a particular type of one-dimensional tensor network states (TNS) [9]. TNSare ansatzes for the quantum state which are characterized by their entanglement content.As one-dimensional TNS, MPS can be used to e�ciently describe the ground states andlow excited states of quantum spin chains, as well as to calculate thermal equilibriumproperties of these models. A great advantage of MPS over the the quantum Monte Carlomethod [10] is that MPS are free of the sign problem [11]. Now MPS/DMRG methodshas become a quasi-exact method for one-dimensional strongly correlated systems.

Recently, the TNS techniques have been applied toWilson’s lattice gauge theory(LGT) [12–20] with Hamiltonian formulation. One particularly interesting case is the Schwinger model[21]. It contains both a fermionic field and a bosonic gauge field, which exhibits featuressimilar to QCD, such as confinement [22] or chiral symmetry breaking [23]. For this reason,the Schwinger model is often used as a benchmark model to explore LGT techniques.

The Schwinger model exhibits a second-order phase transition with a certain back-ground electric field, which is first discussed by Coleman [24]. Close to the quantumcritical point, the correlation length becomes very large. Hence, one has to introduce avery large number of variational parameters in the MPS ansatz to properly capture thee↵ect. A high-precision DMRG calculation with a large number of variational parametershas been done in Ref. [25].

However, for higher-dimensional critical models, the Tensor Network computation witha large number of variational parameters can be computationally very expensive. For ex-ample, in the Projected Entangled Pair States (PEPS) [26, 27], which is a two-dimensionalgeneralization of MPS, the computational e↵ort scales with the D10 (here D represent thesize of the variational matrices). This limits the number of variational parameters that onecan reach in practical computations.

Thus the behavior of Tensor Network algorithm with a small number of variationalparameters at criticality would be very interesting, for it may already provide a qualitativelyestimation for the phase structure of the model. MPS and the Schwinger model are both

5

CHAPTER 1. INTRODUCTION 6

good starting point to investigate this idea.Hence, in this thesis, we try to understand the behavior of MPS with a small number

of variational parameters in the context of the Schwinger Model with background fieldin the critical regime. We consider two approaches to the Schwinger model. In the firstapproach, we integrate out the gauge degree of freedom, leading to nonlocal interactionsto the Hamiltonian. We call it the full model. In the second one, we imposes a truncationin electric flux to represent the states as MPS with finite physical dimension. Hence, wecall it the truncated model. In both models, the behavior of the axial fermion density isinvestigated and used to locate the corresponding critical point.

This thesis is organized as follows. Chapter 2 introduces the theoretical background ofthis work, including the Matrix Product States formalism, the quantum phase transitionand the Schwinger model. Chapter 3 focuses on the numerical approach and results on thebehavior of the order parameter and locating the critical point, as well as the investigationon finite entanglement scaling behavior in the full model. In Chapter 4 we conclude anddiscuss possible future work.

Chapter 2

Theoretical Background

2.1 Matrix Product States

MPS were first introduced as the ground state of the AKLT model in 1987 [28]. Severalyears after the invention of DMRG algorithm in 1992 [1], it was realized that DMRG isa variational ansatz over MPS [29, 30]. Later the connection between MPS ansatz andquantum entanglement was revealed from the quantum information perspective [7, 31].Here we briefly review the formalism and important properties of MPS.

2.1.1 MPS formalism

Let us consider a chain with N sites, an MPS with open boundary condition is a state ofthe form [27]:

| i =d�1X

i1,...,iN=0

Ai11

...AiNN

|i1

, ...iN

i (2.1)

d is the physical dimension of the Hilbert space for each site, and i1

, i2

, ..., in

are indicesof each site. Aik

k

are D ⇥D dimensional complex matrices for 1 < j < N and Ai11

(AiNN

) isa row (column) vector. The parameter D is called the bond dimension.

Fig. 2.1 is a graphical notation of open boundary MPS. The lines between adjacentsites denote the multiplication of matrices of size D.

A simple example of MPS is the Greenberger-Horne-Zeilinger (GHZ) state:

|GHZi = |0i⌦N + |1i⌦N

p2

(2.2)

We can express |GHZi as an MPS with open boundary condition. Choosing the fol-

7

CHAPTER 2. THEORETICAL BACKGROUND 8

Figure 2.1: MPS with open boundary condition.

lowing matrices (vectors for site 1 and site N):

A0

1

=�1

2

� 12N (1, 0) , A1

1

=�1

2

� 12N (0, 1)

A0

k 6=1,N

=�1

2

� 12N

1 00 0

�, A1

k 6=1,N

=�1

2

� 12N

0 00 1

�

A0

N

=�1

2

� 12N

✓10

◆, A1

N

=�1

2

� 12N

✓01

◆

(2.3)

In the case of N = 3, we have

|GHZi = A0

1

A0

2

A0

3

|000i+ A0

1

A0

2

A1

3

|001i+ A0

1

A1

2

A0

3

|010i+ A0

1

A1

2

A1

3

|011i

+A1

1

A0

2

A0

3

|100i+ A1

1

A0

2

A1

3

|101i+ A1

1

A1

2

A0

3

|110i+ A1

1

A1

2

A1

3

|111i

= 1p2

(|000i+ |111i)

(2.4)

This holds for arbitrary N � 2.

2.1.2 Area law

Here we consider a quantum system A and a subset of this system denoted B. The entiresystem has a density matrix ⇢

AB

. The entanglement entropy with respect to the partitionAB is:

S (⇢B

) = �TrB

[⇢B

log ⇢B

] (2.5)

where ⇢B

= TrA

(⇢AB

) is the reduced density matrix of subsystem B.For a generic state, the entanglement entropy S will be proportional to the volume of

the subsystem. However, the area law of the entanglement entropy states that:For ground states of one-dimensional local Hamiltonians with a gap, their entanglement

entropy are bounded by the size of the boundary between A and B.Thus these states are in a ‘corner’ of the one-dimensional Hilbert space.


Figure 2.2: Concept map of the area law

Fig. 2.2 shows the idea of the area law. If the system AB follows the area law, the Smust at most scale with the surface area |@B| of B [32]:

S (⇢) / |@B| (2.6)

MPS obey the one-dimensional area law by construction. More specifically, the entan-glement entropy in MPS is bounded by the logarithm of its bond dimension D [7]:

S (L) k logD (2.7)

where k is a constant parameter.

Figure 2.3: Illustration to show MPS with a larger bond dimension D can represent largerpart of the Hilbert space.

If we increase the bond dimension to D � dbN/2c, finally MPS can represent any quan-tum state of the one-dimensional many-body Hilbert space (See Fig. 2.3). However, due tothe area law of the ground states of local gapped Hamiltonians [33], MPS with relativelysmall bond dimension can already well approximate these ground states and relevant lowexcited states.


2.2 Quantum Phase Transition

2.2.1 Definition and properties

A quantum phase transition is a change in the nature of the ground state due to itsquantum fluctuations at the temperature zero. Quantum phase transitions can be inducedby varying a non-temperature physical parameter. Example of quantum phase transitionis exhibited by the quantum Ising model with transverse field [34]:

HIS

= �NX

i=1

�z

i

�z

i+1

� �NX

i=1

�x

i

(2.8)

This model undergoes a QPT at |�c

| = 1. Like many other cases, this QPT can bedescribed by an order parameter, and in this case the order parameter is the averagemagnetization in the z direction:

Mz

=

*1

V

X

i

�z

i

+(2.9)

In the thermodynamic limit (the system size N ! 1), the behavior of magnetizationwith lambda is shown in Fig. 2.4. It is zero when |�| > 1, and non-zero in the other phase.

Figure 2.4: Illustration of the magnetization Mz

in transversal field Ising model withvarious �.


Many concepts in classical phase transitions are also valid in quantum phase transitions.For example, phase transitions can be classified by its order. In first order phase transitions,the corresponding order parameter has a discontinuous jump. While in the higher orderphase transition, the order parameter itself is continuous however the second derivativeof the order parameters are divergent. Thus these higher order phase transitions are alsocalled the continuous phase transitions.

Continuous phase transitions are classified by their critical exponents, which describethe behavior of the order parameter near the phase transition. For instance, near the phasetransition of the above Ising model, we have the following scaling:

Mz

⇠��2 � �

c

2

��1/8 (2.10)

where 1/8 is the critical exponent. The critical exponents are believed to be the same formodels in the same universality class, which only depends on the dimension of the system,the range of the interaction, and the spin dimension.

For the continuous phase transitions, the correlation length ⇠ will scale with � whenclose to the critical point:

⇠ ⇠ |�� c

|�⌫ (2.11)

The scaling relation Eq. (2.11) plays an important rule in following finite size scalingand finite entanglement scaling methods.

2.2.2 Finite size scaling

The continuous phase transitions actually only occur in the thermodynamic limit. However,for practical calculations, one might be bound to finite system sizes. The idea of FiniteSize Scaling (FSS) [35] is just to analyze finite systems and deduce conclusions for thethermodynamic limit.

In infinite volume, the correlation length ⇠ diverges near the transition point as de-scribed in Eq. (2.11).

However, for finite systems with volume L, when ⇠ ⇠ L the divergence can no longercontinue. When ⇠ � L, L will become the most relevant length-scale. So here the systemhas a pseudo-critical point �

pc

(L) which satisfies

��c

� �Lpc

��c

⇠ L�1/⌫ (2.12)

Equation (2.12) provides a way to extrapolate to the thermodynamic limit from the finitesystem. For a finite system, the order parameter may not diverge, but reach the extremevalue. Thus we can locate the pseudo-critical point using this feature and estimate thecritical point by fitting the data with Eq. (2.12).


2.2.3 Finite entanglement scaling

As shown in Eq. (2.7), the entanglement entropy in MPS is bounded by its bond dimensionD. However, arguments from conformal field theory predict that the entanglement entropyalso scale with the correlation length ⇠ [36]:

S ' c

6log ⇠ (2.13)

where c is the central charge of the underlying conformal field theory. For the class ofquantum Ising model, we have c = 1

2

.Equation (2.7) and Eq. (2.13) hints that, in MPS the finite D bounds the correlation

length with following finite-D scaling relation:

⇠D

= D (2.14)

where = 2 for the Quantum Ising model.The scaling relation Eq. (2.14) was numerically tested by L. Tagliacozzo et al. [37]

using the infinite-MPS algorithm with a high precision. Since the bond dimension boundsthe entanglement entropy described by MPS, this scaling relation Eq. (2.14) is called thefinite entanglement scaling (FES). In next paragraphs, we follow the argument in [37] tointroduce the properties of FES.

In a critical point, one need infinite D for a MPS description of the ground statein the thermodynamic limit, because MPS represent finite correlation length states withexponentially decaying correlations [38]. Thus FES provides an alternative extrapolationmethod to get the value of an observable near to criticality, in which one does not needto work with MPS with matrices of very large bond dimension D. Due to the emergenceof the finite correlation length ⇠

D

with limited D, the critical value will be shifted to aD dependent pseudo-critical point. For the Quantum Ising model with a transverse field,we expect following scaling relation between the true critical point �

c

and pseudo-criticalpoint �D

pc

with finite D (similar to Eq.(2.12)):��D

pc

� �c

��c

⇠ D�/⌫ (2.15)

The FES e↵ect has been demonstrated in several models (e.g., the quantum Ising chain [37],the one-dimensional ANNNI model [39], and quantum three-state Potts model [40]).

Thus, for MPS describing finite systems there will be a correlation length ⇠N

inducedby the finite system size N and another correlation length ⇠

D

induced by the finite bonddimension D. Depending the values of ⇠

N

and ⇠D

, the system is in di↵erent regimes [41]:

For ⇠N

� ⇠D

: FSS regimeFor ⇠

D

� ⇠N

: FES regime

In the FSS (FES) regime, the scaling behavior of the order parameter is mainly domi-nated by N (D). In the crossover regime between FSS and FES, the order parameter willscale with both N and D, which we observe for the Schwinger model in Sec. 3.1.3.


2.3 The Schwinger Model

The Schwinger model, or QED in 1+1 dimensions, is presumably the simplest non-trivialgauge theory with matter. It includes the interaction of fermionic field and photon field, andshows some interesting features that also appear in full QCD model such as confinement [22]and the chiral symmetry breaking [23]. For this reason, Schwinger model is widely used totest numerical approaches to lattice gauge theories.

In this section, we introduce the Schwinger model by closely following Ref. [25].

2.3.1 Continuum formulation

The Lagrangian of the Schwinger model can be written as

L = �1

4Fµ⌫

F µ⌫ + (i�µ

@µ � g�µ

Aµ �m) (2.16)

Here Fµ⌫

= @µ

A⌫

� @⌫

Aµ

is the electromagnetic tensor, and =

✓ upper

lower

◆is a 2-

component Dirac spinor field. The coupling constant, g, has the dimension of mass, thusthe physics of this model is determined by a dimensionless parameter m/g. This modelhas an exact solution for m/g = 0 and m/g = 1, while in general cases there is no exactsolution.

In our calculation, we work with a Hamiltonian formulation in the time-like axial gaugeA

0

= 0. Thus we have F10

= �@0

A1

= E, where E is the one-dimensional electric field.So the Hamiltonian density becomes

H = �i �1@1

+ g �1A1

+m +1

2E2 (2.17)

There is an additional constraint given by Gauss’ law @1

E = g �0 . The Gauss’ lawcan be integrated, and yields:

E (x) = g

Zdxj0 (x) + F (2.18)

where j0 (x) = (x) �0 (x) is the charge density, and the constant of integration Fcan be interpreted as a background field [24].

The background field will lead to the creation of a charge pair if |F | > g/2, thus lowerthe electrostatic energy (shown in Fig. 2.5). The charges then will separate to infinityto lower the background field to |F | < g/2. This behavior at the strong coupling limitm/g ! 0 is shown in Fig. 2.6. Thus the behavior of this model is periodic in F with periodg.


Figure 2.5: The schematic drawing of the creation of a charged pair, which alters theelectric field by amount ±g in between.

Figure 2.6: The schematic drawing of the charged fermions and electric flux at backgroundfield F = g/2 in strong coupling limit m/g ! 0.

2.3.2 Phase structure of the Schwinger model

Coleman [24] investigated the Schwinger model with a background field F . In the weakcoupling regime, m/g ! 1, the vacuum energy of the Schwinger model consists only thestatic electric energy. Thus, due to the periodicity of static electric energy we have:

"0

=

⇢1

2

F 2

�F 1

2

g�

1

2

(g � F )2�1

2

g < F < g� (2.19)

From Eq. (2.19), one can see, that there is a discontinuity in the slope of the energydensity at F = 1

2

g, which corresponds to an first-order transition. However, in the strong-coupling limit m/g = 0 the vacuum energy density will remain constant as a function ofF .

Fig. 2.7 shows the vacuum energy density as a function of F andm/g. Hence, we expectthis first-order transition terminate on a second-order transition at (m/g)

c

[24]. Hamer etal. [42] first located this critical point at (m/g)

c

= 0.325(20) numerically. The most recentDMRG calculation [25] of this critical point gives (m/g)

c

= 0.3335(2).The axial fermion density [43] is usually used as an order parameter to show the critical

behavior:�5 =

⌦i �5 /g

↵0

(2.20)

�5 reaches its minimal at the critical point. This transition corresponds to a sponta-neous breaking of parity symmetry.

2.3.3 Lattice formulation

The Kogut-Susskind lattice formulation [44, 45] is especially useful for our numerical sim-ulation of the Schwinger model. Let a be the lattice spacing and label the sites with aninteger n. The two-component Dirac spinor field (x) is replaced as a form of ‘staggered


Figure 2.7: Vacuum energy "0

of the Schwinger model with various m/g and ↵. Forrelatively large m/g, "

0

experience a phase transition at F = g

2

. The structure of thispicture is inspired by Ref. [25].

fermion’ with field �n

, where fermion components are put on the odd sites and the anti-fermion components are put on the even sites. Between neighboring fermion sites sits alink variable L+

n

= ei✓n .The correspondence between lattice and continuum fields is:

�n

/pa !

⇢ upper

(x) n even lower

(x) n odd(2.21)

1

ag✓n

! �A1 (x) gLn

! E (x) (2.22)

The lattice formulation of the Schwinger Model have the following Hamiltonian:

H = � i

2a

NX

n=1

⇥�†n

ei✓n�n+1

� h.c.⇤+m

NX

n=1

(�1)n�†n

�n

+g2a

2

NX

n=1

(Ln

)2 (2.23)

We use the “compact” formulation, in which ✓n

2 [0, 2⇡] on the lattice. Ln

and ✓n

are


canonical conjugate variables:[✓

n

, Lm

] = i�nm

(2.24)

hence Ln

has integer eigenvalues Ln

= 0,±1,±2, ... .The lattice version of Gauss’ law is:

Ln

� Ln�1

= Qn

= �†n

�n

� 1

2[1� (�1)n] (2.25)

Fig. 2.8 shows this lattice chain.

Figure 2.8: The representation of ‘staggered fermion’. The yellow sites are the fermioncomponents and the brown sites are the antifermion components. The blue sites are gaugesites. The gauge sites are put in between the particle sites to keep the system gaugeinvariant.

Similar to the continuum model, in the open boundary condition case, one can useEq. (2.25) to determine the electric field in the entire chain up to an arbitrary additiveconstant ↵ = F

g

, which represents the background field. Thus we can separate this constantout of the electrostatic energy term in Eq. (2.23):

X

n

(Ln

)2 !X

n

(Ln

+ ↵)2 (2.26)

For our calculation, we use a dimensionless formulation, which is given by

W = � i

2x

NX

n=1

⇥�†n

ei✓n�n+1

� h.c.⇤+

µ

2

NX

n=1

(�1)n�†n

�n

+1

2

NX

n=1

(Ln

)2 (2.27)

where a and m/g in Eq.(2.23) is written as dimensionless parameters x = 1

g

2a

2 and µ = 2m

g

2a

:In the lattice formulation, the basis for MPS representation with open boundary con-

dition has following form:

| i =�� · · · u l d l u l d · · ·

↵(2.28)

where u (d) are fermionic (anti-fermionic) contents on even (odd) sites. l is the gauge linkvalue.

In strong-coupling regime we have m/g ! 0, the term µ

2

NPn=1

(�1)n�†n

�n

in Eq.(2.27)

can be ignored. Thus the strong-coupling ground state has following form: in the latticeformulation has following form:

| i =�� ... + 0 � 0 + 0 � ...

↵(2.29)


where the + (�) represent the fermionic (anti-fermionic) contents on even (odd) sites, andall gauge links l = 0.

In lattice formulation, the order parameter �5 becomes

�5 = � ipx

N

*X

n

(�1)n⇥�†n

ei✓n�n+1

� h.c.⇤+

0

(2.30)

Now the critical behavior at background field ↵ = 1

2

in the lattice formulation matchesprecisely with the continuum formulation.

2.3.4 Full model and truncated model

In our exploration, we consider two particular models. One is the exact Schwinger modelwith open boundary condition, which we call the full model. Another one imposes a trun-cation in electric flux, thus we call it the truncated model.

The full model

Let us first look at the Eq. (2.25) again. If we know the electric flux Li

for an arbitrarysite i, and Fermionic content on every site, we can use Eq. (2.25) to determine the electricflux for every site:

Ln

= Li

+nX

k=i+1

Qk

(2.31)

In our numerics, we work with background field ↵ = 1

2

in the subspace of vanishingtotal charge (which means

Pn

Qn

=Pn

��†n

�n

� 1

2

[1� (�1)n]�= 0). Thus the electric flux

in the boundary L0

= LN

. Fig. 2.9 shows the possible maximal and minimal values of theelectric flux in this case.

Plugging Eq. (2.31) to the Hamiltonian Eq. (2.23) gives the resulting Hamiltonian:

H = � i

2a

NX

n=1

⇥�†n

ei✓n�n+1

� h.c.⇤+m

NX

n=1

(�1)n�†n

�n

+g2a

2

NX

n=1

nX

i=1

Qi

+ L0

!2

(2.32)

Thus this eliminates the gauge degrees of freedoms (which makes it feasible to e�cientlysimulate this model), and introduces nonlocal interactions to the Hamiltonian. In followingnumerical calculation, we are interested in states with zero total charge (

Pn

Qn

= 0), so we

consider chains with even N .

The truncated model

The truncated model we consider corresponds to the proposal to simulate compact QEDusing fermionic and bosonic atoms trapped in an optical superlattice [46].


Figure 2.9: Possible maximal value (blue lines) and minimal value (yellow lines) of theelectric field L

n

at each site (red points) for the full model. The number of sites N is even.

The two kinds of bosons A and B have the creation operators a†n

and b†n

, thus thenumber of particles N

0

= a†n

an

+ b†n

bn

. This allows to generate a Hamiltonian of the formof Eq. (2.23) with the link operators

L+

n

= i2a†

n

bnp

N0

(N0

+ 2)Ln

=1

2

�a†n

an

� b†n

bn

�(2.33)

The link operators are angular momentum operators in the Schwinger representation.In this model, a truncation in electric flux |L

n

| Lmax

to represent the states as MPS withthe physical dimension of link variables d

link

= 2Lmax

+ 1 = N0

+ 1. Fig. 2.10 shows thepossible maximal and minimal values of the electric flux in this case. There are also otherkinds of truncation for the Schwinger model, see [47] for an example.

In the experimental realization of this model, the Gauss’ law is ensured via angularmomentum conservation in the scattering between the fermions bosons, thus does nothave to be imposed manually (usually by adding a penalty term [48–51]). In spite of thefinite link dimension, this model has the same gauge symmetry as the Schwinger model.When N

0

! 1, the link operators become pure phases (L+

n

! ei✓n) and this model recoverthe behavior of the lattice formulation of the Schwinger model Eq. (2.23).


Figure 2.10: Possible maximal value (blue lines) and minimal value (yellow lines) of theelectric field L

n

at each site (red points) for the truncated dlink

= 5 model. The numberof sites N is even. The flux truncation (green lines) limit the |L| L

max

= 2.

2.3.5 Lattice spin formulation of the Schwinger model

The lattice spin formulation of the Schwinger model is used in our numerical calculation.It can be obtained the Jordan-Wigner transformation [52]:

�n

=Y

l<n

⇥i�3

l

⇤��n

�†n

=Y

l<n

⇥�i�3

l

⇤�+

n

(2.34)

Plugging this into Eq. (2.23) yields the lattice spin Hamiltonian of the Schwinger model:

H =1

2a

X

n

��+

n

ei✓n��n+1

+ h.c.�+

1

2mX

n

�1 + (�1)n�3

n

�+

ga2

2

X

n

(Ln

)2 (2.35)

The Gauss’ law (Eq. (2.25)) becomes:

Ln

� Ln�1

=1

2

⇥�3

n

+ (�1)n⇤

(2.36)

And the �5 can be written in spin variables as following:

�5 =

px

N

*X

n

(�1)n⇥�+

n

ei✓n��n+1

+ h.c.⇤+

0

(2.37)


In the lattice spin formulation, the basis for MPS representation with open boundarycondition has following form:

| i =�� ... s

e

l so

l se

l so

...↵

(2.38)

where se

(so

) are spin value on even (odd) sites. l is the gauge link value.The strong-coupling ground state in the lattice spin formulation has following form:

| i =�� ... # 0 " 0 # 0 " ...

↵(2.39)

where the spin direction of even sites and odd sites are opposite to each other, and allgauge links l = 0.

Chapter 3

Numerical Approach

In our calculation, we use the lattice spin formulation of the Schwinger model with back-ground field ↵ = 1

2

in the subspace of vanishing total charge. The ground state is computedby standard MPS variational ansatz with open boundary condition (the algorithm is de-scribed in Ref. [7]) and a small number of variational parameters (which means smallbond dimension D in MPS). We studied the behavior of the axial fermion density �5, anduse �5 as the order parameter to estimate the critical point at D = 20 for both the fullmodel and the truncated model (See Appendix. 5.3 for the detail of choosing the orderparameter). We work in fixed-volumes vol = N/

px 2 [6, 30], and parameter x 2 [1, 400].

For the truncated model, we observe the e↵ect of flux truncation on order parameter, andhow this e↵ect possibly sets a limit on the ability to locate the true critical point.

We also show the (D,N) scaling behavior of the pseudo-critical point, to explore theinterplay of finite-size scaling and finite-entanglement scaling in the full model.

21

CHAPTER 3. NUMERICAL APPROACH 22

3.1 Results of the Full model

3.1.1 Behavior of the axial fermion density

In the continuum model, �5 is zero for m/g smaller than the critical mass (m/g)c

. Directlyat the critical point, m/g = (m/g)

c

, it has a discontinuity and jumps to a finite value,which is subsequently decreasing for m/g > (m/g)

c

. Numerically demonstration of thisbehavior can be found in Ref. [25]1.

Let us have a look at the behavior of �5 versus di↵erent m/g for di↵erent D in a finitesystem (Fig. 3.1).

Figure 3.1: �5 with various m/g with N = 50 and x = 25 (vol = N/px = 10) for the full

model. Data for di↵erent D are denoted by corresponding colors in the figure. The errorbars are smaller than the points size. As a guide for the eye, the data points are connected.

Compared with the result in Ref. [25], in Fig. 3.1 the value of �5 does not stay at 0when m/g < (m/g)

c

, and the discontinuous decrease of �5 at (m/g)c

is smoothed. InFig. 3.1, �5 will reach a minimum at the pseudo-critical point[43].

We can observe the e↵ect of small D in Fig. 3.1: the data for di↵erent D vary a lotnear the critical region, while keep close to each other at large m/g. The m/g value at thelocation of the minimum of every curve corresponds to the pseudo-critical point (m/g)

pc

for corresponding (N, x,D).

1In our calculation, the sign convention for �

5is the opposite to Ref. [25]. Thus in Ref. [25], �

5

discontinuously increase at (m/g)c and decrease steadily towards zero at large m/g.


3.1.2 Estimating the critical point for small number of parame-ters

In the lattice numerical calculation, one needs to extract the continuum limit from thefinite system. One can do the extraction by applying the corresponding scaling hypothesisfitting (see Sec. 2.2.2). For estimating the critical point of the full model with D = 20, wedo the following steps to extrapolate to the critical point:

Step 1 Locate the pseudo-critical point for every fixed pair (x, vol). We calculate theorder parameter �5 at di↵erent m/g, and choose the position of the minimal �5 as thepseudo-critical point (m/g)(x,vol)

pc

. The behavior of �5 is previously shown in Fig. 3.1.

Step 2 For every fixed volume vol, apply a linear fit to the data (m/g)(x,vol)pc

as a functionof 1/

px and extrapolate the value to 1/

px ! 0. Thus we get the pseudo-critical points

(m/g)(vol)pc

for every fixed volume vol.

Figure 3.2: Continuum extrapolations of the pseudo-critical point (m/g)(x,vol)pc

towards

1/px ! 0 for the full model. The colored dots are extrapolated results (m/g)(x,vol)

pc

fromstep 1, while the solid lines are the linear fitting using data inside the “fitting region” onthe figure. Error bars are determined by considering the ‘flatness’ of the critical region (seeAppendix 5.2 for the details).


Fig. 3.2 shows a typical example for Step 2. Possibly due to small D e↵ect, (m/g)(x,vol)pc

will deviate from the linear trend in large x regime (we discuss the reason of this phe-nomenon in Sec. 3.2.2). Thus in practice, we fit the data in an intermediate regime (the“fitting region” shown in Fig. 3.2). The volumes we use are vol = 6, 10, 14, 20, 30, and thefitting region is 1/

px 2 [0.15, 0.65]. In this region, the linear fit can well describe the data,

which is good enough to satisfy the accuracy requirement in our calculation.

Step 3 Apply a linear fit to the results (m/g)(vol)pc

and extrapolate the value to theinfinite volume limit, 1/vol ! 0. Thus we get the estimation for the critical point (m/g)

c

at D = 20. Fig. 3.3 show this procedure.

Figure 3.3: Continuum extrapolation for the pseudo-critical point (m/g)(vol)pc

towards

1/vol ! 0. The green dots are the extrapolated results (m/g)(vol)pc

from step 2, whilethe blue line is a linear fit of the data. As a reference, the red dashed line shows theaccurate DMRG [25] result (m/g)

c

⇠ 0.33. The error bars are smaller than the points size.


From Fig. 3.3, we can read out the critical point (m/g)c

= 0.389 (2). Considering thesmall D used, it is remarkably close to the true result (m/g)

c

= 0.3335(2) from Ref. [25],which supports the idea that small D qualitatively captures the critical behavior of thefull model.

3.1.3 Searching for FES in full model

For the full model, we investigate the scaling of (m/g)pc

with D, trying to probe the finiteentanglement scaling regime and the FSS/FES crossover regime. For small values of D,near criticality, MPS miss the system size N , and thus it is D that scales, while if D is largeenough, one can capture the e↵ect of the true size of the system, thus it is N that scales(see Sec. 2.2.3 for details). Here we consider open boundary condition, which is di↵erentfrom Ref. [37].

First, we can see the behavior of �5 for various D and N (in Fig. 3.4). From this graph,we see the data do not show a dependence with D when D > 20, while for smaller D thereis a big variation. Thus we expect that only at much smaller D values the FES e↵ects willbe visible.

Figure 3.4: The behavior of �5 for di↵erent (1/N, 1/D) at x = 16,m/g = 0.32. The surfaceis created by interpolating between the blue data points.


Therefore, we have performed calculations with very small D values. Here D 2[4, 7, 8, 9, 10, 11, 12, 13, 14, 16, 20], and we extrapolate the pseudo-critical points (m/g)

pc

with fixed (D,N) pairs for several lattice spacing x 2 [16, 100, 225]. We observed thatstarting the variational search from random MPS with such small values of D, results infrequent local minima thus get wrong ground states. For this problem, we apply processinspired by adiabatic evolution to find the correct ground state thus keep our numericsstable under small D (see Appendix 5.1).

Figure 3.5: (a) The behavior of (m/g)pc

for di↵erent N and D values at x = 16. Thesurface is created by doing interpolation between the blue data points. (b) the (m/g)

pc

asa function of N at various D. (c) The (m/g)

pc

as a function of D at various N .


Subfigure (a) in Fig. 3.5 shows m/g versus 1/D and 1/N at x = 16. In subfigure (b),when we increase D, the value for (m/g)

pc

will basically increase, then saturate at somepoint. This saturation indicates that we are entering the FSS regime, where the correlationlength ⇠

D

induced by D is much larger than the correlation length ⇠N

induced by N (e.g.⇠D

� ⇠N

), thus (m/g)pc

does not depend much on D. In subfigure (c), we also see the(m/g)

pc

increases with N when N is small, and reaches a stable value when N is large.The saturation shows that we are entering FES regime, where ⇠

N

� ⇠D

.The FES regime and FSS regime both exist in Subfigure (a). Moreover, there is a

FES/FSS crossover regimes where (m/g)pc

depends on both N and D.


for di↵erent N and D values at x = 100. Thesurface is created by doing interpolation to the blue data points. (b) The (m/g)

pc

as afunction of N at various D. (c) The (m/g)

pc



Subfigure (a) in Fig. 3.6 shows m/g versus 1/D and 1/N at x = 16 at x = 100.Compared to the x = 16 case, subfigure (a) shows that for small D values, the (m/g)

pc

are shifted to smaller values. This is reflected in subfigure (b), in which there is a “middleplateau” when we increase D, and then go to a higher plateau if we further increase D.


for di↵erent N and D values at x = 225. Thesurface is created by doing interpolation to the blue data points. (b) The (m/g)

pc

as afunction of N at various D. The black arrow points to the rough location of the thresholdD

th

. (c) The (m/g)pc



Fig. 3.7 shows⇣1/D, 1/N, (m/g)

pc

⌘at x = 225. Compared to previous cases, the

(m/g)pc

values at smallD are shifted more towards zero, and the “plateau” feature becomesmore clear. In this case we can roughly observe a threshold D

th

in subfigure (b). Here Dmust be larger than D

th

to reach the “plateau” and get reliable (m/g)pc

values.From these data, we see the FES/FSS crossover regime appear at small D and small

x. In the case of small x, the (m/g)pc

� 1/N, 1/D surface is more smooth, while for largex case a sharp transition happens near to a threshold D

th

. These behaviors possibly set alimit of applying this technique to be used in our MPS calculation of the Schwinger model.Further investigation in two-dimensional models will be interesting because in that case,the computation is limited in small D regime.


3.2 Results of the truncated model

3.2.1 Behavior of the axial fermion density

In the truncated model, the maximal electric flux Ln

is limited by |Ln

| Lmax

, and thevalue of L

max

determines the dimension of the link degrees of freedom, dlink

= 2Lmax

+ 1.The flux truncation leads to a deviation of the result from the full model. This e↵ect isdirectly reflected in the behavior of the order parameter �5.

Fig. 3.8 shows the behavior of �5 for di↵erent D with dlink

= 7. Compared to the fullmodel case (Fig. 3.1), we can observe that the data also tend to vary more when near tothe critical region, and the minima of �5 getting shifted towards smaller (m/g)

pc

.

Figure 3.8: �5 with various m/g with system size N = 50 and lattice spacing x = 25 forthe truncated d

link

= 7 model. Data with di↵erent D are denoted by corresponding colorsin the figure.

The flux truncation also plays an important rule to modify the behavior of �5 withm/g, which is shown in Fig. 3.9.


Figure 3.9: �5 with various m/g with N = 50, x = 25 and D = 20 for di↵erent truncatedmodels. Data with di↵erent link dimension d

link

are denoted by corresponding colors inthe figure.

From Fig. 3.9, di↵erent truncated dlink

models cause a big di↵erence on �5 when m/gis close to the critical region. The truncation shifts the pseudo-critical point towardsm/g = 0.

By Comparing Fig. 3.8 and Fig. 3.9, we can observe that in this regime, varying dlink

cause a much stronger di↵erence on �5 than varying D. This phenomenon holds for allcases in our exploration, which demonstrates that the behavior of truncated model is verysensitive to d

link

.

3.2.2 Estimating the critical point for small number of parame-ters

The procedure of estimating the critical point for the truncated model is the same as forthe full model, we first locate the pseudo-critical point (m/g)(x,vol)

pc

by the position of theminimal �5. Then we do the 1/

px ! 0 extrapolation and 1/vol ! 0 extrapolation.

As we already saw in Fig. 3.9, the flux truncation will shift the pseudo-critical pointtowards smaller (m/g)

pc

values. This is reflected in the extrapolation to 1/px ! 0 step.

(See Fig. 3.10).


Figure 3.10: Continuum extrapolations for the pseudo-critical point (m/g)(x,vol)pc

for the

truncated dlink

= 7 model. The colored dots are extrapolated results (m/g)(x,vol)pc

fromstep 1, while the solid lines are the linear fit using data inside the“fitting region” 1/

px 2

(0.4, 1.01) on the figure. Error bars are determined by considering the ’flatness’ of thecritical region see Appendix 5.2 for the detail.

In Fig. 3.10, the data can be basically divided into two regimes. For small values of1/px, the (m/g)(x,vol)

pc

shows a linear trend when we increase the system size. For relatively

small values of 1/px, the (m/g)(x,vol)

pc

begin to heavily shift down toward zero.Thus for very reduced lattice spacings and correspondingly large system sizes, the small

D MPS with open boundary condition is not capturing the “true” physics of these models.This e↵ect set a limit to the range of parameters we can explore with our numerics (whichis not computationally demanding).

Hence we have to limit our linear fit in the intermediate x region. With this intermediateextrapolation method, we can qualitatively locate the pseudo-critical point (m/g)(vol)

pc

fordi↵erent volumes, further do the linear extrapolation to the 1/vol ! 0. This techniqueworks for the truncated models with d

link

� 5, for the truncated dlink

= 3 model the(m/g)(x,vol)

pc

get shifted down to zero so heavily that we can not find a large enough fittingregion.


Figure 3.11: The critical point (m/g)c

with di↵erent link dimension dlink

estimated atD = 20. The blue dots are the extrapolated results for di↵erent truncated models. As areference, the red dashed line shows the accurate DMRG [25] result (m/g)

c

⇠ 0.33.

Similar to Fig. 3.3, Fig. 3.11 demonstrates that the extrapolation procedure includ-ing the intermediate region extrapolation, can give us a good qualitative estimation forthe critical point in truncated models. The value of (m/g)

c

for the critical point will in-crease with d

link

, and the trend is consistently towards the value of the full model result(m/g)

c

= 0.389 (2) during 1/dlink

! 0.

The following table (3.1) shows the results of (m/g)c

for various of dlink

.

Table 3.1: (m/g)c

for various dlink

models with D = 20.

dlink

5 7 9 11 15(m/g)

c

0.273(8) 0.33(1) 0.35(2) 0.358(5) 0.37(1)

Chapter 4

Conclusion and Outlook

In this thesis, we investigated the phase structure of the Schwinger model using MatrixProduct States with small bond dimension (D ⇠ 20). For the full model and the truncatedmodels with various d

link

, the behavior of axial fermion density �5 is investigated and usedto locate the critical point in both cases. We fix D = 20 and perform fixed-volume extrapo-lation in di↵erent volumes to get pseudo-critical points at continuum limit 1/

px ! 0, then

extrapolate to infinite volume to get the critical point. In the 1/px ! 0 extrapolation,

we overcome the problem of criticality behavior disappearing at large x in our numerics bychoosing an intermediate region to do extrapolation. We also took first steps towards thefinite entanglement scaling behavior for the full model.

For the full model, we observed the behavior of �5 and compared with results in [25]to see the finite system e↵ects. We also showed that varying D can a↵ect the value of�5 more when close to the critical region. Our estimation of the critical point for the fullmodel yields (m/g)

c

= 0.389 (2). The pseudo-critical points scale with D mainly for thesmall D regime (D 20), and smooth behavior could only be obtained with small x cases,thus qualitatively revealing the finite entanglement scaling regime for the full model.

For the truncated model, the value of �5 with di↵erent D also varies more when closeto the critical region. The flux truncation e↵ect heavily shifts the pseudo-critical pointtowards smaller m/g. We estimated critical points for the truncated models with variousflux truncation d

link

(for example, we get (m/g)c

= 0.331 (9) for truncated dlink

= 7model). The trend of (m/g)

c

values is consistently towards the value of the full modelresult (m/g)

c

= 0.389 (2) during 1/dlink

! 0.Compared to the reference value (m/g)

c

⇠ 0.33, our results show that small MPSgive good qualitative estimations for the critical point in the full model and that in thetruncated models.

Further possible investigations like the e↵ect of flux truncation, and the reason of criticalbehavior disappearing at large x in our numerics, would be very interesting. Ultimately,a most interesting direction for research would be to explore how these results extend tothe two-dimensional case of PEPS [26, 27]. Given the complexity of numerics, in thatcase, it would be extremely useful to be able to extract reliable information about higherdimensional models from small D simulations.

34

Chapter 5

Appendix: Numerical Techniques

The results in this thesis were obtained using a variational search for the ground state oversets of MPS with small bond dimension D, an algorithm which is similar to DMRG [7].The particular application to our problem requires nevertheless specific techniques. In thisappendix, we describe some of the practical details in the calculations.

5.1 Searching for the ground state

In the numerics, the small bond dimension sometimes will cause the MPS variationalground state search get trapped in a local minima of energy. Thus we apply an adiabatic-inspired process [53] to produce the ground state in this case. It works as follows:

Step 1 For the calculation at the set of parameters (N, x,D), we start the numericalprogram for very large (m/g)

init

(in practice, (m/g)init

= 200), and set the strong couplingground state (see Sec. 2.3.1) as the initial state for the variational algorithm. In this weakcoupling regime, we know the program can reliably find the ground state |G

0

i.

Step 2 Using the resulting ground state |G0

i in Step 1 as the initial state, we start theprogram to find the ground state of the model with the same set of (N, x,D) and a slightlysmaller m/g. Since |G

0

i is a reliable ground state for parameters in Step 1, it should be agood guess for the system in Step 2, where the parameters are almost the same with thatin Step 1.

Step 3 Similar to the Step 2, we iteratively use the state |Gn�1

i as the initial state tofind the ground state |G

n

i of the model with slightly smaller m/g for each time. At theend we get reliable data for small m/g case.

35

CHAPTER 5. APPENDIX: NUMERICAL TECHNIQUES 36

Figure 5.1: The order parameter �5 for di↵erentm/g inD = 20, x = 400, N = 400 case. (a)Results computed by MPS using random state as the initial state. (b) Results computedby adiabatic process as described in text.

Fig. 5.1 clearly shows the improvement of data quality using this approach.


5.2 Estimating the error of locating pseudo-critical

point

Since our calculations are done in systems with finite (N, x,D) rather than in the ther-modynamic and continuum limit, the critical feature of the model may not be evident.This corresponds to the minimum of the �5 � m/g curve becoming flat. Let us considera (m/g)

2

near to the pseudo-critical point (m/g)pc

. If the value of �5 at (m/g)2

is within(1± ") · �5

min

(here " is the relative numerical fluctuation), then it becomes possible that(m/g)

2

is the true (m/g)pc

. Thus we adopt the distance of (m/g)2

and (m/g)pc

as the errorin our calculation. This is illustrated in Fig.5.2.

Figure 5.2: Example of locating the pseudo-critical point (m/g)(x,vol)pc

(here D = 20, vol =

10, x = 36). The position of the red point is (m/g)(x,vol)pc

and surrounding line correspondsto the its error bar. In this case, the errorbar is chosen by the range of points within thebox |�5| 2 (1± 0.01) |�5

min

|.

If no (m/g)2

be found to have a �5 within (1± ") ·�5

min

, we adopt the distance betweenthe minimum (m/g)

pc

to its nearest two other points m/g as the error. This is illustratedin Fig. 5.3.


Figure 5.3: Example of locating the pseudo-critical point (m/g)(x,vol)pc

(here D = 20, vol =

10, x = 225). The position of the red point is (m/g)(x,vol)pc

and surrounding line correspondsto the its error bar. In this case, the error bar is chosen by the distance to the nearest twopoints.

This method is used to get the error bars in Fig. 3.10 and Fig. 3.2 when locating thepseudo-critical point. We set " = 10�5 in our calculation.


5.3 Choosing the best order parameter

There are three order parameters used in [25] to locate the critical point of the Schwingermodel. They are:

• The axial fermion density : �5 = � i

px

N

⌧Pn

(�1)n⇥�†n

ei✓n�n+1

� h.c.⇤�

0

(lattice form)

• The average electric field : �↵ = 1

N

⌧Pn

(Ln

+ ↵)

�

0

(lattice form);

• The energy gap between ground state and the first excited state : � = E1

� E0

.

In continuum and thermodynamic limit, �↵ and �5 are supposed to be 0 when m/g <(m/g)

c

, and �↵ discontinuously increases at (m/g)c

while �5 discontinuously decreases at(m/g)

c

. �5 then increase steadily towards zero at large m/g, whereas �↵ approaches theexpected value of 1

2

. The gap � is suppose to be zero at (m/g)c

, and will increase bothtowards m/g ! 0 and m/g ! 1. A numerical demonstration of these behaviors can befound in Ref. [25]1.

In our calculation, we can see the behavior of this three parameters for various bonddimension:

Fig. 5.4 is an example of the behavior of this three order parameters in our calculation.Since we have various kinds of truncation e↵ects (in general finite N , D, x, also for thetruncated model we have finite d

link

), the behaviors of order parameters deviate to theprecise results in Ref. [25] in following aspects:

In subfigure (a), the value of �5 does not stay at 0 when m/g < (m/g)c

, and thediscontinuity feature of �5 at (m/g)

c

is not clear. However, since �5 monotonically increasesto zero at large m/g, we can use the position of minimal �5 to locate (m/g)(x,vol)

pc

.In subfigure (b), the value of �↵ also does not stay at 0 when m/g < (m/g)

c

, and thediscontinuity feature of �↵ at (m/g)

c

is not clear. Due to the fact that the entire curve ismonotonic, there is no simple clear feature for �↵ to locate (m/g)(x,vol)

pc

.In subfigure (c), the position of minimal � can be considered as a potential feature to

locate the pseudo-critical point. However, compared to the minimum of �5 in subfigure(a), the curve is much more flat, thus will yield a larger error for (m/g)(x,vol)

pc

according tothe method mentioned in Sec. 5.2.

Thus in our calculation, we choose the axial fermion density �5 as our order parameter,which provides the most clear and accurate way to locate (m/g)(x,vol)

pc

.

1In our calculation, the sign convention for �

5is the opposite to Ref. [25]. Thus in Ref. [25], �

5

discontinuously increase at (m/g)c and decrease steadily towards zero at large m/g.


Figure 5.4: The order-parameters versus m/g curve with di↵erent bond dimension D(x = 25, N = 50). (a) The behavior of �5. In our calculation, the sign convention for �5 isthe opposite to Ref. [25]. (b) The behavior of �↵. (c) The behavior of the gap �.

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