hybrid monte-carlo simulations of electronic properties of graphene [ arxiv:1206.0619]

40
Hybrid Monte-Carlo simulations electronic properties of graphe [ArXiv:1206.0619] P. V. Buividovich (Regensburg University)

Upload: everly

Post on 24-Feb-2016

34 views

Category:

Documents


0 download

DESCRIPTION

Hybrid Monte-Carlo simulations of electronic properties of graphene [ ArXiv:1206.0619]. P. V. Buividovich ( Regensburg University). Graphene ABC. Graphene : 2D carbon crystal with hexagonal lattice a = 0.142 nm – Lattice spacing π orbitals are valence orbitals (1 electron per atom) - PowerPoint PPT Presentation

TRANSCRIPT

Page 1: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Hybrid Monte-Carlo simulations of electronic properties of graphene

[ArXiv:1206.0619]P. V. Buividovich

(Regensburg University)

Page 2: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Graphene ABC• Graphene: 2D carbon crystal with hexagonal lattice • a = 0.142 nm – Lattice spacing• π orbitals are valence orbitals (1 electron per atom)• Binding energy κ ~ 2.7 eV• σ orbitals create chemical bonds

Page 3: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]
Page 4: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Two simplerhombic

sublattices А and В

Geometry of hexagonal lattice

Periodic boundary conditions on the Euclidean torus:

Page 5: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

The “Tight-binding” Hamiltonian

YXYX aa ,',', }ˆ,ˆ{ Fermi statistics

“Staggered” potential m distinguishes even/odd lattice sites

Page 6: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Physical implementation of staggered potential

Boron Nitride

Graphene

Page 7: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Spectrum of quasiparticles in grapheneConsider the non-Interacting tight-binding model !!!

Eigenmodes are just the plain waves:

Eigenvalues:

3

1

)(a

eki aek

One-particleHamiltonian

Page 8: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Spectrum of quasiparticles in graphene

Close to the «Dirac points»:

“Staggered potential” m = Dirac mass

Page 9: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Spectrum of quasiparticles in graphene

Dirac points are only covered by discrete lattice momenta if the lattice size is a multiple of three

Page 10: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

2 Fermi-points Х 2 sublattices= 4 components of the Dirac spinor

),,,( RLRL

),,,( BABA

Chiral U(4) symmetry (massless fermions): right left

Discrete Z2 symmetry between sublattices

А В

Symmetries of the free Hamiltonian

U(1) x U(1) symmetry: conservation of currents with different spins

Page 11: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

• Each lattice site can be occupied by two electrons (with opposite spin)

• The ground states is electrically neutral

• One electron (for instance ) at each lattice site

• «Dirac Sea»: hole = absence of electron in the state

Particles and holes

Page 12: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Lattice QFT of Graphene

Redefined creation/annihilation operators

Charge operator

Standard QFT vacuum

Page 13: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Electromagnetic interactions

Link variables (Peierls Substitution)

Conjugate momenta = Electric field

Lattice Hamiltonian(Electric part)

Page 14: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Electrostatic interactions

rerV)1(

2)(2

Dielectric permittivity:

• Suspended graphene ε = 1.0• Silicon Dioxide SiO2

ε ~ 3.9• Silicon Carbide SiC ε ~ 10.01

2

Effective Coulomb coupling constantα ~ 1/137 1/vF ~ 2 (vF ~ 1/300)

Strongly coupled theory!!!Magnetic+retardation effects suppressed

Page 15: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Discretization of Laplacian on the hexagonal latticereproduces Coulomb potential with a good precision

Electrostatic interactions on the lattice

Page 16: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Main problem: the spectrum of excitations in interacting graphene

Lattice simulations, Schwinger-Dyson equations

Renormalization,Large N,Experiment [Manchester group, 2012]

Spontaneous breaking of sublattice symmetry= mass gap = condensate formation =

= decrease of conductivity

???

Page 17: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Numerical simulations: Path integral representation

∫𝒅Ψ|Ψ>¿Ψ|= 𝑰 ¿Ψ >¿ ∏¿𝑋𝑌 > ¿¿ 𝜃 𝑋𝑌>¿η 𝑋>¿ ¿ ¿

¿

�̂�𝑋𝑌∨𝜃𝑋𝑌 >¿ 𝜃𝑋𝑌∨𝜃𝑋𝑌 >¿

ψ̂ 𝑋∨η𝑋≥η𝑋∨η𝑋>¿

Decomposition of identity

Eigenstates of the gauge field

Fermionic coherent states (η – Grassman variables)

�̂�=∏𝑋𝛿(∑

𝑌𝜋 𝑋𝑌− �̂�𝑋) Gauss law constraint

(projector on physical space)

Technical details

Page 18: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Numerical simulations: Path integral representation

: • Electrostatic potential field• Lagrange multiplier for the Gauss’ law• Analogue of the Hubbard-Stratonovich field

Technical details

Page 19: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Lattice action for fermions (no doubling!!!):

Path integral weight:

Numerical simulations: Path integral representation

Positive weight due to two spin components!

Technical details

Page 20: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Hybrid Monte-Carlo: a brief introductionProblem:

generate field configurations φ(x) with probability

For graphene – nonlocal action due to fermion determinant

Metropolis algorithmPropose new field configurations with probability

Accept/reject with probability

• Exact algorithm• Local updates of fieldsBUT:• Fermion determinant recalculation

Page 21: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Hybrid Monte-Carlo: a brief introductionMolecular Dynamics

• Global updates of fields ϕ(x)

• 100% acceptance rateBUT: • Energy non-conservation

for numerical integrators

Classical motion with

If ergodic:

π(x) – conjugate momentum for φ(x)

Page 22: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Hybrid Monte-Carlo = Molecular Dynamics + Metropolis

• Use numerically integrated Molecular Dynamics trajectories as Metropolis proposals

• Numerical error is corrected by accept/reject

• Exact algorithm

• Ψ-algorithm [Technical]: Represent determinant as Gaussian integralMolecular Dynamics

Trajectories

Page 23: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

• Hexagonal lattice• Noncompact U(1) gauge field• Fast heatbath algorithm outside of graphene plane• Geometry: graphene on the substrate

Numerical simulations using the Hybrid Monte-Carlo method

Page 24: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Breaking of lattice symmetry

• Anti-ferromagnetic state (Gordon-Semenoff 2011)• Kekule dislocations (Araki 2012)• Point defects

Intuition from relativistic QFTs (QCD): Symmetry breaking = = gap in the spectrum

Page 25: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Spontaneous sublattice symmetry breaking in graphene

Order parameter:The difference between the number of particles on А and В sublattices

ΔN = NA – NB“Mesons”: particle-hole bound state

Page 26: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Differences of particle numbers

Page 27: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Differences of particle numbers on lattices of different size

Extrapolation to zero mass

Page 28: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Susceptibility of particle number differences 0

m

NN m

Page 29: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Conductivity of grapheneCurrent operator: = charge, flowing through lattice links

Retarded propagator and conductivity:

Page 30: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Current-current correlators in Euclidean space:

Green-Kubo relations:

Thermal integral kernel:

Technical detailsConductivity of graphene:

Green-Kubo relations

Page 31: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Conductivity of grapheneσ(ω) – dimensionless quantity (in a natural system of units), in SI: ~ e2/h

Conductivity from Euclidean correlator: an ill-posed problem Maximal Entropy Method

Approximate calculation of σ(0):

AC conductivity, averaged over w ≤ kT

)0()()()2/sinh(

22

)2/( 2

0

kTwwwdwG ∫

Technical details

Page 32: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Conductivity of graphene:free theory

For small frequencies (Dirac limit):

Threshold value w = 2 m

Universal limiting value at κ >> w >> m: σ0 = π e2/2 h=1/4 e2/ħ

At w = 2 m:σ = 2 σ0

Page 33: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Conductivity of graphene:Free theory

Page 34: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Current-current correlators: numerical results

κ Δτ = 0.15, m Δτ = 0.01, κ/(kT) = 18, Ls = 24

Page 35: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Conductivity of graphene σ(0): numerical results

(approximate definition)

Page 36: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Direct measurements of the density of states

• Experimentally motivated definition

• Valid for non-interacting fermions

• Finite μ is introduced in observables only (partial quenching)

Page 37: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Direct measurements of the density of states

m/κ = 0.1

Page 38: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Direct measurements of the density of states

m/κ = 0.5

Page 39: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Conclusions• Electronic properties of graphene at half-filling can be

studied using the Hybrid Monte-Carlo algorithm. • Sign problem is absent due to the symmetries of the

model.• Signatures of insulator-semimetal phase transition for

monolayer graphene. • Order parameter: difference of particle numbers on two simple sublattices• Spontaneous breaking of sublattice symmetry is

accompanied by a decrease of conductivity• Direct measurements of the density of states indicate

increasing Fermi velocity

see ArXiv:1206.0619

Page 40: Hybrid  Monte-Carlo simulations  of  electronic  properties of  graphene [ ArXiv:1206.0619]

Outlook• Lattice simulations by independent groups: Spontaneous symmetry breaking in graphene at α ~ 1 (ε ~ 4 – SiO2) [Drut, Lahde; Hands; Rebbi; ITEP Group; PB]• Experimentally: suspended graphene is

conducting, no signature of a gap in the spectrum [Elias et al. 2011]• What are we missing? Mass? Finite volume?• Our strategy works not so well as for Lattice QCD

• Another interesting case: double-layered graphene,