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)
• Binding energy κ ~ 2.7 eV
• σ orbitals create chemical bonds
Geometry of hexagonal lattice
Two simple
rhombic
sublattices
А and В
Periodic boundary
conditions on the
Euclidean torus:
The “Tight-binding” Hamiltonian

{ aˆ  , X , aˆ  ',Y }    ' X ,Y
Fermi statistics
“Staggered” potential m distinguishes
even/odd lattice sites
Physical implementation
of staggered potential
Graphene
Boron Nitride
Spectrum of quasiparticles in graphene
Consider the non-Interacting tight-binding model !!!
One-particle
Hamiltonian
Eigenmodes are just the plain waves:
3
 (k ) 
e
a 1
Eigenvalues:
 
i k e a
Spectrum of quasiparticles in graphene
Close to the «Dirac points»:
“Staggered potential” m = Dirac mass
Spectrum of quasiparticles in graphene
Dirac points are only covered by discrete lattice
momenta if the lattice size is a multiple of three
Symmetries of the free Hamiltonian
2 Fermi-points
Х
2 sublattices
=
4 components of the
Dirac spinor
Chiral U(4) symmetry
(massless fermions):
right
left
( L, R, L , R )
(  A , B , A , B )
Discrete Z2 symmetry
between sublattices
А
В
U(1) x U(1) symmetry: conservation
of currents with different spins
Particles and holes
• 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
Lattice QFT of Graphene
Redefined creation/
annihilation operators
Charge
operator
Standard QFT vacuum
Electromagnetic interactions
Link variables
(Peierls Substitution)
Conjugate momenta
= Electric field
Lattice
Hamiltonian
(Electric part)
Electrostatic interactions
Effective Coulomb coupling constant
α ~ 1/137 1/vF ~ 2 (vF ~ 1/300)
Strongly coupled theory!!!
Magnetic+retardation effects suppressed
Dielectric permittivity:
• Suspended graphene
ε = 1.0
• Silicon Dioxide SiO2
ε ~ 3.9
V (r ) 
2e
2
(   1) r
 
2
 1
• Silicon Carbide SiC
ε ~ 10.0
Electrostatic interactions on the lattice
Discretization of Laplacian on the hexagonal lattice
reproduces Coulomb potential with a good precision
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
Numerical simulations:
Path integral representation
𝑯
𝑻𝒓 𝐞𝐱 𝐩 −
= 𝑻𝒓
𝒌𝑻
=
𝒆𝒙𝒑 −𝑯 ∆𝝉 =
𝒌=𝟏…𝒏
𝒅Ψ𝟏 … 𝒅Ψ𝒏 Ψ𝟏 |𝑷 𝒆𝒙𝒑 −𝑯 ∆𝝉 |Ψ𝟐 Ψ𝟐 |𝑷 𝒆𝒙𝒑 −𝑯 ∆𝝉 |Ψ𝟑 …
𝒅Ψ Ψ >< Ψ = 𝑰
|Ψ > =
|𝜃𝑋𝑌 > |η𝑋 >
<𝑋𝑌>
Decomposition
of identity
𝜃𝑋𝑌 |𝜃𝑋𝑌 > = 𝜃𝑋𝑌 |𝜃𝑋𝑌 >
Eigenstates of the gauge field
ψ𝑋 |η𝑋 >= η𝑋 |η𝑋 >
Fermionic coherent states
(η – Grassman variables)
𝑃=
Gauss law constraint
(projector on physical space)
𝛿(
𝑋
𝜋𝑋𝑌 − 𝑞𝑋 )
𝑌
Numerical simulations:
Path integral representation
𝝋𝑿 : • Electrostatic potential field
• Lagrange multiplier for the Gauss’ law
• Analogue of the Hubbard-Stratonovich field
Numerical simulations:
Path integral representation
Lattice action for fermions (no doubling!!!):
Path integral weight:
Positive weight due to two spin components!
Hybrid Monte-Carlo: a brief introduction
Problem:
generate field configurations φ(x) with probability
𝑷 𝝋 𝒙 ~𝒆𝒙𝒑(−𝑺[𝝋(𝒙)])
For graphene – nonlocal action due to fermion determinant
Metropolis algorithm
Propose new field
configurations with probability
𝑷 𝝋 𝒙 → 𝝋′ 𝒙
Accept/reject with probability
𝜶 = 𝒆𝒙𝒑 −𝑺 𝝋′ 𝒙 + 𝑺 𝝋 𝒙
• Exact algorithm
• Local updates of fields
BUT:
• Fermion determinant recalculation
Hybrid Monte-Carlo: a brief introduction
Molecular Dynamics
Classical motion with
𝛑𝟐 𝒙
𝑯=
+𝑺 𝝋 𝒙
𝟐
𝒙
• Global updates of fields
ϕ(x)
• 100% acceptance rate
BUT:
• Energy non-conservation
for numerical integrators
If ergodic:
𝑷 𝝋 𝒙 ~𝒆𝒙𝒑(−𝑺[𝝋(𝒙)])
𝟏 𝑻
=
𝒅𝝉 𝜹 𝝋 𝒙 , 𝝋 𝒙, 𝝉
𝑻 𝟎
π(x) – conjugate
momentum for φ(x)
Hybrid Monte-Carlo
= Molecular Dynamics + Metropolis
• Use numerically
integrated Molecular
Dynamics trajectories as
Metropolis proposals
• Numerical error is
corrected by
accept/reject
• Exact algorithm
Molecular Dynamics
Trajectories
• Ψ-algorithm [Technical]:
Represent determinant
as Gaussian integral
Numerical simulations using
the Hybrid Monte-Carlo method
•
•
•
•
Hexagonal lattice
Noncompact U(1) gauge field
Fast heatbath algorithm outside of graphene plane
Geometry: graphene on the substrate
Breaking of lattice symmetry
Intuition from relativistic QFTs (QCD):
Symmetry breaking =
= gap in the spectrum
• Anti-ferromagnetic state
(Gordon-Semenoff 2011)
• Kekule dislocations
(Araki 2012)
• Point defects
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
Differences of particle numbers
Differences of particle numbers
on lattices of different size
Extrapolation to zero mass
Susceptibility of particle
number differences
N 
 N
m
m0
Conductivity of graphene
Current operator:
= charge, flowing
through lattice links
Retarded propagator and conductivity:
Conductivity of graphene:
Green-Kubo relations
Current-current correlators in Euclidean space:
Green-Kubo relations:
Thermal integral kernel:
Conductivity of graphene
σ(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):

G ( / 2) 
dw
 2
0
2w
sinh(  w / 2 )
 ( w )   ( kT )  ( 0 )
AC conductivity, averaged over w ≤ kT
2
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
Conductivity of graphene:
Free theory
Current-current correlators:
numerical results
κ Δτ = 0.15, m Δτ = 0.01, κ/(kT) = 18, Ls = 24
Conductivity of graphene σ(0):
numerical results
(approximate definition)
Direct measurements of the density of states
• Experimentally
motivated definition
• Valid for non-interacting
fermions
• Finite μ is introduced in
observables only
(partial quenching)
Direct measurements of the density of states
m/κ = 0.1
Direct measurements of the density of states
m/κ = 0.5
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
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, 𝑬 =
𝒌𝟐
𝟐𝒎