Post on 24-Feb-2016
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)• Binding energy κ ~ 2.7 eV• σ orbitals create chemical bonds
Two simplerhombic
sublattices А and В
Geometry of hexagonal lattice
Periodic boundary conditions on the Euclidean torus:
The “Tight-binding” Hamiltonian
YXYX aa ,',', }ˆ,ˆ{ Fermi statistics
“Staggered” potential m distinguishes even/odd lattice sites
Physical implementation of staggered potential
Boron Nitride
Graphene
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
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
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
• 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
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
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
Discretization of Laplacian on the hexagonal latticereproduces Coulomb potential with a good precision
Electrostatic interactions on the lattice
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
∫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
Numerical simulations: Path integral representation
: • Electrostatic potential field• Lagrange multiplier for the Gauss’ law• Analogue of the Hubbard-Stratonovich field
Technical details
Lattice action for fermions (no doubling!!!):
Path integral weight:
Numerical simulations: Path integral representation
Positive weight due to two spin components!
Technical details
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
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)
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
• 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
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
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 0
m
NN m
Conductivity of grapheneCurrent operator: = charge, flowing through lattice links
Retarded propagator and conductivity:
Current-current correlators in Euclidean space:
Green-Kubo relations:
Thermal integral kernel:
Technical detailsConductivity of graphene:
Green-Kubo relations
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
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,