Hybrid MonteCarlo simulations of electronic properties of graphene

Hybrid Monte-Carlo simulations of electronic properties of graphene [Ar. Xiv: 1206. 0619] P. V. Buividovich (Regensburg University)

Graphene ABC • Graphene: 2 D 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 e. V • 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 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: Eigenvalues:

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 Discrete Z 2 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/v. F ~ 2 (v. F ~ 1/300) Strongly coupled theory!!! Magnetic+retardation effects suppressed Dielectric permittivity: • Suspended graphene ε = 1. 0 • Silicon Dioxide Si. O 2 ε ~ 3. 9 • Silicon Carbide Si. C 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

l ca ni ch Te Numerical simulations: Path integral representation ils ta de Decomposition of identity Eigenstates of the gauge field Fermionic coherent states (η – Grassman variables) Gauss law constraint (projector on physical space)

l ca ni ch Te Numerical simulations: Path integral representation ils ta de • • • Electrostatic potential field Lagrange multiplier for the Gauss’ law Analogue of the Hubbard-Stratonovich field

l ca ni ch Te Numerical simulations: Path integral representation ils ta de Lattice action for fermions (no doubling!!!): Path integral weight: Positive weight due to two spin components!

Hybrid Monte-Carlo: a brief introduction Metropolis algorithm

Hybrid Monte-Carlo: a brief introduction Molecular Dynamics • Global updates of fields ϕ(x) • 100% acceptance rate BUT: • Energy non-conservation for numerical integrators

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 = N A – N B “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

Conductivity of graphene Current operator: = charge, flowing through lattice links Retarded propagator and conductivity:

l ca ni ch Te Conductivity of graphene: Green-Kubo relations: Thermal integral kernel: ils ta de Current-current correlators in Euclidean space:

l ca ni ch Te Conductivity of graphene Conductivity from Euclidean correlator: an ill-posed problem Maximal Entropy Method Approximate calculation of σ(0): AC conductivity, averaged over w ≤ k. T ils ta de σ(ω) – dimensionless quantity (in a natural system of units), in SI: ~ e 2/h

Conductivity of graphene: free theory For small frequencies (Dirac limit): Threshold value w = 2 m Universal limiting value at κ >> w >> m: At w = 2 m: σ0 = π e 2/2 h=1/4 e 2/ħ σ = 2 σ 0

Conductivity of graphene: Free theory

Current-current correlators: numerical results κ Δτ = 0. 15, m Δτ = 0. 01, κ/(k. T) = 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 Ar. Xiv: 1206. 0619

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