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

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

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Two simple rhombicsublattices А and В Geometry of hexagonal lattice Periodic boundary conditions on the Euclidean torus:

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The “Tight-binding” Hamiltonian Fermi statistics “Staggered” potential m distinguishes even/odd lattice sites

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Physical implementation of staggered potential Boron Nitride Graphene

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Spectrum of quasiparticles in graphene Consider the non-Interacting tight-binding model !!! Eigenmodes are just the plain waves: Eigenvalues: One-particleHamiltonian

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Spectrum of quasiparticles in graphene Close to the «Dirac points»: “Staggered potential” m = Dirac mass

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

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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 Symmetries of the free Hamiltonian U(1) x U(1) symmetry: conservation of currents with different spins

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Each lattice site can be occupied by two electrons (with opposite spin) Each lattice site can be occupied by two electrons (with opposite spin) The ground states is electrically neutral The ground states is electrically neutral One electron (for instance ) One electron (for instance ) at each lattice site at each lattice site «Dirac Sea»: «Dirac Sea»: hole = hole = absence of electron absence of electron in the state in the state Particles and holes

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Lattice QFT of Graphene Redefined creation/ annihilation operators Chargeoperator Standard QFT vacuum

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Electromagnetic interactions Link variables (Peierls Substitution) Conjugate momenta = Electric field LatticeHamiltonian (Electric part)

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Electrostatic interactions Dielectric permittivity: Suspended graphene Suspended graphene ε = 1.0 ε = 1.0 Silicon Dioxide SiO 2 Silicon Dioxide SiO 2 ε ~ 3.9 ε ~ 3.9 Silicon Carbide SiC Silicon Carbide SiC ε ~ 10.0 ε ~ 10.0 Effective Coulomb coupling constant α ~ 1/137 1/v F ~ 2 (v F ~ 1/300) Strongly coupled theory!!! Magnetic+retardation effects suppressed

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Discretization of Laplacian on the hexagonal lattice reproduces Coulomb potential with a good precision Electrostatic interactions on the lattice

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Main problem: the spectrum of excitations in interacting graphene Lattice simulations, Schwinger-Dysonequations Renormalization, Large N, Experiment [Manchester group, 2012] Spontaneous breaking of sublattice symmetry = mass gap = condensate formation = = decrease of conductivity ???

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Numerical simulations: Path integral representation Decomposition of identity of identity Eigenstates of the gauge field Fermionic coherent states (η – Grassman variables) Gauss law constraint (projector on physical space) Technical details

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Numerical simulations: Path integral representation Electrostatic potential fieldElectrostatic potential field Lagrange multiplier for the Gauss’ lawLagrange multiplier for the Gauss’ law Analogue of the Hubbard-Stratonovich fieldAnalogue of the Hubbard-Stratonovich field Technical details

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Lattice action for fermions (no doubling!!!): Path integral weight: Numerical simulations: Path integral representation Positive weight due to two spin components! Technical details

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Hybrid Monte-Carlo: a brief introduction Metropolis algorithm

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Hybrid Monte-Carlo: a brief introduction Molecular Dynamics Global updates of fields ϕ (x)Global updates of fields ϕ (x) 100% acceptance rate100% acceptance rateBUT: Energy non-conservation for numerical integratorsEnergy non-conservation for numerical integrators

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Hybrid Monte-Carlo = Molecular Dynamics + Metropolis Use numerically integrated Molecular Dynamics trajectories as Metropolis proposalsUse numerically integrated Molecular Dynamics trajectories as Metropolis proposals Numerical error is corrected by accept/rejectNumerical error is corrected by accept/reject Exact algorithmExact algorithm Ψ-algorithm [Technical]:Ψ-algorithm [Technical]: Represent determinant Represent determinant as Gaussian integral as Gaussian integral Molecular Dynamics Trajectories

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Hexagonal lattice Hexagonal lattice Noncompact U(1) gauge field Noncompact U(1) gauge field Fast heatbath algorithm outside of graphene plane Fast heatbath algorithm outside of graphene plane Geometry: graphene on the substrate Geometry: graphene on the substrate Numerical simulations using the Hybrid Monte-Carlo method

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Breaking of lattice symmetry Anti-ferromagnetic state Anti-ferromagnetic state (Gordon-Semenoff 2011) (Gordon-Semenoff 2011) Kekule dislocations Kekule dislocations (Araki 2012) (Araki 2012) Point defects Point defects Intuition from relativistic QFTs (QCD): Symmetry breaking = = gap in the spectrum

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Spontaneous sublattice symmetry breaking in graphene Order parameter: The difference between the number of particles on А and В sublattices Δ N = N A – N B “Mesons”: particle-hole bound state

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Differences of particle numbers

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on lattices of different size Extrapolation to zero mass

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Susceptibility of particle number differences

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

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Current-current correlators in Euclidean space: Green-Kubo relations: Thermal integral kernel: Technical details Conductivity of graphene: Green-Kubo relations

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Conductivity of graphene σ(ω) – dimensionless quantity (in a natural system of units), in SI: ~ e 2 /h (in a natural system of units), in SI: ~ e 2 /h Conductivity from Euclidean correlator: an ill-posed problem Maximal Entropy Method Maximal Entropy Method Approximate calculation of σ(0): AC conductivity, averaged over w ≤ kT Technical details

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

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Conductivity of graphene: Free theory

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Current-current correlators: numerical results κ Δτ = 0.15, m Δτ = 0.01, κ/(kT) = 18, L s = 24

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Conductivity of graphene σ(0): numerical results (approximate definition)

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Direct measurements of the density of states Experimentally motivated definitionExperimentally motivated definition Valid for non-interacting fermionsValid for non-interacting fermions Finite μ is introduced in observables only (partial quenching)Finite μ is introduced in observables only (partial quenching)

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Direct measurements of the density of states m/κ = 0.1

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Direct measurements of the density of states m/κ = 0.5

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Conclusions Electronic properties of graphene at half-filling can be studied using the Hybrid Monte-Carlo algorithm. 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. Sign problem is absent due to the symmetries of the model. Signatures of insulator-semimetal phase transition for monolayer graphene. Signatures of insulator-semimetal phase transition for monolayer graphene. Order parameter: Order parameter: difference of particle numbers on two simple sublattices difference of particle numbers on two simple sublattices Spontaneous breaking of sublattice symmetry is accompanied by a decrease of conductivity Spontaneous breaking of sublattice symmetry is accompanied by a decrease of conductivity Direct measurements of the density of states indicate increasing Fermi velocity Direct measurements of the density of states indicate increasing Fermi velocity see ArXiv:

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Outlook

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