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Graphene, topological insulators and Weyl semimetals Pavel Buividovich (Regensburg) Why these condmat systems? They are very similar to relativistic strongly coupled QFT • Dirac/Weyl points • Quantum anomalies • Strong coupling • Spontaneous symmetry breaking • • • • Much simpler than QCD (the most interesting SC QFT) Relatively easy to realize in practice (table-top vs LHC) We (LQCD) can contribute to these fields of CondMat We can learn something new new lattice actions new algorithms new observables/analysis tools BUT BEWARE: ENTROPY QCD Small (Log 1) CondMat Large (all materials) VS COMPLEXITY Large (Millenium problem) Small (mean-field often enough) Instantaneous approximation Typical values of vF ~ c/300 (Graphene) Typical sample size ~ 100 nm (1000 lattice units) Propagation time ~ 10-16 s (Typical energy ~ 100 eV) Magnetic interactions ~ vF2 Coulomb interactions are more important by factor ~1/vF2 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: Tight-binding model of Graphene Or The Standard Model of Graphene “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 Dirac fermions Dirac fermions Near the Dirac points Dirac fermions «Valley» magnetic field` Mechanical strain: hopping amplitudes change «Valley» magnetic field [N. Levy et. al., Science 329 (2010), 544] 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 Lattice simulations of the tight-binding model Lattice Hamiltonian from the beginning Fermion doubling is physical Perturbation theory in 1D (Euclidean time) • No UV diverging diagrams • Renormalization is not important • Not so important to have exact chirality • No sign problem at neutrality • HMC simulations are possible Chiral symmetry breaking in graphene Symmetry group of the low-energy theory is U(4). Various channels of the symmetry breaking are possible. Two of them are studied at the moment. They correspond to 2 different nonzero condensates: - antifferromagnetic condensate - excitonic condensate From microscopic point of view, these situations correspond to different spatial ordering of the electrons in graphene. Antiferromagnetic condensate: opposite spin of electrons on different sublattices Excitonic condensate: opposite charges on sublattices Chiral symmetry breaking in graphene: analytical study 1) E. V. Gorbar et. al., Phys. Rev. B 66 (2002), 045108. αс = 1,47 2) O. V. Gamayun et. al., Phys. Rev. B 81 (2010), 075429. αс = 0,92 3), 4)..... reported results in the region αс = 0,7...3,0 D. T. Son, Phys. Rev. B 75 (2007) 235423: large-N analysis: Excitonic condensate P. V. Buividovich et. al., Phys. Rev. B 86 (2012), 045107. Joaquín E. Drut, Timo A. Lähde, Phys. Rev. B 79, 165425 (2009) All calculations were performed on the lattice with 204 sites Graphene conductivity: theory and experiment Lattice calculations: phase transition at ε=4 Experiment: D. C. Elias et. al., Nature Phys, 7, (2011), 701; No evidence of the phase transition Path integral representation Partition function: Introduction of fermionic coherent states: Using the following relations: and Hubbard-Stratonovich transformation: Fermionic action and (no) sign problem No sign problem! At half-filling Antiferromagnetic phase transition P. V. Buividovich, M. I. Polikarpov, Phys. Rev. B 86 (2012) 245117 Comparison of the potentials «Screening» of Coulomb interaction at small distances Condensate with modified potentials Screened potential Coulomb potential Ulybyshev, Buividovich, Katsnelson, Polikarpov, Phys. Rev. Lett. 111, 056801 (2013) Phase diagram Short-range interaction Influence of the short-range interactions on the excitonic phase transition: O.V. Gamayun et. al. Phys. Rev. B 81, 075429 (2010). Short-range repulsion suppresses formation of the excitonic condensate. Antiferromagnetic phase Excitonic phase Long-range interaction Graphene with vacancies • Hoppings are equal to zero for all links connecting vacant site with its neighbors. • Charge of the site is also zero. • Approximately corresponds to Hydrogen adatoms. • Midgap states, power-law decay of wavefunctions Nonzero density of states near Fermi-points: • Cooper instability • AFM/Excitonic condensates What about other defects? ??? Electron spin near vacancies Graphene in strong magnetic fields What is the relevant ground state for B ~ 15 T? Spin is not polarized… Kekule distortion: superlattice structure Skyrmions