1. Pavel Buividovich (Regensburg)

2. They are very similar to relativistic strongly coupled QFT Dirac/Weyl points Dirac/Weyl points Quantum anomalies Quantum anomalies Strong coupling Strong coupling Spontaneous symmetry breaking Spontaneous symmetry breaking Much simpler than QCD (the most interesting SC QFT) Much simpler than QCD (the most interesting SC QFT) Relatively easy to realize in practice (table-top vs LHC) Relatively easy to realize in practice (table-top vs LHC) We (LQCD) can contribute to these fields of CondMat We (LQCD) can contribute to these fields of CondMat We can learn something new We can learn something new new lattice actions new lattice actions new algorithms new algorithms new observables/analysis tools new observables/analysis tools BUT BEWARE: ENTROPY VS COMPLEXITY QCD Small (Log 1) Large (Millenium problem) CondMat Large (all materials) Small (mean-field often enough)

3. Typical values of v F ~ c/300 (Graphene) Typical sample size ~ 100 nm (1000 lattice units) Propagation time ~ 10 -16 s (Typical energy ~ 100 eV) Magnetic interactions ~ v F 2 Coulomb interactions are more important by factor ~1/v F 2

5. Graphene ABC Graphene: 2D carbon crystal with hexagonal lattice Graphene: 2D carbon crystal with hexagonal lattice a = 0.142 nm – Lattice spacing a = 0.142 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

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

7. Or The Standard Model of Graphene “Staggered” potential m distinguishes even/odd lattice sites

8. Physical implementation of staggered potential Boron Nitride Graphene

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

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

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

12. Dirac fermions

13. Near the Dirac points

14. Dirac fermions

15. «Valley» magnetic field` Mechanical strain: hopping amplitudes change

16. «Valley» magnetic field [N. Levy et. al., Science 329 (2010), 544]

17. 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

18. 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

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

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

21. 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

22. 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 No UV diverging diagrams Renormalization is not important Renormalization is not important Not so important to have exact chirality Not so important to have exact chirality No sign problem at neutrality No sign problem at neutrality HMC simulations are possible HMC simulations are possible

23. 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 - antifferromagnetic condensate - excitonic 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

24. Chiral symmetry breaking in graphene: analytical study 1) E. V. Gorbar et. al., Phys. Rev. B66 (2002), 045108. α с = 1,47 2) O. V. Gamayun et. al., Phys. Rev. B 81 (2010), 075429. α с = 0,92 3), 4)..... reported results in the region α с = 0,7...3,0 D. T. Son, Phys. Rev. B 75 (2007) 235423: large-N analysis:

25. 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 20 4 sites

26. Graphene conductivity: theory and experiment Experiment: D. C. Elias et. al., Nature Phys, 7, (2011), 701; No evidence of the phase transition Lattice calculations: phase transition at ε=4

27. Path integral representation Partition function: Introduction of fermionic coherent states: Using the following relations: and Hubbard-Stratonovich transformation:

28. Fermionic action and (no) sign problem No sign problem! At half-filling

29. Antiferromagnetic phase transition P. V. Buividovich, M. I. Polikarpov, Phys. Rev. B 86 (2012) 245117

30. «Screening» of Coulomb interaction at small distances Comparison of the potentials

31. Condensate with modified potentials Ulybyshev, Buividovich, Katsnelson, Polikarpov, Phys. Rev. Lett. 111, 056801 (2013) Screened potential Coulomb potential

32. Long-range interaction Short-range interaction Excitonic phase Antiferromagnetic phase Phase diagram 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.

33. Graphene with vacancies Hoppings are equal to zero for all links connecting vacant site with its neighbors.Hoppings are equal to zero for all links connecting vacant site with its neighbors. Charge of the site is also zero.Charge of the site is also zero. Approximately corresponds to Hydrogen adatoms.Approximately corresponds to Hydrogen adatoms. Midgap states, power-law decay of wavefunctionsMidgap states, power-law decay of wavefunctions Nonzero density of states near Fermi-points: Cooper instabilityCooper instability AFM/Excitonic condensatesAFM/Excitonic condensates What about other defects? What about other defects? ???

34. Electron spin near vacancies

35. Graphene in strong magnetic fields What is the relevant ground state for B ~ 15 T? Spin is not polarized… Kekule distortion: superlattice structure Skyrmions