1 / 23

Fundamental physics with two-dimensional carbon

Fundamental physics with two-dimensional carbon . Igor Herbut (Simon Fraser University, Vancouver). Chi-Ken Lu (Indiana) Bitan Roy (Maryland) Vladimir Juricic (Utrecht) Oskar Vafek (Florida) Gordon Semenoff (UBC)

adin
Download Presentation

Fundamental physics with two-dimensional carbon

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Fundamental physics with two-dimensional carbon Igor Herbut (Simon Fraser University, Vancouver) Chi-Ken Lu (Indiana) Bitan Roy (Maryland) Vladimir Juricic (Utrecht) Oskar Vafek (Florida) Gordon Semenoff (UBC) Fakher Assaad (Wurzburg) Vieri Mastropietro (Milan)

  2. Single layer of graphite: graphene (Geim and Novoselov, 2004) Two triangular sublattices: A and B; one electron per site (half filling) Tight-binding model ( t = 2.5 eV ): (Wallace, PR 1947) The sum is complex => two equations for two variables for zero energy => Dirac points (no Fermi surface)

  3. Brillouin zone: Two inequivalent (Dirac) points at : +K and -K Dirac fermion: 4 components/spin component “Low - energy” Hamiltonian: i=1,2 , (isotropic, v = c/300 = 1, in our units). Neutrino-like in 2D!

  4. Symmetry:emergent and exact 1) Cone’s isotropy (rotational symmetry); Lorentz invariance! 2) Chiral (valley, or pseudospin): = • , 3) Time reversal (exact) 4) Spin rotational invariance (exact)

  5. Does not Coulomb repulsion matter: yes, but! with the interaction term, (Hubbard + Coulomb) Long-range part is not screened, and it may matter.

  6. The Fermi velocity depends on the scale: (Gonzalez et al, NPB 1993) To the leading order: and the Fermi velocity increases! It goes to where which is at the velocity (in units of velocity of light): = >

  7. The ultimate low-energy theory: reduced QED3 (matter in 2+1 D + gauge fields in 3+1 D) Gauge field propagator: and the fine structure constant is scale invariant!! Dirac fermions are massless, with a velocity of light. Experiment: (Ellias et al, Nature 2011)

  8. Back to reality: should we not worry about finite-range pieces of Coulomb interaction? Yes, in principle: (Grushin et al, PRB 2012) (IH, PRL 2006) At large interaction some symmetry gets broken.

  9. Masses (symmetry breaking order parameters): 1) “Charge-density-wave” (Semenoff, PRL 1984 (CDW); IH, PRL 2006 (SDW)) 2) Kekule bond-density-wave (Hou, Chamon, Mudry, PRL 2007) Chiral triplet, Lorentz singlets, time-reversal invariant!

  10. 3) Topological insulator(Haldane, PRL 1988) Lorentz and chiral singlet, breaks time-reversal. 4)+ all these in spin triplet versions (+ 4 superconducting states)

  11. Gross-Neveu-Yukawa theory: epsilon-expansion, epsilon = 3-d(IH, Juricic, Vafek, PRB 2009) Neel order parameter: (Higgs field) Field theory: (for SDW transition, only) With Coulomb long-range interaction:

  12. RG flow, leading order: CDW (SDW) Exponents: Long-range “charge”: and marginally irrelevant !

  13. Emergent relativity:if wedefine a small deviation of velocity it is (the leading) irrelevantperturbation close to d=3 : Consequence: universal ratio of specific heats and of bosonic and fermionic masses. Transition: from gapless (fermions) to gapless (bosons)!

  14. Finite size scaling in quantum Monte Carlo: near d=3,

  15. Crossing point and the critical interaction (from magnetization) This suggests: (in Hubbard) Uc = 3.78 (F. Assaad and IH, PRX 2013)

  16. Anything at low U?? Meet the artificial graphene: (Gomes, 2012)

  17. Hubbard model with a flat band: (Roy, Assaad, IH, PRX 2014) “Global antiferromagnet”

  18. Back to Dirac masses: SO(5) symmetry An example: (SDW (3 masses), Kekule (2 masses)) These 5 masses and the 2 (alpha) matrices in the Dirac Hamiltonian form a maximal anticommuting set of dimension 8: Clifford algebra C(2,5) This remains true even if all the possible superconducting masses are included: (Ryu et al, PRB 2010)

  19. To include superconducting masses Nambu doubling is necessary: matrices become 16x16, but (antilinear!) particle-hole symmetry restricts: 1) Masses to be purely imaginary 2) Alpha matrices to be purely real This separation leads to C(2,5) as the maximal algebra. 16x16 representation, however, is “quaternionic”:

  20. Real representations of C(p,q): (IH, PRB 2012, Okubo, JMP 1991, ABS 1964)

  21. Example: U(1) superconducting vortex (s-wave, singlet) (IH, PRL 2010) : {CDW, Kekule BDW1, Kekule BDW2} : {Haldane-Kane-Mele TI (triplet)} Lattice: 2K component External staggered potential Core is insulating ! (Ghaemi, Ryu, Lee, PRB 2010)

  22. Conclusions: Honeycomb lattice: playground for interacting electrons Coulomb interaction: 1) long ranged, not screened, but ultimately innocuous, 2) short range leads to symmetry breaking Novel (Higgs) quantum phase transition in the Hubbard model; global antiferromagnet under strain 4) SO(5) symmetry of the Dirac masses (order parameters) implies duality relations and non-trivial topological defects

More Related