(Simon Fraser University, Vancouver) 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)

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)

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!

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)

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

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): = >

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)

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.

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!

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

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:

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

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

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

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

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

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

(SDW (3 masses), Kekule (2 masses)) 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)

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”:

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

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)

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