1. (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)

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: 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: Exponents: Long-range “charge”:
CDW (SDW)
Exponents:
Long-range “charge”:
and marginally irrelevant !

13. 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)!

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. (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)

20. 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”:

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

22. 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)

23. 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