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)

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)

Single layer of graphite: graphene (Geim and Novoselov, 2004)

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!

1) Cone’s isotropy (rotational symmetry); Lorentz invariance!

2) Chiral (valley, or pseudospin):

=

,

3) Time reversal (exact)

4) Spin rotational invariance (exact)

Symmetry: emergent and 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)

Field theory: (for SDW transition, only)

Neel order parameter: (Higgs field)

With Coulomb long-range interaction:

RG flow, leading order:

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 :

and of bosonic and fermionic masses.

Transition: from gapless (fermions) to gapless (bosons)!

Consequence: universal ratio of specific heats

Finite size scaling in quantum Monte Carlo:

near d=3,

Uc = 3.78

Crossing point and the critical interaction (from magnetization)

This suggests: (in Hubbard)

(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”

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 componentExternal staggered potential

Core is insulating ! (Ghaemi, Ryu, Lee, PRB 2010)

Conclusions:

1) Honeycomb lattice: playground for interacting electrons

2) Coulomb interaction: 1) long ranged, not screened, but ultimately innocuous, 2) short range leads to symmetry breaking

3) 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