Theory of interacting electrons on the honeycomb lattice

Abstract

The low-energy theory of electrons interacting via repulsive short-range interactions on graphene's honeycomb lattice at half filling is presented. The exact symmetry of the Lagrangian with local quartic terms for the Dirac field dictated by the lattice is D_2 x U_c(1) x (time reversal), where D_2 is the dihedral group, and U_c(1) is a subgroup of the SU_c(2) "chiral" group of the non-interacting Lagrangian, that represents translations in Dirac language. The Lagrangian describing spinless particles respecting this symmetry is parameterized by six independent coupling constants. We show how first imposing the rotational, then Lorentz, and finally chiral symmetry to the quartic terms, in conjunction with the Fierz transformations, eventually reduces the set of couplings to just two, in the "maximally symmetric" local interacting theory. We identify the two critical points in such a Lorentz and chirally symmetric theory as describing metal-insulator transitions into the states with either time-reversal or chiral symmetry being broken. In the site-localized limit of the interacting Hamiltonian the low-energy theory describes the continuous transitions into the insulator with either a finite Haldane's (circulating currents) or Semenoff's (staggered density) masses, both in the universality class of the Gross-Neveu model. The picture of the metal-insulator transition on a honeycomb lattice emerges at which the residue of the quasiparticle pole at the metallic and the mass-gap in the insulating phase both vanish continuously as the critical point is approached. We argue that the Fermi velocity is non-critical as a consequence of the dynamical exponent being fixed to unity by the emergent Lorentz invariance. Effects of long-range interaction and the critical behavior of specific heat and conductivity are discussed.Comment: 16 revtex pages, 4 figures; typos corrected, new and updated references; published versio

Artificial honeycomb lattices for electrons, atoms and photons

“…An "extended" Hubbard model involving an additional interaction term describing nearest neighbor repulsions of strength V has been studied analytically by employing renormalization-group (RG) techniques in the limit of a large number N f of fermion flavors [84,85] (we remind the reader that N f = 4 for electrons in graphene). It has been found [84,85] that sufficiently large values of V /t stabilize charge-density-wave phases over semimetallic or Mott insulating phases.…”

“…It has been found [84,85] that sufficiently large values of V /t stabilize charge-density-wave phases over semimetallic or Mott insulating phases. Numerical RG calculations [86] have qualitatively confirmed these results but also discovered that second-neighbor repulsions favor topological states with spontaneously broken time-reversal symmetry (i.e.…”

Chirald-wave superconductivity in doped graphene

“…[4,5,6,7,8,9,10,11,12,13,14,15,17,16]). There, the predominant ordering tendencies are in the particle-hole channel, and usually superconductivity is not among the leading candidates.…”

“…A large number of exotic states of matter has been proposed theoretically, see e.g. [4,5,6,7,8,9,10,11,12,13,14,15,16,17], but most have not been experimentally observed as of yet. One exception is multi-layer graphene systems where there are now experimental reports of an energy gap opening at low temperatures, which has been ascribed to interactions effects [18,19,20,21,22,23,24,25].…”

Inducing topological order in a honeycomb lattice

“…20 Extensive studies on interacting graphene have shown a variety of competing phases. [22][23][24] Below we discuss the effects of producing the longer range interactions required for topological order using the Friedel oscillations and Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions occurring in a metallic (Fermi-gas) environment. Our analysis considers the possibility of forming both topological phases as well as other competing phases.…”