Interactions and Phase Transitions on Graphene’s Honeycomb Lattice

Abstract

The low-energy theory of interacting electrons on graphene's two-dimensional honeycomb lattice is derived and discussed. In particular, the Hubbard model in the large-N limit is shown to have a semi-metal -antiferromagnetic insulator quantum critical point in the universality class of the Gross-Neveu model. The same equivalence is conjectured to hold in the physical case N = 2, and its consequences for various physical quantities are examined. The effects of the long-range Coulomb interaction and of the magnetic field are discussed.A graphite monolayer, or graphene, emerged recently as the new frontier in physics of electronic systems with reduced dimensionality [1]. Such two-dimensional, or quasi-two-dimensional systems have led to some of the most startling discoveries in the condensed matter physics in the recent past, the quantum Hall effects and the metal-insulator transitions in silicon-MOSFETS and Ga-As heterostructures, and the high-temperature superconductivity in cuprates being prime examples. What makes graphene qualitatively new is its semi-metallic nature with low-energy quasiparticles behaving as 'relativistic' Dirac spinors over a good portion of the conducting band. The spinor structure is a general consequence of the bipartite nature of the honeycomb lattice The relativistic spectrum and the concomitant linearly vanishing density of states at the Fermi level, similarly as in the superconducting state of cuprates, provide graphene's quasiparticles with an additional protection against the effects of interactions. Nevertheless, a sufficiently strong repulsion is expected to turn the semimetallic state into a gapped insulator, possibly breaking the translational and/or the rotational symmetry in the process. Within the simplest interacting theory defined by the Hubbard model there is convincing numerical evidence for the quantum phase transition at a large Hubbard U into an antiferromagnet (AF) [6]. On the other hand, the long-range Coulomb interaction remains unscreened in the semi-metal (SM) [7], and has been argued to favor the charge-density-wave (CDW) at strong coupling [8]. The competition between different instabilities, the universality class, or even the order, of the SM -insulator transition, and the interplay of interactions with the Landau quantization in the external magnetic field present some of the basic open problems. Although graphene in its natural state may not be near a critical point [9], one can conceive mechanical deformations that would pull it deeper into the strong-coupling regime [10]. Finally, the outcome of the competition between different interactions should have consequences for the selection of the ground state in the magnetic field, even at weak coupling [11].In the present communication some of these issues are addressed by considering the half-filled Hubbard model on a honeycomb lattice, complemented with the additional long-range Coulomb interaction between electrons. The analysis is based on a useful decomposition of Hubbard's on-site interaction on a bipartit...

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.…”

A functional perspective on emergent supersymmetry

“…in specific condensed matter systems such as graphene [2][3][4]. The emergence of internal symmetries has also been discussed both in fermionic [5] as well as bosonic [6,7] systems.…”

“…As we have already seen, the beta function β h is identical in (8.2) and (6.2), so it only remains to establish that we can reconstruct β u of (6.2) from (8.2) using the identification u = (w ) 2 2 . In order to do so, we begin by taking a derivative of β w with respect to the field.…”

Quantum critical point in graphene approached in the limit of infinitely strong Coulomb interaction

“…(5), we have not included any contact four-Fermi interactions, which are irrelevant at weak coupling. At strong coupling, however, these interactions develop nontrivial fixed points [11]. We shall assume that these four-Fermi interactions start out with small couplings and flow to the trivial fixed point.…”