Quantum Hall ferromagnetism in graphene: SU(4) bosonization approach

Abstract

We study the quantum Hall effect in graphene at filling factors = 0 and = ± 1, concentrating on the quantum Hall ferromagnetic regime, within a nonperturbative bosonization formalism. We start by developing a bosonization scheme for electrons with two discrete degrees of freedom ͑spin-1 / 2 and pseudospin-1 / 2͒ restricted to the lowest Landau level. Three distinct phases are considered, namely, the so-called spinpseudospin, spin, and pseudospin phases. The first corresponds to a quarter-filled ͑ =−1͒ lowest Landau level, while the others to a half-filled ͑ =0͒ lowest Landau level. In each case, we show that the elementary neutral excitations can be treated approximately as a set of n-independent kinds of boson excitations. The boson representations of the projected electron density, the spin, pseudospin, and mixed spin-pseudospin density operators are derived. We then apply the developed formalism to the effective continuous model, which includes SU͑4͒ symmetry breaking terms, recently proposed by Alicea and Fisher ͓Phys. Rev. B 74, 075422 ͑2006͔͒. For each quantum Hall state, an effective interacting boson model is derived and the dispersion relations of the elementary excitations are analytically calculated. We propose that the charged excitations ͑quantum Hall skyrmions͒ can be described as a coherent state of bosons. We calculate the semiclassical limit of the boson model derived from the SU͑4͒ invariant part of the original fermionic Hamiltonian and show that it agrees with the results of Arovas et al. ͓Phy. Rev. B 59, 13147 ͑1999͔͒ for SU͑N͒ quantum Hall skyrmions. We briefly discuss the influence of the SU͑4͒ symmetry breaking terms in the skyrmion energy.

Theoretical expectations for a fractional quantum Hall effect in graphene

“…This fourfold spin-valley symmetry is described in the framework of the SU(4) group which covers the two copies of the SU(2) spin and the SU(2) valley isospin. Lattice effects break this SU(4) symmetry at an energy scale V C (a/l B ) ≃ 0.1B[T]/ǫ meV [10,11,12,13], which is roughly on the same order of magnitude as the expected Zeeman effect in graphene [9]. Other symmetry-breaking mechanisms have been proposed [14,15,16] but happen to be equally suppressed with respect to the leading interaction energy scale V C .…”

Phase diagram for theν=0quantum Hall state in monolayer graphene

“…Projected Hamiltonian for n = 0 LL When addressing the many-body aspects of the ν = 0 state, as a starting point, one may neglect the contributions from n = 0 LLs and restrict oneself to the dynamics within n = 0 LL, described in terms of the field (26). From the form (24) ofψ 0 (r), we obtain…”

“…Substitutingψ(r) in the form (23) into Eqs. (10), (11), and (17) and retaining only the n = 0 LL component (24), we obtain the bare projected Hamiltonian in terms of the field (26),…”

“…Further on, when discussing n = 0 LL below, we refer to this 2D degree of freedom (ψ 0KAσ ,ψ 0K Bσ ) as just the "KK valley isospin". Accordingly, we join the nonvanishing components of the n = 0 LL operator (24) in a 4D vector (26) in the isospin-spin space KK ⊗ s.…”

Tilted-Cone-Induced Easy-Plane Pseudo-Spin Ferromagnet and Kosterlitz–Thouless Transition in Massless Dirac Fermions

“…[12][13][14][15][16][17][18][19] Generically, the ferromagnetic ordering may be understood within an interaction model with no explicit spin or pseudo-spin symmetry breaking; in order to minimize their exchange energy, the global N-particle wave function should be fully antisymmetric in its orbital part, the (pseudo-)spin part needs to be fully symmetric in order to fulfil fermionic statistics. Whereas in a normal metal this ordering is only partial, due to the increase in the kinetic energy, a single Landau level may be viewed as an infinitely flat energy band, and the ferromagnetic ordering may therefore be complete.…”