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

We study the quantum Hall effect in graphene at filling factors $\nu =0$ and $\nu =\pm 1$, 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-$1∕2$) restricted to the lowest Landau level. Three distinct phases are considered, namely, the so-called spin-pseudospin, spin, and pseudospin phases. The first corresponds to a quarter-filled $(\nu =-1)$ lowest Landau level, while the others to a half-filled $(\nu =0)$ 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 SU(4) symmetry breaking terms, recently proposed by Alicea and Fisher [Phys. Rev. B 74, 075422 (2006)]. 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 skyrmions) can be described as a coherent state of bosons. We calculate the semiclassical limit of the boson model derived from the SU(4) invariant part of the original fermionic Hamiltonian and show that it agrees with the results of Arovas et al. [Phy. Rev. B 59, 13147 (1999)] for $\mathrm{SU}\left(N\right)$ quantum Hall skyrmions. We briefly discuss the influence of the SU(4) symmetry breaking terms in the skyrmion energy.

Figure 1

(Color online) Schematic representation of the four highly degenerate lowest Landau levels when $E_Z>E_P$. The state ∣SPFM⟩ is obtained by completely filling the energy level (1). b1, b2, and b3 are the elementary neutral excitations which are related to the density operators ${\rho }_{21}\left(\mathbf{q}\right)$, ${\rho }_{31}\left(\mathbf{q}\right)$, and ${\rho }_{41}\left(\mathbf{q}\right)$, respectively.

Figure 2

(Color online) Schematic representation of the four highly degenerate lowest Landau levels when $E_Z>E_P$. The state ∣SFM⟩ is obtained by completely filling the energy levels (1) and (3). b1, b2, b3, and b4 are the elementary neutral excitations which are related to the density operators ${\rho }_{21}\left(\mathbf{q}\right)$, ${\rho }_{23}\left(\mathbf{q}\right)$, ${\rho }_{41}\left(\mathbf{q}\right)$, and ${\rho }_{43}\left(\mathbf{q}\right)$, respectively.

Figure 3

(Color online) Schematic representation of the four highly degenerate lowest Landau levels when $E_Z<E_P$. The state ∣PFM⟩ is obtained by completely filling the energy levels (1) and (2). b1, b2, b3, and b4 are the elementary neutral excitations which are related to the density operators ${\rho }_{31}\left(\mathbf{q}\right)$, ${\rho }_{32}\left(\mathbf{q}\right)$, ${\rho }_{41}\left(\mathbf{q}\right)$, and ${\rho }_{42}\left(\mathbf{q}\right)$, respectively.

Figure 4

(Color online) Schematic representation of the honeycomb lattice. The triangular sublattices A and B are respectively represented by blue (filled) and red (empty) circles. $\mathbf{a}$ and $\mathbf{b}$ are the primitive vectors of the underline triangular sublattice A and $\mathbf{d}$ is the basis vector.