Lowest Landau level bosonization

We develop a bosonization scheme for the two-dimensional electron gas in the presence of a uniform magnetic field perpendicular to the two-dimensional system when the filling factor is one $(\nu =1)$. We show that the elementary neutral excitations of this system, known as magnetic excitons, can be treated approximately as bosons and we apply the method to the interacting system. We show that the Hamiltonian of the fermionic system is mapped into an interacting bosonic Hamiltonian, where the dispersion relation of the bosons agrees with previous calculations of Kallin and Halperin. The interaction term accounts for the formation of bound states of two-bosons. We discuss a possible relation between these excitations and the skyrmion-antiskyrmion pair, in analogy with the ferromagnetic Heisenberg model. Finally, we analyze the semiclassical limit of the interacting bosonic Hamiltonian and show that the results are in agreement with those derived from the model of Sondhi et al. for the quantum Hall skyrmion.

Figure 1

Schematic representation of (a) the 2DEG ground state at $\nu =1$ (quantum Hall ferromagnet) and (b) the elementary neutral excitation (the electron-hole pair). $g$ is the Zeeman energy.

Figure 2

Schematic representation of a two-bosons state: (a) $\mid l{q}_1\mid$,$\mid l{q}_2\mid <1$ and (b) $\mid l{q}_3\mid$,$\mid l{q}_4\mid ⪢1$.

Figure 3

Dispersion relation of the bosons [Eq. (51), with $g=0$], in units of the Coulomb energy $e^2∕\left(ϵl\right)$ as a function of the momentum $\mathbf{q}$.

Figure 4

Dispersion relation of the state of two-bosons, in units of the Coulomb energy $e^2∕\left(ϵl\right)$ as a function of the total momentum $\mathbf{P}$ for $g=0$. Solid line: lowest energy bound state; shaded area: continuum of the scattering states; dashed line: energy of the quasielectron-quasihole pair plus one spin wave [Eq. (61)]. See text for details.

Figure 5

Schematic representation of the quasiparticles considered by Rezayi in Ref. 37: (a) the quantum Hall ferromagnet plus one spin down electron and (b) the spin down electron plus a spin-wave excitation.