Hopf Term and the Effective Lagrangian for the Skyrmions in a Two-Dimensional Electron Gas at SmallgFactor

Abstract

We study interacting electrons in two dimensions moving in the lowest Landau level under the condition that the Zeeman energy is much smaller than the Coulomb energy and the filling factor is one. In this case, Skyrmion quasiparticles play an important role. Here, we present a simple and transparent derivation of the corresponding effective Lagrangian. In its kinetic part, we find a nonzero Hopf term, the prefactor of which we determine rigorously. In the Hamiltonian part, we calculate, by means of a gradient expansion, the Skyrmion-Skyrmion interaction completely up to fourth order in spatial derivatives. [S0031-9007(97)02620-3] PACS numbers: 73.20.Dx, 71.35.Ji, 73.20.Mf, 75.30.Et Two-dimensional electron gases, as manufactured in GaAs heterostructures, show a rich variety of features as the strength of a magnetic field or the particle density is varied. Most prominent among these are the quantum Hall effects (QHEs) at integer and fractional filling factors n (n N e ͞N F , where N e and N F are the number of electrons and the orbital degeneracy of the Landau level, respectively). The existence of a well-defined single-particle spectrum of discrete, spin-split Landau levels allows one to explain the integer QHE. Until recently it was accepted that when in such a system the characteristic Coulomb energy is less than the cyclotron energy, the ordinary low-lying excitations are electron-hole pairs of opposite spins (spin excitons [1,2]). These have a nonzero kinetic energy with a strong k dispersion due to the electron-electron interaction. In a completely filled Landau level, the energy gap for creating a widely separated quasielectron-quasihole pair, a large spin exciton (i.e., one with k !`), is apart from the Zeeman splitting governed by the exchange energy associated with a hole, and is equal to jgjm B B 1 q p 2 e 2 kl H . Here, g is the effective single particle Landé factor and l H ͑ch͞eB͒ 1͞2 is the magnetic length. This is in qualitative agreement with experiments on the temperature-dependent longitudinal resistance in the thermally activated regime [3,4].Recent theoretical investigations [5,6] near filling factor n 1, however, revealed that the interplay between Zeeman and Coulomb interaction results in a more complex type of excitations with unusual spin order which can be described as Skyrmions. It was shown [5] that the energy gap required to create a widely separated Skyrmion-anti-Skyrmion pair, is only half of the gap required to create a large spin exciton. Skyrmions appeared originally in condensed matter physics in the context of the Heisenberg ferromagnet as solutions of the O͑3͒ nonlinear sigma model in two dimensions for nonzero values of the topological charge [7]. Provided now that the Zeeman energy is less than the Coulomb energy, the elec-tron gas is equivalent to an isotropic itinerant ferromagnet. The latter can then be described by a three-component order parameter in a 2D coordinate space, i.e., just by the O͑3͒ nonlinear sigma model [7][8][9] in which one finds topologically nontrivi...

Hopf Term for a Two-Dimensional Electron Gas

“…(3), Q p, as suggested in Ref. [1]. Strictly speaking, integral (4) has to be somewhat modified for this model, because k x and k y are not good quantum numbers in the magnetic field simultaneously.…”

“…In a recent Letter [1], Apel and Bychkov presented a microscopic calculation of the prefactor Q of the topological Hopf term in the effective action of a twodimensional electron gas in a magnetic field at the filling factor n 1. This derivation is questionable.…”

“…(14) of Ref. [1] does not contain the Hopf term at all. This zero result may be an artifact of the parametrization of the rotation matrices and the unit vector n by the Euler angles u, f, and c, which is potentially dangerous, because f and c are ill defined when cos u 61.…”

“…Recently, Iordanskii and Plyasunov [8] derived Q microscopically for the model of Ref. [1] following the approach of Ref. [2].…”

Bag formation in quantum Hall ferromagnets

“…There has been a lot of effort to derive an effective field theory for the quantum Hall ferromagnet [12,13,14,15,16,7]. The result in terms of a gradient expansion of the unit magnetization vectorm is essentially an O(3) model, i.e.…”