Nuclear spin relaxation in integral and fractional quantum Hall systems

Abstract

We report on a numerical study of the relaxation rates of nuclear spins coupled through the hyperfine interaction to a two dimensional electron gas (2DEG) at magnetic fields corresponding to both fractional and integral Landau level (LL) fillings $\ensuremath{\nu}.$ The Hamiltonians of up to 20 interacting electrons are diagonalized exactly in the spherical geometry, neglecting finite layer width, disorder, and LL mixing. The spectral functions ${\ensuremath{\tau}}^{\ensuremath{-}1}(E)$ describing response of the 2DEG to the reversal of an embedded localized spin are calculated. In a (locally) incompressible $\ensuremath{\nu}=1$ or $\frac{1}{3}$ state, the finite Coulomb energy of short spin waves, together with the small nuclear Zeeman energy, prevent nuclear spin relaxation even in the limit of vanishing electron Zeeman energy ${(E}_{\mathrm{Z}}).$ However, we find that the nuclear spins can couple to the internal excitations of mobile finite-size skyrmions that appear in the 2DEG at sufficiently low ${E}_{\mathrm{Z}}$ and at $\ensuremath{\nu}$ slightly different from 1 or $\frac{1}{3}.$ The experimentally observed dependence of nuclear spin relaxation rate on ${E}_{\mathrm{Z}}$ and $\ensuremath{\nu}$ is qualitatively explained in terms of the occurrence of skyrmions and antiskyrmions of various topological charge.