Hopf invariant for long-wavelength skyrmions in quantum Hall systems for integer and fractional fillings

Abstract

We show that a Hopf term exists in the effective action of long-wavelength skyrmions in quantum Hall systems for both odd integer and fractional filling factors $ν= 1/(2s +1)$, where $s$ is an integer. We evaluate the prefactor of the Hopf term using Green function method in the limit of strong external magnetic field using model of local interaction. The prefactor ($N$) of the Hopf term is found to be equal to $ν$. The spin and charge densities and hence the total spin and charge of the skyrmion are computed from the effective action. The total spin is found to have a dominant contribution from the Berry term in the effective action and to increase with the size of the skyrmion. The charge and the statistics of the skyrmion, on the other hand, are completely determined by the prefactor of the Hopf term. Consequently, the skyrmions have charge $νe$ and are Fermions (anyons) for odd integer (fractional) fillings. We also obtain the effective action of the skyrmions at finite temperature. It is shown that at finite temperature, the value of the prefactor of the Hopf term depends on the order in which the zero-momentum and zero-frequency limits are taken.