Non-renormalization of the Hall viscosity of integer and Jain fractional quantum Hall phases by Coulomb interactions

Abstract

We proof the non-renormalization of the Hall viscosity by Coulomb interactions for integer and Jain fractional quantum Hall states in the presence of Galilean invariance building on previous results obtained for the Hall conductivity. We employ Wigner-Weyl calculus in order to represent the Hall viscosity in terms of a novel Green function expression supplemented by covariant derivatives. The Green function based expression is derived within the free field theory of electrons and explicitly calculated for this case as well as in the mean field approximation of the composite fermion theory Jain states employing the model by Lopez and Fradkin. The topological orbital spin of composite fermions distinguishes their mean field treatment from that of electrons resulting in an additional topological contribution. We furthermore proof the topological invariance of the newly introduced Green function expression for the Hall viscosity for the leading contribution in powers of the filling factor with and without Galilean boost symmetry. The latter condition is known to already imply a Hall viscosity per emergent quasiparticle number density quantized in units of one half times the average quasiparticle orbital spin or one quarter times the Wen-Zee shift. The Wen-Zee shift features a contribution from the composite fermion topological orbital spin relative to that of electrons which we highlight within our formalism and connect to Haldane's geometric Hall viscosity framework.