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

TL;DR

This work addresses whether the Hall viscosity in integer and Jain fractional quantum Hall states remains unrenormalized by Coulomb interactions. It develops a Weyl-symbol, Green-function–based topological invariant framework within the Lopez–Fradkin composite fermion model to express and compute the Hall viscosity, deriving an invariant $\mathcal{N}_{\eta_H}$ that yields $\eta_H = \frac{1}{2\pi}\mathcal{N}_{\eta_H} B_{eff}$ with $B_{eff}$ the interaction-renormalized field; it shows perturbative non-renormalization and accounts for a topological-spin–driven shift $\Delta^{s_{top}}\eta_H = \frac{sp}{2}\frac{B_{eff}}{2\pi}$ for Jain states. The analysis unifies mean-field and fully interacting cases, proving the topological character of the Hall viscosity and its robustness under Coulomb interactions, while clarifying how the Wen–Zee shift is modified by composite fermion orbital spin. The results have significant implications for the geometric response of quantum Hall fluids and extend non-renormalization insights previously established for the Hall conductivity to the Hall viscosity, with potential applications to inhomogeneous or thermal settings.

Abstract

We proof the non-renormalization of the Hall viscosity by Coulomb interactions for integer and fractional quantum Hall Jain states 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 topological invariant comprised of Green functions and work within the composite fermion field theory model of Jain states of the fractional quantum Hall fluid presented by Lopez and Fradkin. The topological expression is first 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. The topological orbital spin of composite fermions distinguishes their mean field treatment from that of electrons resulting in an additional topological contribution. We then argue that the introduction of Coulomb interactions does not lead to perturbative corrections of the Hall viscosity in both integer and fractional quantum Hall fluids. The proof relies on the assumptions of homogeneity and rotational invariance of an underlying sample modulo the vector potential giving rise to the homogeneous external magnetic field. These conditions 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 latter features a contribution from the composite fermion topological orbital spin relative to that of electrons.