Hall viscosity and putative quantum Hall states without positive-definite K-matrix

TL;DR

This paper explores specific quantum Hall states with a particular K-matrix, analyzing their wave functions, Hall viscosity, and clustering behavior on the torus, revealing geometry and particle number dependencies.

Contribution

It introduces a generalized monodromy matching method to construct and analyze these states, highlighting issues like clustering and non-positive-definite K-matrices.

Findings

Hall viscosity depends on geometry and particle number

Clustering persists despite anti-symmetrization

Wave functions are fixed by boundary conditions and modular invariance

Abstract

We investigate putative quantum Hall effect states, labeled by their K-matrix equal to (1 1 3), by defining them on the torus and computing their Hall viscosity. Such states have been introduced on the sphere as a phase distinct from Pfaffian and anti-Pfaffian ones. This was done in order to explain certain results on thermal Hall conductivity in favor of particle-hole symmetric Pfaffian topological order in presence of Landau level mixing. The requirements of boundary conditions, modular invariance and ground state degeneracy are enough to uniquely fix the form of the proposed wave functions. We generalize a method to enforce them which we call monodromy matching and check our results on wave functions and Hall viscosity against realizations on the torus of Laughlin and hierarchical states. We highlight the issues in the realization of these states, which turn out to exhibit the formation of clusters. We show that the effect of anti-symmetrization on the system is not enough to prevent clustering; we compute the Hall viscosity for the Halperin version of these states and the fully anti-symmetrized one and we find them being dependent on the geometry and the particle number.