Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states and p_x+ip_y paired superfluids

TL;DR

This work shows that trial wavefunctions built from conformal blocks can realize non-Abelian statistics and Hall viscosity in quantum Hall states and px+ip_y paired superfluids. By analyzing adiabatic transport, modular transformations, and the plasma-screening structure of conformal-block overlaps, the authors prove that holonomy equals monodromy in gapped (massive IR) phases, deriving explicit MR-state statistics for two and four quasiholes and calculating Hall viscosity across Laughlin and paired-state families. They develop general criteria linking CFT perturbations, RG flow, and IR fixed points to determine when conformal-block holonomies accurately reproduce topological data, and demonstrate that non-unitary RCFTs typically cannot describe gapped topological phases. The results connect CFT data (conformal weights, fusion rules, and modular matrices) to bulk topological invariants (statistics, Hall viscosity, and modular transformations), highlighting the conditions under which conformal-block wavefunctions encode robust topological order with potential applications to topological quantum computation.

Abstract

Many trial wavefunctions for fractional quantum Hall states in a single Landau level are given by functions called conformal blocks, taken from some conformal field theory. Also, wavefunctions for certain paired states of fermions in two dimensions, such as p_x+ip_y states, reduce to such a form at long distances. Here we investigate the adiabatic transport of such many-particle trial wavefunctions using methods from two-dimensional field theory. One context for this is to calculate the statistics of widely-separated quasiholes, which has been predicted to be non-Abelian in a variety of cases. The Berry phase or matrix (holonomy) resulting from adiabatic transport around a closed loop in parameter space is the same as the effect of analytic continuation around the same loop with the particle coordinates held fixed (monodromy), provided the trial functions are orthonormal and holomorphic in the parameters so that the Berry vector potential (or connection) vanishes. We show that this is the case (up to a simple area term) for paired states (including the Moore-Read quantum Hall state), and present general conditions for it to hold for other trial states (such as the Read-Rezayi series). We argue that trial states based on a non-unitary conformal field theory do not describe a gapped topological phase, at least in many cases. By considering adiabatic variation of the aspect ratio of the torus, we calculate the Hall viscosity, a non-dissipative viscosity coefficient analogous to Hall conductivity, for paired states, Laughlin states, and more general quantum Hall states. Hall viscosity is an invariant within a topological phase, and is generally proportional to the "conformal spin density" in the ground state.