Quasi-adiabatic Continuation for Disordered Systems: Applications to Correlations, Lieb-Schultz-Mattis, and Hall Conductance

TL;DR

This paper introduces a new framework for analyzing disordered quantum systems, proving decay of correlations and quantization of Hall conductance under a mobility gap, and presents an optimized quasi-adiabatic continuation method with improved bounds.

Contribution

It defines a mobility gap for many-body systems, constructs corrected quasi-adiabatic operators, and proves key properties like correlation decay and Hall conductance quantization, including for systems with boundaries.

Findings

Decay of correlation functions established.

Hall conductance quantization proven under mild assumptions.

An optimized quasi-adiabatic operator with subexponential decay bounds.

Abstract

We present a possible definition of a mobility gap for a many-body quantum system, in analogy to definitions of dynamical localization for single particle systems. Using this definition, we construct "corrected" quasi-adiabatic continuation operators. Under an appropriate definition of a unique ground state, we show how to introduce virtual fluxes. Armed with these results, we can directly carry over previous results in the case of a spectral gap. We present a proof of decay of correlation functions and we present a proof of Hall conductance quantization under very mild density-of-states assumptions defined later. We also generalize these definitions to the case of a "bulk mobility gap", in the case of a system with boundaries, and present a proof of Hall conductance quantization on an annulus under appropriate assumptions. Further, we present a new "optimized" quasi-adiabatic continuation operator which simplifies previous estimates and tightens bounds in certain cases. This is presented in an appendix which can be read independently of the rest of the paper as it also improves estimates in the case of systems with a spectral gap. This filter function used decays in time at least as fast as ${\cal O}(\exp(-t^\alpha))$ for all $\alpha<1$, a class of decay called subexponential (a tighter description of what is possible is below). Using this function it is possible to tighten recent estimates of the Hall conductance quantization for gapped systems\cite{hall} to a decay which is subexponential in system size.