A many-body index for quantum charge transport

TL;DR

This paper introduces a new integer-valued index for many-body quantum systems that quantifies charge transport and unifies several known phenomena, including quantum Hall conductance and the Lieb-Schultz-Mattis theorem.

Contribution

It defines a stable, perturbation-resistant index for charge transport in many-body systems, connecting various physical phenomena under a unified framework.

Findings

The index reduces to known cases like non-interacting fermions and charge density.

It provides a new proof of quantized Hall conductance in interacting systems.

It recovers the Lieb-Schultz-Mattis theorem as a special case.

Abstract

We propose an index for pairs of a unitary map and a clustering state on many-body quantum systems. We require the map to conserve an integer-valued charge and to leave the state, e.g. a gapped ground state, invariant. This index is integer-valued and stable under perturbations. In general, the index measures the charge transport across a fiducial line. We show that it reduces to (i) an index of projections in the case of non-interacting fermions, (ii) the charge density for translational invariant systems, and (iii) the quantum Hall conductance in the two-dimensional setting without any additional symmetry. Example (ii) recovers the Lieb-Schultz-Mattis theorem, and (iii) provides a new and short proof of quantization of Hall conductance in interacting many-body systems.