A many-body Fredholm index for ground state spaces and Abelian anyons

TL;DR

This paper introduces a many-body index extending Fredholm theory to charge-conserving topologically ordered systems, providing new proofs of quantized Hall conductance and constructing operators for fractional Abelian anyons.

Contribution

It defines a fractional many-body index applicable to topologically ordered systems, linking index theory with anyon braiding and fractional charge creation.

Findings

Provides a new proof of Hall conductance quantization.

Constructs Wilson loop operators for Abelian anyons.

Defines a fractional index for topologically ordered ground states.

Abstract

We propose a many-body index that extends Fredholm index theory to many-body systems. The index is defined for any charge-conserving system with a topologically ordered $p$-dimensional ground state sector. The index is fractional with the denominator given by $p$. In particular, this yields a new short proof of the quantization of the Hall conductance and of Lieb-Schulz-Mattis theorem. In the case that the index is non-integer, the argument provides an explicit construction of Wilson loop operators exhibiting a non-trivial braiding and that can be used to create fractionally charged Abelian anyons.