Extraction of many-body Chern number from a single wave function

TL;DR

This paper introduces a method to determine the many-body Chern number from a single wave function, simplifying topological invariant calculations in quantum Hall systems without needing the full Hamiltonian or wave function family.

Contribution

The authors present a novel approach to extract the many-body Chern number from one wave function, requiring only an additional integer invariant for fractional quantum Hall states.

Findings

Validated method through extensive numerical simulations

Successfully extracted Chern numbers for IQH and FQH states

Provided a way to determine topological invariants without Hamiltonian knowledge

Abstract

The quantized Hall conductivity of integer and fractional quantum Hall (IQH and FQH) states is directly related to a topological invariant, the many-body Chern number. The conventional calculation of this invariant in interacting systems requires a family of many-body wave functions parameterized by twist angles in order to calculate the Berry curvature. In this paper, we demonstrate how to extract the Chern number given a single many-body wave function, without knowledge of the Hamiltonian. For FQH states, our method requires one additional integer invariant as input: the number of $2\pi$ flux quanta, $s$, that must be inserted to obtain a topologically trivial excitation. As we discuss, $s$ can be obtained in principle from the degenerate set of ground state wave functions on the torus, without knowledge of the Hamiltonian. We perform extensive numerical simulations involving IQH and FQH states to validate these methods.