Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions

TL;DR

This paper introduces a method to compute multi-wavefunction overlaps and multi-entropy in (2+1)-dimensional topological ground states, revealing topological invariants and bounds related to conformal field theory and free fermion systems.

Contribution

It develops a framework to calculate multi-wavefunction overlaps and multi-entropy for topological states using bulk-boundary correspondence and correlator methods, linking these to topological invariants and central charge.

Findings

Multi-wavefunction overlaps encode topological invariants as higher Berry phases.

Multi-entropy differences are bounded by the central charge of ungappable modes.

The correlator method enables numerical computation of multi-entropy in free fermion models.

Abstract

Multi-wavefunction overlaps -- generalizations of the quantum mechanical inner product for more than two quantum many-body states -- are valuable tools for studying many-body physics. In this paper, we investigate the multi-wavefunction overlap of (2+1)-dimensional gapped ground states, focusing particularly on symmetry-protected topological (SPT) states. We demonstrate how these overlaps can be calculated using the bulk-boundary correspondence and (1+1)-dimensional edge theories, specifically conformal field theory. When applied to SPT phases, we show that the topological invariants, which can be thought of as discrete higher Berry phases, can be extracted from the multi-wavefunction overlap of four ground states with appropriate symmetry actions. Additionally, we find that the multi-wavefunction overlap can be expressed in terms of the realignment of reduced density matrices. Furthermore, we illustrate that the same technique can be used to evaluate the multi-entropy -- a quantum information theoretical quantity associated with multi-partition of many-body quantum states -- for (2+1)-dimensional gapped ground states. Combined with numerics, we show that the difference between multi-entropy for tripartition and second R\'enyi entropies is bounded from below by $(c_{{\it tot}}/4)\ln 2$ where $c_{{\it tot}}$ is the central charge of ungappable degrees of freedom. To calculate multi-entropy numerically for free fermion systems (such as Chern insulators), we develop the correlator method for multi-entropy.