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

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.