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

TL;DR

This work develops a bulk-boundary framework to compute multi-wavefunction overlaps and multi-entropy for (2+1)D gapped ground states, with a focus on symmetry-protected topological phases. By mapping multipartite overlaps to (1+1)D edge-CFT amplitudes and vertex states, it extracts topological invariants from four-state overlaps via F-symbols and group cohomology, while realignment of reduced density matrices provides a unifying real-space picture. The authors derive a universal bound on the finite-size correction kappa in terms of the total central charge, and they implement a covariance-matrix method to numerically evaluate multi-entropy for free-fermion systems like Chern insulators, finding results consistent with the bound and exhibiting characteristic finite-size scaling near phase transitions. They also extend the formalism to include symmetry-twist defects, discuss charged entanglement, and outline avenues to apply the approach to non-invertible topological orders, highlighting the broader significance for characterizing multipartite entanglement in topological matter.

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 Rényi 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.

Figures (8)

• Figure 1: (a) The regular quantum mechanical inner product between two states $\Psi_{\alpha}=\sum_i\psi_{i\alpha}|i\rangle$ and $\Psi_{\beta}=\sum_i\psi_{i\beta}|i\rangle$, $\langle \Psi_{\beta} | \Psi_{\alpha}\rangle = \sum_i \psi^*_{\beta i}\psi_{\alpha i}$. The inner product can also be written in terms of a maximally entangled state that is represented as a cup on the right-hand side. entangled state. (b) The triple inner product for three states. Each state is defined in a bipartite Hilbert space, e.g., $\Psi_{\alpha}=\sum_{i,j}\psi_{ij, \alpha}|i\rangle|j\rangle$. (c) The quadruple inner product. Each state is defined in a tripartite Hilbert space, e.g., $\Psi_{\alpha}=\sum_{i,j,k}\psi_{ijk, \alpha}|i\rangle|j\rangle|k\rangle$. (d) The multi-wavefunction overlap relevant to multi-entropy.
• Figure 2: (a) The tripartition of a (2+1)d gapped ground state into three regions $A, B, C$. (b) The representation of the tripartite ground state using the bulk-boundary correspondence and the vertex state in edge conformal field theory. (c) The path integral representation of the reduced density matrix $\rho_{AB}$. (d) The path integral representation of the realignment $\rho^R_{AB}$.
• Figure 3: (a) The MPS representations of (1+1)d invertible states $\Psi_{\alpha}$; $\Psi_{\alpha\beta}$ is a MPS representation in the mixed gauge; (b) The triple inner product of $\Psi_{\alpha},\Psi_{\beta},\Psi_{\gamma}$; (c) The triple inner product in the mixed gauge.
• Figure 4: The quadruple inner product in (2+1) dimensions. (a) The path integral for the quadruple inner product; the red lines represent symmetry twist defects (the arrows indicate $g$ or $g^{-1}$ and do not represent the orientation of the surface.) (b) The defect line network for the sphere part of the path integral. (c) The relevant path integrals for the reduced density matrices, with partial symmetry operations. The second path integral can be obtained by taking realignment. (d) The sphere part of the path integral that appears in the quadruple inner product.
• Figure 5: (a) The path integral representation of multipartition function $Z(A{\,:\,}B{\,:\,}C)$. (b) The spherical part of the multipartition function.