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arXiv:2410.08284 · Physical review. B./Physical review. B

Multiwavefunction overlap and multientropy for topological ground states in ( 2 + 1 ) dimensions

Bowei Liu, Junjia Zhang, Shuhei Ohyama, Yuya Kusuki, Shinsei Ryu

Published Aug 12, 2025• 11 citations
arXivHTMLPDF

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.

DOI
10.1103/tjcf-yryh

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We discuss multipartitions of the gapped ground states of (2+1)(2+1)(2+1)-dimensional topological liquids into three (or more) spatial regions that are adjacent to each other and meet at points. By considering the reduced density matrix obtained by tracing over a subset of the regions, we compute various correlation measures, such as entanglement negativity, reflected entropy, and associated spectra. We utilize the bulk-boundary correspondence to show that such multipartitions can be achieved by using what we call vertex states in (1+1)(1+1)(1+1)-dimensional conformal field theory: these are a type of state used to define an interaction vertex in string field theory and can be thought of as a proper generalization of conformal boundary states. This approach allows an explicit construction of the reduced density matrix near the entangling boundaries. We find the fingerprints of topological liquid in these quantities, such as (universal pieces in) the scaling of the entanglement negativity, and a nontrivial distribution of the spectrum of the partially transposed density matrix. For reflected entropy, we test the recent claim that states the difference between reflected entropy and mutual information is given, once short-range correlations are properly removed, by (c/3)ln2(c/3)ln2(c/3)ln2 where ccc is the central charge of the topological liquid that measures ungappable edge degrees of freedom. As specific examples, we consider topological chiral ppp-wave superconductors and Chern insulators. We also study a specific lattice-fermion model realizing Chern insulator phases and calculate the correlation measures numerically, both in its gapped phases and at critical points separating them.
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