Multipartitioning topological phases by vertex states and quantum entanglement

TL;DR

This paper investigates multipartitions of (2+1)D topological liquids into multiple regions, using vertex states and entanglement measures to reveal universal topological features and their relation to conformal field theory.

Contribution

It introduces a novel approach using vertex states in conformal field theory to explicitly construct reduced density matrices for multipartitions in topological phases.

Findings

Universal scaling of entanglement negativity reveals topological signatures.

Spectrum of partially transposed density matrix shows non-trivial distribution.

Reflected entropy relates to central charge and edge degrees of freedom.

Abstract

We discuss multipartitions of the gapped ground states of (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)-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 non-trivial 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 $(c/3)\ln 2$ where $c$ is the central charge of the topological liquid that measures ungappable edge degrees of freedom. As specific examples, we consider topological chiral $p$-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.