Tensor Network Implementation of Bulk Entanglement Spectrum

TL;DR

This paper introduces a tensor network framework to analyze topological phase transitions via the bulk entanglement spectrum, revealing critical models within ground states like the AKLT wavefunction.

Contribution

It develops a method to compute the bulk entanglement spectrum using tensor networks, linking quantum criticality to classical statistical models in higher dimensions.

Findings

Derived universality classes of topological phase transitions.

Identified critical vertex models within the AKLT wavefunction.

Provided a tensor network approach for analyzing boundary-mediated topological transitions.

Abstract

Many topologically nontrivial states of matter possess gapless degrees of freedom on the boundary, and when these boundary states delocalize into the bulk, a phase transition occurs and the system becomes topologically trivial. We show that tensor networks provide a natural framework for analyzing such topological phase transitions in terms of the boundary degrees of freedom which mediate it. To do so, we make use of a correspondence between a topologically nontrivial ground state and its phase transition to a trivial phase established in [1]. This involved computing the bulk entanglement spectrum (BES) of the ground state, upon tracing out an extensive subsystem. This work implements BES via tensor network representations of ground states. In this framework, the universality class of the quantum critical entanglement Hamiltonian in d spatial dimensions is either derived analytically or mapped to a classical statistical model in d + 1 dimensions, which can be studied using Monte Carlo or tensor renormalization group methods. As an example, we analytically derive the universality classes of topological phase transitions from the spin-1 chain Haldane phase and demonstrate that the AKLT wavefunction (and its generalizations) remarkably contains critical six-vertex (and in general eight-vertex) models within it.