# Entanglement phases as holographic duals of anyon condensates

## TL;DR

This paper explores how anyon condensation in topological phases can be understood through the entanglement spectrum of PEPS, revealing a correspondence with boundary symmetries and characterizing phase transitions.

## Contribution

It establishes a novel framework linking anyon condensation to the entanglement spectrum in PEPS, including detailed analysis of abelian double models and phase transitions.

## Key findings

- Anyon condensation corresponds to boundary entanglement spectrum behavior.
- Characterization of all abelian double model condensations via entanglement structure.
- Numerical phase diagram including non-anyon condensation transitions.

## Abstract

Anyon condensation forms a mechanism which allows to relate different topological phases. We study anyon condensation in the framework of Projected Entangled Pair States (PEPS) where topological order is characterized through local symmetries of the entanglement. We show that anyon condensation is in one-to-one correspondence to the behavior of the virtual entanglement state at the boundary (i.e., the entanglement spectrum) under those symmetries, which encompasses both symmetry breaking and symmetry protected (SPT) order, and we use this to characterize all anyon condensations for abelian double models through the structure of their entanglement spectrum. We illustrate our findings with the Z4 double model, which can give rise to both Toric Code and Doubled Semion order through condensation, distinguished by the SPT structure of their entanglement. Using the ability of our framework to directly measure order parameters for condensation and deconfinement, we numerically study the phase diagram of the model, including direct phase transitions between the Doubled Semion and the Toric Code phase which are not described by anyon condensation.