Non-Invertible Duality Transformation Between SPT and SSB Phases

TL;DR

This paper generalizes the Kennedy-Tasaki transformation to closed chains by sacrificing unitarity, revealing a non-invertible duality that connects symmetry-breaking and SPT phases, and offers a systematic way to construct SPT phases.

Contribution

It introduces a non-unitary, non-invertible transformation on closed chains that links SPT and SSB phases, expanding the understanding of dualities in topological phases.

Findings

Defines a non-unitary transformation on closed chains.

Connects SPT phases with symmetry-breaking phases via non-invertible fusion rules.

Provides a framework for systematic construction of SPT phases.

Abstract

In 1992, Kennedy and Tasaki constructed a non-local unitary transformation that maps between a $\mathbb{Z}_2\times \mathbb{Z}_2$ spontaneously symmetry breaking phase and the Haldane gap phase, which is a prototypical Symmetry-Protected Topological phase in modern framework, on an open spin chain. In this work, we propose a way to define it on a closed chain, by sacrificing unitarity. The operator realizing such a non-unitary transformation satisfies non-invertible fusion rule, and implements a generalized gauging of the $\mathbb{Z}_2\times \mathbb{Z}_2$ global symmetry. These findings connect the Kennedy-Tasaki transformation to numerous other concepts developed for SPT phases, and opens a way to construct SPT phases systematically using the duality mapping.