Noninvertible duality transformation between symmetry-protected topological and spontaneous symmetry breaking phases

Abstract

In 1992, Kennedy and Tasaki constructed a nonlocal unitary transformation that maps between a ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ spontaneously symmetry breaking phase and the Haldane gap phase, which is a prototypical symmetry-protected topological (SPT) 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 nonunitary transformation satisfies noninvertible fusion rule and implements a generalized gauging of the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ 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.