Parametrized topological phases in 1d and T-duality

TL;DR

This paper explores parametrized topological phases in 1+1 dimensions, revealing a novel T-duality that relates different families of phases via mathematical structures like bundles and circle actions.

Contribution

It introduces a new T-duality for parametrized topological phases, connecting different parameter spaces and higher Berry classes using advanced algebraic and tensor network techniques.

Findings

Existence of a global MPS parametrization linked to bundle triviality.

Introduction of a T-duality relating different families of topological phases.

Mathematical realization of T-duality via gauging circle actions on trace algebras.

Abstract

There are families of physical systems that cannot be adiabatically evolved to the trivial system uniformly across the parameter space, even if each system in the family belongs to the trivial phase. The obstruction is measured by higher Berry class. We analyze families of topological systems in 1+1d using families of invertible TQFTs and families of RG fixed states of spin chains. We use the generalized matrix-product states to describe RG fixed points of all translation invariant pure splits states on spin chains. Families of such fixed points correspond to bundles of Hilbert-Schmidt operators. There exists a global MPS parametrization of the family if and only if the latter bundle is trivial. We propose a novel duality of parametrized topological phases which is an avatar of the T-duality in string theory. The duality relates families with different parameter spaces and different higher Berry classes. Mathematically, the T-duality is realized by gauging the circle action on the continuous trace algebra generated by parametrized matrix-product tensors.