Quantum Phase Transitions Between a Class of Symmetry Protected Topological States

TL;DR

This paper investigates quantum phase transitions between symmetry protected topological states, revealing boundary phenomena, lattice realizations, and the nature of critical points, including fractionalized excitations and various transition types.

Contribution

It introduces a framework connecting boundary phase transitions to bulk SPTs and demonstrates how to realize these transitions in lattice models with additional non-local symmetries.

Findings

Boundary phase transitions can be induced by symmetry breaking fields.

Transitions can be realized in lattice models with tuning parameters.

Unusual excitations may appear at continuous transitions.

Abstract

The subject of this paper is the phase transition between symmetry protected topological states (SPTs). We consider spatial dimension $d$ and symmetry group $G$ so that the cohomology group, $H^{d+1}(G,U(1))$, contains at least one $Z_{2n}$ or $Z$ factor. We show that the phase transition between the trivial SPT and the root states that generate the $ Z_{2n} $ or $Z$ groups can be induced on the boundary of a d+1 dimensional $G\times Z_2^T$-symmetric SPT by a $Z_2^T$ symmetry breaking field. Moreover we show these boundary phase transitions can be "transplanted" to d dimensions and realized in lattice models as a function of a tuning parameter. The price one pays is for the critical value of the tuning parameter there is an extra non-local (duality-like) symmetry. In the case where the phase transition is continuous, our theory predicts the presence of unusual (sometimes fractionalized) excitations corresponding to delocalized boundary excitations of the non-trivial SPT on one side of the transition. This theory also predicts other phase transition scenarios including first order transition and transition via an intermediate symmetry breaking phase.