One-Dimensional Symmetry Protected Topological Phases and their Transitions

TL;DR

This paper offers a unified framework for understanding one-dimensional symmetry protected topological phases and their phase transitions, connecting various models and proposing a conjecture relating edge modes to critical behavior.

Contribution

It maps different SPT models to a Kitaev chain framework and conjectures a lower bound on the central charge for phase transitions involving edge modes.

Findings

Unified description of fermionic and bosonic SPTs

Identification of new properties in known models

Conjecture relating edge mode dimension to critical point central charge

Abstract

We present a unified perspective on symmetry protected topological (SPT) phases in one dimension and address the open question of what characterizes their phase transitions. In the first part of this work we use symmetry as a guide to map various well-known fermionic and spin SPTs to a Kitaev chain with coupling of range $\alpha \in \mathbb Z$. This unified picture uncovers new properties of old models --such as how the cluster state is the fixed point limit of the Affleck-Kennedy-Lieb-Tasaki state in disguise-- and elucidates the connection between fermionic and bosonic phases --with the Hubbard chain interpolating between four Kitaev chains and a spin chain in the Haldane phase. In the second part, we study the topological phase transitions between these models in the presence of interactions. This leads us to conjecture that the critical point between any SPT with $d$-dimensional edge modes and the trivial phase has a central charge $c \geq \log_2 d$. We analytically verify this for many known transitions. This agrees with the intuitive notion that the phase transition is described by a delocalized edge mode, and that the central charge of a conformal field theory is a measure of the gapless degrees of freedom.