Time’s Arrow and the Fragility of Topological Phases Authors: Max McGinley and Nigel R. Cooper arXiv:2003.08120 Recommended with a commentary by Michael Zaletel, University of California, Berkeley |View Commentary (pdf)| DOI: 10.36471/JCCM_April_2020_03 https://doi.org/10.36471/JCCM_April_2020_03

Journal Club for Condensed Matter PhysicsApr 30, 20200 citations

arXiv:2003.08120 · Nature Physics

Fragility of time-reversal symmetry protected topological phases

Max McGinley, Nigel R. Cooper

Published Jul 6, 2020• 47 citations

arXiv HTML PDF

DOI: 10.1038/s41567-020-0956-z

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\mathrm{PSU}(N)

and time-reversal symmetry. These states can be described by a principal chiral model with a topological $\ensuremath{\Theta}$ term. As long as the time-reversal symmetry and

\mathrm{PSU}(N)

symmetry are both preserved, the (1+1)-dimensional boundary of this system must be either gapless or degenerate. We will also construct a wave function of a spin-1 system on the honeycomb lattice, which is a candidate for the symmetry-protected topological phase with both SO(3) and time-reversal symmetry.

Based on the recently established parafermionic matrix product states, we study the classification of one-dimensional gapped phases of parafermions with time-reversal (TR) symmetry satisfying

{T}^{2}=1

. Without extra symmetry, it has been found that

{\mathbb{Z}}_{p}

parafermionic gapped phases can be classified as topological phases, spontaneous symmetry breaking (SSB) phases, and a trivial phase, which are uniquely labeled by the divisors

n

p

. In the presence of TR symmetry, however, the enriched classification is characterized by three indices

n

, $\ensuremath{\kappa}$, and $\ensuremath{\mu}$, where $\ensuremath{\kappa}\ensuremath{\in}{\mathbb{Z}}_{2}$ denotes the linear or projective TR actions on the edges, and $\ensuremath{\mu}\ensuremath{\in}{\mathbb{Z}}_{2}$ indicates the commutation relations between the TR and (fractionalized) charge operator. For the

{\mathbb{Z}}_{r}

-symmetric parafermionic ground states, where

r=p

for trivial or topological phases, and

r=p/n

for SSB phases, each original gapped phase with odd

r

is divided into two phases, while each phase with even

r

is further separated into four phases. The gapped parafermionic phases with the TR symmetry include the symmetry protected topological phases, symmetry enriched topological phases, and the SSB coexisting symmetry protected topological phases. From analyzing the structures and symmetries of their reduced density matrices of these resulting topological phases, we can obtain the topologically protected degeneracies of their entanglement spectra.