Two dimensional Symmetry Protected Topological Phases with PSU(N) and time reversal symmetry

TL;DR

This work identifies and characterizes two-dimensional bosonic SPT phases with $PSU(N)\times\mathbf{Z}^T_2$ symmetry using a semiclassical field-theory description based on a $2+1$-dimensional principal chiral model with a topological $\Theta$-term for an order parameter in $\mathrm{U}(N)$. At $\Theta=2\pi$ the bulk is fully gapped but the 1+1D boundary is constrained to be either gapless or degenerate due to an induced boundary $\Theta'$-term with value $\pi$, signaling a nontrivial SPT phase. The authors provide a concrete lattice realization for $N=2$, $m=1$ (i.e., $SO(3)\times\mathbf{Z}_2^T$) on a honeycomb lattice using a two-color slave-fermion construction, yielding robust edge states described by an $SU(2)_1$ CFT and a gapped bulk with no topological degeneracy. They argue that at least $N$ inequivalent phases exist for $PSU(N)\times\mathbf{Z}^T_2$, and they discuss generalizations to arbitrary even spatial dimensions through the relevant homotopy groups, with potential connections to external gauge responses awaiting future work.

Abstract

Symmetry protected topological phase is one type of nontrivial quantum disordered many-body state of matter. In this work we study one class of symmetry protected topological phases in two dimensional space, with both PSU(N) and time reversal symmetry. These states can be described by a principal chiral model with a topological Theta-term. As long as the time-reversal symmetry and 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.

Figures (2)

• Figure 1: ($a$) We map Eq. \ref{['Philag']} to a one dimensional tight-binding model. Hopping between nearest neighbor sites corresponds to changing $\Phi$ by $1/2$. The $\Theta$-term grants two types of monopoles a factor $\exp(i\Theta/2)$ and $\exp(- i\Theta/2)$ respectively, which forbids nearest neighbor hopping when $\Theta = \pi$ due to destructive interference between these two types of monopoles. ($b$) The SU($N$) spin localized at the domain wall Eq. \ref{['domainwall']}.
• Figure 2: Lattice construction for $N=2, m=1$ and its edge state. (a) the mean field Hamiltonian Eq. \ref{['mf2']} consists of the nearest-neighbor hopping $t$ and the "color"-orbit interaction $\lambda$ and $t'$. The dashed lines represent the usual "color"-orbit coupling $\sim \lambda$ in the Kane-Mele model. It is equivalent to the Kane-Mele model except for an additional $x$-directional "color"-orbit couplings $\sim t'$ (represented by the dotted lines) on top of the usual "color"-orbit term $\sim \lambda$. This anisotropic coupling breaks the gauge symmetry down to SU(2) and the lattice rotational symmetry which is irrelevant for the symmetry protected phase. (b) the edge theory consists of right-moving ($\tau^{z} = 1$) and left-moving ($\tau^{z} = -1$) spin-$1/2$ doublets. The time-reversal symmetry maps a right-mover into a left-mover with the opposite spin state, $i.e.$ the color and the spin are both flipped by the time-reversal symmetry. (c) the domain wall in $\phi_{0}(x)$ Eq.(24) traps a localized spin-$1/2$. This can be derived from the WZW term written in terms of $(\phi_{0}, \phi_{\mu}), \mu = 1,2,3$ as discussed below Eq. \ref{['o3theta']}.