Quantized topological terms in weak-coupling gauge theories with a global symmetry and their connection to symmetry-enriched topological phases

We study the quantized topological terms in a weak-coupling gauge theory with gauge group $G_g$ and a global symmetry $G_s$ in $d$ space-time dimensions. We show that the quantized topological terms are classified by a pair $(G,{\nu }_d)$, where $G$ is an extension of $G_s$ by $G_g$ and ${\nu }_d$ an element in group cohomology ${\mathcal{H}}^d(G,\mathbb{R}/\mathbb{Z})$. When $d=3$ and/or when $G_g$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e., gapped long-range-entangled phases with symmetry). Thus, those SET phases are classified by ${\mathcal{H}}^d(G,\mathbb{R}/\mathbb{Z})$, where $G/G_g=G_s$. We also apply our theory to a simple case $G_s=G_g=Z_2$, which leads to 12 different SET phases in $2+1$ dimensions [(2+1)D], where quasiparticles have different patterns of fractional $G_s=Z_2$ quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry $G_s$, which may lead to different fractionalizations of $G_s$ quantum numbers and different fractional statistics [if in ($2+1$)D].

Figure 1

(a) The possible gapped phases for a class of Hamiltonians $H(g_1,g_2)$ without any symmetry. (b) The possible gapped phases for the class of Hamiltonians $H_{\mathrm{symm}}(g_1,g_2)$ with symmetry. The yellow regions in (a) and (b) represent the phases with long-range entanglement. Each phase is labeled by its entanglement properties and symmetry breaking properties. SRE stands for short-range entanglement, LRE for long-range entanglement, SB for symmetry breaking, SY for no symmetry breaking. SB-SRE phases are the Landau symmetry breaking phases. The SY-SRE phases are the SPT phases. The SY-LRE phases are the SET phases.

Figure 2

Two branched simplices with opposite orientations. (a) A branched simplex with positive orientation and (b) a branched simplex with negative orientation.