SPT Indices Emerging From Translation Invariance in Two-Dimensional Quantum Spin Systems

We consider SPT-phases with on-site G (where G is any finite group) symmetry for two-dimensional quantum spin systems. We then impose translation invariance in one direction and observe that on top of the \(H^3(G,{\mathbb {T}})\) $H^3(G,\mathbb{T})$ -valued index constructed in Ogata (Forum Math Pi 9(e13):1–62, 2021), an additional \(H^2(G,{\mathbb {T}})\) $H^2(G,\mathbb{T})$ -valued index emerges. We also show that if we impose translation invariance in two directions, on top of the expected \(H^3(G,{\mathbb {T}})\oplus H^2(G,{\mathbb {T}})\oplus H^2(G,{\mathbb {T}})\) $H^3(G,\mathbb{T})\oplus H^2(G,\mathbb{T})\oplus H^2(G,\mathbb{T})$ valued index, an additional \(H^1(G,{\mathbb {T}})\) $H^1(G,\mathbb{T})$ -valued index emerges.