Motivated by the $\mathrm{\Omega }$-spectrum proposal of unique gapped ground states by Kitaev, we study adiabatic cycles in gapped quantum spin systems from various perspectives. We give a few exactly solvable models in one and two spatial dimensions and discuss how nontrivial adiabatic cycles are detected. For one spatial dimension, we study the adiabatic cycle in detail with the matrix product state and show that the symmetry charge can act on the space of matrices without changing the physical states, which leads to nontrivial loops with symmetry charges. For generic spatial dimensions, based on the Bockstein isomorphism $H^d(G,U\left(1\right))\cong H^{d+1}(G,\mathbb{Z})$, we study a group cohomology model of the adiabatic cycle that pumps a symmetry-protected topological phase on the boundary by one period. It is shown that the spatial texture of the adiabatic Hamiltonian traps a symmetry-protected topological phase in one dimension lower.

• Received 26 October 2021
• Revised 25 August 2022
• Accepted 26 August 2022

DOI:https://doi.org/10.1103/PhysRevB.106.125108