On adiabatic cycles of quantum spin systems

Abstract

Motivated by the $\Omega$-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 $H^d(G,U(1))\cong H^{d+1}(G,\mathbb{Z})$, 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.

arXiv commentSec.II.F was rewritten

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• Charting the space of ground states with tensor networks
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• Higher Berry curvature from matrix product states
• Charge pumps, boundary modes, and the necessity of unnecessary criticality
• Modulating Hamiltonian Approach to Quantum Many-Body Systems and Crystalline Topological Phases Protected by Generalized Magnetic Translations
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation
• Generalized Josephson effect with arbitrary periodicity in quantum magnets