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<p>For any locality-preserving action of a group <i>G</i> on a quantum spin chain one can define an anomaly index taking values in the group cohomology of <i>G</i>. The anomaly index is a kinematic quantity, it does not depend on the Hamiltonian. We prove that a nonzero anomaly index prohibits any <i>G</i>-invariant Hamiltonian from having <i>G</i>-invariant gapped ground states. Lieb–Schultz–Mattis-type theorems are a special case of this result when <i>G</i> involves translations. In the case when the symmetry group <i>G</i> is a Lie group, we define an anomaly index which takes values in the differentiable group cohomology as defined by J.-L. Brylinski and prove a similar result.</p>

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Anomalous Symmetries of Quantum Spin Chains and a Generalization of the Lieb–Schultz–Mattis Theorem

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • Anton Kapustin,
  • Nikita Sopenko

摘要

For any locality-preserving action of a group G on a quantum spin chain one can define an anomaly index taking values in the group cohomology of G. The anomaly index is a kinematic quantity, it does not depend on the Hamiltonian. We prove that a nonzero anomaly index prohibits any G-invariant Hamiltonian from having G-invariant gapped ground states. Lieb–Schultz–Mattis-type theorems are a special case of this result when G involves translations. In the case when the symmetry group G is a Lie group, we define an anomaly index which takes values in the differentiable group cohomology as defined by J.-L. Brylinski and prove a similar result.