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October 9, 2025Open Access

Anomaly diagnosis via symmetry restriction in two-dimensional lattice systems

Key Points

A nontrivial index indicates a failure to construct a symmetric Hamiltonian without a gap, impacting physical realizability.
The cohomology group H^4(G,U(1)) provides a framework for defining anomalies in quantum systems under symmetry actions.
The method is exemplified with the symmetry group G as a product of Z2 groups, demonstrating practical applicability.
A diagrammatic interpretation of the anomaly formula suggests connections to advanced categorical structures in quantum theory.

Abstract

We describe a method for computing the anomaly of any finite unitary symmetry group G acting by finite-depth quantum circuits on a two-dimensional lattice system. The anomaly is characterized by an index valued in the cohomology group H⁴ (G, U (1) ), which generalizes the Else-Nayak index for locality preserving symmetries of quantum spin chains. We show that a nontrivial index precludes the existence of a trivially gapped symmetric Hamiltonian; it is also an obstruction to ``onsiteability" of the symmetry action. We illustrate our method via a simple example with G=Z₂₂₂₂. Finally, we provide a diagrammatic interpretation of the anomaly formula which hints at a higher categorical structure.

Anomaly diagnosis via symmetry restriction in two-dimensional lattice systems

Key Points

Abstract

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