Anomaly diagnosis via symmetry restriction in two-dimensional lattice systems

Kyle Kawagoe; Wilbur Shirley

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arXiv · 2507.07430

Anomaly diagnosis via symmetry restriction in two-dimensional lattice systems

Kyle Kawagoe·Wilbur Shirley

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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 $H^4(G,U(1))$, 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 $G=\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$. Finally, we provide a diagrammatic interpretation of the anomaly formula which hints at a higher categorical structure.

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Kyle Kawagoe, Wilbur Shirley. 2025-07-10. Anomaly diagnosis via symmetry restriction in two-dimensional lattice systems. https://arxiv.org/abs/2507.07430

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