Failure of the Crystalline Equivalence Principle for Weak Free Fermions

TL;DR

The paper interrogates whether the crystalline equivalence principle (CEP) extends from interacting bosonic SPTs to free-fermion crystalline phases, distinguishing weak versus strong regimes. It contrasts cohomology-based CEP, which applies to certain interacting classifications, with K-theory-based CEP for free fermions, showing that CEP fails for weak crystalline free fermions but can hold for strong crystalline free fermions even though K-theory is not Borel. It highlights the distinct roles of spatial and internal symmetries (G vs H) and contrasts fixed-point vs homotopy fixed-point constructions, illustrating how non-Borel frameworks can still yield CEP in special cases. The authors also propose alternative interacting classifications without CEP and discuss implications for choosing between group-cohomology and bordism-based theories, emphasizing the need for careful physical interpretation when interactions are present.

Abstract

Interacting crystalline SPT phases were first classified by Thorngren and Else through the crystalline equivalence principle, suggesting that a spatial group symmetry can be treated as if it were an internal symmetry group. Using techniques from topology we elucidate how this principle holds for one of the proposals for the interacting bosonic case and yet fails for weak free fermion phases. Last we show how a variant of the principle does hold for strong crystalline free fermion phases, showing that it is not necessary for the cohomology theory to be of Borel type.