Failure of the Crystalline Equivalence Principle for Weak Free Fermions

TL;DR

This paper investigates the limitations of the crystalline equivalence principle in classifying interacting crystalline SPT phases, revealing it fails for weak free fermions but holds for strong crystalline free fermions.

Contribution

It demonstrates the failure of the crystalline equivalence principle for weak free fermion phases and shows a variant of it holds for strong crystalline free fermion phases.

Findings

Crystalline equivalence principle fails for weak free fermions.

A variant of the principle holds for strong crystalline free fermions.

The cohomology theory need not be of Borel type for the principle to hold.

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.