Rational magnetic equivariant K-theory

TL;DR

The paper defines magnetic equivariant K-theory $\mathcal{K}_G(X)$ for magnetic groups $G$ with a surjection to $\mathbb{Z}_2$ and kernel $G_0$, encoding vector bundles that are complex-linear on $G_0$ and complex-anti-linear off it. It proves a fundamental rational reduction: the restriction map $\iota^*: \mathcal{K}_G^*(X) \to K_{G_0}^*(X)$ lands in the $\mathbb{Z}_2$-invariant part and induces a rational isomorphism $\iota^* \otimes \mathbb{Q}: \mathcal{K}_G^*(X) \otimes \mathbb{Q} \cong K_{G_0}^*(X)^{\mathbb{Z}_2} \otimes \mathbb{Q}$, with a similar statement for twisted theories. This reduces non-torsion information to conjugation-invariant complex K-theory calculations and generalizes to twisted magnetic K-theory. As an application, for 2D altermagnets with $C_4\mathbb{T}$ symmetry and spin $S_z$, the spin-up Chern number provides an integer bulk invariant, whose parity yields the $\mathbb{Z}_2$ bulk invariant, aligning with prior work on bulk classifications. The results supply a practical tool to extract non-torsion invariants in magnetic topological phases and point to pathways for extending the framework to broader twisted magnetic symmetries.

Abstract

We introduce the magnetic equivariant K-theory groups as the K-theory groups associated to magnetic groups and their respective magnetic equivariant complex bundles. We restrict the magnetic group to its subgroup of elements that act complex linearly, and we show that this restriction induces a rational isomorphism with the conjugation invariant part of the complex equivariant K-theory of the restricted group. This isomorphism allows to calculate the torsion free part of the magnetic equivariant K-theory groups reducing it to known calculations in complex equivariant K-theory