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Magnetic Equivariant K-theory

Higinio Serrano, Bernardo Uribe, Miguel A. Xicoténcatl

TL;DR

This work develops magnetic equivariant K-theory to classify topological features in crystals with time-reversal and spatial symmetries. It generalizes Atiyah–Segal real and equivariant K-theories via magnetic groups, defines magnetic vector bundles, and constructs the Grothendieck K-theory $\mathbf{K}_G(X)$, including Bott periodicity, Thom isomorphism, and a Clifford-module description of coefficients. It also establishes the reduced theory, the AHSS framework, and twistings by central extensions, enabling concrete computations. The paper then applies the theory to topological phases of matter, notably spin-orbit coupling scenarios and a detailed calculation for a 2D torus with $C_4\mathbb{T}$ symmetry, illustrating 4-periodicity and bulk invariants that distinguish trivial and topological phases. Overall, it provides a calculational toolkit for magnetic-equivariant invariants with direct relevance to altermagnets and topological crystalline phases.

Abstract

We present the fundamental properties of the K-theory groups of complex vector bundles endowed with actions of magnetic groups. In this work we show that the magnetic equivariant K-theory groups define an equivariant cohomology theory, we determine its coefficients, we show Bott's, Thom's and the degree shift isomorphism, we present the Atiyah-Hirzeburh spectral sequence, and we explicitly calculate two magnetic equivariant K-theory groups in order to showcase its applicability. These magnetic equivariant K-theory groups are relevant in condensed matter physics since they provide topological invariants of gapped Hamiltonians in magnetic crystals.

Magnetic Equivariant K-theory

TL;DR

This work develops magnetic equivariant K-theory to classify topological features in crystals with time-reversal and spatial symmetries. It generalizes Atiyah–Segal real and equivariant K-theories via magnetic groups, defines magnetic vector bundles, and constructs the Grothendieck K-theory , including Bott periodicity, Thom isomorphism, and a Clifford-module description of coefficients. It also establishes the reduced theory, the AHSS framework, and twistings by central extensions, enabling concrete computations. The paper then applies the theory to topological phases of matter, notably spin-orbit coupling scenarios and a detailed calculation for a 2D torus with symmetry, illustrating 4-periodicity and bulk invariants that distinguish trivial and topological phases. Overall, it provides a calculational toolkit for magnetic-equivariant invariants with direct relevance to altermagnets and topological crystalline phases.

Abstract

We present the fundamental properties of the K-theory groups of complex vector bundles endowed with actions of magnetic groups. In this work we show that the magnetic equivariant K-theory groups define an equivariant cohomology theory, we determine its coefficients, we show Bott's, Thom's and the degree shift isomorphism, we present the Atiyah-Hirzeburh spectral sequence, and we explicitly calculate two magnetic equivariant K-theory groups in order to showcase its applicability. These magnetic equivariant K-theory groups are relevant in condensed matter physics since they provide topological invariants of gapped Hamiltonians in magnetic crystals.

Paper Structure

This paper contains 21 sections, 59 theorems, 314 equations, 1 figure.

Table of Contents

  1. Magnetic groups and their representations
  2. Magnetic groups
  3. Representations of magnetic groups
  4. Morphisms of representations
  5. Classification of irreducible representations
  6. Magnetic Equivariant K-theory
  7. Magnetic equivariant vector bundles
  8. Definition of Magnetic Equivariant K-theory
  9. Topological properties
  10. Magnetic K-theory as a cohomology theory
  11. Products and Bott periodicity
  12. Clifford Modules
  13. Coefficients of Magnetic Equivariant K-theory
  14. Twistings and degree shift
  15. Complexes of vector bundles and Thom isomorphism
  16. ...and 6 more sections

Key Result

Proposition 1.9

Every representation of a finite magnetic group is a sum of irreducible representations.

Figures (1)

  • Figure 1: $\mathbb{Z}/4$-CW decomposition of the torus $T^2=S^1\times S^1$. The 0-skeleton consists of the points $\Gamma, A, A',X$, the 1-skeleton adds the 1-cells generated by the orbits of $\gamma$ and $\sigma$, and the 2-skeleton adds the 2-cell generated by the orbit of $\Omega$. The group

Theorems & Definitions (168)

  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Example 1.4
  • Remark 1.5
  • Example 1.6
  • Definition 1.7
  • Example 1.8
  • Proposition 1.9
  • Lemma 1.10: Maschke
  • ...and 158 more