Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle

Arun Debray

Abstract

Freed-Hopkins give a mathematical ansatz for classifying gapped invertible phases of matter with a spatial symmetry in terms of Borel-equivariant generalized homology. We propose a slight generalization of this ansatz to account for cases where the symmetry type mixes nontrivially with the spatial symmetry, such as crystalline phases with spin-1/2 fermions. From this ansatz, we prove as a theorem a "fermionic crystalline equivalence principle," as predicted in the physics literature. Using this and the Adams spectral sequence, we compute classifications of some classes of phases with a point group symmetry; in cases where these phases have been studied by other methods, our results agree with the literature.

Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle

Abstract

Freed-Hopkins give a mathematical ansatz for classifying gapped invertible phases of matter with a spatial symmetry in terms of Borel-equivariant generalized homology. We propose a slight generalization of this ansatz to account for cases where the symmetry type mixes nontrivially with the spatial symmetry, such as crystalline phases with spin-1/2 fermions. From this ansatz, we prove as a theorem a "fermionic crystalline equivalence principle," as predicted in the physics literature. Using this and the Adams spectral sequence, we compute classifications of some classes of phases with a point group symmetry; in cases where these phases have been studied by other methods, our results agree with the literature.

Paper Structure

This paper contains 76 sections, 111 theorems, 195 equations, 24 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Reader's guide to the different sections
  3. Phases on a $G$-space: the general principle
  4. Invertible phases on a space
  5. Invertible phases on a $G$-space
  6. Mixing internal and crystalline symmetries
  7. The fermionic crystalline equivalence principle
  8. Case $H = \mathrm{Spin}$
  9. Case $H = {\mathrm{Spin}^c}$
  10. Putting it together
  11. Computations in examples: summary of results and some generalities
  12. Some interesting phases to study
  13. Methods of computation
  14. The Adams spectral sequence
  15. The Atiyah-Hirzebruch spectral sequence
  16. ...and 61 more sections

Key Result

Theorem 1

Let $A_4$ act on $\mathbb R^3$ as the orientation-preserving symmetries of a tetrahedron. Then $\mathit{Ph}_0^{A_4}(\mathbb R^3; f)$ vanishes, where $f$ is the local system of symmetry types for either spinless or spin-$1/2$ fermions in both Altland-Zirnbauer classes D and A.

Figures (24)

  • Figure 1: Left: the $\mathcal{A}(1)$-module structure on $\widetilde{H}^*((BD_{2n})^{2-V_\lambda};\mathbb Z/2)$ in low degrees, when $n\equiv 2\bmod 4$. Here $\alpha\coloneqq x^4y+y^5$. The submodule pictured here contains all elements of degree at most $5$. Right: the $E_2$-page of the corresponding Adams spectral sequence computing $\mathit{ko}$-theory.
  • Figure 2: Left: the low-degree mod $2$ cohomology of $(BD_{2n})^{2-V_\lambda}$ over $\mathcal{A}(1)$, $n\equiv 0\bmod 4$. This summand contains all elements in degrees $5$ and below. The dashed line indicates that the $\mathbb Z/2^r$ Bockstein maps $U y$ to $U w$, which we need in \ref{['first_dih_differential']}. Right: the $E_2$-page of the Adams spectral sequence computing $\widetilde{\mathit{ko}}_*((BD_{2n})^{2-V_\lambda})_2^\wedge$. See \ref{['first_dih_differential']} for how to address the differential in topological degree $2$ and \ref{['D0mod4_nodiff']} to show the differential in topological degree $5$ vanishes.
  • Figure 3: The map $(BD_{2n})^{2-V_\lambda}\to \Sigma^2\mathit{MTO}_2$ induces a map between the Adams spectral sequences computing their $\mathit{ko}$-theory groups. We use this in \ref{['D0mod4_nodiff']} to show the pictured $d_2$ vanishes, as the square of pink arrows in the above figure is commutative. The right-hand side of this figure, which displays $\mathop{\mathrm{Ext}}\nolimits(\widetilde{H}^*(\Sigma^2\mathit{MTO}_2;\mathbb Z/2))$, is adapted from Campbell Cam17.
  • Figure 4: Left: the $\mathcal{A}(1)$-module structure on $\widetilde{H}^*(M_n;\mathbb Z/2)$ in low degrees. The pictured summand contains all elements in degrees $4$ and below. Right: the Ext of this module, which is the $E_2$-page of the Adams spectral sequence converging to $\widetilde{\mathit{ko}}_*(M_n)$. See the proof of \ref{['spinless_D0mod4_thm']} for more information.
  • Figure 5: Left: the $\mathcal{E}(1)$-module structure on $\widetilde{H}^*((BD_{2n})^{2-V_\lambda};\mathbb Z/2)$, $n\equiv 2\bmod 4$, in low degrees. The pictured submodule contains all elements in degrees $5$ and below. Right: the Adams $E_2$-page computing $\widetilde{\mathit{ku}}_*((BD_{2n})^{2-V_\lambda})$.
  • ...and 19 more figures

Theorems & Definitions (233)

  • Theorem
  • Definition 1.1
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.8
  • Remark 1.11
  • Definition 1.12
  • Definition 1.13
  • ...and 223 more