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Weak topological phases in the presence of interactions

Omar Antolín Camarena, Arun Debray, Cameron Krulewski, Natalia Pacheco-Tallaj, Daniel Sheinbaum, Luuk Stehouwer

Abstract

We investigate the stability of weak symmetry-protected topological phases (SPTs) in the presence of short-range interactions, focusing on the tenfold way classification. Using Atiyah's Real $\mathit{KR}$-theory and Anderson-dualized bordism, we classify free and interacting weak phases across all Altland-Zirnbauer symmetry classes in low dimensions. Extending the free-to-interacting map of Freed-Hopkins, we mathematically compute how the behavior of free weak SPTs changes when interactions are introduced as well as predict intrinsically-interacting weak phases in certain classes. Our mathematical techniques involve T-duality and the James splitting of the torus. Our results provide a mathematical framework for understanding the persistence of weak SPTs under interactions, with potential implications for experimental and theoretical studies of these phases.

Weak topological phases in the presence of interactions

Abstract

We investigate the stability of weak symmetry-protected topological phases (SPTs) in the presence of short-range interactions, focusing on the tenfold way classification. Using Atiyah's Real -theory and Anderson-dualized bordism, we classify free and interacting weak phases across all Altland-Zirnbauer symmetry classes in low dimensions. Extending the free-to-interacting map of Freed-Hopkins, we mathematically compute how the behavior of free weak SPTs changes when interactions are introduced as well as predict intrinsically-interacting weak phases in certain classes. Our mathematical techniques involve T-duality and the James splitting of the torus. Our results provide a mathematical framework for understanding the persistence of weak SPTs under interactions, with potential implications for experimental and theoretical studies of these phases.

Paper Structure

This paper contains 22 sections, 32 theorems, 88 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Ansatz for the Weak Free-to-Interacting Map
  3. Fermionic symmetry groups
  4. K-theory classifications of free fermion phases
  5. Bordism classifications of interacting phases
  6. The classification of reflection-positive invertible field theories
  7. Spacetime symmetry groups for the tenfold way
  8. What changes for weak phases?
  9. Freed-Hopkins' free-to-interacting map for strong phases
  10. ABS Orientation
  11. Anderson Self-Duality of $K$-Theory
  12. Free-to-Interacting Maps
  13. The free-to-interacting map constrains the spectrum of SPT phases
  14. T-duality
  15. Splitting the generalized cohomology of tori
  16. ...and 7 more sections

Key Result

Theorem 12

The group of $A$-symmetric free SPT phases is isomorphic to $\mathit{KO}_{2}(A)$.

Theorems & Definitions (56)

  • Example 3
  • Definition 11
  • Theorem 12: neutralluuk
  • Remark 13
  • Example 14
  • Corollary 19
  • Remark 20
  • Definition 22
  • Definition 23
  • Theorem 25: Freed-Hopkins freed_reflection_2021, Grady Gra23
  • ...and 46 more