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Symmetry protected topological phases in non-interacting fermion systems

Xiao-Gang Wen

TL;DR

This work provides a unified, $K$-theory based framework for classifying gapped, symmetry-protected topological phases of non-interacting fermions across all allowed full symmetry groups $G_f$ that include $U(1)$, time-reversal, charge conjugation, and fermion parity. By employing Kitaev’s $K$-theory and the associated classifying spaces $C_p$ and $R_p$, the authors derive explicit $\,\pi_0$ classifications for complex and real fermion classes, extend them to translation-symmetric systems, and analyze topological defects via $\pi_{d-1}$ of these spaces. The paper shows how the full symmetry group determines the phase structure, demonstrates Bott periodicity, and provides comprehensive tables linking symmetry groups to topological classes and defect classifications; it also emphasizes that all 10 classes can be realized in electron systems. These results yield a dimension-independent, structurally unified description of non-interacting fermion topological phases, with clear implications for boundary modes, defects, and material realizations. Overall, the framework generalizes prior classifications and offers a robust lens for predicting and interpreting topological phenomena in solid-state systems.

Abstract

Symmetry protected topological (SPT) phases are gapped quantum phases with a certain symmetry, which can all be smoothly connected to the same trivial product state if we break the symmetry. For non-interacting fermion systems with time reversal (T), charge conjugation (C), and/or U(1) (N) symmetries, the total symmetry group can depend on the relations between those symmetry operations, such as T N T^{-1}= N or T N T^{-1}= -N. As a result, the SPT phases of those fermion systems with different symmetry groups have different classifications. In this paper, we use Kitaev's K-theory approach to classify the gapped free fermion phases for those possible symmetry groups. In particular, we can view the U(1) as a spin rotation. We find that superconductors with the S_z spin rotation symmetry are classified by Z in even dimensions, while superconductors with the time reversal plus the S_z spin rotation symmetries are classified by Z in odd dimensions. We show that all 10 classes of gapped free fermion phases can be realized by electron systems with certain symmetries. We also point out that to properly describe the symmetry of a fermionic system, we need to specify its full symmetry group that includes the fermion number parity transformation (-)^N. The full symmetry group is actually a projective symmetry group.

Symmetry protected topological phases in non-interacting fermion systems

TL;DR

This work provides a unified, -theory based framework for classifying gapped, symmetry-protected topological phases of non-interacting fermions across all allowed full symmetry groups that include , time-reversal, charge conjugation, and fermion parity. By employing Kitaev’s -theory and the associated classifying spaces and , the authors derive explicit classifications for complex and real fermion classes, extend them to translation-symmetric systems, and analyze topological defects via of these spaces. The paper shows how the full symmetry group determines the phase structure, demonstrates Bott periodicity, and provides comprehensive tables linking symmetry groups to topological classes and defect classifications; it also emphasizes that all 10 classes can be realized in electron systems. These results yield a dimension-independent, structurally unified description of non-interacting fermion topological phases, with clear implications for boundary modes, defects, and material realizations. Overall, the framework generalizes prior classifications and offers a robust lens for predicting and interpreting topological phenomena in solid-state systems.

Abstract

Symmetry protected topological (SPT) phases are gapped quantum phases with a certain symmetry, which can all be smoothly connected to the same trivial product state if we break the symmetry. For non-interacting fermion systems with time reversal (T), charge conjugation (C), and/or U(1) (N) symmetries, the total symmetry group can depend on the relations between those symmetry operations, such as T N T^{-1}= N or T N T^{-1}= -N. As a result, the SPT phases of those fermion systems with different symmetry groups have different classifications. In this paper, we use Kitaev's K-theory approach to classify the gapped free fermion phases for those possible symmetry groups. In particular, we can view the U(1) as a spin rotation. We find that superconductors with the S_z spin rotation symmetry are classified by Z in even dimensions, while superconductors with the time reversal plus the S_z spin rotation symmetries are classified by Z in odd dimensions. We show that all 10 classes of gapped free fermion phases can be realized by electron systems with certain symmetries. We also point out that to properly describe the symmetry of a fermionic system, we need to specify its full symmetry group that includes the fermion number parity transformation (-)^N. The full symmetry group is actually a projective symmetry group.

Paper Structure

This paper contains 14 sections, 88 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Gapped free fermion phases -- the complex classes
  3. The $d=0$ case
  4. The properties of classifying spaces
  5. The $d\neq 0$ cases
  6. Gapped free fermion phases -- the real classes
  7. The $d=0$ case -- the symmetry groups
  8. The $d=0$ case -- the classifying spaces
  9. The properties of classifying spaces
  10. The $d\neq 0$ cases
  11. A general discussion
  12. Classification with translation symmetry
  13. Defects in $d$-dimensional gapped free fermion phases with symmetry $G_f$
  14. Summery