Interaction effects on 1D fermionic symmetry protected topological phases

TL;DR

This paper investigates how interactions affect the classification of 1D fermionic symmetry protected topological phases, revealing that weak interactions reduce the classification from Z to Z_4, with all phases realizable by free fermions.

Contribution

It demonstrates that interactions reduce the classification of 1D fermionic SPT phases from Z to Z_4 and shows all these phases can be realized with free fermions, extending understanding of topological phase stability.

Findings

Weak interactions reduce classification from Z to Z_4.

All four phases in the reduced classification can be realized with free fermions.

Classification depends on the symmetry group, with Z_n leading to Z_4 or Z_2 classifications.

Abstract

In free fermion systems with given symmetry and dimension, the possible topological phases are labeled by elements of only three types of Abelian groups, Z_1, Z_2, or Z. For example non-interacting 1D fermionic superconducting phases with S_z spin rotation and time-reversal symmetries are classified by Z. We show that with weak interactions, this classification reduces to Z_4. Using group cohomology, one can additionally show that there are only four distinct phases for such 1D superconductors even with strong interactions. Comparing their projective representations, we find all these four symmetry protected topological phases can be realized with free fermions. Further, we show that 1D fermionic superconducting phases with Z_n discrete S_z spin rotation and time-reversal symmetries are classified by Z_4 when n=even and Z_2 when n=odd; again, all these strongly interacting topological phases can be realized by non-interacting fermions. Our approach can be applied to systems with other symmetries to see which 1D topological phases can be realized by free fermions.