Dimensional hierarchy of fermionic interacting topological phases

TL;DR

This paper introduces a dimensional reduction method to understand how interactions can trivialize fermionic topological phases, reducing their classification from an integer to a finite cyclic group.

Contribution

It provides a general framework for classifying interacting fermionic topological phases via dimensional reduction and explicitly constructs interactions that trivialize certain phases.

Findings

Reduces non-interacting $\

Establishes a condition for symmetry-preserving interactions to trivialize phases.

Derives a $\

Abstract

We present a dimensional reduction argument to derive the classification reduction of fermionic symmetry protected topological phases in the presence of interactions. The dimensional reduction proceeds by relating the topological character of a $d$-dimensional system to the number of zero-energy bound states localized at zero-dimensional topological defects present at its surface. This correspondence leads to a general condition for symmetry preserving interactions that render the system topologically trivial, and allows us to explicitly write a quartic interaction to this end. Our reduction shows that all phases with topological invariant smaller than $n$ are topologically distinct, thereby reducing the non-interacting $\mathbb{Z}$ classification to $\mathbb{Z}_n$.