Unwinding Fermionic SPT Phases: Supersymmetry Extension

TL;DR

This paper demonstrates how 1+1D fermionic SPT phases can be trivialized by extending symmetries and Hilbert space, revealing supersymmetry structures that remove certain fermionic anomalies and enable new classifications.

Contribution

It introduces a method to unwind fermionic SPT phases through symmetry extension, incorporating supersymmetry, and provides explicit examples and potential generalizations.

Findings

Fermionic SPTs can be trivialized via symmetry extension.

Supersymmetry can be realized on the boundary of fermionic SPTs.

Certain fermionic anomalies can be removed using supersymmetry extensions.

Abstract

We show how 1+1-dimensional fermionic symmetry-protected topological states (SPTs, i.e. nontrivial short-range entangled gapped phases of quantum matter whose boundary exhibits 't Hooft anomaly and whose bulk cannot be deformed into a trivial tensor product state under finite-depth local unitary transformations only in the presence of global symmetries), indeed can be unwound to a trivial state by enlarging the Hilbert space via adding extra degrees of freedom and suitably extending the global symmetries. The extended projective global symmetry on the boundary can become supersymmetric in a specific sense, i.e., it contains group elements that do not commute with the fermion number parity $(-1)^F$, while the anti-unitary time-reversal symmetry becomes fractionalized. This also means we can uplift and remove certain exotic fermionic anomalies (e.g., "parity" anomaly in time-reversal or reflection symmetry) via appropriate supersymmetry extensions in terms of group extensions. We work out explicit examples for multi-layers of 1+1d Majorana fermion chains, then comment on models with Sachdev-Ye-Kitaev (SYK) interactions, intrinsic fermionic gapless SPTs protected by supersymmetry, and generalizations to higher spacetime dimensions via a cobordism theory.