Fermionic Matrix Product States and One-Dimensional Short-Range Entangled Phases with Anti-Unitary Symmetries

TL;DR

This paper extends Matrix Product States to classify one-dimensional fermionic systems with complex symmetries, providing a framework for understanding their entangled phases and symmetry actions.

Contribution

It introduces a formalism for fermionic MPS with anti-unitary and orientation-reversing symmetries, classifies phases via invariants, and establishes a group law for stacking phases.

Findings

Classified fermionic short-range entangled phases using three invariants.

Connected fermionic MPS at RG fixed points to equivariant algebras.

Derived a group law for stacking fermionic phases with symmetry.

Abstract

We extend the formalism of Matrix Product States (MPS) to describe one-dimensional gapped systems of fermions with both unitary and anti-unitary symmetries. Additionally, systems with orientation-reversing spatial symmetries are considered. The short-ranged entangled phases of such systems are classified by three invariants, which characterize the projective action of the symmetry on edge states. We give interpretations of these invariants as properties of states on the closed chain. The relationship between fermionic MPS systems at an RG fixed point and equivariant algebras is exploited to derive a group law for the stacking of fermionic phases protected by general fermionic symmetry groups.