Generalized Thouless Pumps in 1+1-dimensional Interacting Fermionic Systems

TL;DR

This paper explores the generalization of Thouless pumps for fermion parity in 1+1D interacting fermionic systems, using matrix product states to characterize and establish topological invariants of these pumps.

Contribution

It introduces a framework for analyzing generalized Thouless pumps in 1+1D interacting fermionic systems using fermionic matrix product states and defines new topological invariants.

Findings

Constructed non-trivial pumps in trivial and non-trivial phases.

Proved stability of pumps against interactions.

Defined topological invariants consistent with existing results.

Abstract

The Thouless pump is a phenomenon in which $\mathrm{U}(1)$ charges are pumped from an edge of a fermionic system to another edge. The Thouless pump has been generalized in various dimensions and for various charges. In this paper, we investigate the generalized Thouless pumps of fermion parity in both trivial and non-trivial phases of $1+1$-dimensional interacting fermionic short range entangled (SRE) states. For this purpose, we use matrix product states (MPSs). MPSs describe many-body systems in $1+1$ dimensions, and can characterize SRE states algebraically. We prove fundamental theorems for fermionic MPSs (fMPSs) and use them to investigate the generalized Thouless pumps. We construct non-trivial pumps in both the trivial and non-trivial phases and we show the stability of the pumps against interactions. Furthermore, we define topological invariants for the generalized Thouless pumps in terms of fMPSs and establish consistency with existing results. These are invariants of the family of SRE states that are not captured by the higher dimensional Berry curvature. We also argue a relation between the topological invariants of the generalized Thouless pump and the twist of the $K$-theory in the Donovan-Karoubi formulation.