Topological invariants for interacting systems: from twisted boundary condition to center-of-mass momentum

TL;DR

This paper establishes a connection between topological invariants derived from twisted boundary conditions and center-of-mass momentum states in interacting multi-particle quantum systems, validated through numerical analysis of the Aubry-André-Harper model.

Contribution

It demonstrates the equivalence of topological invariants obtained via TBC and c.m. momentum approaches in interacting systems, providing a new framework for studying multi-particle topological states.

Findings

Berry phase from TBC equals that from c.m. momentum states

Chern number is the winding number of the Berry phase

Numerical results confirm consistency between TBC and c.m. approaches

Abstract

Beyond the well-known topological band theory for single-particle systems, it is a great challenge to characterize the topological nature of interacting multi-particle quantum systems. Here, we uncover the relation between topological invariants defined through the twist boundary condition (TBC) and the center-of-mass (c.m.) momentum state in multi-particle systems. We find that the Berry phase defined through TBC can be equivalently obtained from the multi-particle Wilson loop formulated by c.m. momentum states. As the Chern number can be written as the winding of the Berry phase, we consequently prove the equivalence of Chern numbers obtained via TBC and c.m. momentum state approaches. As a proof-of-principle example, we study topological properties of the Aubry-Andr{\'e}-Harper (AAH) model. Our numerical results show that the TBC approach and c.m. approach are well consistent with each other for both many-body case and few-body case. Our work lays a concrete foundation and provides new insights for exploring multi-particle topological states.