Degeneracy and Consistency Condition of Berry phases: Gap Closing under the Twist

TL;DR

This paper establishes a consistency condition for Berry phases in many-body systems that predicts gap closings under gauge twists, linking to the Lieb-Schultz-Mattis theorem and symmetry considerations.

Contribution

It introduces a new degeneracy and gap closing condition based on Berry phases and symmetries, extending previous theorems to broader cases.

Findings

Predicts inevitable gap closing in half-integer spin chains.

Relates Berry phase conditions to the Lieb-Schultz-Mattis theorem.

Extends analysis to degenerate multiplets and fermionic systems.

Abstract

We have discussed a consistency condition of Berry phases defined by a local gauge twist and spatial symmetries of the many body system. It imposes a non trivial gap closing condition under the gauge twist in both finite- and infinite-size systems. It also implies a necessary condition for the gapped and unique ground state. As for the simplest case, it predicts an inevitable gap closing in the Heisenberg chain of half integer spins. Its relation to the Lieb-Schultz-Mattis theorem is discussed based on the symmetries of the twisted Hamiltonian. The discussion is also extended to the (approximately) degenerated multiplet and fermion cases. It restricts the number of the states in the low energy cluster of the spectrum by the filling of the fermions. Constraints by the reflection symmetry are also discussed.