Lieb-Schultz-Mattis theorem with a local twist for general one-dimensional quantum systems

TL;DR

This paper extends the Lieb-Schultz-Mattis theorem to a local twist version applicable to a wide class of one-dimensional quantum systems with U(1) symmetry, showing that non-integer filling leads to gapless excitations and no long-range order.

Contribution

It introduces a local twist formulation of the theorem for general 1D quantum systems, broadening the scope beyond previous global symmetry assumptions.

Findings

Non-integer filling implies low-lying excitations grow with system size.

Ground states without long-range order are incompatible with a spectral gap.

The theorem applies without requiring time-reversal or inversion symmetry.

Abstract

We formulate and prove the local twist version of the Yamanaka-Oshikawa-Affleck theorem, an extension of the Lieb-Schultz-Mattis theorem, for one-dimensional systems of quantum particles or spins. We can treat almost any translationally invariant system wth global $U(1)$ symmetry. Time-reversal or inversion symmetry is not assumed. It is proved that, when the "filling factor" is not an integer, a ground state without any long-range order must be accompanied by low-lying excitations whose number grows indefinitely as the system size is increased. The result is closely related to the absence of topological order in one-dimension. The present paper is written in a self-contained manner, and does not require any knowledge of the Lieb-Schultz-Mattis and related theorems.