Geometric approach to Lieb-Schultz-Mattis theorem without translation symmetry under inversion or rotation symmetry

TL;DR

This paper introduces a geometric approach to the Lieb-Schultz-Mattis theorem applicable to systems with inversion or rotation symmetry but lacking translation symmetry, revealing degeneracies and constraints on gapped ground states.

Contribution

It extends the Lieb-Schultz-Mattis theorem to non-translation symmetric systems using a geometric symmetry-twisting method, broadening its applicability.

Findings

Inversion-symmetric systems with half-integer spins have doubly degenerate spectra under symmetry-twisting.

Rotation-symmetric models with projective representations exhibit similar degeneracies.

Unique symmetric gapped ground states are forbidden in the systems studied.

Abstract

We propose a geometric {approach to Lieb-Schultz-Mattis theorem for} quantum many-body systems with discrete spin-rotation symmetries and lattice inversion or rotation symmetry, but without translation symmetry assumed. Under symmetry-twisting on a $(d-1)$-dimensional plane, we find that any $d$-dimensional inversion-symmetric spin system possesses a doubly degenerate spectrum when it hosts a half-integer spin at the inversion-symmetric point. We also show that any rotation-symmetric generalized spin model with a projective representation at the rotation center has a similar degeneracy under symmetry-twisting. We argue that these degeneracies imply that {a unique symmetric gapped ground state that is smoothly connected to product states} is forbidden in the original untwisted systems -- generalized inversional/rotational Lieb-Schultz-Mattis theorems without lattice translation symmetry imposed. The traditional Lieb-Schultz-Mattis theorems with translations also fit in the proposed framework.