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

We propose a geometric approach to the 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 or rotational Lieb-Schultz-Mattis theorems without lattice translation symmetry imposed. The traditional Lieb-Schultz-Mattis theorems with translations also fit in the proposed framework.

Automated summary

We propose a geometric approach to the 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, we find that doubly degenerate spectra exist in d-dimensional inversion-symmetric spin systems with half-integer spins at the inversion-symmetric point. We also observe a similar degeneracy in rotation-symmetric generalized spin models with projective representations at the rotation center. These degeneracies imply the absence of a unique symmetric gapped ground state connected to product states in the original untwisted systems without lattice translation symmetry imposed.