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

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.

arXiv comment6 pages, 7 figures

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• Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains
• Duality viewpoint of criticality
• Generalized Kramers-Wannier Duality from Bilinear Phase Map
• Lieb-Schultz-Mattis Theorem for 1D Quantum Magnets with Antiunitary Translation and Inversion Symmetries
• Projectification of point group symmetries with a background flux and Lieb-Schultz-Mattis theorem
• Anisotropy-induced spin parity effects