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arXiv:2505.00927 · Physical review. B./Physical review. B

Projectification of point group symmetries with a background flux and the Lieb-Schultz-Mattis theorem

Yasuhiro Tada, Masaki Oshikawa

Published Feb 23, 2026• 3 citations
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Abstract

We discuss the Lieb-Schultz-Mattis (LSM) theorem in two-dimensional spin systems with on-site ${\mathrm U}(1)\rtimes {\mathbb Z}_2$ spin rotation symmetry and point group $C_{2v}$ symmetry about a site. We ``twist" the point group symmetry by introducing a small uniform U(1) flux to obtain a projective symmetry, similarly to the familiar magnetic translation symmetry. The LSM theorem is proved in presence of the flux and then it is demonstrated that the theorem holds also for the flux-free system. Besides, the uniform flux enables us to show the LSM theorem for the time-reversal symmetry and the site-centered $C_2$-rotation symmetry.

DOI
10.1103/8gq5-d954

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We discuss the Lieb-Schultz-Mattis (LSM) theorem in two-dimensional spin systems with on-site U(1)⋊Z2{\mathrm U}(1)\rtimes {\mathbb Z}_2U(1)⋊Z2​ spin rotation symmetry and point group C2vC_{2v}C2v​ symmetry about a site. We ``twist" the point group symmetry by introducing a small uniform U(1) flux to obtain a projective symmetry, similarly to the familiar magnetic translation symmetry. The LSM theorem is proved in presence of the flux and then it is demonstrated that the theorem holds also for the flux-free system. Besides, the uniform flux enables us to show the LSM theorem for the time-reversal symmetry and the site-centered C2C_2C2​-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 $(d\ensuremath{-}1)$-dimensional plane, we find that any ddd-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.
We prove the Lieb-Schultz-Mattis theorem on the energy spectrum of a general two- or three-dimensional quantum many-body system with the U(1) particle number conservation and translation symmetry. Especially, it is demonstrated that the theorem holds in a system with long-range interactions. To this end, we introduce approximate magnetic translation symmetry under the total magnetic flux Φ=2π instead of the exact translation symmetry, and explicitly construct low energy variational states. The energy spectrum at Φ=2π is shown to agree with that at Φ=0 in the thermodynamic limit, which concludes the Lieb-Schultz-Mattis theorem.
In this study, we consider one-dimensional (1D) quantum spin systems with translation and discrete symmetries (spin reversal, space inversion, and time reversal symmetries). By combining the continuous U(1) symmetry with the discrete symmetries and using the extended Lieb–Schultz–Mattis (LSM) theorem [E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 (1961); K. Nomura, J. Morishige, and T. Isoyama, J. Phys. A 48, 375001 (2015)], we investigate the relation between the ground states, energy spectra, and symmetries. For half-integer spin cases, we generalize the dimer and Néel concepts using the discrete symmetries, and we can reconcile the LSM theorem with the dimer or Néel states, since there was a subtle dilemma. Furthermore, a part of discrete symmetries is enough to classify possible phases. Thus we can deepen our understanding of the relation between the LSM theorem and discrete symmetries.
Z2×Z2Z_2\times Z_2Z2​×Z2​ and SU(2) symmetry with Lieb-Schultz-Mattis theorem
In this study, we consider one-dimension (1D) quantum spin systems with the translation and discrete symmetries (spin reversal, space inversion and time reversal symmetries). By combining the continuous U(1) symmetry with the discrete symmetries and using the extended Lieb-Schultz-Mattis theorem \cite{Lieb-Schultz-Mattis-1961}\cite{Nomura-Morishige-Isoyama-2015}, we investigate the relation between the ground states, energy spectra and symmetries. For half-integer spin cases, we generalize the dimer and Neel concepts using the discrete symmetries, and we can reconcile the LSM theorem with the dimer or Neel states, since there was a subtle dilemma. Furthermore, a part of discrete symmetries is enough to classify possible phases. Thus we can deepen our understanding of the relation between the LSM theorem and the discrete symmetries.
We construct a Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2Z2​×Z2​ gauge theory coupled to matter on a one-dimensional chain, aiming to study the ground-state physics in the Gauss law subspace. We show that the theory in the Gauss law subspace has a U(1)(1)(1) symmetry whose generator commutes with lattice translations, but anticommutes with the lattice reflection operator. This leads to a Lieb-Schultz-Mattis (LSM) theorem that always rules out a trivial gapped ground state in the Gauss law subspace, if the hamiltonian is invariant under translations and reflection. Any point in the parameter space must realize either a spontaneously symmetry broken (SSB) ground state, or a gapless ground state. Imposing the Gauss law is pivotal for the existence of the U(1)(1)(1) symmetry, and hence of the LSM theorem. We thus demonstrate a novel mechanism to obtain an LSM-type theorem, wherein the symmetry responsible for the theorem originates from the kinematic constraints of a gauge theory. We identify a point in the parameter space at which the system is gapless. At the gapless point, the excitations admit a description in terms of free Dirac fermions with a constraint on the total fermion number. The asymptotic behavior of the two-point correlation function of the simplest local gauge-invariant quantity at the gapless point is found to be $ \propto \cos{(πr)}\,r^{-2/9}$, where rrr is the lattice separation between the two points. This model is also a natural platform to study phase diagram topological defects residing in families of SSB phases.
Lieb-Schultz-Mattis (LSM) theorem and its various generalizations provide powerful guidance towards the search for novel phases of matter. Here, the authors propose and prove a generalized LSM theorem suitable for 2+1D lattice models of interacting bosons or spins, with both magnetic flux and fractional spin in the unit cell. One outcome of this theorem is that, under certain conditions, the gapped ground states preserving all symmetries must be a nontrivial symmetry-protected topological (SPT) phase. Such symmetry-enforced SPTs display a dyonic character in that they associate charge with symmetry flux, which is demonstrated using quasiexactly solvable models constructed by decorating quantum dimer models.