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arXiv:2605.13606

Lieb-Schultz-Mattis theorem from gauge constraints

Bhandaru Phani Parasar

Published May 13, 2026• 0 citations
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Abstract

We construct a $\mathbb{Z}_2 \times \mathbb{Z}_2$ 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$(1)$ 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$(1)$ 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 $r$ 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.

DOI
10.48550/arxiv.2605.13606

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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 ∝cos⁡(πr) r−2/9 \propto \cos{(πr)}\,r^{-2/9}∝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.
The authors construct a Lieb-Schultz-Mattis (LSM)-type theorem applicable for abelian gauge theories without matter. This theorem is based on generalized global symmetries acting on Wilson lines. It provides strong constraints on possible phases, realized in lattice gauge theories. The generalized LSM theorem is demonstrated on the example of a U(1) gauge theory that simulates the quantum dimer model on a bipartite lattice. The authors find a mixed `t Hooft anomaly involving the generalized global symmetry at the Rokhsar-Kivelson critical point that corresponds to the LSM constraint.
The Lieb-Schultz-Mattis (LSM) theorem and its higher-dimensional extensions forbid the existence of a unique, symmetric, and gapped ground state at fractional fillings in quantum many-body systems with a conserved particle number (or spin angular momentum) and the conventional translation symmetry of Euclidean lattices. In this work, we propose a generalization of the LSM theorem to quantum many-body systems on hyperbolic lattices, i.e., regular tessellations of two-dimensional negatively curved space. By leveraging concepts from hyperbolic band theory in a many-body setting, we adapt Oshikawa's flux-threading argument to periodic hyperbolic lattices with a non-Euclidean (Fuchsian) translation symmetry and compute a lower-bound to the ground-state degeneracy as a function of filling and lattice geometry. We explore the consequences of LSM constraints for gapped phases of hyperbolic quantum matter and suggest frustrated spin models on hyperbolic analogs of the square and triangular lattices as promising platforms for realizing symmetric spin liquids in hyperbolic space.
The Lieb-Schultz-Mattis (LSM) theorem dictates that emergent low-energy states from a lattice model cannot be a trivial symmetric insulator if the filling per unit cell is not integral and if the lattice translation symmetry and particle number conservation are strictly imposed. In this paper, we compare the one-dimensional gapless states enforced by the LSM theorem and the boundaries of one-higher dimensional strong symmetry-protected topological (SPT) phases from the perspective of quantum anomalies. We first note that they can both be described by the same low-energy effective field theory with the same effective symmetry realizations on low-energy modes, wherein non-on-site lattice translation symmetry is encoded as if it were an internal symmetry. In spite of the identical form of the low-energy effective field theories, we show that the quantum anomalies of the theories play different roles in the two systems. In particular, we find that the chiral anomaly is equivalent to the LSM theorem, whereas there is another anomaly that is not related to the LSM theorem but is intrinsic to the SPT states. As an application, we extend the conventional LSM theorem to multiple-charge multiple-species problems and construct several exotic symmetric insulators. We also find that the (3+1)d chiral anomaly provides only the perturbative stability of the gaplessness local in the parameter space.
Symmetry provides powerful non-perturbative constraints in quantum many-body systems. A prominent example is the Lieb-Schultz-Mattis (LSM) anomaly -- a mixed 't Hooft anomaly between internal and translational symmetries that forbids a trivial symmetric gapped phase. In this work, we investigate lattice translation operators in systems with an LSM anomaly. We construct explicit lattice models in two and three spatial dimensions and show that, after gauging the full internal symmetry, translation becomes non-invertible and fuses into defects of the internal symmetry. The result is supported by the anomaly-inflow in view of topological field theory. Our work extends earlier one-dimensional observations to a unified higher-dimensional framework and clarifies their origin in mixed anomalies and higher-group structures, highlighting a coherent interplay between internal and crystalline symmetries.
The Lieb-Schultz-Mattis (LSM) theorem and its descendants impose strong constraints on the low-energy behavior of interacting quantum systems. In this paper, we formulate LSM-type constraints for lattice translation invariant systems with generalized U(1) symmetries which have recently appeared in the context of fracton phases: U(1) polynomial shift and subsystem symmetries. Starting with a generic interacting system with conserved dipole moment, we examine the conditions under which it supports a symmetric, gapped, and nondegenerate ground state, which we find requires that both the filling fraction and the bulk charge polarization take integer-values. Similar constraints are derived for systems with higher moment conservation laws or subsystem symmetries, in addition to lower bounds on the ground-state degeneracy when certain conditions are violated. Finally, we discuss the mapping between LSM-type constraints for subsystem symmetries and the anomalous symmetry action at boundaries of subsystem symmetric topological (SSPT) states.
The Lieb-Schultz-Mattis (LSM) theorem provides a general constraint on quantum many-body systems and plays a significant role in the Haldane gap phenomena and topological phases of matter. Here, we extend the LSM theorem to open quantum systems and establish a general theorem that restricts the steady state and spectral gap of Liouvillians based solely on symmetry. Specifically, we demonstrate that the unique gapped steady state is prohibited when translation invariance and U(1) symmetry are simultaneously present for noninteger filling numbers. As an illustrative example, we find that no dissipative gap is open in the spin-1/2 dissipative Heisenberg model, while a dissipative gap can be open in the spin-1 counterpart-an analog of the Haldane gap phenomena in open quantum systems. Furthermore, we show that the LSM constraint manifests itself in a quantum anomaly of the dissipative form factor of Liouvillians. We also find the LSM constraints due to symmetry intrinsic to open quantum systems, such as Kubo-Martin-Schwinger symmetry. Our work leads to a unified understanding of phases and phenomena in open quantum systems.