Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems

TL;DR

This work develops a comprehensive topological framework for Lieb–Schultz–Mattis type constraints in quantum spin systems with spatial and internal symmetries. It recasts the problem in terms of anomalous textures and defect networks, and then translates these into computable invariants via equivariant homology, enabling anomaly matching between microscopic textures and possible gapped symmetric ground states. The authors provide explicit descent-based calculations for translations and point groups, prove a rigorous result in two dimensions under an in-cohomology approximation, and show that lattice homotopy suffices to capture traditional LSM constraints in the tested quantum-magnet settings. They also discuss SPT–LSM theorems, bulk–boundary interpretations, and directions for extending the framework to fermions and beyond-cohomology phases, highlighting practical computational paths and open questions.

Abstract

The Lieb-Schultz-Mattis (LSM) theorem states that a spin system with translation and spin rotation symmetry and half-integer spin per unit cell does not admit a gapped symmetric ground state lacking fractionalized excitations. That is, the ground state must be gapless, spontaneously break a symmetry, or be a gapped spin liquid. Thus, such systems are natural spin-liquid candidates if no ordering is found. In this work, we give a much more general criterion that determines when an LSM-type theorem holds in a spin system. For example, we consider quantum magnets with arbitrary space group symmetry and/or spin-orbit coupling. Our criterion is intimately connected to recent work on the general classification of topological phases with spatial symmetries and also allows for the computation of an "anomaly" associated with the existence of an LSM theorem. Moreover, our framework is also general enough to encompass recent works on "SPT-LSM" theorems where the system admits a gapped symmetric ground state without fractionalized excitations, but such a ground state must still be non-trivial in the sense of symmetry-protected topological (SPT) phases.

Figures (10)

• Figure 1: An example of a fusion move. The overall symmetry group is $G = G_{\mathrm{space}} \times G_{\mathrm{int}}$, for some internal symmetry group $G_{\mathrm{int}}$, and where $G_{\mathrm{space}}$ is generated by a three-fold rotation and a reflection (i.e. $G_{\mathrm{space}} = D_3$). The site symmetry group $G_s$ for each site on the left-hand side is $\mathbb{Z}_2 \times G_{\mathrm{int}}$. Under lattice homotopy, one can fuse a $G$ orbit comprising three of these points into a single point whose site symmetry group is enlarged to $G$. We need a fusion rule giving a map $\mathcal{H}^2(\mathbb{Z}_2 \times G_{\mathrm{int}}, \mathrm{U}(1)) \to \mathcal{H}^2(G, \mathrm{U}(1)))$ to describe the impact of such a fusion on the anomalous texture.
• Figure 2: A cell decomposition of the plane. In an invertible-defect network, 2-cells carry a 2-dimensional topological phase, 1-cells carry invertible gapped interfaces between topological phases, and 0-cells carry invertible gapped junctions between interfaces.
• Figure 3: A degree-0 anomalous defect network in a bosonic system in two dimensions. All the 2-cells and 1-cells carry gapped invertible phases and interfaces respectively, but 0-cells (points) carry emergent degenerate modes transforming projectively under their respective isotropy groups $G_s$.
• Figure 4: An anomalous texture can appear at the boundary of a 1-skeletal defect network in one higher dimension.
• Figure 5: In a spin system which has $SO(3)$ spin rotation symmetry and translation symmetry, a system with two spin-half particles per unit cell gives rise to an anomalous texture which can be cancelled by a collection of Haldane chains.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4