Lieb-Schultz-Mattis constraints from stratified anomalies of modulated symmetries

We introduce stratified symmetry operators and stratified anomalies in quantum lattice systems as generalizations of onsite symmetry operators and onsite projective representations. A stratified symmetry operator is a symmetry operator that factorizes into mutually independent subsystem symmetry operators; its stratified anomaly is defined as the collection of anomalies associated with these subsystem operators. We develop a cellular chain complex formalism for stratified anomalies of internal symmetries and show that, in the presence of crystalline symmetries, they give rise to Lieb-Schultz-Mattis (LSM) constraints. This includes LSM anomalies and SPT-LSM theorems. We apply this framework to modulated $G$ symmetries, which are symmetries whose total symmetry group is ${G_\mathrm{tot} = G \rtimes G_\mathrm{s}}$, with $G_\mathrm{s}$ the crystalline symmetry group. Notably, a nonzero stratified anomaly within a fundamental domain of $G_\mathrm{s}$ (e.g., a unit cell) does not always imply an LSM anomaly for modulated symmetries. Instead, the existence of an LSM anomaly also depends on how $G_\mathrm{s}$ acts on $G$. When $G_\mathrm{s}$ is the lattice translation group, we find an explicit criterion for when a stratified anomaly causes an LSM anomaly, and classify LSM anomalies using homology groups of $G_\mathrm{s}$-invariant cellular chains. We illustrate this through examples of exponential and dipole symmetries with stratified anomalies, both in ${(1+1)}$D and ${(2+1)}$D, and construct a stabilizer code model of a modulated SPT subject to an SPT-LSM theorem.

💡 Research Summary

The paper introduces “stratified symmetry operators” and “stratified anomalies” as a broad generalization of onsite symmetry actions and their projective representations in quantum lattice systems. A stratified symmetry operator factorizes the global symmetry into a product of mutually independent subsystem symmetry operators, each acting on a distinct region (or stratum) of the lattice. The collection of anomalies associated with these subsystems—captured by projective phases in each dimension—constitutes a stratified anomaly. The authors formalize this structure using a cellular chain complex: each d‑cell contributes a chain group, and the anomalies appear as co‑cycles in this complex. An equivalence relation is defined by stacking with ancilla systems that admit symmetric short‑range‑entangled (SPT) phases and by local unitary changes of basis, yielding equivalence classes that represent physical anomalies.

The work then incorporates crystalline (spatial) symmetries G_s, focusing on the group extension 1 → G → G_tot → G_s → 1. In the split‑extension case G_tot = G ⋊ G_s, the spatial symmetry acts non‑trivially on the internal symmetry via a homomorphism ρ: G_s → Aut(G). Such “modulated” symmetries are the central object of study. The authors derive a precise criterion for when a non‑zero stratified anomaly defined on a fundamental domain (e.g., a unit cell) leads to a Lieb‑Schultz‑Mattis (LSM) anomaly: the anomaly must be invariant under the action of ρ on the G_s‑invariant cellular chains. When G_s consists solely of lattice translations, the classification of LSM anomalies reduces to the homology groups H_n(C_*^{G_s‑inv}) of the G_s‑invariant chain complex. If the criterion fails, the stratified anomaly instead yields an SPT‑LSM theorem, i.e., an obstruction to a trivial SPT rather than a conventional LSM constraint.

Concrete examples illustrate the theory. In (1+1) D, Z_N dipole symmetries always produce LSM anomalies from their stratified anomalies, while exponential (U(1) ⋊ ℤ) symmetries may or may not, depending on ρ; the latter case leads to an SPT‑LSM theorem. The authors construct a stabilizer‑code model that realizes an exponential symmetry SPT subject to such a theorem. In (2+1) D, Z_N dipole symmetries exhibit a richer behavior: their 1‑stratum anomalies can produce LSM anomalies only when compatible with the translation action. The paper also connects stratified anomalies to the Atiyah‑Hirzebruch spectral sequence, showing how the formalism fits into the equivariant generalized homology classification of anomalies.

Overall, the work provides a systematic, mathematically rigorous framework for understanding how spatially modulated internal symmetries generate Lieb‑Schultz‑Mattis constraints, extending the traditional LSM paradigm to a broad class of lattice models with subsystem and higher‑form symmetries. It opens new avenues for designing exotic SPT phases and for classifying anomalies in systems where internal and crystalline symmetries intertwine.