Interacting-cluster spin liquids with robust flat bands evolving into higher-rank half-moon phases and topological Lifshitz transitions

Classical spin liquids are disordered magnetic phases, governed by local constraints that often give rise to flat-band ground states. When constraints take the form of a zero-divergence field within a cluster of spins, the spin liquid is often described by an emergent Coulomb gauge theory. Here we introduce an interaction $η$ between these clusters of spins which compete with the zero-divergence field. Using a framework embracing both the connectivity matrices of graph theory and the topology of band structures, we develop a generic theory of interacting-cluster Hamiltonians. We show how flat bands remain at zero energy up to finite interaction $η$, until a dispersive band becomes negative, stabilizing a spiral spin liquid with a hypersurface of ground-state manifold in reciprocal space. This hypersurface can be interpreted as an effective Fermi surface in the spectrum of the parent system, acting as a tunable energy selector despite the absence of particle filling. This effective Fermi surface serves as a mold for the apparition of the half-moon patterns in the equal-time structure factor. Our generic approach enables to extend the notion of half moons to the perturbation of higher-rank Coulomb fields and pinch-line spin liquids. In particular, multi-fold half moons appear when unconventional gauge charges, such as potential fractons, are stabilized in the ground state. Finally, half-moon phases can be tuned across the equivalent of a Lifshitz transition, when the hypersurface manifold changes topology.

💡 Research Summary

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The authors develop a comprehensive theory of interacting‑cluster classical spin liquids, focusing on how a cluster‑cluster coupling η competes with the zero‑divergence (Gauss‑law) constraints that normally give rise to flat‑band ground states. Starting from a generic cluster Hamiltonian H = ∑ₙ|Cₙ|², where each cluster “constrainer’’ Cₙ is a weighted sum of the Heisenberg spins within the cluster, they rewrite the problem in Fourier space using constraint vectors Lₓ(q) and a connectivity matrix hᵥ←c(q). Within the Luttinger‑Tisza approximation, any spin mode orthogonal to all constraint vectors has zero energy, leading to a number of flat bands equal to the difference between the number of sublattices (nₛ) and the number of distinct cluster types (n_c). This flat‑band count can also be read directly from the kernel of the connectivity matrix, linking the problem to graph‑theoretic concepts.

The key innovation is the addition of an inter‑cluster interaction term η∑⟨n,m⟩Cₙ·C_m. When η = 0 the system remains in a Coulomb phase characterized by pinch‑point singularities in the static structure factor. As η is increased, the dispersive bands shift upward in energy. The authors identify a critical value η_c at which the lowest dispersive band touches the flat‑band manifold and then moves below it. At η > η_c the ground‑state manifold is no longer the flat‑band subspace but a hypersurface in reciprocal space where the formerly dispersive modes become energetically favorable. This hypersurface plays the role of an “effective Fermi surface’’: although there are no fermions, the selected energy level acts as a tunable Fermi level, determining which wave‑vectors belong to the ground state.

The emergence of this hypersurface dramatically reshapes the equal‑time structure factor. Pinch points disappear and are replaced by half‑moon (half‑moon) intensity patterns that trace the contour of the effective Fermi surface. The authors show that the radius and orientation of the half‑moon arcs can be continuously tuned by η, providing a direct experimental signature of the spiral spin‑liquid phase that replaces the Coulomb phase.

Beyond the simplest rank‑1 (vector) gauge fields, the paper extends the analysis to higher‑rank U(1) gauge theories. When the constraint‑vector space shrinks at a band‑touching point of order 2n, the low‑energy description involves a rank‑n tensor electric field obeying a higher‑order Gauss law. Such systems exhibit pinch‑line singularities rather than point‑like pinch points. Introducing η in these higher‑rank models generates multi‑fold half‑moon patterns: several concentric or overlapping half‑moon arcs appear, reflecting the presence of unconventional gauge charges such as fractons. Thus, the half‑moon phenomenology becomes a diagnostic for exotic gauge sectors.

Finally, the authors discuss topological Lifshitz transitions of the hypersurface. As η is varied, the topology of the effective Fermi surface can change (e.g., from a closed sphere to a torus or to a multiply‑connected surface). These changes are accompanied by abrupt modifications of the band structure and the structure factor, analogous to electronic Lifshitz transitions but occurring in a purely bosonic spin system without particle filling. The transition points are identified by tracking topological invariants (e.g., Berry‑phase integrals or Euler characteristics) of the hypersurface. Numerical simulations on several two‑ and three‑dimensional lattices (kagome, pyrochlore, octagonal kagome, bipyramidal kagome, etc.) confirm the analytical predictions and illustrate the sequence of phases: flat‑band Coulomb spin liquid → spiral spin liquid with single half‑moon → multi‑fold half‑moon (higher‑rank) → Lifshitz‑transitioned spiral phases.

In summary, the paper provides: (1) a graph‑theoretic and band‑topology framework for classifying flat‑band cluster spin liquids; (2) a clear mechanism—controlled by η—for destabilizing the flat‑band manifold and creating an interaction‑tuned effective Fermi surface; (3) a unified description of half‑moon patterns as fingerprints of spiral spin liquids, including their extension to higher‑rank gauge fields and fracton charges; and (4) a demonstration that these systems can undergo topological Lifshitz transitions driven solely by the strength of inter‑cluster coupling. The work bridges concepts from condensed‑matter band topology, gauge‑field theory, and classical frustrated magnetism, and suggests concrete experimental routes (e.g., neutron scattering on kagome or pyrochlore materials with tunable exchange pathways) to observe the predicted half‑moon signatures and Lifshitz transitions.