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

TL;DR

The paper develops a generic interacting-cluster framework that marries constraint-vector and connectivity-matrix formalisms to study flat-band spin liquids under inter-cluster coupling. By deriving a polynomial dispersion $\Lambda(\mathbf{q})$ of parent-cluster bands and identifying a tunable effective Fermi surface $\lambda_c$ that governs ground-state selection, it explains how half-moon patterns emerge in the equal-time structure factor and how these signatures evolve through Lifshitz-like transitions. It further extends the theory to higher-rank Coulomb fields, yielding multifold half moons and pinch-line–to–half-moon transitions that signal fracton-like gauge charges in the ground state. The framework provides design principles for flat-band engineering, unifying spin-liquid physics with higher-rank gauge theories and topological transitions, and suggests robust, tunable signatures for experimental exploration of spiral spin liquids and related fracton phenomena.

Abstract

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.

Figures (9)

• Figure 1: Illustration of the mapping between the cluster Hamiltonian (\ref{['Eq: general cluster H']}) and the connectivity matrices of Eq. (\ref{['Eq: connectivity matrix expression for H(1)']}). First, we consider a cluster at random, that is arbitrarily depicted as an hexagon here. Let us place a virtual site at its center; these central sites form the premedial lattice. Second, let us link all cluster sites $i$ to the central site $n$ (represented as curved oriented lines here). This is precisely the definition of the connectivity matrix $h^{v \leftarrow c}$ with coefficients $h^{v \leftarrow c}_{in}=\gamma_i^n$. Finally, let us link the center site $n$ with all cluster sites $i$ via the transpose of the connectivity matrix $h^{c \leftarrow v} = (h^{v \leftarrow c})^T$. The resulting matrix $H^{(1)} = h^{v \leftarrow c} h^{c \leftarrow v}$ thus pairs, within each cluster, all cluster sites two-by-two, and with the appropriate coupling constant as defined in Hamiltonian (\ref{['Eq: general cluster H']}).
• Figure 2: Schematic representation of Eq. (\ref{['Eq: connectivity matrix expression for H(2)']}). The scalar product of two constrainers associated with two neighboring clusters can be translated in term of connectivity matrices by introducing two different types of virtual bonds. The first types are the one already used for cluster Hamiltonians. These virtual bonds that link all vertices sites from a cluster to its virtual central site, are here depicted as black lines, and are associated with the connectivity matrix $h^{v \leftarrow c}$ as already discussed. The second type of virtual links are bonding the central sites from neighboring clusters together, they are depicted in red and can be encoded in the incident matrix $h^{c \leftarrow c}$ which coefficients $h^{c \leftarrow c}_{mn}$ are unity if the central sites $m$ and $n$ belong to two interacting clusters and zero otherwise.
• Figure 3: Effective Fermi-level illustration. The band structure of an interacting-cluster system can be written as a polynomial function $\Lambda(\lambda)$ of the band structure of the parent cluster system (see Eq. (\ref{['Eq: Dispersion H(2)']})). (a) Since the parent spectrum contains only positive energies, a positive value of the linear coefficient $\alpha - \eta \Xi / \Omega$ results in a simple, non-uniform rescaling of the parent band structure. (b-c) When the linear coefficient $\alpha - \eta \Xi / \Omega$ becomes negative, the polynomial ceases to be strictly increasing, and the minima of the polynomial select the intersections with the parent band structure as the new ground-state manifold. (b) Just beyond the transition, the new ground-state manifold $\mathcal{Q}$ consists of closed contours encircling the contact points of the parent system. (c) When the polynomial minimum $\lambda_c$ crosses a saddle point of the parent-system band structure, the topology of the ground-state manifold changes, corresponding to a Lifshitz transition. Note that the flat bands of the parent system (depicted in red), which are located at zero energy, remain pinned at zero energy in the interacting-cluster spectrum, since the polynomial $\Lambda(\lambda)$ has no constant term.
• Figure 4: (a) Cluster structure of the generalized checkerboard model. The plain and crossed circles depict the two sublattices, while blue, red and yellow circles respectively appear with coefficient 1, $\gamma_1$ and $\gamma_2$ in the constrainer $\bm{\mathcal{C}}_\square$ of Eq. (\ref{['Cckb']}). (b) Effective strength of the cluster interactions associated with the kernel $K_\mathbf{R}$ (\ref{['Eq: Relevant Kernel']}) for the generalized checkerboard, assuming $\gamma_2 = 0$ for clarity of the figure. The vector between two clusters is defined as $\mathbf{R}_{np}$. The strength of each cluster interaction $(n,p)$ is determined by all the combinations of pairs of sites $i$ and $j$ belonging to sublattice $\mu$, whose positions in the cluster respect $\mathbf{r}_{i,\mu} - \mathbf{r}_{j,\mu} = \mathbf{R}_{np}$. Each of these combinations is associated with the products of the weights $\gamma_{i, \mu} \gamma_{j, \mu}/\omega$, with $\omega = 1$ here. Since the two sublattices are equivalent, let us apply our reasoning on one arbitrary sublattice. One has a weight $1$ associated with vectors $\pm \mathbf{e}_1$, and $\gamma_1$ associated with vectors $\pm \mathbf{u}_1, \pm \mathbf{u}_2$. The weight of a cluster-to-cluster link is thus always a combination of terms proportional to either $1^2$, $1\times\gamma_1$ or $(\gamma_1)^2$ as depicted in blue. Because of the two possible orientations of the bond vector $\mathbf{R}_{np}$ each cluster-to-cluster link weight must be proportional to two. Note that some links weights can get contributions from two sublattices, as it is the case for the bonds linking first or second neighbors clusters.
• Figure 5: Generalized checkerboard model Figures (a,b) are obtained with constrainer parameters $\gamma_1 = -1/2, \gamma_2 =0, \alpha=1$, which fixes $\Xi = 6$ and thus $\zeta_c = -6$. The parameter $\zeta$ is thus set to $\zeta = -4 > \zeta_c$. The figure (a) presents the structure factor, showing fourfold half moons, in good agreement with figure (b) representing the weight $W_2(\mathbf{q})$ of the dispersive band and the minimum of this dispersive band depicted as a red line. Figures (c-d) are obtained with $\gamma_1 = 1, \gamma_2 = 1/3$, imposing $\Xi = 112/9$ and $\zeta_c = -112/9 \simeq -12.4$, suggesting the choice $\zeta = -10$. The structure factor (c) presents sixfold half moons, in good agreement with the weight $W_2(\mathbf{q})$ depicted on panel (d) that shows sixfold pinch points. For both cases calculations of the structure factors are performed via SCGA (see Appendix \ref{['Appendix : SCGA']}) with inverse temperature $\beta = 20$ and with the $q_i$ in $a^{-1}$ units.