Percolation and criticality of systems with competing interactions on Bethe lattices: limitations and potential strengths of cluster schemes

TL;DR

This work formalizes a generalized FK--CK cluster framework for frustrated systems and solves three Bethe-lattice models (isotropic SALR, ANNNI, and RBIM) with cavity methods to assess whether cluster-based sampling can weaken critical slowing down. It shows that, due to negative bond weights, constructive FK--CK–type clusters cannot be sampled usefully in frustrated regimes, even as they correctly reproduce spin correlations and Ising-like critical universality on the Bethe lattice. The study also demonstrates that, for Ising-like transitions, FK--CK and α-parameter cluster percolation thresholds coincide with Tc within numerical precision, while convergence deteriorates near Lifshitz points and multicritical Nishimori points. These results place strong limits on constructive cluster schemes for frustrated systems, though they also open avenues for alternative cluster-generation methods, including learning-based approaches, to exploit correlation structures without relying on positive bond probabilities. Overall, the paper advances understanding of why cluster schemes struggle with frustration and clarifies the precise percolation–criticality relationships in mean-field-like Bethe-lattice settings.

Abstract

The random clusters introduced by Fortuin and Kasteleyn (FK) and analyzed by Coniglio and Klein (CK) for Ising and related models have led first Swendsen and Wang and then Wolff to formulate remarkably efficient Markov chain Monte Carlo sampling schemes that weaken the critical slowing down. In frustrated models, however, no standard way to produce a comparable gain at small frustration -- let alone efficiently sample the large frustration regime -- has yet been identified. In order to understand why formulating appropriate cluster criteria for frustrated models has thus far been elusive, we here study minimal short-range attractive and long-range repulsive as well as spin-glass models on Bethe lattices. Using a generalization of the CK approach and the cavity-field method, the appropriateness and limitations of the FK--CK type clusters are identified. We find that a standard, constructive cluster scheme is then inoperable, and that the frustration range over which generalized FK--CK clusters are even definable is finite. These results demonstrate the futility of seeking constructive cluster schemes for frustrated systems but leaves open the possibility that alternate approaches could be devised.

Figures (13)

• Figure 1: Chain of four Ising spins with ferromagnetic interactions, $J_{ij}=J>0$, under periodic boundary conditions, in its minimal energy spin configuration $(s_t, s_r, s_b, s_l) = (\uparrow, \uparrow, \uparrow, \uparrow)$. All possible clusters are shown, with zero to four links (solid red lines), thus identifying spins as being either part (filled circle) or not (unfilled circle) of a cluster. Cluster multiplicities are 1, 4, 6, 4, and 1, respectively, and FK--CK cluster weights $W_\mathrm{FK}$ (center) are expressed for bonding probability $\alpha=e^{-2\beta J}\in(0,1]$.
• Figure 2: Configurations with up (black) and down (white) spins corresponding to the various cavity fields of the isotropic SALR model on a Bethe lattice with connectivity $c+1=5$. $\eta_\mathrm{cav}(s, s')$ is the (local) configuration probability for the current site $s$ and the backward (cavity) site $s'$. The upper labels $E, \, F, \, O, \, R$ equal $\eta_\mathrm{cav}(s, s')$ in each of the four cases and are used as short-hand notation for the latter in the text. Here and in what follows we will denote with an up arrow, $\uparrow \ $, the spins pointing up and with downarrow, $\downarrow \ $, the spins pointing down.
• Figure 3: Free energy (right axis, red) and leading eigenvalue (left axis, blue) of the isotropic SARL model for $\kappa=0.22$ on the $c+1=3$ Bethe lattice. The hysteresis of the heating (solid line) and cooling (dashed line) curves hints at the presence of a (weakly) first-order paramagnetic-to-ferromagnetic phase transition -- as suggested in Ref. charbonneau2021solution. The end of the metastability range of these curves, $T_{\mathrm{heat}}=0.06977$ and $T_{\mathrm{cool}}=0.06137$ (dashed vertical lines), coincides with $\lambda_\mathrm{max} = 1$ (horizontal dotted line) for the complete expression in Eq. \ref{['eq:lsa_matrix']} (solid blue line) and its homogeneous reduction in Eq. \ref{['eq:homogeneouslambdamax']} (dashed blue line), respectively. These results bound the thermodynamic transition temperature, $T_c=0.06560$, where the free energy curves cross.
• Figure 4: (a) Probability that spins $i$ and $j$ are parallel and part of a same cluster, $\langle\gamma_{ij}^\parallel\rangle$, for the case $d=1$, as determined from the generalized FK--CK cluster scheme (see Eqs. \ref{['eq:perc_1d_1']}--\ref{['eq:perc_1d_9']} and Appendix \ref{['sec:appendix_tm']}) for $\kappa = 0.1$ and $\beta = 1, 1.5, 2$, and $4$ (colored lines, from bottom to top). The decay of the numerical results is well described by an exponential from, $\langle\gamma^\parallel_{ij}\rangle=e^{-\lvert i-j\rvert/\xi}$ (dashed black lines) with fitted correlation length $\xi$. (b) The correlation length results from $\langle\gamma_{ij}^\parallel\rangle$ (markers) match those from $\langle s_i s_j\rangle$ (lines) for $\kappa = 0.1, 0$ and $-0.1$ (from bottom to top) in Ref. zheng2022weakening. The lengths correspond at all temperatures and are consistent with the scaling $\xi=e^{2\beta J(1-2\kappa)}/2$, here extrapolated to $\beta\rightarrow0$ (dashed lines).
• Figure 5: Configurations with up (black) and down (white) spins corresponding to the cavity fields of ANNNI models defined in (a) non-axial and (b) axial directions on a Bethe lattice with $c+1=4$.