The multirange Ising conjecture: does adding multiplicative interaction ranges on $\mathbb{Z}^d$ drive the critical coupling to that of a higher-dimensional Ising model?

For the ferromagnetic Ising model on $\mathbb{Z}^d$ ($d\geq 2$) with interactions along the principal axes at ranges $1, k_1, k_1k_2, \dots, k_1\cdots k_n$ (each range an integer multiple of the previous one) and uniform coupling, simulations by do Amaral, de Lima, Dickman and Atman [Phys. Rev. E 101, 052138 (2020)] show that for $d=2$ with $n=1$ (ranges $1, m$) the inverse critical temperature $K_c(m)$ decreases monotonically toward the Ising value on $\mathbb{Z}^4$, and with $n=2$ (ranges $1, m, m^2$) toward the value on $\mathbb{Z}^6$. The authors conjecture, by analogy with a proven theorem for multirange percolation, that this holds in full generality. The open problem is to prove or refute this multirange Ising conjecture. The numerics further raise associated characterization questions — the conjectured monotonicity of $K_c$ in each multiplicative parameter, the rate and universality of the finite-range corrections, and the theoretical reason convergence appears to require integer-multiple ranges — which remain open alongside the conjecture but are not part of it. The conjecture remains open.

Background

The Ising model places spins $\sigma_i\in\{-1,+1\}$ on the vertices of a lattice with configurational energy $H=-J\sum_{(i,j)}\sigma_i\sigma_j$ summed over interacting pairs; for ferromagnetic coupling $J>0$ it orders below a critical temperature $T_c$, more conveniently written as the inverse critical coupling $K_c=J/(k_B T_c)$. $K_c$ is known exactly only on the square lattice ($d=2$; Onsager), while in $d\geq 3$ only high-precision numerical estimates exist (e.g. $K_c(\mathbb{Z}^4)\approx 0.1496947$, from Lundow and Markström's four-dimensional simulations).

The multirange construction modifies the interaction graph rather than the spin variables. Starting from $\mathbb{Z}^d$, one adds bonds of lengths $m_1=k_1$, $m_2=k_1k_2$, ..., $m_n=k_1\cdots k_n$ (all $k_i\geq 2$ integers), in each of the $2d$ principal lattice directions, so every spin interacts with its nearest neighbors and, with the same coupling $J$, with the spins at each of the added distances. For Bernoulli percolation (bonds open independently of each other, in both the bond and site variants) on this graph, de Lima, Sanchis and Silva [Stoch. Process. Appl. 121, 2043 (2011)] proved rigorously that, for $d\geq 2$ and any $n\geq 1$, the critical probability converges as every $k_i\to\infty$ to the critical probability of percolation on $\mathbb{Z}^{d(n+1)}$; they conjectured the convergence to be monotone and nonincreasing in each $k_i$, which numerical work supported for $d=2$, $n=1,2$ (do Amaral et al., Physica A 535, 122383 (2019)). The mechanism has an exactly solvable precedent in one dimension: Turban [J. Phys. A 49, 355002 (2016); J. Phys. A 50, 205001 (2017)] showed that a one-dimensional Ising or Potts chain with pair couplings at ranges $1$ and $m$ is equivalent, by a change of spin variables, to the two-dimensional nearest-neighbor model on a rectangular lattice of width $m$ (the two couplings acting as the couplings in the two lattice directions), so that in the double limit of infinite chain length and $m\to\infty$ the two-dimensional critical behavior is reproduced exactly.

do Amaral, de Lima, Dickman and Atman [Phys. Rev. E 101, 052138 (2020)] simulated the corresponding Ising model on the square lattice with the Wolff cluster algorithm (a Monte Carlo method that flips whole clusters of aligned, correlated spins at each step). With two ranges ($n=1$, lengths $1$ and $m$, $m\in\{2,5,8,10,13,16,19\}$, system sizes $768\leq L\leq 2048$), the finite-size-estimated $K_c(m)$, obtained from maxima of the magnetic susceptibility (the variance of the total magnetization) and of the derivative of the Binder cumulant (a dimensionless ratio of magnetization moments whose derivative at criticality diverges with system size), decreases monotonically in $m$ and extrapolates to $K_c(\mathbb{Z}^4)$; fits of the form $K_c(m)=a\,m^b+c$ give $c\approx 0.14965$ (consistent with $0.1496947$) and $b\approx -2.07$. With three ranges ($n=2$, lengths $1,m,m^2$, $m\in\{2,\dots,6\}$, system sizes $1280\leq L\leq 2560$), the extrapolated limit is consistent with $K_c(\mathbb{Z}^6)\approx 0.09229$ with $b\approx -3.25$. When the largest range is fixed (at $u=15$) and the intermediate one varies over values that need not divide it, the effective critical coupling varies irregularly and non-monotonically, mirroring percolation. The authors state the general conjecture that, as in percolation, the critical coupling of the multirange Ising model on $\mathbb{Z}^d$ with $n$ multiplicative added ranges converges to that of the Ising model on $\mathbb{Z}^{d(n+1)}$ — but for interacting spins no proof exists, the percolation proof relying on independence arguments unavailable here. Beyond the bare limit statement, the numerics raise three characterization questions: monotonicity of $K_c$ in each $k_i$; the rate and universality of the finite-range corrections (are the fitted exponents universal, or dependent on $d$, $n$, or the observable used?); and a theoretical explanation of why convergence appears to require each range to be an integer multiple of the previous one.

Problem Statement

Resolve the multirange Ising conjecture posed by do Amaral, de Lima, Dickman and Atman (2020): for the ferromagnetic Ising model with Hamiltonian $H=-J\sum_{(i,j)\in E_G}\sigma_i\sigma_j$ on the graph $G$ obtained from $\mathbb{Z}^d$ ($d\geq 2$) by adding, along all principal lattice directions, bonds of lengths $m_1=k_1$, $m_2=k_1k_2$, ..., $m_n=k_1\cdots k_n$ ($k_i\geq 2$ integers, uniform coupling $J$ on all bonds), prove or refute that the inverse critical temperature converges, as every $k_i\to\infty$, to the inverse critical temperature of the Ising model on the hypercubic lattice of dimension $d(n+1)$:

$$\lim_{k_1,\dots,k_n\to\infty} K_c(k_1,\dots,k_n) \;=\; K_c\!\left(\mathbb{Z}^{d(n+1)}\right).$$

A complete answer establishes (or disproves) this correspondence in the stated generality. The associated characterization questions raised by the numerics — (a) whether $K_c$ is monotone (nonincreasing) in each multiplicative parameter $k_i$; (b) the rate and universality of the finite-range corrections, i.e. whether the empirically fitted power-law decay $K_c(m)-K_c^{\infty}\sim a\,m^{b}$, where $K_c^{\infty}$ denotes the limiting critical coupling (with $b\approx -2.07$ for $n=1$ and $b\approx -3.25$ for $n=2$ in $d=2$), has universal exponents or ones depending on $d$, $n$, or the observable; and (c) the theoretical characterization of admissible range choices — an explanation of why each range must be an integer multiple of the previous one for clean convergence, and of the irregular, non-monotonic behavior observed when intermediate ranges are not divisors of longer ones — remain open alongside the conjecture, but are not required components of an answer to it and do not narrow it. The authors also raise the broader question whether the same multirange-to-higher-dimensional correspondence between number of interaction ranges and effective dimension holds for other models with local interactions exhibiting phase transitions, such as the $q$-state Potts model; that is a separate outlook question, not part of the objective stated here.

References

1. Charles S. do Amaral, Bernardo N. B. de Lima, Ronald Dickman, A. P. F. Atman, "Multirange Ising model on the square lattice", Physical Review E 101, 052138 (2020). DOI: 10.1103/PhysRevE.101.052138; arXiv:1910.01115. https://arxiv.org/abs/1910.01115
2. Bernardo N. B. de Lima, Rémy Sanchis, Roger W. C. Silva, "Critical point and percolation probability in a long range site percolation model on $\mathbb{Z}^d$", Stochastic Processes and their Applications 121, 2043–2048 (2011). DOI: 10.1016/j.spa.2011.05.009; arXiv:1105.4558. https://arxiv.org/abs/1105.4558
3. C. S. do Amaral, M. Schnabel, B. N. B. de Lima, A. P. F. Atman, "Multi-range percolation model on the square lattice", Physica A: Statistical Mechanics and its Applications 535, 122383 (2019). DOI: 10.1016/j.physa.2019.122383. https://doi.org/10.1016/j.physa.2019.122383
4. Loïc Turban, "One-dimensional Ising model with multispin interactions", Journal of Physics A: Mathematical and Theoretical 49, 355002 (2016). DOI: 10.1088/1751-8113/49/35/355002; arXiv:1605.05199. https://arxiv.org/abs/1605.05199
5. Loïc Turban, "One-dimensional q-state Potts model with multi-site interactions", Journal of Physics A: Mathematical and Theoretical 50, 205001 (2017). DOI: 10.1088/1751-8121/aa6ad1; arXiv:1701.09058. https://arxiv.org/abs/1701.09058
6. Bernardo N. B. de Lima, Leonardo T. Rolla, Daniel Valesin, "Monotonicity and phase diagram for multirange percolation on oriented trees", Random Structures & Algorithms 55, 160–172 (2018). DOI: 10.1002/rsa.20805; arXiv:1702.03841. https://arxiv.org/abs/1702.03841
7. Bernardo N. B. de Lima, Réka Szabó, Daniel Valesin, "Multirange percolation on oriented trees: Critical curve and limit behavior", Random Structures & Algorithms 62, 519–541 (2022). DOI: 10.1002/rsa.21104; arXiv:2103.05316. https://arxiv.org/abs/2103.05316
8. Olivier Couronné, "Characterization of the threshold for multi-range percolation on oriented trees" (2023). arXiv:2307.01554. https://arxiv.org/abs/2307.01554
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10. P. H. Lundow, K. Markström, "Critical behavior of the Ising model on the four-dimensional cubic lattice", Physical Review E 80, 031104 (2009). DOI: 10.1103/PhysRevE.80.031104. https://doi.org/10.1103/PhysRevE.80.031104