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Ising Criticality on Sparse Power-Law Shortcut Networks in Three Dimensions: Crossover Analysis and Open Questions

Štěpán Sekanina

Published Apr 24, 2026• 0 citations

Abstract

We study the Ising phase transition on a three-dimensional cubic lattice augmented with sparse, distance-dependent shortcuts whose density decays as P(r) proportional to r^(-2). This network ensemble, characterized by geometric embedding, sparse additive quenched shortcuts, and power-law distance dependence, falls outside the scope of existing results for both uniform small-world networks (Barrat-Weigt, Herrero) and dense long-range Ising models (Fisher-Ma-Nickel, Sak). We present partial analytical results. The effective propagator exhibits three regimes: a Wilson-Fisher intermediate regime (standard Laplacian dominant), a Levy IR regime (wormhole term dominant), and a biharmonic UV regime, with propagator crossover at approximately 125 lattice units. A scaling estimate suggests the nonlocal operator is only marginally relevant at Wilson-Fisher, implying a potentially very slow RG flow. A mean free path analysis yields a finite effective signal velocity of approximately 2.4 times the local propagation speed, with the exponent alpha = 2 lying at the apparent boundary for finite-speed propagation on sparse shortcut networks. We propose a concrete Monte Carlo simulation protocol (approximately 500 CPU-hours) using Wolff cluster algorithm and finite-size scaling with Binder cumulant analysis to definitively determine the universality class. The key observable is U4*, which unambiguously distinguishes 3D Ising (U4* approximately 0.61) from mean-field (U4* approximately 0.47) universality. Control simulations on pure 3D lattice and uniform shortcut networks are specified. The problem is relevant to discrete quantum gravity models with holographic connectivity, network science, and the theory of phase transitions on complex geometries. All analytical results are clearly labeled as scaling estimates or conjectures where appropriate, and five explicit unresolved issues are identified.

DOI: 10.5281/zenodo.19730581

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We study the Ising phase transition on a three-dimensional cubic lattice augmented with sparse, distance-dependent shortcuts whose density decays as P(r) proportional to r^(-2). This network ensemble, characterized by geometric embedding, sparse additive quenched shortcuts, and power-law distance dependence, falls outside the scope of existing results for both uniform small-world networks (Barrat-Weigt, Herrero) and dense long-range Ising models (Fisher-Ma-Nickel, Sak). We present partial analytical results. The effective propagator exhibits three regimes: a Wilson-Fisher intermediate regime (standard Laplacian dominant), a Levy IR regime (wormhole term dominant), and a biharmonic UV regime, with propagator crossover at approximately 125 lattice units. A scaling estimate suggests the nonlocal operator is only marginally relevant at Wilson-Fisher, implying a potentially very slow RG flow. A mean free path analysis yields a finite effective signal velocity of approximately 2.4 times the local propagation speed, with the exponent alpha = 2 lying at the apparent boundary for finite-speed propagation on sparse shortcut networks. We propose a concrete Monte Carlo simulation protocol (approximately 500 CPU-hours) using Wolff cluster algorithm and finite-size scaling with Binder cumulant analysis to definitively determine the universality class. The key observable is U4*, which unambiguously distinguishes 3D Ising (U4* approximately 0.61) from mean-field (U4* approximately 0.47) universality. Control simulations on pure 3D lattice and uniform shortcut networks are specified. The problem is relevant to discrete quantum gravity models with holographic connectivity, network science, and the theory of phase transitions on complex geometries. All analytical results are clearly labeled as scaling estimates or conjectures where appropriate, and five explicit unresolved issues are identified.

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The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.

A Monte Carlo method is applied to simulate the static critical behavior of a cubic-lattice 3D Ising model for systems with quenched disorder. Numerical results are presented for the spin concentrations of p = 1.0, 0.95, 0.9, 0.8, 0.6 on L × L × L lattices with L = 20–60 under periodic boundary conditions. The critical temperature is determined by the Binder cumulant method. A finite-size scaling technique is used to calculate the static critical exponents α, β, γ, and ν (for specific heat, susceptibility, magnetization, and correlation length, respectively) in the range of p under study. Universality classes of critical behavior are discussed for three-dimensional diluted systems.

We study the critical behavior of the Ising model in three dimensions on a lattice with site disorder by using Monte Carlo simulations. The disorder is either uncorrelated or long-range correlated with correlation function that decays according to a power law ${r}^{\ensuremath{-}a}$. We derive the critical exponent of the correlation length $\ensuremath{\nu}$ and the confluent correction exponent $\ensuremath{\omega}$ in dependence of

a

by combining different concentrations of defects $0.05\ensuremath{\le}{p}_{d}\ensuremath{\le}0.4$ into one global fit ansatz and applying finite-size scaling techniques. We simulate and study a wide range of different correlation exponents $1.5\ensuremath{\le}a\ensuremath{\le}3.5$ as well as the uncorrelated case $a=\ensuremath{\infty}$ and are able to provide a global picture not yet known from previous works. Additionally, we perform a dedicated analysis of our long-range correlated disorder ensembles and provide estimates for the critical temperatures of the system in dependence of the correlation exponent

a

and the concentration of defects

{p}_{d}

. We compare our results to known results from other works and to the conjecture of Weinrib and Halperin: $\ensuremath{\nu}=2/a$, and discuss the occurring deviations.

We study the critical behavior of a generalized icosahedral model on the simple cubic lattice. The field variable of the icosahedral model might take 1 of 12 vectors of unit length which are given by the normalized vertices of the icosahedron as value. Similar to the Blume-Capel model, where in addition to $\ensuremath{-}1$ and 1, as in the Ising model, the spin might take the value 0, we add in the generalized model (0,0,0) as allowed value. There is a parameter

D

that controls the density of these voids. For a certain range of

D

, the model undergoes a second-order phase transition. On the critical line,

\text{O}(3)

symmetry emerges. Furthermore, we demonstrate that within this range, similar to the Blume-Capel model on the simple cubic lattice, there is a value of

D

, where leading corrections to scaling vanish. We perform Monte Carlo simulations for lattices of a linear size up to

L=400

by using a hybrid of local Metropolis and cluster updates. The motivation to study this particular model is mainly of technical nature. Less memory and CPU time are needed than for a model with

\text{O}(3)

symmetry at the microscopic level. As the result of a finite-size scaling analysis we obtain $\ensuremath{\nu}=0.711\phantom{\rule{0.16em}{0ex}}64(10)$, $\ensuremath{\eta}=0.037\phantom{\rule{0.16em}{0ex}}84(5)$, and $\ensuremath{\omega}=0.759(2)$ for the critical exponents of the three-dimensional Heisenberg universality class. The estimate of the irrelevant renormalization group eigenvalue that is related with the breaking the

\text{O}(3)

symmetry is ${y}_{\mathrm{ico}}=\ensuremath{-}2.19(2)$.