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arXiv:2609.17166

Numerical Study of Stability of Clean Critical Points across Aperiodic, Topological, and Uncorrelated Disorder

Shuhao Fan, Lu Liu, Wenan Guo

Published Sep 15, 2026• 0 citations
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Abstract

Using the two-dimensional Ashkin-Teller (AT) model, we compare criticality on three non-periodic lattices: the Smith-hat aperiodic tiling, Voronoi-Delaunay (VD) random triangulations, and uncorrelated diluted square lattices. The decay of the block-averaged coordination fluctuation $σ_Q$ with exponent $α$ is used to describe the connectivity disorder. The first two lattices share the same exponent $α$, which differs from that of the third. We consider the regime where the correlation-length exponent $ν<1$, where randomness is relevant according to the Harris criterion $d ν\le 2$, but should be irrelevant in cases of the Smith-hat tiling and VD triangulations, where $αν>1$, according to the Harris--Barghathi--Vojta (HBV) criterion. For the diluted lattice, we indeed find that the clean universality behavior breaks down along the entire critical line, indicating a crossover to a fixed line dominated by disorder, in line with both the Harris criterion and the HBV criterion. In contrast, both VD and Smith-hat lattices display critical exponents consistent with the clean AT universality class, as validated by a Coulomb-gas self-consistency check, violating the Harris criterion while conforming to the HBV criterion.

DOI
10.48550/arxiv.2609.17166

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Using the two-dimensional Ashkin-Teller (AT) model, we compare criticality on three non-periodic lattices: the Smith-hat aperiodic tiling, Voronoi-Delaunay (VD) random triangulations, and uncorrelated diluted square lattices. The decay of the block-averaged coordination fluctuation σQσ_QσQ​ with exponent ααα is used to describe the connectivity disorder. The first two lattices share the same exponent ααα, which differs from that of the third. We consider the regime where the correlation-length exponent ν<1ν<1ν<1, where randomness is relevant according to the Harris criterion dν≤2d ν\le 2dν≤2, but should be irrelevant in cases of the Smith-hat tiling and VD triangulations, where αν>1αν>1αν>1, according to the Harris--Barghathi--Vojta (HBV) criterion. For the diluted lattice, we indeed find that the clean universality behavior breaks down along the entire critical line, indicating a crossover to a fixed line dominated by disorder, in line with both the Harris criterion and the HBV criterion. In contrast, both VD and Smith-hat lattices display critical exponents consistent with the clean AT universality class, as validated by a Coulomb-gas self-consistency check, violating the Harris criterion while conforming to the HBV criterion.
We study the effects of topological (connectivity) disorder on phase transitions. We identify a broad class of random lattices whose disorder fluctuations decay much faster with increasing length scale than those of generic random systems, yielding a wandering exponent of ω=(d-1)/(2d) in d dimensions. The stability of clean critical points is thus governed by the criterion (d+1)ν>2 rather than the usual Harris criterion dν>2, making topological disorder less relevant than generic randomness. The Imry-Ma criterion is also modified, allowing first-order transitions to survive in all dimensions d>1. These results explain a host of puzzling violations of the original criteria for equilibrium and nonequilibrium phase transitions on random lattices. We discuss applications, and we illustrate our theory by computer simulations of random Voronoi and other lattices.
The Voronoi construction is ubiquitous across the natural sciences and engineering. In statistical mechanics, however, only its dual, the Delaunay triangulation, has been considered in the investigation of critical phenomena. In this paper we set to fill this gap by studying three prominent systems of classical statistical mechanics, the equilibrium spin-1/2 Ising model, the nonequilibrium contact process, and the conserved stochastic sandpile model on two-dimensional random Voronoi graphs. Particular motivation comes from the fact that these graphs have vertices of constant coordination number, making it possible to isolate topological effects of quenched disorder from node-intrinsic coordination number disorder. Using large-scale numerical simulations and finite-size scaling techniques, we are able to demonstrate that all three systems belong to their respective clean universality classes. Therefore, quenched disorder introduced by the randomness of the lattice is irrelevant and does not influence the character of the phase transitions. We report the critical points to considerable precision and, for the Ising model, also the first correction-to-scaling exponent.
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We show that a broad class of quantum critical points can be stable against locally correlated disorder even if they are unstable against uncorrelated disorder. Although this result seemingly contradicts the Harris criterion, it follows naturally from the absence of a random-mass term in the associated order-parameter field theory. We illustrate the general concept with explicit calculations for quantum spin-chain models. Instead of the infinite-randomness physics induced by uncorrelated disorder, we find that weak locally correlated disorder is irrelevant. For larger disorder, we find a line of critical points with unusual properties such as an increase of the entanglement entropy with the disorder strength. We also propose experimental realizations in the context of quantum magnetism and cold-atom physics.
The critical properties of the spin-1 Blume-Capel model in two dimensions is studied on Voronoi-Delaunay random lattices with quenched connectivity disorder. The system is treated by applying Monte Carlo simulations using the heat-bath update algorithm together with single histograms re-weighting techniques. We calculate the critical temperature as well as the critical exponents as a function of the crystal field $\mathrm{\ensuremath{\Delta}}$. It is found that this disordered system exhibits phase transitions of first- and second-order types that depend on the value of the crystal field. For values of $\mathrm{\ensuremath{\Delta}}\ensuremath{\le}3$, where the nearest-neighbor exchange interaction JJJ has been set to unity, the disordered system presents a second-order phase transition. The results suggest that the corresponding exponent ratio belongs to the same universality class as the regular two-dimensional ferromagnetic model. There exists a tricritical point close to ${\mathrm{\ensuremath{\Delta}}}_{t}=3.05(4)$ with different critical exponents. For ${\mathrm{\ensuremath{\Delta}}}_{t}\ensuremath{\le}\mathrm{\ensuremath{\Delta}}<3.4$ this model undergoes a first-order phase transition. Finally, for $\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}3.4$ the system is always in the paramagnetic phase.
We study the effects of uncorrelated quenched disorder to the phase diagram and continuous transitions of three-dimensional lattice Z_{2} gauge Higgs models. For this purpose, we consider two types of quenched disorder, associated with the sites and plaquettes of the cubic lattice. In both cases, for sufficiently weak disorder, the phase diagram remains similar to that of the pure system, showing two different phases (one of them being a topologically ordered phase), separated by two different continuous transition lines. However, the quenched disorder changes the universality classes of the critical behaviors along some of the transition lines. The random-plaquette disorder turns out to be relevant along the topological Z_{2} gauge transition line, so the critical behaviors belong to the different random-plaquette Z_{2} gauge (RPZ_{2}G) universality class with length-scale exponent ν=ν_{rp}≈0.82; on the other hand, it turns out to be irrelevant along the other Ising^{×} transition line (a variant of the Ising transitions with a gauge-dependent order parameter), leaving unchanged its asymptotic critical behaviors with ν=ν_{I}≈0.63. The random-site disorder leads to a substantially different scenario: it destabilizes the Ising^{×} critical behaviors of the pure model, changing them into those of the randomly dilute Ising^{×} (RDI^{×}) universality class with ν=ν_{RDI}≈0.68, while the critical behaviors along the other Z_{2} gauge topological transition line remains stable, with ν=ν_{I}≈0.63.