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arXiv:2505.04282 · Physical review. E

Critical behavior analysis of pure dipolar triangular lattice via equilibrium and nonequilibrium Monte Carlo simulations

S. Ismailzadeh, M. D. Niry

Published Nov 7, 2025• 0 citations
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Abstract

Magnetic thin films and two-dimensional (2D) arrays of magnetic nanoparticles exhibit unique physical properties that make them valuable for a wide range of technological applications. In such systems, dipolar interactions play a crucial role in determining their physical behavior. However, due to the anisotropic and long-range nature of dipolar interactions, conventional Monte Carlo (MC) methods face challenges in investigating these systems near criticality. In this study, we examine the critical behavior of a triangular lattice of XY dipoles using the optimized Tomita MC algorithm tailored for dipolar interactions. We employ two independent computational approaches to estimate the critical temperature and exponents: equilibrium MC simulations with histogram reweighting and the nonequilibrium relaxation method. Notably, both approaches demonstrate that this XY dipolar system might be in a new universality class very close to the 2D Ising universality class.

DOI
10.1103/51b7-xhs4

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Magnetic thin films and two-dimensional (2D) arrays of magnetic nanoparticles exhibit unique physical properties that make them valuable for a wide range of technological applications. In such systems, dipolar interactions play a crucial role in determining their physical behavior. However, due to the anisotropic and long-range nature of dipolar interactions, conventional Monte Carlo (MC) methods face challenges in investigating these systems near criticality. In this study, we examine the critical behavior of a triangular lattice of XY dipoles using the optimized Tomita MC algorithm tailored for dipolar interactions. We employ two independent computational approaches to estimate the critical temperature and exponents: equilibrium MC simulations with histogram reweighting and the nonequilibrium relaxation method. Notably, both approaches demonstrate that this XY dipolar system might be in a new universality class very close to the 2D Ising universality class.
Motivated by a recent experiment on a square-lattice Rydberg atom array realizing a long-range dipolar XY model [C. Chen et al., Nature (London) 616, 691 (2023)], we numerically study the model's equilibrium properties. We obtain the phase diagram, critical properties, entropies, variance of the magnetization, and site-resolved correlation functions. We consider both ferromagnetic and antiferromagnetic interactions and apply quantum Monte Carlo and pseudo-Majorana functional renormalization group techniques, generalizing the latter to a U(1)U(1)U(1) symmetric setting. Our simulations perform extensive thermometry in dipolar Rydberg atom arrays and establish conditions for adiabaticity and thermodynamic equilibrium. On the ferromagnetic side of the experiment, we determine the entropy per particle $S/N\ensuremath{\approx}0.5$, close to the one at the critical temperature, Sc/N=0.585(15){S}_{c}/N=0.585(15)Sc​/N=0.585(15). The simulations suggest the presence of an out-of-equilibrium plateau at large distances in the correlation function, thus motivating future studies on the nonequilibrium dynamics of the system.
Critical properties of the Ising model on a stacked triangular lattice, with antiferromagnetic first- and second-neighbor in-plane interactions, are studied by extensive histogram Monte Carlo simulations. The results, in conjunction with the recently determined phase diagram, strongly suggest that the transition from the period-3 ordered state to the paramagnetic phase remains in the XY universality class. This conclusion is in contrast with a previous suggestion of mean-field tricritical behavior.
Critical properties of the fully frustrated XY models on square and triangular lattices are investigated by means of the nonequilibrium relaxation (NER) method. We examine the validity of the conclusion on the universality class of the chiral transition previously obtained by the NER method, in which that belongs to a non-Ising-type one. To clarify it, we analyze the NER of fluctuations for longer Monte Carlo steps on lager lattices than those performed previously. The calculation is made at the chiral transition temperatures estimated carefully by the use of recently improved dynamical scaling analysis. The result indicates that the asymptotic behavior of the time-dependent exponents, which should converge to the critical exponents, show the same tendency as those obtained previously in shorter times. We also apply the improved dynamical scaling analysis to the estimation of Kosterlitz-Thouless (KT) transitions and confirm the existence of the double transitions for the chiral and KT phases with more reliable estimations.
We study the thermal transition between the paramagnetic and antiferromagnetic phases of interacting magnetic dipoles on square lattices in two dimensions. By Monte Carlo simulations, we find strong long-range order in the low temperature phase and a critical behavior that is, as in the fourfold anisotropic XY model, nonuniversal, in that the single-site quadrupolar anisotropy h4{h}_{4}h4​ is a relevant variable. However, the critical indices differ markedly from the ones for the anisotropic XY model. They vary from Ising type, within errors, for h4=0{h}_{4}=0h4​=0, to a critical behavior in the ${h}_{4}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit that is nearly as abrupt as a first-order transition.
The phase transitions (PTs) and critical properties of the antiferromagnetic Ising model on a layered (stacked) triangular lattice have been studied by the Monte Carlo method using a replica algorithm with allowance for the next-nearest-neighbor interactions. The character of PTs is analyzed using the histogram technique and the method of Binder cumulants. It is established that the transition from the disordered to paramagnetic phase in the adopted model is a second-order PT. Static critical exponents of the heat capacity (α), susceptibility (γ), order parameter (β), and correlation radius (ν) and the Fischer exponent η are calculated using the finite-size scaling theory. It is shown that (i) the antiferromagnetic Ising model on a layered triangular lattice belongs to the XY universality class of critical behavior and (ii) allowance for the intralayer interactions of next-nearest neighbors in the adopted model leads to a change in the universality class of critical behavior.
In this work we have used extensive Monte Carlo simulations and finite size scaling theory to study the phase transition in the dipolar planar rotator model (dPRM), also known as dipolar XY model. The true long-range character of the dipolar interactions was taken into account by using the Ewald summation technique. Our results for the critical exponents do not fit those from known universality classes. We observed that the specific heat is apparently non-divergent and the critical exponents are ν = 1.277(2), β = 0.2065(4) and γ = 2.218(5). The critical temperature was found to be T(c) = 1.201(1). Our results are clearly distinct from those of a recent renormalization group study from Maier and Schwabl (2004 Phys. Rev. B 70 134430) and agrees with the results from a previous study of the anisotropic Heisenberg model with dipolar interactions in a bilayer system using a cut-off in the dipolar interactions (Mól and Costa 2009 Phys. Rev. B 79 054404).