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arXiv:2605.11573

A Monte Carlo Study of the Dipolar Universality Class in Three Dimensions

Akira Matsumoto, Yu Nakayama, Toshiki Onagi, Slava Rychkov

Published May 12, 2026• 0 citations
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Abstract

The dipolar universality class describes the phase transition in 3D ferromagnets with strong dipolar interactions, as first discussed by Aharony and Fisher in the 1970s. While this universality class has been studied theoretically using renormalization group methods, as well as experimentally, little is known about it from Monte Carlo simulations. In this paper, we aim to bridge this gap. We introduce a lattice model that faithfully implements the transverse constraint on the order parameter. We introduce a Markov Chain Monte Carlo algorithm which involves a combination of local Metropolis updates preserving the constraint, and a global update of the zero mode. We perform simulations on cubic lattices up to volume $48\times 48 \times 48$. We observe a continuous phase transition between the disordered and ordered phases. We obtain estimates of the critical exponent $ν$ and the critical Binder ratio, and compare them with results from other techniques. We also study whether rotational invariance emerges at the critical point, and discuss the influence of the toroidal geometry on this question.

DOI
10.48550/arxiv.2605.11573

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The dipolar universality class describes the phase transition in 3D ferromagnets with strong dipolar interactions, as first discussed by Aharony and Fisher in the 1970s. While this universality class has been studied theoretically using renormalization group methods, as well as experimentally, little is known about it from Monte Carlo simulations. In this paper, we aim to bridge this gap. We introduce a lattice model that faithfully implements the transverse constraint on the order parameter. We introduce a Markov Chain Monte Carlo algorithm which involves a combination of local Metropolis updates preserving the constraint, and a global update of the zero mode. We perform simulations on cubic lattices up to volume 48×48×4848\times 48 \times 4848×48×48. We observe a continuous phase transition between the disordered and ordered phases. We obtain estimates of the critical exponent ννν and the critical Binder ratio, and compare them with results from other techniques. We also study whether rotational invariance emerges at the critical point, and discuss the influence of the toroidal geometry on this question.
Aharony and Fisher showed that non-local dipolar effects in magnetism destabilize the Heisenberg fixed point in real ferromagnets, leading to a new fixed point, called the dipolar fixed point. The non-perturbative nature of the new fixed point, however, has not been uncovered for many decades. Inspired by the recent understanding that the dipolar fixed point is scale-invariant but not conformal invariant, we perform the Monte Carlo simulation of the local Heisenberg-dipolar model on the lattice of 40340^3403 by introducing the local cost function parameterized by a parameter λλλ and study its critical exponents, which should become identical to the dipolar fixed point of Aharony and Fisher in the infinite coupling limit λ=∞λ= \inftyλ=∞. We find that the critical exponents become noticeably different from those of the Heisenberg fixed point for a finite coupling constant λ=8λ=8λ=8 (e.g. ν=0.601(2)(−2+0)ν=0.601(2)(^{+0}_{-2})ν=0.601(2)(−2+0​) in the local Heisenberg-dipolar model while ν=0.712(1)(−0+3)ν=0.712(1)(^{+3}_{-0})ν=0.712(1)(−0+3​) in the Heisenberg model), and the spin correlation function has a feature that it becomes divergence-free, implying the lack of conformal invariance.
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).
Aharony and Fisher showed that nonlocal dipolar effects in magnetism destabilize the Heisenberg fixed point in real ferromagnets, leading to a new fixed point called the dipolar fixed point. The nonperturbative nature of the new fixed point, however, has not been uncovered for many decades. Inspired by the recent understanding that the dipolar fixed point is scale invariant but not conformal invariant, we perform the Monte Carlo simulation of the local Heisenberg-dipolar model on the lattice of 403{40}^{3}403 by introducing the local cost function parameterized by a parameter $\ensuremath{\lambda}$ and study its critical exponents, which should become identical to the dipolar fixed point of Aharony and Fisher in the infinite coupling limit $\ensuremath{\lambda}=\ensuremath{\infty}$. We find that the critical exponents become noticeably different from those of the Heisenberg fixed point for a finite coupling constant $\ensuremath{\lambda}=8$ [e.g., $\ensuremath{\nu}=0.601(2){(}_{\ensuremath{-}2}^{+0}){{(}_{\ensuremath{-}4}^{+5})}_{{\ensuremath{\beta}}_{c}}$ in the local Heisenberg-dipolar model while $\ensuremath{\nu}=0.712(1){(}_{\ensuremath{-}0}^{+3}){{(}_{\ensuremath{-}1}^{+1})}_{{\ensuremath{\beta}}_{c}}$ in the Heisenberg model], and the spin correlation function has a feature that it becomes divergence free, implying the lack of conformal invariance.
In this master thesis the criticality of the 3D XY model with cubic anisotropy has been examined by the means of Monte Carlo simulation. The 3D XY model is a special case of the O(n)O(n)O(n) spin-vector model which describe numerous phenomena's. Here, nnn is the number of vector dimensions and for the XY model n=2n=2n=2. The O(n)O(n)O(n)-model is invariant for global rotations. This means that it is arbitrary which direction in the symmetry breaking occurs. In real system one should always expect perturbations from a perfect system, here it is done by adding a cubic anisotropy (order p=4p=4p=4) then the symmetry of the Hamiltonian is lowered and it will be advantageous for the spins to be ordered in directions 90o90^o90o apart. This symmetry is called C4C_{4}C4​, which is a part of the cyclic group, and it restricts the symmetry breaking to one of these four directions. \n\nThe criticality of this model is when the spins suddenly align as the temperature is lowered to a critical temperature. Sufficiently close to the critical temperature quantities like the magnetization, which is the sum of spins, will show a power law growth characterised by critical exponents specified by the class of universality of the transition. The critical exponents can be obtained by finite-size scaling. By applying renormalization theory on the critical region one can show that some of parameters in the Hamiltonian are: relevant and effect the scaling, irrelevant and does not effect the scaling and dangerously irrelevant meaning that they only affect some of the critical exponents. Previous Monte Carlo and renormalization calculus show that anisotropy which is relevant for p<4p<4p<4 becomes dangerously irrelevant for pgeq4p\\geq4pgeq4. The question of interest is what kind of effect the dangerously irrelevant parameter in the Hamiltonian has on the scaling. This is interesting because the effect of dangerously irrelevant parameters is hardly studied and anisotropy arise in many crystal systems because of the lattice geometry. \n\nThe results are obtained by Monte Carlo simulations and the critical slowing down close to the phase transition is solved by the method of parallel tempering. This method was chosen because it is simple to implement in a parallel code.\n\nThe Binder cumulant for different quantities and the peak of the magnetic susceptibility was used support a critical temperature Tc=2.2019textnormalkT_c = 2.2019 \\ \\textnormal kTc​=2.2019textnormalk for no field applied and no anisotropy. The known critical exponents for the no field and no anisotropy was used. The scaling made a good collapse, meaning that the right TcT_cTc​ and the correct critical exponents where used. The scaling did not take into account a magnetic field and therefore a good collapse of curves where not obtained in this case. When the cubic anisotropy was added the critical exponent for the magnetic susceptibility and the cubic order parameter changed, while the other exponents where unchanged. This supports the view that cubic anisotropy is dangerously irrelevant for the 3D XY model. The scaling of the magnetic susceptibility showed a possible new kind of scaling, a scaling which is different on the two sides of the transition. Although a more accurate determination of the critical exponent in the ordered phase require more extensive simulations. It appears that this exponent is larger than the exponent in the disordered phase, where one recovered the isotropic exponent. The difference in the critical exponents has been predicted to increase with ppp and an investigation of the hexagonal anisotropy p=6p=6p=6 has been started.
We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.
In this work, the phase diagram of the ferromagnetic Ising model with dipole interactions is revisited with the aim of determining the nature of the phase transition between stripe-ordered phases with width nnn (hn)({h}_{n})(hn​) and tetragonal liquid (TL) phases. Extensive Monte Carlo simulations are performed in order to study the short-time dynamic behavior of the observables for selected values of the ratio between the ferromagnetic exchange and dipolar constants $\ensuremath{\delta}$. The obtained results indicate that the h1{h}_{1}h1​-TL phase transition line is continuous up to $\ensuremath{\delta}=1.2585$, while for the h2{h}_{2}h2​-TL line a weak first-order character is found in the interval $1.2585\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1.36$ and becomes continuous for $1.37\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1.9$. This result suggests the existence of a tricritical point close to $\ensuremath{\delta}=1.37$. When it is appropriate, the complete set of critical exponents is obtained, and in all the studied cases they depend on $\ensuremath{\delta}$ but do not belong to the Ising universality class. Furthermore, short-time dynamic studies reveal that at the point where the mentioned lines meet the ${h}_{1}\ensuremath{-}{h}_{2}$ line, i.e., at $\ensuremath{\delta}=1.2585$, the critical phase corresponding to the ${h}_{1}\ensuremath{-}\mathrm{TL}$ transition coexists with the h2{h}_{2}h2​ phase.