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Physical review. B./Physical review. B

Monte Carlo study of the Heisenberg model with local dipolar interaction

Etsuko Itou, Akira Matsumoto, Yu Nakayama, Toshiki Onagi

Published Sep 5, 2025• 1 citation
Publisher

Abstract

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 ${40}^{3}$ 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.

DOI
10.1103/95xx-m6xg

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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 \ensuremathλ\ensuremath{\lambda}\ensuremathλ and study its critical exponents, which should become identical to the dipolar fixed point of Aharony and Fisher in the infinite coupling limit \ensuremathλ=\ensuremath∞\ensuremath{\lambda}=\ensuremath{\infty}\ensuremathλ=\ensuremath∞. We find that the critical exponents become noticeably different from those of the Heisenberg fixed point for a finite coupling constant \ensuremathλ=8\ensuremath{\lambda}=8\ensuremathλ=8 [e.g., \ensuremathν=0.601(2)(\ensuremath−2+0)(\ensuremath−4+5)\ensuremathβc\ensuremath{\nu}=0.601(2){(}_{\ensuremath{-}2}^{+0}){{(}_{\ensuremath{-}4}^{+5})}_{{\ensuremath{\beta}}_{c}}\ensuremathν=0.601(2)(\ensuremath−2+0​)(\ensuremath−4+5​)\ensuremathβc​​ in the local Heisenberg-dipolar model while \ensuremathν=0.712(1)(\ensuremath−0+3)(\ensuremath−1+1)\ensuremathβc\ensuremath{\nu}=0.712(1){(}_{\ensuremath{-}0}^{+3}){{(}_{\ensuremath{-}1}^{+1})}_{{\ensuremath{\beta}}_{c}}\ensuremathν=0.712(1)(\ensuremath−0+3​)(\ensuremath−1+1​)\ensuremathβc​​ in the Heisenberg model], and the spin correlation function has a feature that it becomes divergence free, implying the lack of conformal invariance.
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.
We revisit critical phenomena in isotropic ferromagnets with strong dipolar interactions. The corresponding renormalization group fixed point, dipolar fixed point, was first studied in 1973 by Aharony and Fisher [Phys. Rev. Lett. 30, 559 (1973]. It is distinct from the Heisenberg fixed point, although the critical exponents are close. On the theoretical side, we discuss scale invariance without conformal invariance realized by this fixed point. We elucidate the nonrenormalization of the virial current due to a shift symmetry, and show that the same mechanism is at work in all other known local fixed points which are scale but not conformal invariant. On the phenomenological side, we discuss the relative strength of dipolar and short-range interactions. In some materials, like the europium compounds, dipolar interactions are strong, and the critical behavior is dipolar. In others, like Fe or Ni, dipolar interactions are weaker, and the Heisenberg critical behavior in a range of temperatures is followed by the dipolar behavior closer to the critical point. Some of these effects have been seen experimentally.
A dipolar fixed point introduced by Aharony and Fisher is a physical example of interacting scale-invariant but nonconformal field theories. We find that the perturbative critical exponents computed in ε expansions violate the conformal bootstrap bound. We formulate the functional renormalization group equations Wetterich and Polchinski to study the fixed point. We present some results in three dimensions within (uncontrolled) local potential approximations (with or without perturbative anomalous dimensions). Published by the American Physical Society 2024
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.
The critical properties of the antiferromagnetic Heisenberg model on the three-dimensional stacked-triangular lattice are studied by means of a large-scale Monte Carlo simulation in order to get insight into the controversial issue of the criticality of the noncollinear magnets with the $O(3)\ifmmode\times\else\texttimes\fi{}O(2)$ symmetry. The maximum size studied is 3843{384}^{3}3843, considerably larger than the sizes studied by the previous numerical works on the model. Availability of such large-size data enables us to examine the detailed critical properties including the effect of corrections to the leading scaling. Strong numerical evidence of the continuous nature of the transition is obtained. Our data indicate the existence of significant corrections to the leading scaling. Careful analysis by taking account of the possible corrections yields critical exponents estimates, $\ensuremath{\alpha}=0.44(3), \ensuremath{\beta}=0.26(2), \ensuremath{\gamma}=1.03(5), \ensuremath{\nu}=0.52(1), \ensuremath{\eta}=0.02(5)$, and the chirality exponents, ${\ensuremath{\beta}}_{\ensuremath{\kappa}}=0.40(3)$ and ${\ensuremath{\gamma}}_{\ensuremath{\kappa}}=0.77(6)$, supporting the existence of the O(3)O(3)O(3) chiral [or $O(3)\ifmmode\times\else\texttimes\fi{}O(2)$] universality class governed by a ``chiral'' fixed point. We also obtain an indication that the underlying fixed point is of the focus type, characterized by the complex-valued correction-to-scaling exponent, $\ensuremath{\omega}=0.{1}_{\ensuremath{-}0.05}^{+0.4}+i\phantom{\rule{4pt}{0ex}}0.{7}_{\ensuremath{-}0.4}^{+0.1}$. The focus-like nature of the chiral fixed point accompanied by the spiral-like renormalization-group (RG) flow is likely to be the origin of the apparently complicated critical behavior. The results are compared and discussed in conjunction with the results of other numerical simulations, several distinct types of RG calculations including the higher-order perturbative massive and massless RG calculations and the nonperturbative functional RG calculation, and the conformal-bootstrap program.
The magnetic phase transition in a Heisenberg fluid is studied by means of the finite size scaling technique. We find that even for larger systems, considered in an ensemble with fixed density, the critical exponents show deviations from the expected lattice values similar to those obtained previously. This puzzle is clarified by proving the importance of the leading correction to the scaling that appears due to Fisher renormalization with the critical exponent equal to the absolute value of the specific heat exponent alpha. The appearance of such new corretions to scaling is a general feature of systems with constraints.