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arXiv:2503.09389 · Physical review. E

Canonical equilibrium of mean-field O ( n ) models in the presence of random fields

Soumya Kanti Pal, Shamik Gupta

Published Jun 11, 2025• 0 citations
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Abstract

We study canonical-equilibrium properties of random field O(n) models involving classical continuous vector spins of n components with mean-field interactions and subject to disordered fields acting on individual spins. To this end, we employ two complementary approaches: the mean-field approximation, valid for any disorder distribution, and the replica trick, applicable when the disordered fields are sampled from a Gaussian distribution. On the basis of an exact analysis, we demonstrate that when replica symmetry holds, both the approaches yield identical expression for the free energy per spin of the system. As consequences, we study the case of n=2 (XY spins) and that of n=3 (Heisenberg spins) for two representative choices of the disorder distribution, namely, a Gaussian and a symmetric bimodal distribution. For both n=2 and n=3, we demonstrate that while the magnetization exhibits a continuous phase transition as a function of temperature for the Gaussian case, the transition could be either continuous or first order with an emergent tricriticality when the disorder distribution is bimodal. We also discuss in the context of our models the issue of self-averaging of extensive variables near the critical point of a continuous phase transition.

DOI
10.1103/nhkq-sb63

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We study canonical-equilibrium properties of random field O(n) models involving classical continuous vector spins of n components with mean-field interactions and subject to disordered fields acting on individual spins. To this end, we employ two complementary approaches: the mean-field approximation, valid for any disorder distribution, and the replica trick, applicable when the disordered fields are sampled from a Gaussian distribution. On the basis of an exact analysis, we demonstrate that when replica symmetry holds, both the approaches yield identical expression for the free energy per spin of the system. As consequences, we study the case of n=2 (XY spins) and that of n=3 (Heisenberg spins) for two representative choices of the disorder distribution, namely, a Gaussian and a symmetric bimodal distribution. For both n=2 and n=3, we demonstrate that while the magnetization exhibits a continuous phase transition as a function of temperature for the Gaussian case, the transition could be either continuous or first order with an emergent tricriticality when the disorder distribution is bimodal. We also discuss in the context of our models the issue of self-averaging of extensive variables near the critical point of a continuous phase transition.
We define a replica field theory describing finite dimensional site-disordered spin systems by introducing the notion of grand canonical disorder, where the number of spins in the system is random but quenched. A general analysis of this field theory is made using the Gaussian variational or Hartree - Fock method, and illustrated with several specific examples. Irrespective of the form of interaction between the spins this approximation predicts a spin-glass phase. We discuss the replica symmetric phase at length, explicitly identifying the correlator that diverges at the spin-glass transition. We also discuss the form of continuous replica symmetry breaking found just below the transition. Finally we show how an analysis of ferromagnetic ordering indicates a breakdown of the approximation.
We consider classical spin models of two- and three-dimensional spins with continuous symmetry and investigate the effect of a symmetry-breaking unidirectional quenched disorder on the magnetization of the system. We work in the mean-field regime. We show by perturbative calculations and numerical simulations that although the continuous symmetry of the magnetization is lost due to disorder, the system still magnetizes in specific directions, albeit with a lower value as compared to the case without disorder. The critical temperature at which the system starts magnetizing and the magnetization at low- and high-temperature limits in the presence of disorder are estimated. Moreover, we treat the SO(n)n\mathrm{SO}(n) nSO(n)n-component spin model to obtain the generalized expressions for the near-critical scalings, which suggest that the effect of disorder in magnetization increases with increasing dimension. We also study the behavior of magnetization of the classical XYXYXY spin model in the presence of a constant magnetic field, in addition to the quenched disorder. We find that in the presence of the uniform magnetic field, disorder may enhance the component of magnetization in the direction that is transverse to the disordered field.
A classical spherical random-field Hamiltonian with long-range (power-law) interactions is investigated by means of the replica theory. Both ferromagnetic and anti-ferromagnetic interactions are considered. The use of continuous variables instead of Ising variables in the spherical version of the model allows one to calculate the free energy exactly. The existence of an equilibrium phase transition is investigated based on the replica-symmetric solution. The results are applied to the Coulomb-glass model of interacting localized electrons in a disordered solid. This model is shown not to have an equilibrium phase transition for spatial dimensions D<or=4. For D>4 the model has a phase transition to an ordered phase; however, it does not have a phase transition to a 'glassy' phase.
We obtain a representation of the free energy of an XY model on a fully connected graph with spins subjected to a random crystal field of strength D and with random orientation α. Results are obtained for an arbitrary probability distribution of the disorder using large deviation theory, for any D. We show that the critical temperature is insensitive to the nature and strength of the distribution p(α), for a large family of distributions which includes quadriperiodic distributions, with p(α)=p(α+π/2), which includes the uniform and symmetric bimodal distributions. The specific heat vanishes as temperature T→0 if D is infinite, but approaches a constant if D is finite. We also studied the effect of asymmetry on a bimodal distribution of the orientation of the random crystal field and obtained the phase diagram comprising four phases: a mixed phase (in which spins are canted at angles which depend on the degree of disorder), an x-Ising phase, a y-Ising phase, and a paramagnetic phase, all of which meet at a tetracritical point. The canted mixed phase is present for all finite D, but vanishes when D→∞.
We study the random-field Ising model with long-range interactions and show the exactness of the mean-field theory under certain mild conditions. This is a generalization of the result of Mori for the non-random and spin-glass cases. To treat random fields, we evoke the self-averaging property of a function of random fields, without recourse to the replica method. The result is that the mean-field theory gives the exact expression of the canonical free energy for systems with power-decaying interactions if the power is smaller than or equal to the spatial dimension.
The form of the field distribution, in a random-field Ising model, is known to play an important role in determining the order of the phase transition and the existence of a tricritical point. We have studied a random-field Ising model with a trimodal distribution (sum of three \ensuremath{\delta} functions), which simulates a Gaussian distribution for a certain range of its parameters, within the mean-field renormalization-group scheme. We have considered clusters of one, two, and four spins, and the recursion relations for the determination of the fixed points are numerically analyzed. Using this analysis we have constructed the phase diagram of the system in the space of coupling (K), field (h), and zero-field probability (p). It is shown that the line of tricritical points ends for values of p near (2/3, very different from that obtained in the usual mean-field approximation. Mean-field renormalization-group (p\ensuremath{\simeq}(2/3) results are found to be consistent with the equivalence of the trimodal and Gaussian (absence of tricritical points) distributions until the fourth-order moments for p=(2/3.