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arXiv:2201.09498 · Physical review. E

Phase transitions in XY models with randomly oriented crystal fields

Sumedha, Mustansir Barma

Published Feb 8, 2022• 2 citations
arXivHTMLPDF

Abstract

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→∞.

DOI
10.1103/physreve.105.024111

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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→∞.
Abstract We use large deviation theory to obtain the free energy of the XY model on a fully connected graph on each site of which there is a randomly oriented field of magnitude h . The phase diagram is obtained for two symmetric distributions of the random orientations: (a) a uniform distribution and (b) a distribution with cubic symmetry. In both cases, the disorder–averaged ordered state reflects the symmetry of the underlying distribution. The phase boundary has a multicritical point (MCP) which separates a locus of continuous transitions (for small values of h ) from a locus of first order transitions (for large h ). The free energy is a function of a single variable in case (a) and a function of two variables in case (b), leading to different characters of the MCPs in the two cases. We find that the locus of continuous transitions is given by the same equation for a family of quadriperiodic distributions, which includes the distributions (a) and (b). However, the location of the MCP and the nature of ordered state depend on the form of the distribution. The disorder-averaged ground state energy is found exactly, and the specific heat is shown to approach a constant as temperature approaches zero.
We determine the arrangement of spins in the ground state of the XY model with quenched, random fields, on a fully connected graph. Two types of disordered fields are considered, namely, randomly oriented magnetic fields and randomly oriented crystal fields. Orientations are chosen from a uniformly isotropic distribution, but disorder fluctuations in each realization of a finite system lead to a breaking of rotational symmetry. The result is an interesting pattern of spin orientations found by solving a system of coupled, nonlinear equations within perturbation theory and also by exact numerical continuation. All spins lie within a cone for small enough ratio of field to coupling strength, with an interesting distribution of spin orientations, with peaks at the cone edges. The orientation of the cone depends strongly on the realization of disorder, but the opening angle does not. In the case of random magnetic fields, the cone angle widens as the ratio increases till a critical value at which there is a first-order phase transition and the cone disappears. With random crystal fields, there is no phase transition and the cone angle approaches 180^{∘} for large values of the ratio. At finite low temperatures, Monte Carlo simulations show that the formation of a cone and its subsequent alignment along the equilibrium direction occur on two different timescales.
The mean-field approximation is used to calculate the phase diagrams of the spin-3/2 random transverse crystal field (RTCF) model. A bimodal distribution of the RTCF is employed in which one of the modes is in competition with the other mode whose strength is adjusted with a controllable parameter α. Thus, some sites in the lattice are under the effect of RTCF Dx with probability p and the rest of the sites are under the effect of αDx with 1 − p. The phase diagrams of the model are obtained on the RTCF-temperature planes by varying p on square lattices. Both second- and first-order phase transitions are displayed and thus tricritical points are also observed. The critical end points and end points for the first-order phase transition lines were also found.
We numerically investigate the critical properties of the two-dimensional XY model with four-fold and eight-fold crystal fields. Using extensive Monte Carlo simulations, we compute the specific heat, magnetization, orientational magnetization, and correlation length ratio to examine how these anisotropies influence the nature of phase transitions. The critical temperature and the critical exponent are systematically mapped across both compatible and competing field regimes, spanning from weak to strong field strengths. The numerical results reveal that the critical exponent exhibits a linear dependence on the inverse of the four-fold field strength in the strong field regime, while remaining finite and non-divergent in the weak field regime. Furthermore, the orientational magnetization effectively identifies and characterizes Ising-type phase transitions in the competing field regime.
Abstract The critical behavior of mixed spin Ising model is investigated when the transverse fields are randomly distributed. The calculation shows that when the transverse fields are both bimodally distributed, the transition temperature decreases with increasing transverse field and eventually vanishes at a finite field. When one of the transverse fields is bimodally and the other trimodally distributed, the temperature decreases but does not vanish. When both transverse fields are trimodally distributed, the transverse field dependence of temperature breaks off and a tricritical point appears due to the interaction between quantum fluctuations and disorder.
The bimodal random crystal field(?) effects are investigated on the phase diagrams of spin-3/2 Ising model by using the effective-field theory with correlations based on two approximations:the general van der Waerden identity and the approximated van der Waerden identity.In our approach,the crystal field is either turned on or turned off randomly for a given probability p or q = 1- p,respectively.Then the phase diagrams are constructed on the(?,kT /J) and(p,kT /J)planes for given p and ?,respectively,when the coordination number is z = 3.Furthermore,the effect of randomization of the crystal field is illustrated on the(?,kT /J) plane for p = 0.5 when z = 3,4,and 6.All these are carried out for both approximations and then the results are compared to point out the differences.In addition to the lines of second-order phase transitions,the model also exhibits first-order phase transitions and the lines of which terminate at the isolated critical points for high p values.