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arXiv:2403.19939 · Physical review. E

Spin cones in random-field X Y models

Rajiv G. Pereira, Ananya Janardhanan, Mustansir Barma

Published Jul 26, 2024• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.110.014141

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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.
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 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.
We study the two-dimensional XY\mathrm{XY}XY model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to T=0T=0T=0 in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.
We investigate equilibrium statistical properties of the isotropic quantum XY spin-1/2 model in an external magnetic field when the interaction and field parts are subjected to quenched or annealed disorder or both. The randomness present in the system are termed annealed or quenched depending on the relation between two different time scales---the time scale associated with the equilibration of the randomness and the time of observation. Within a mean-field framework, we study the effects of disorders on spontaneous magnetization, both by perturbative and numerical techniques. Our primary interest is to understand the differences between quenched and annealed cases, and also to investigate the interplay when both of them are present in a system. We find that the magnetization survives in the presence of a unidirectional random field, irrespective of its nature, i.e., whether it is quenched or annealed. However, the field breaks the circular symmetry of the magnetization, and the system magnetizes in specific directions, parallel or transverse to the applied magnetic field. Interestingly, while the transverse magnetization is affected by the annealed disordered field, the parallel one remains unfazed by the same. Moreover, the annealed disorder present in the interaction term does not affect the system's spontaneous magnetization and the corresponding critical temperature, irrespective of the presence or absence of quenched or annealed disorder in the field term. We carry out a comparative study of these and all other different combinations of the disorders in the interaction and field terms, and point out their generic features.
We investigate the effect of a unidirectional quenched random field on the anisotropic quantum spin-1/2 XYXYXY model, which magnetizes spontaneously in the absence of the random field. We adopt a mean-field approach for this analysis. In general, the models considered have Ising symmetry, and as such they exhibit ferromagnetic order in two and three dimensions in the presence of not too large disorder. Even in the special case when the model without disorder has U(1)\text{U}(1)U(1) symmetry, a small U(1)\text{U}(1)U(1)-symmetry-breaking random field induces ferromagnetic long-range order in two dimensions. The mean-field approach, consequently, provides a rather good qualitative and even quantitative description when applied not too close to the criticality. We show that spontaneous magnetization persists even in the presence of the random field, but the magnitude of magnetization gets suppressed due to disorder, and the system magnetizes in the directions parallel and transverse to the random field. Our results are obtained via analytical calculations within a perturbative framework and by numerical simulations. Interestingly, we show that it is possible to enhance a component of magnetization in the presence of the disorder field provided that we apply an additional constant field in the XYXYXY plane. Moreover, we derive generalized expressions for the critical temperature and the scalings of the magnetization near the critical point for the XYXYXY spin system with arbitrary fixed quantum spin angular momentum.