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.
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 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 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 study the correlated-disorder-driven zero-temperature equilibrium phase transition of the random-field Ising magnet (RFIM) using exact numerical ground-state calculations for cubic lattices. We consider correlations of the quenched disorder decaying proportionally to ra, where r is the distance between two lattice sites and a<0. To obtain exact ground states, we use a well-established mapping to the graph-theoretical maximum-flow problem, which allows us to study large system sizes of more than $2\ifmmode\times\else\texttimes\fi{}{10}^{6}$ spins. We use finite-size-scaling analyses for values $a={\ensuremath{-}1,\ensuremath{-}2,\ensuremath{-}3,\ensuremath{-}7}$ to calculate the critical point and the critical exponents characterizing the behavior of the specific heat, magnetization, susceptibility, and the correlation length close to the critical point. We find basically the same critical behavior as for the RFIM with $\ensuremath{\delta}$-correlated disorder, except for the finite-size exponent of the susceptibility and for the case $a\ensuremath{\ge}\ensuremath{-}2$, where the results are also compatible with a phase transition at infinitesimal disorder strength. We numerically confirm earlier predictions.
We investigate the magnetic and glassy transitions of the square-lattice XY model in the presence of random phase shifts. We consider two different random-shift distributions: the Gaussian distribution and a slightly different distribution (cosine distribution) which allows the exact determination of the Nishimori line where magnetic and overlap correlation functions are equal. We perform Monte Carlo simulations for several values of the temperature and of the variance of the disorder distribution, in the paramagnetic phase close to the magnetic and glassy transition lines. We find that, along the transition line separating the paramagnetic and the quasi-long-range order phases, magnetic correlation functions show a universal Kosterlitz-Thouless behavior as in the pure XY model, while overlap correlations show a disorder-dependent critical behavior. This behavior is observed up to a multicritical point which, in the cosine model, lies on the Nishimori line. Finally, for large values of the disorder variance, we observe a universal zero-temperature glassy critical transition, which is in the same universality class as that occurring in the gauge-glass model.
We investigate the temperature-disorder (T-$\ensuremath{\sigma}$) phase diagram of a three-dimensional gauge glass model, which is a cubic-lattice nearest-neighbor XY model with quenched random phase shifts Axy at the bonds. We consider the uncorrelated phase-shift distribution $P({A}_{\mathit{xy}})~\mathrm{exp}[(\mathrm{cos}{A}_{\mathit{xy}})/\ensuremath{\sigma}]$, which has the pure XY model, and the uniform distribution of random phase shifts as extreme cases at $\ensuremath{\sigma}=0$ and $\ensuremath{\sigma}\ensuremath{\rightarrow}\ensuremath{\infty}$, respectively, and which gives rise to equal magnetic and overlap correlation functions when $T=\ensuremath{\sigma}$. Our study is mostly based on numerical Monte Carlo simulations. While the high-temperature phase is always paramagnetic, at low temperatures there is a ferromagnetic phase for weak disorder (small $\ensuremath{\sigma}$) and a glassy phase at large disorder (large $\ensuremath{\sigma}$). These three phases are separated by transition lines with different magnetic and glassy critical behaviors. The disorder induced by the random phase shifts turns out to be irrelevant at the paramagnetic-ferromagnetic transition line, where the critical behavior belongs to the 3D XY universality class of pure systems; disorder gives rise only to very slowly decaying scaling corrections. The glassy critical behavior along the paramagnetic-glassy transition line belongs to the gauge-glass universality class, with a quite large exponent $\ensuremath{\nu}=3.2(4)$. These transition lines meet at a multicritical point M, located at ${T}_{M}={\ensuremath{\sigma}}_{M}=0.7840(2)$. The low-temperature ferromagnetic and glassy phases are separated by a third transition line, from M down to the T=0 axis, which is slightly reentrant.
Abstract We discuss the finite-size scaling of the ferromagnetic Ising model on random regular graphs. These graphs are locally tree-like, and in the limit of large graphs, the Bethe approximation gives the exact free energy per site. In the thermodynamic limit, the Ising model on these graphs show a phase transition. This transition is rounded off for finite graphs. We verify the scaling theory prediction that this rounding off is described in terms of the scaling variable [ T / T c − 1] S 1/2 (where T and T c are the temperature and the critical temperature respectively, and S is the number of sites in the graph), and not in terms of a power of the diameter of the graph, which varies as log S . We determine the theoretical scaling functions for the specific heat capacity and the magnetic susceptibility of the absolute value of the magnetization in closed form and compare them to Monte Carlo simulations.