Presence Of Random Field Research Articles

Critical behaviours of anisotropic XY ferromagnet in the presence of random field

The anisotropic XY ferromagnet has been studied by Monte Carlo simulation in a three dimensional simple cubic lattice. The increase in critical temperature (ferro-para transition) has been noticed with increasing the strength of anisotropy. The effects of random fields (both with full circular symmetry and in angular window) on the critical temperature are investigated systematically in the anisotropic XY ferromagnet in three dimensions. Reduction of the critical temperature of anisotropic XY ferromagnet has been observed in the presence of random field. The compensating field (the required amount of field which preserves the critical temperature for isotropic XY ferromagnet) has been studied as a function of the strength of anisotropy. The compensating field was found to depend linearly on the strength of anisotropy. We have also studied the effects of random field confined in the angular window and observed the reduction of the critical temperature with increase of the angular extension. The critical behaviours are formalised by the usual finite size analysis and the estimation of critical exponents for the susceptibility and the specific heat.

Prediction of a first-order phase transition in two-dimensional ferromagnets in the presence of random fields

Random magnetic fields suppress the long-range magnetic ordering through the formation of the magnetic domains. The size of the domains is determined by the competition among the exchange interaction, the magnetic anisotropy, and the strength of the random field. Here we theoretically investigate the temperature dependence of the magnetization of the two-dimensional domains for the anisotropic Heisenberg model with a random magnetic field. We find that magnetization of the domains displays a first-order phase transition in which the magnetization is discontinuous at a critical temperature. Moreover, the first-order transition persists even in the presence of an external magnetic field whose magnitude is smaller than the strength of the disorder. The above unusual first-order phase transition can be experimentally tested with doped two-dimensional magnets.

Effects of Random Excitations on the Dynamical Response of Duffing Systems

We study the dynamics of a Duffing oscillator excited by correlated random perturbations for both fixed and periodically modulated stiffness. In the case of fixed stiffness we see that Poincare map gets distorted due to the random excitation and, the distortion increases with the increase of correlation of the field. In a strongly correlated field, however, the map becomes purely random. We analyse the maximum value of the Lyapunov exponent and see that the random response competes with the chaotic motion to increase the stability of the system. In the case of periodically modulated stiffness, the periodic parametric excitation causes the Duffing system to execute dynamics of two fixed-point attractors. These attractors remain non-chaotic even in the presence of random field but can get merged due to induced fluctuation in the trajectory. It is seen that the random field can change the status of the system from transit to stable state.

The effects of the one-step replica symmetry breaking on the Sherrington–Kirkpatrick spin glass model in the presence of random field with a joint Gaussian probability density function for the exchange interactions and random fields

The Sherrington–Kirkpatrick Ising spin glass model, in the presence of a random magnetic field, is investigated within the framework of the one-step replica symmetry breaking. The two random variables (exchange integral interaction Jij and random magnetic field hi) are drawn from a joint Gaussian probability density function characterized by a correlation coefficient ρ, assuming positive and negative values. The thermodynamic properties, the three different phase diagrams and system’s parameters are computed with respect to the natural parameters of the joint Gaussian probability density function at non-zero and zero temperatures. The low temperature negative entropy controversy, a result of the replica symmetry approach, has been partly remedied in the current study, leading to a less negative result. In addition, the present system possesses two successive spin glass phase transitions with characteristic temperatures.

Spin-glass phase transition and behavior of nonlinear susceptibility in the Sherrington-Kirkpatrick model with random fields

The behavior of the nonlinear susceptibility $\chi_3$ and its relation to the spin-glass transition temperature $T_f$, in the presence of random fields, are investigated. To accomplish this task, the Sherrington-Kirkpatrick model is studied through the replica formalism, within a one-step replica-symmetry-breaking procedure. In addition, the dependence of the Almeida-Thouless eigenvalue $\lambda_{\rm AT}$ (replicon) on the random fields is analyzed. Particularly, in absence of random fields, the temperature $T_f$ can be traced by a divergence in the spin-glass susceptibility $\chi_{\rm SG}$, which presents a term inversely proportional to the replicon $\lambda_{\rm AT}$. As a result of a relation between $\chi_{\rm SG}$ and $\chi_3$, the latter also presents a divergence at $T_f$, which comes as a direct consequence of $\lambda_{\rm AT}=0$ at $T_f$. However, our results show that, in the presence of random fields, $\chi_3$ presents a rounded maximum at a temperature $T^{*}$, which does not coincide with the spin-glass transition temperature $T_f$ (i.e., $T^* > T_f$ for a given applied random field). Thus, the maximum value of $\chi_3$ at $T^*$ reflects the effects of the random fields in the paramagnetic phase, instead of the non-trivial ergodicity breaking associated with the spin-glass phase transition. It is also shown that $\chi_3$ still maintains a dependence on the replicon $\lambda_{\rm AT}$, although in a more complicated way, as compared with the case without random fields. These results are discussed in view of recent observations in the LiHo$_x$Y$_{1-x}$F$_4$ compound.

Anisotropic four-state clock model in the presence of random fields.

A four-state clock ferromagnetic model is studied in the presence of different configurations of anisotropies and random fields. The model is considered in the limit of infinite-range interactions, for which the mean-field approach becomes exact. Both representations of Cartesian spin components and two Ising variables are used, in terms of which the physical properties and phase diagrams are discussed. The random fields follow bimodal probability distributions and the richest criticality is found when the fields, applied in the two Ising systems, are not correlated. The phase diagrams present new interesting topologies, with a wide variety of critical points, which are expected to be useful in describing different complex phenomena.

Statistical mechanics of the spherical hierarchical model with random fields

We study analytically the equilibrium properties of the spherical hierarchical model in the presence of random fields. The expression for the critical line separating a paramagnetic from a ferromagnetic phase is derived. The critical exponents characterising this phase transition are computed analytically and compared with those of the corresponding D-dimensional short-range model, leading to conclude that the usual mapping between one dimensional long-range models and D-dimensional short-range models holds exactly for this system, in contrast to models with Ising spins. Moreover, the critical exponents of the pure model and those of the random field model satisfy a relationship that mimics the dimensional reduction rule. The absence of a spin-glass phase is strongly supported by the local stability analysis of the replica symmetric saddle-point as well as by an independent computation of the free-energy using a renormalization-like approach. This latter result enlarges the class of random field models for which the spin-glass phase has been recently ruled out.

Localization of spin waves in disordered quantum rotors

We study the dynamics of excitations in a system of $O(N)$ quantum rotors in the presence of random fields and random anisotropies. Below the lower critical dimension ${d}_{\mathrm{lc}}=4$ the system exhibits a quasi-long-range order with a power-law decay of correlations. At zero temperature the spin waves are localized at the length scale ${L}_{\mathrm{loc}}$ beyond which the quantum tunneling is exponentially suppressed, $c\ensuremath{\sim}{e}^{\ensuremath{-}{(L/{L}_{\mathrm{loc}})}^{2(\ensuremath{\theta}+1)}}$. At finite temperature $T$ the spin waves propagate by thermal activation over energy barriers that scale as ${L}^{\ensuremath{\theta}}$. Above ${d}_{\mathrm{lc}}$ the system undergoes an order-disorder phase transition with activated dynamics such that the relaxation time grows with the correlation length $\ensuremath{\xi}$ as $\ensuremath{\tau}\ensuremath{\sim}{e}^{C{\ensuremath{\xi}}^{\ensuremath{\theta}}/T}$ at finite temperature and as $\ensuremath{\tau}\ensuremath{\sim}{e}^{{C}^{\ensuremath{'}}{\ensuremath{\xi}}^{2(\ensuremath{\theta}+1)}/{\ensuremath{\hbar}}^{2}}$ in the vicinity of the quantum critical point.

Quenched disorder forbids discontinuous transitions in nonequilibrium low-dimensional systems

Quenched disorder affects significantly the behavior of phase transitions. The Imry-Ma-Aizenman-Wehr-Berker argument prohibits first-order or discontinuous transitions and their concomitant phase coexistence in low-dimensional equilibrium systems in the presence of random fields. Instead, discontinuous transitions become rounded or even continuous once disorder is introduced. Here we show that phase coexistence and first-order phase transitions are also precluded in nonequilibrium low-dimensional systems with quenched disorder: discontinuous transitions in two-dimensional systems with absorbing states become continuous in the presence of quenched disorder. We also study the universal features of this disorder-induced criticality and find them to be compatible with the universality class of the directed percolation with quenched disorder. Thus, we conclude that first-order transitions do not exist in low-dimensional disordered systems, not even in genuinely nonequilibrium systems with absorbing states.

Percolation and Schramm–Loewner evolution in the 2D random-field Ising model

The presence of random fields is well known to destroy ferromagnetic order in Ising systems in two dimensions. When the system is placed in a sufficiently strong external field, however, the size of clusters of like spins diverges. There is evidence that this percolation transition is in the universality class of standard site percolation. It has been claimed that, for small disorder, a similar percolation phenomenon also occurs in zero external field. Using exact algorithms, we study ground states of large samples and find little evidence for a transition at zero external field. Nevertheless, for sufficiently small random-field strengths, there is an extended region of the phase diagram, where finite samples are indistinguishable from a critical percolating system. In this regime we examine ground-state domain walls, finding strong evidence that they are conformally invariant and satisfy Schramm–Loewner evolution ( SLE κ ) with parameter κ = 6 . These results add support to the hope that at least some aspects of systems with quenched disorder might be ultimately studied with the techniques of SLE and conformal field theory.

Magnetic Properties of the Metamagnet Ising Model in a three-dimensional Lattice in a Random and Uniform Field

The Monte Carlo technique is here used for study the behavior of metamagnet Ising model in presence of random field. The phase diagram is obtained by employing Glauber's algorithm in a cubic lattice of linear size L with values ranging from 16 to 42 and with periodic boundary conditions.

Probabilistic optimal design in the presence of random fields

This article describes a methodology to incorporate a random field in a probabilistic optimization problem. The approach is based on the extraction of the features of a random field using a reduced number of experimental observations. This is achieved by proper orthogonal decomposition. Using Lagrange interpolation, a modified random field is obtained by changing the contribution of each feature. The contributions are controlled using scalar parameters, which can be considered as random variables. This allows one to perform a random-field-based probabilistic optimization with few random variables. The methodology is demonstrated on a tube impacting a rigid wall for which a random field modifies the planarity of the tube’s wall.

Long-range correlated random field and random anisotropyO(N)models: A functional renormalization group study

We study the long-distance behavior of the $O(N)$ model in the presence of random fields and random anisotropies correlated as $\ensuremath{\sim}1∕{x}^{d\ensuremath{-}\ensuremath{\sigma}}$ for large separation $x$ using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double $\ensuremath{\epsilon}=d\ensuremath{-}4$ and $\ensuremath{\sigma}$ expansion. We find that the long-range disorder correlator remains analytic but generates short-range disorder whose correlator develops the usual cusp. This allows us to obtain the phase diagrams in $(d,\ensuremath{\sigma},N)$ parameter space and compute the critical exponents to first order in $\ensuremath{\epsilon}$ and $\ensuremath{\sigma}$. We show that the standard renormalization group methods with a finite number of couplings used in previous studies of systems with long-range correlated random fields fail to capture all critical properties. We argue that our results may be relevant to the behavior of $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{A}$ in aerogel.

Random-energy model in random fields

The random-energy model is studied in the presence of random fields. The problem is solved exactly both in the microcanonical ensemble, without recourse to the replica method, and in the canonical ensemble using the replica formalism. The phase diagrams for bimodal and Gaussian random fields are investigated in detail. In contrast to the Gaussian case, the bimodal random field may lead to a tricritical point and a first-order transition. An interesting feature of the phase diagram is the possibility of a first-order transition from paramagnetic to mixed phase.

Dynamic Dielectric Nonlinearity in Epitaxial Thin-Film Relaxors and Ferroelectrics

The dynamic dielectric response of epitaxial thin-film PbMg1/3Nb2/3O3, PbSc0.5Nb0.5O3, and Ba0.4Sr0.6TiO3 was experimentally studied as a function of temperature, frequency, and amplitude of ac electric field (to 10 MV/m). In thin-film relaxors at all amplitudes, a glasslike behavior was evidenced by Vogel-Fulcher relationship, scaling of the dielectric peak, and broadening of the relaxation time spectra on cooling. Both this and the nonlinear dielectric response were consistent with reorientation of randomly interacting dipoles at presence of random fields. In thin-film ferroelectrics, a relaxorlike nonlinear dielectric behavior was found. As a reason for this, possible coexistence of polar and nonpolar regions was discussed.

Spin dynamics of the quantum XY chain and ladder in a random field

We investigate the Hamiltonian dynamics of two low-dimensional quantum spin systems in a random field, at the infinite-temperature limit: the XY chain and the two-leg XY ladder with interchain Ising interactions. We determine the longitudinal spin autocorrelation functions of the spin- 1 2 XY chain and ladder in the presence of disordered fields by using the method of recurrence relations. The first six basis vectors for the chain and the first four basis vectors for the ladder of the dynamic Hilbert spaces of σ j z ( t), as well as the corresponding recurrents and moments of the time-dependent autocorrelation function, are analytically computed for bimodal distributions of the fields. We did find a remarkable result in the disordered models. Cases with a fraction of p sites under field B B and a fraction of 1− p sites under the field B A have the same longitudinal dynamics as those with p sites under field B A and 1− p sites under the field B B . We also find that both the XY chain and the two-leg XY ladder with Ising interchain coupling in the presence of random fields are sensitive to the percentage of disorder but not to the intensity of the fields.

Glass behavior of a relaxor Cd2Nb2O7 in a weak DC field

The temperature behavior of the dielectric response ehis(T) at a frequency of 1 kHz (110 K< T<300 K), δe(T)=[ehis(T)-eZFC(T)], and of the order parameter q(T), characterizing e(T) in a relaxor, is investigated for the relaxor Cd2Nb2O7 on samples with different thermal and electric histories in a dc field much weaker than the polarization saturation field. In a weak field (E dc=0.95 kV/cm), the behavior of δe(T) ∝ χnl changes in the region T f =184 K and the ehis(T) curves diverge monotonically at lower temperatures, indicating the development of the glassy state in the system. Analysis of the behavior of q(T) in the framework of the model of spin and dipole glasses shows that random interactions of polar microscopic regions in the presence of random fields play a dominating role in the phase formed below T f in the ZFC and FH/ZFC regimes. The relative contribution of random fields increases in the FC regime in a weak field and is manifested in the ZFH/FC regime both below and above T f .

Diffusion-limited reaction in the presence of random fields and transition rates

The diffusion-limited reaction was studied on a one-dimensional lattice in the presence of random fields and transition rates using Monte Carlo simulations. In the case of transition rates the hopping probabilities at a site are distributed according to the power law p(y)=νyν−1 with 0&amp;lt;ν⩽1 and 0&amp;lt;y⩽1. The density of the reactants decays according to a power-law, C(t)∼t−α(ν) for A+A→0 and A+B→0 annihilation reactions. The exponent α(ν) depends on the disorder exponent ν. For A+A→0, we found α(ν)=ν/(1+ν). For A+B→0, we observed α=0.25 at ν&amp;gt;0.4 and α decreases monotonically for ν&amp;lt;0.4. In the case of the random fields the density decays according to C(t)∼[b(E)/log(t)]2 regardless of the strength of the random fields E for A+A→0 and A+A→A reactions, where b(E)∼log[(1+E)/(1−E)]. The diffusion-limited coagulation A+A→A belongs to the same universality class as the A+A→0 reaction under the random fields. For A+B→0 annihilations we observe that the density decays according to C(t)∼b(E)/log(t) in the presence of the random fields.

A conjectured scenario for order-parameter fluctuations in spin glasses

We study order-parameter fluctuations (OPF) in disordered systems by considering the behavior of some recently introduced paramaters $G,G_c$ which have proven very useful to locate phase transitions. We prove that both parameters G (for disconnected overlap disorder averages) and $G_c$ (for connected disorder averages) take the respective universal values 1/3 and 13/31 in the $T\to 0$ limit for any {\em finite} volume provided the ground state is {\em unique} and there is no gap in the ground state local-field distributions, conditions which are met in generic spin-glass models with continuous couplings and no gap at zero coupling. This makes $G,G_c$ ideal parameters to locate phase transitions in disordered systems much alike the Binder cumulant is for ordered systems. We check our results by exactly computing OPF in a simple example of uncoupled spins in the presence of random fields and the one-dimensional Ising spin glass. At finite temperatures, we discuss in which conditions the value 1/3 for G may be recovered by conjecturing different scenarios depending on whether OPF are finite or vanish in the infinite-volume limit. In particular, we discuss replica equivalence and its natural consequence $\lim_{V\to\infty}G(V,T)=1/3$ when OPF are finite. As an example of a model where OPF vanish and replica equivalence does not give information about G we study the Sherrington-Kirkpatrick spherical spin-glass model by doing numerical simulations for small sizes. Again we find results compatible with G=1/3 in the spin-glass phase.

Effects of random fields in an antiferromagnetic Ising spin glass

The effects of random fields on the two-sublattice infinite-ranged Ising spin-glass model are investigated. This model is expected to be appropriate as a mean-field description of antiferromagnetic spin glasses such as FexMn1-xTiO3. Within replica-symmetric calculations, we study the influence of Gaussian and bimodal random fields on the phase transitions and phase diagrams. It is shown that, in the presence of random fields, the first-order transitions are weakened and may become continuous. Also, the antiferromagnetic phases are always destroyed by sufficiently strong random fields. A qualitative comparison with existing experimental results and the limitations of the present calculations are discussed.