Random Ferromagnet Research Articles

Disordered Potts model on the diamond hierarchical lattice: Numerically exact treatment in the large-qlimit

We consider the critical behavior of the random $q$-state Potts model in the large-$q$ limit with different types of disorder leading to either the nonfrustrated random ferromagnet regime or the frustrated spin-glass regime. The model is studied on the diamond hierarchical lattice for which the Migdal-Kadanoff real-space renormalization is exact. It is shown to have a ferromagnetic and a paramagnetic phase and the phase transition is controlled by four different fixed points. The state of the system is characterized by the distribution of the interface free energy $P(I)$ which is shown to satisfy different integral equations at the fixed points. By numerical integration we have obtained the corresponding stable laws of nonlinear combination of random numbers and obtained numerically exact values for the critical exponents.

Mechanism of Slow Relaxation due to Screening Effect in a Frustrated System

We study a slow relaxation process in a frustrated spin system in which a type of screening effect due to a frustrated environment plays an important role. This screening effect is attributed to the highly degenerate configurations of the frustrated environment. This slow relaxation is due to an entropy effect and is different from those due to the energy barrier observed in systems such as random ferromagnets. In the present system, even if there is no energy gap, the slow relaxation still takes place. Thus, we call this phenomenon ``entropic slowing down''. Here, we study the mechanism of entropic slowing down quantitatively in an Ising spin model with the so-called decorated bonds. The spins included in decorated bonds (decoration spins) cause a peculiar density of states, which causes on entropy-induced screening effect. We analytically estimate the time scale of the system that increases exponentially with the number of decoration spins. We demonstrate the scaling of relaxation processes using the time scale at all temperatures including the critical point.

Persistence and memory in patchwork dynamics for glassy models

Slow dynamics in disordered materials prohibits direct simulation of their rich nonequilibrium behavior at large scales. ``Patchwork dynamics'' is introduced to mimic relaxation over a very broad range of time scales by directly equilibrating or optimizing on successive length scales. This dynamics is used to study coarsening and to replicate memory effects for spin glasses and random ferromagnets. It is also used to find, with high confidence, correct ground states in large toroidal samples.

Non-equilibrium dynamics of polymers and interfaces in random media: conjecture ψ = ds/2 for the barrier exponent

We consider various random models (directed polymer, random ferromagnets, spin glasses) in their disorder-dominated phases, where the free-energy cost F(L) of an excitation of length L presents fluctuations that grow as a power-law ΔF(L) ∼ Lθ with the ‘droplet’ exponent θ. Within the droplet theory, the energy and entropy of such excitations present fluctuations that grow as where ds is the dimension of the surface of the excitation. These systems usually present a positive ‘chaos’ exponent ζ = ds/2 − θ > 0, meaning that the free-energy fluctuation of order Lθ is a near cancellation of much bigger energy and entropy fluctuations of order . Within the standard droplet theory, the dynamics is characterized by a barrier exponent ψ satisfying the bounds θ ⩽ ψ ⩽ d − 1. In this paper, we argue that a natural value for this barrier exponent is ψ = ds/2: (i) for the directed polymer where ds = 1, this corresponds to ψ = 1/2 in all dimensions; (ii) for disordered ferromagnets where ds = d − 1, this corresponds to ψ = (d − 1)/2; (iii) for spin glasses where interfaces have a non-trivial dimension ds known numerically, our conjecture ψ = ds/2 gives numerical predictions in d = 2 and d = 3. We compare these values with the available numerical results for each case, in particular with the measure ψ ≃ 0.49 of Kolton–Rosso–Giamarchi (Kolton et al 2005 Phys. Rev. Lett. 95 180604) for the non-equilibrium dynamics of a directed elastic string.

Superaging in two-dimensional random ferromagnets

We study the aging properties, in particular the two-time autocorrelations, of the two-dimensional randomly diluted Ising ferromagnet below the critical temperature via Monte Carlo simulations. We find that the autocorrelation function displays additive aging C(t,t{w})=C{st}(t)+C{ag}(t,t{w}), where the stationary part Cst} decays algebraically. The aging part shows anomalous scaling C{ag}(t,t{w})=C[h(t)h(t{w})], where h(u) is a nonhomogeneous function excluding a t/t{w} scaling.

NONEQUILIBRIUM PHASE TRANSITIONS IN MODEL FERROMAGNETS: A REVIEW

The thermodynamical behaviors of ferromagnetic systems in equilibrium are well studied. However, the ferromagnetic systems far from equilibrium became an interesting field of research in last few decades. Recent exploration of ferromagnetic systems in the presence of a steady magnetic field are also studied by using standard tools of equilibrium statistical physics. The ferromagnet in the presence of time-dependent magnetic field, shows various interesting phenomena. An usual response of a ferromagnet in the presence of a sinusoidally oscillating magnetic field is the hysteresis. Apart from this hysteretic response, the nonequilibrium dynamic phase transition is also a very interesting phenomenon. In this chapter, the nonequilibrium dynamic phase transitions of the model ferromagnetic systems in presence of time-dependent magnetic field are discussed. For this kind of nonequilibrium phase transition, one cannot employ the standard techniques of equilibrium statistical mechanics. The recent developments in this direction are mainly based on numerical simulation (Monte Carlo). The Monte Carlo simulation of kinetic Ising model, in presence of sinusoidally oscillating (in time but uniform over space) magnetic field, is extensively performed to study the nonequilibrium dynamic phase transition. The temperature variations of dynamic order parameter, dynamic specific heat, dynamic relaxation time etc. near the transition point are discussed. The appearance and behaviors of a dynamic length scale and a dynamic time scale near the transition point are also discussed. All these studies indicate that this proposed dynamic transition is a nonequilibrium thermodynamic phase transition. The disorder (quenched) induced zero temperature (athermal) dynamic transition is studied in random field Ising ferromagnet. The dynamic transition in the Heisenberg ferromagnet is also studied. The nature of this transition in the Heisenberg ferromagnet depends on the anisotropy and the polarisation of the applied time varying magnetic field. The anisotropic Heisenberg ferromagnet in the presence of elliptically polarised magnetic field shows multiple dynamic transitions. This multiple dynamic transitions in anisotropic Heisenberg ferromagnet are discussed here. Recent experimental evidences of dynamic transitions are also discussed very briefly.

Growing Length Scales during Aging in 2dDisordered Systems

The non-equilibrium dynamics of three paradigmatic models for two-dimensional systems with quenched disorder is studied with a focus on the existence and analysis of a growing length scale during aging at low temperatures: 1) The random bond Ising ferromagnet, 2) the Edwards-Anderson model for a spin glas, 3) the solid-on-solid model on a disordered substrate (equivalent to the sine-Gordon model with random phase shifts). Interestingly, we find in all three models a length scale that grows algebraically with time (up to the system size in cases 1 and 3, up to the finite equilibrium length in case 2) with a temperature dependent growth exponent. Whereas in cases 1 and 2 this length scale characterizes a coarsening process, it represents in case 3 the growing size of fluctuations during aging.

One-phonon assisted energy transfer in periodic arrays of optically active ions

Energy transfer in periodic arrays of optically active ions is investigated. It is assumed that the transfer of excitation between ions takes place incoherently via a one-phonon process where the wavelength of the phonon is large in comparison with the inter-ion spacing. It is pointed out that the energy transfer can only occur where there is a mismatch in energy between the excited states of the two ions. The energy diffusion constant is calculated in an effective medium approximation assuming a Gaussian lineshape with no correlation between the excited state energies of neighboring ions. The temperature dependence of the diffusion constant is established for cubic lattices with nearest-neighbor transfer. Energy transfer in one-dimensional arrays with nearest-neighbor transfer is studied using an extension of the coherent exchange approximation developed for random ferromagnets. The coherent exchange approximation reproduces the asymptotic non-diffusive behavior found in numerical studies of fluorescence line narrowing in one-dimensional systems.

Is there a spin-glass phase in the random temperature Ising ferromagnet?

In this paper we study the phase diagram of the disordered Ising ferromagnet. Within the framework of the Gaussian variational approximation it is shown that in systems with a finite value of the disorder in dimensions D=4 and D < 4 the paramagnetic and ferromagnetic phases are separated by a spin-glass phase. The transition from paramagnetic to spin-glass state is continuous (second-order), while the transition between spin-glass and ferromagnetic states is discontinuous (first-order). It is also shown that within the considered approximation there is no replica symmetry breaking in the spin-glass phase. The validity of the Gaussian variational approximation for the present problem is discussed, and we provide a tentative physical interpretation of the results.

Freezing in random graph ferromagnets.

Using T=0 Monte Carlo and simulated annealing simulation, we study the energy relaxation of ferromagnetic Ising and Potts models on random graphs. In addition to the expected exponential decay to a zero energy ground state, a range of connectivities for which there is power law relaxation and freezing to a metastable state is found. For some connectivities this freezing persists even using simulated annealing to find the ground state. The freezing is caused by dynamic frustration in the graphs, and is a feature of the local search nature of the Monte Carlo dynamics used. The implications of the freezing on agent-based complex system models are briefly considered.

Magnon-fracton crossover in quenched random site-diluted ferromagnets

The strong departure from the Bloch ${T}^{3/2}$ law behavior of magnetization in dilute amorphous ${(T}_{p}{\mathrm{Ni}}_{1\ensuremath{-}p}{)}_{80}{\mathrm{B}}_{16}{\mathrm{Si}}_{4}$ $(T=\mathrm{Fe},\mathrm{Co})$ alloys is shown to be a manifestation of the crossover from hydrodynamic to critical spin wave dynamics induced by the diverging correlation length near percolation threshold. An appropriate choice of the density of states for magnetic excitations on self-similar (fractal) percolation network permits an accurate determination of the magnon-to-fracton crossover line in the magnetic phase diagram of quenched random site-diluted ferromagnets. By unambiguously demonstrating that the fracton dimensionality ${d}_{f}\ensuremath{\simeq}4/3$ and the conductivity percolation critical exponent ${\ensuremath{\sigma}}_{p}&lt;2,$ the present results vindicate the Alexander-Orbach conjecture and the Golden inequality for percolating network with Euclidean dimension $d=3.$

Self-organized criticality in the hysteresis of the Sherrington–Kirkpatrick model

We study hysteretic phenomena in random ferromagnets. We argue that the angle-dependent magnetostatic (dipolar) terms introduce frustration and long-range interactions in these systems. This makes it plausible that the Sherrington–Kirkpatrick model may be able to capture some of the relevant physics of these systems. We use scaling arguments, replica calculations and large scale numerical simulations to characterize the hysteresis of the zero temperature SK model. By constructing the distribution functions of the avalanche sizes, magnetization jumps and local fields, we conclude that the system exhibits self-organized criticality everywhere on the hysteresis loop.

Numerical Renormalization Group Study of Random Transverse Ising Models in One and Two Space Dimensions

The quantum critical behavior and the Griffiths-McCoy singularities of random quantum Ising ferromagnets are studied by applying a numerical implementation of the Ma-Dasgupta-Hu renormalization group scheme. We check the procedure for the analytically tractable one-dimensional case and apply our code to the quasi-one-dimensional double chain. For the latter we obtain identical critical exponents as for the simple chain implying the same universality class. Then we apply the method to the two-dimensional case for which we get estimates for the exponents that are compatible with a recent study in the same spirit.

Self-Organized Criticality in the Hysteresis of the Sherrington-Kirkpatrick Model

We study hysteretic phenomena in random ferromagnets. We argue that the angle dependent magnetostatic (dipolar) terms introduce frustration and long range interactions in these systems. This makes it plausible that the Sherrington - Kirkpatrick model may be able to capture some of the relevant physics of these systems. We use scaling arguments, replica calculations and large scale numerical simulations to characterize the hysteresis of the zero temperature SK model. By constructing the distribution functions of the avalanche sizes, magnetization jumps and local fields, we conclude that the system exhibits self-organized criticality everywhere on the hysteresis loop.

Application of a continuous time cluster algorithm to the two-dimensional random quantum Ising ferromagnet

A cluster algorithm formulated in continuous (imaginary) time is presented for Ising models in a transverse field. It works directly with an infinite number of time-slices in the imaginary time direction, avoiding the necessity to take this limit explicitly. The algorithm is tested at the zero-temperature critical point of the pure two-dimensional (2d) transverse Ising model. Then it is applied to the 2d Ising ferromagnet with random bonds and transverse fields, for which the phase diagram is determined. Finite size scaling at the quantum critical point as well as the study of the quantum Griffiths-McCoy phase indicate that the dynamical critical exponent is infinite as in 1d.

Comment on “Order-Disorder Transition in an External Field in Random Ferromagnets and Nematic Elastomers”

A Comment on the Letter by S. V. Fridrikh and E. M. Terentjev, Phys. Rev. Lett. 79, 4661 (1997). The authors of the Letter offer a Reply.Received 30 November 1998DOI:https://doi.org/10.1103/PhysRevLett.82.3380©1999 American Physical Society

Microscopic model of relaxor phenomena in Pb containing mixed oxides

The pulsed neutron atomic pair-density function (PDF) analysis of Pb containing ferroelectric oxides suggests that the local structure around Pb ion is surprisingly independent of composition and crystal structure, and has a large electric dipole due to off-centering of Pb in the PbO12 cluster. This leads to a picture of interacting local dipolar moments through dipolar and strain fields. Randomness of the interaction and the local strain fields due to mixed B-site ions can result in the relaxor behavior. Drawing upon the knowledge on the random anisotropy ferromagnet and glass transition of atomic glasses the origin of the relaxor ferroelectric phenomena will be discussed.

Griffiths singularities and the replica instantons in the random ferromagnet

The problem of existence of non-analytic (Griffith-like) contributions to the free energy of weakly disordered Ising ferromagnet is studied from the point of view of the replica theory. The consideration is done in terms of the usual random temperature Ginzburg-Landau Hamiltonian in space dimensions $D < 4$ in the zero external magnetic field. It is shown that in the paramagnetic phase, at temperatures not too close to $T_{c}$ (where the behaviour of the pure system is correctly described by the Gaussian approximation), the free energy of the system has additional non-perturbative contribution of the form $\exp{-(const) \frac{\tau^{(4-D)/2}}{u}}$ (where $\tau = (T-T_{c})/T_{c}$), which has essential singularity in the parameter $u \to 0$ which describes the strength of the disorder. It is demonstrated that this contribution appears due to non-linear localized (instanton-like) solutions of the mean-field stationary equations which are characterized by the special type of the replica symmetry breaking . It is argued that physically these replica instantons describe the contribution from rare spatial "ferromagnetic islands" in which local (random) temperature is below $T_{c}$

Critical Behavior and Griffiths-McCoy Singularities in the Two-Dimensional Random Quantum Ising Ferromagnet

We study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one-dimension. At the critical point, the dynamical exponent is infinite and the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point there are Griffiths-McCoy singularities, characterized by a single, continuously varying exponent, z', which diverges at the critical point, as in one-dimension. Consequently, the zero temperature susceptibility diverges for a RANGE of parameters about the transition.

A non-equilibrium percolation transition in random Ising ferromagnets

We study the geometrical properties of the domain structure that results when a random Ising ferromagnet is quenched to zero temperature. We define a novel connectivity transition between like spins in the frozen configuration, and show that it is in a new universality class, distinct from that of random percolation. The scaling is robust and independent of model details. The relevance of this transition to spinodal decomposition is discussed.