We apply to the Random Field Ising Model at zero temperature (T= 0) the perturbative loop expansion around the Bethe solution. A comparison with the standard epsilon-expansion is made, highlighting the key differences that make the new expansion much more appropriate to correctly describe strongly disordered systems, especially those controlled by a T = 0 RG fixed point. This new loop expansion produces an effective theory with cubic vertices. We compute the one-loop corrections due to cubic vertices, finding new terms that are absent in the epsilon-expansion. However, these new terms are subdominant with respect to the standard, supersymmetric ones, therefore dimensional reduction is still valid at this order of the loop expansion.
We present exact expressions for hysteresis loops in the ferromagnetic random field Ising model in the limit of zero temperature and zero driving frequency for an arbitrary initial state of the model on a Bethe lattice. This work extends earlier results that were restricted to an initial state with all spins pointing parallel to each other.
In the simple [hyper]cubic five dimension near neighbor interaction Ising ferromagnet, extensive simulation measurements are made of the link overlap and the spin overlap distributions. These "two replica" measurements are standard in the Spin Glass context but are not usually recorded in ferromagnet simulations. The moments and moment ratios of these distributions (the variance, the kurtosis and the skewness) show clear critical behaviors at the known ordering temperature of the ferromagnet. Analogous overlap effects can be expected quite generally in Ising ferromagnets in any dimension. The …
In the paper the Curie temperatures of selectively diluted planar Ising ferromagnet on the triangular lattice are calculated vs. concentration of magnetic atoms. Various analytical approaches are compared with the exact numerical calculations for finite clusters, as well as with the exact analytical solutions for the triangular and honeycomb lattices.
We study the free energy distribution function of weakly disordered Ising ferromagnet in terms of the D-dimensional random temperature Ginzburg-Landau Hamiltonian. It is shown that besides the usual Gaussian "body" this distribution function exhibits non-Gaussian tails both in the paramagnetic and in the ferromagnetic phases. Explicit asymptotic expressions for these tails are derived. It is demonstrated that the tails are strongly asymmetric: the left tail (for large negative values of the free energy) is much more slow than the right one (for large positive values of the free energy). It is …
We study hysteresis in anti-ferromagnetic random-field Ising model at zero temperature. The external field is cycled adiabatically between -$\infty$ and $\infty$. Two different distributions of the random-field are considered, (i) a uniform distribution of width $2Δ$ centered at the origin, and (ii) a Gaussian distribution with average value zero and standard deviation $σ$. In each case the hysteresis loop is determined exactly in one dimension and compared with numerical simulations of the model.
We investigate the ferromagnetic-glassy transitions which separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice +-J (Edwards-Anderson) Ising model with bond distribution $P(J) = p δ(J - 1) + (1-p) δ(J + 1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly re…
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 Grif…
A ferromagnetic Ising chain which is endowed with a single-spin-flip Glauber dynamics is investigated. For an arbitrary annealing protocol, we derive an exact integral equation for the domain wall density. This integral equation admits an asymptotic solution in the limit of extremely slow cooling. For instance, we extract an asymptotic of the density of domain walls at the end of the cooling procedure when the temperature vanishes. Slow annealing is usually studied using a Kibble-Zurek argument; in our setting, this argument leads to approximate predictions which are in good agreement with exa…
The explicit form of the Griffiths singularity in the random ferromagnetic Ising model in external magnetic field is derived. In terms of the continuous random temperature Ginzburg-Landau Hamiltonian it is shown that in the paramagnetic phase away from the critical point the free energy as the function of the external magnetic field h in the limit h -> 0 has the essential singularity of the form $\exp[-(const)h^{-D/3}]$ (where 1<D<4 is the space dimensionality). It is demonstrated that in terms of the replica formalism this contribution to the free energy comes due to off-perturbative replica …
We study the $\pm J$ three-dimensional Ising model with a spatially uniaxially anisotropic bond randomness on the simple cubic lattice. The $\pm J$ random exchange is applied in the $xy$ planes, whereas in the z direction only a ferromagnetic exchange is used. After sketching the phase diagram and comparing it with the corresponding isotropic case, the system is studied, at the ferromagnetic-paramagnetic transition line, using parallel tempering and a convenient concentration of antiferromagnetic bonds ($p_z=0 ; p_{xy}=0.176$). The numerical data point out clearly to a second-order ferromagnet…
It is shown by Monte Carlo method that the finite size scaling (FSS) holds in the two dimensional random-coupled Ising ferromagnet. It is also demonstrated that the form of universal FSS function constructed via novel FSS scheme depends on the strength of the random coupling for strongly disordered cases. Monte Carlo measurements of thermodynamic (infinite volume limit) data of the correlation length ($ξ$) up to $ξ\simeq 200$ along with measurements of the fourth order cumulant ratio (Binder's ratio) at criticality are reported and analyzed in view of two competing scenarios. It is demonstrate…
The random field Ising model driven by a slowly varying uniform external field at zero temperature provides a caricature of several threshold activated systems. In this model, the non-equilibrium response of the system can be obtained analytically on a Bethe lattice if the initial state of the system has all spins aligned parallel to each other. We consider ferromagnetic as well as anti-ferromagnetic interactions. The ferromagnetic model exhibits avalanches and non-equilibrium critical behavior. The anti-ferromagnetic model is marked by the absence of these features. The ferromagnetic model is…
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 dive…
In this Monte Carlo study we concentrated on the influence of non-magnetic impurities arranged as the lines with random orientation on paramagnetic-to-ferromagnetic phase transition in the 3D Ising model. Special emphasize is given to the long-distance decay of the impurity-impurity pair correlation function. It is shown that for the lattice sizes considered (L=10-96) and for the two different impurity distributions (purely random and mutually avoiding lines) the function is governed by the power law of 1/r^a with an universal exponent a \simeq 2. This result supports our findings about the nu…
We discuss analytical approximation schemes for the dynamics of diluted spin models. The original dynamics of the complete set of degrees of freedom is replaced by a hierarchy of equations including an increasing number of global observables, which can be closed approximately at different levels of the hierarchy. We illustrate this method on the simple example of the Ising ferromagnet on a Bethe lattice, investigating the first three possible closures, which are all exact in the long time limit, and which yield more and more accurate predictions for the finite-time behavior. We also investigat…
The critical behaviour of three-dimensional semi-infinite Ising ferromagnets at planar surfaces with (i) random surface-bond disorder or (ii) a terrace of monatomic height and macroscopic size is considered. The Griffiths-Kelly-Sherman correlation inequalities are shown to impose constraints on the order-parameter density at the surface, which yield upper and lower bounds for the surface critical exponent $β_1$. If the surface bonds do not exceed the threshold for supercritical enhancement of the pure system, these bounds force $β_1$ to take the value $β_1^{ord}$ of the latter system's ordinar…
We study sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework we derive explicit expressions for the probability distribution function of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energies is Gaussian, and the typical sample-to-sample fluctuations as well as the average value scale with the system size $L$ like $\sim L \ln\ln(L)$. In contrast, the specific heat is shown to be self-ave…
We consider the behavior of an Ising ferromagnet obeying the Glauber dynamics under the influence of a fast switching, random external field. Analytic results for the stationary state are presented in mean-field approximation, exhibiting a novel type of first order phase transition related to dynamic freezing. Monte Carlo simulations performed on a quadratic lattice indicate that many features of the mean field theory may survive the presence of fluctuations.
We consider the behavior of an Ising ferromagnet obeying the Glauber dynamics under the influence of a fast switching, random external field. In Part I, we introduced a general formalism for describing such systems and presented the mean field theory. In this article we derive results for the one dimensional case, which can be only partially solved. Monte Carlo simulations performed on a square lattice indicate that the main features of the mean field theory survive the presence of strong fluctuations.
We study the second-moment correlation length and the reduced susceptibility of two ferromagnetic Ising models with zero-temperature ordering. By introducing a scaling variable motivated by high-temperature series expansions, we are able to scale data for the one-dimensional Ising ferromagnet rigorously over the entire temperature range. Analogous scaling expressions are then applied to the two-dimensional fully frustrated Villain model where excellent finite-size scaling over the entire temperature range is achieved. Thus we broaden the applicability of the extended scaling method to Ising sy…
We investigate the effects of quenched bond randomness on the critical properties of the two-dimensional ferromagnetic Ising model embedded in a triangular lattice. The system is studied in both the pure and disordered versions by the same efficient two-stage Wang-Landau method. In the first part of our study we present the finite-size scaling behavior of the pure model, for which we calculate the critical amplitude of the specific heat's logarithmic expansion. For the disordered system, the numerical data and the relevant detailed finite-size scaling analysis along the lines of the two well-k…
We study the zero temperature static properties of dissipative ensembles of quantum Ising spins arranged on periodic one dimensional finite clusters and on an infinite chain. The spins interact ferro-magnetically with nearest-neighbour pure and random couplings. They are subject to a transverse field and coupled to an Ohmic bath of quantum harmonic oscillators. We analyze the coupled system using Monte Carlo simulations of the classical two-dimensional counterpart model. The coupling to the bath enhances the extent of the ordered phase, as found in mean-field spin-glasses. In the case of finit…
Frequency evolutions of hysteresis loop area and hysteresis tools such as remanence and coercivity of a kinetic Ising model in the presence of quenched bond dilution are investigated in detail. The kinetic equation describing the time dependence of the magnetization is derived by means of effective-field theory with single-site correlations. It is found that the frequency dispersions of hysteresis loop area, remanence and coercivity strongly depend on the quenched bond randomness, as well as applied field amplitude and oscillation frequency. In addition, the shape of the hysteresis curves for …
We consider the behavior of an Ising ferromagnet obeying the Glauber dynamics under the influence of a fast switching, random external field. After introducing a general formalism for describing such systems, we consider here the mean-field theory. A novel type of first order phase transition related to spontaneous symmetry breaking and dynamic freezing is found. The non-equilibrium stationary state has a complex structure, which changes as a function of parameters from a singular-continuous distribution with Euclidean or fractal support to an absolutely continuous one.
We introduce transverse ferromagnetic interactions, in addition to a simple transverse field, to quantum annealing of the random-field Ising model to accelerate convergence toward the target ground state. The conventional approach using only the transverse-field term is known to be plagued by slow convergence when the true ground state has strong ferromagnetic characteristics for the random-field Ising model. The transverse ferromagnetic interactions are shown to improve the performance significantly in such cases. This conclusion is drawn from the analyses of the energy eigenvalues of instant…
Hamiltonians for general multi-state spin-glass systems with Ising symmetry are derived for both sequential and synchronous updating of the spins. The possibly different behaviour caused by the way of updating is studied in detail for the (anti)-ferromagnetic version of the models, which can be solved analytically without any approximation, both thermodynamically via a free-energy calculation and dynamically using the generating functional approach. Phase diagrams are discussed and the appearance of two-cycles in the case of synchronous updating is examined. A comparative study is made for the…
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-…
The Dyson hierarchical version of the quantum Ising chain with Long-Ranged power-law ferromagnetic couplings $J(r) \propto r^{-1-σ}$ and pure or random transverse fields is studied via real-space renormalization. For the pure case, the critical exponents are explicitly obtained as a function of the parameter $σ$, and are compared with previous results of other approaches. For the random case, the RG rules are numerically applied and the critical behaviors are compared with previous Strong Disorder Renormalization results.
With up to 7 million spins, the existence of spontaneous magnetization of Ising spins on directed Barabasi-Albert networks is investigated by Monte Carlo simulations. We confirm our earlier result that the magnetization for different temperatures T decays after a characteristic time tau(T), which we extrapolate to diverge at zero temperature by a modified Arrhenius law,or perhaps a power law.
We check the existence of a spontaneous magnetisation of Ising and Potts spins on semi-directed Barabasi-Albert networks by Monte Carlo simulations. We verified that the magnetisation for different temperatures $T$ decays after a characteristic time $τ(T)$, which we extrapolate to diverge at positive temperatures $T_c(N)$ by a Vogel-Fulcher law, with $T_c(N)$ increasing logarithmically with network size $N$.
As a result of the interplay between the intrinsic off-diagonal terms of the dipolar interaction and an applied {\it transverse} field $H_t$, the diluted ${\rm LiHo_xY_{1-x}F_4}$ system at x$> 0.5$ is equivalent to a ferromagnet in a longitudinal random field (RF). At low $H_t$ the quantum fluctuations between the Ising like doublet states are negligible, while the effective induced RF is appreciable. This results in a practically exact equivalence to the classical RF Ising model. By tuning $H_t$, the applied longitudinal field, and the dilution, the Ising model can be realized in the presence…
We study the Glauber dynamics of Ising spin models with random bonds, on finitely connected random graphs. We generalize a recent dynamical replica theory with which to predict the evolution of the joint spin-field distribution, to include random graphs with arbitrary degree distributions. The theory is applied to Ising ferromagnets on randomly diluted Bethe lattices, where we study the evolution of the magnetization and the internal energy. It predicts a prominent slowing down of the flow in the Griffiths phase, it suggests a further dynamical transition at lower temperatures within the Griff…
We consider the random-bond +- J Ising model on a square lattice as a function of the temperature T and of the disorder parameter p (p=1 corresponds to the pure Ising model). We investigate the critical behavior along the paramagnetic-ferromagnetic transition line at low temperatures, below the temperature of the multicritical Nishimori point at T*= 0.9527(1), p*=0.89083(3). We present finite-size scaling analyses of Monte Carlo results at two temperature values, T=0.645 and T=0.5. The results show that the paramagnetic-ferromagnetic transition line is reentrant for T<T*, that the transitions …
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: the dynamical exponent is infinite and, at the critical point, the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point, there may be different exponents for the divergence of the average and typical correlation lengths, again as in one-dimension, but the evidence for this is less strong.
A Green's function formalism is used to calculate the spectrum of localized modes of an impurity layer implanted within a ferromagnetic thin film. The equations of motion for the Green's functions are determined in the framework of the Ising model in a transverse field. We show that depending on the thickness, exchange and effective field parameters, there is a ``crossover'' effect between the surface modes and impurity localized modes. For thicker films the results show that the degeneracy of the surface modes can be lifted by the presence of an impurity layer.
We use the replica method in order to obtain an expression for the variational free energy of an Ising ferromagnet on a Viana-Bray lattice in the presence of random external fields. Introducing a global order parameter, in the replica-symmetric context, the problem is reduced to the analysis of the solutions of a nonlinear integral equation. At zero temperature, and under some restrictions on the form of the random fields, we are able to perform a detailed analysis of stability of the replica-symmetric solutions. In contrast to the behaviour of the Sherrington-Kirkpatrick model for a spin glas…
The four-dimensional +-J random-bond Ising model is studied using ground-state calculations. System sizes up to N=6^4 spins are considered. Here it is found that the ferromagnetic-spin glass transition occurs at a critical concentration p_c=0.28(1) of the antiferromagnetic bonds, which is comparable to values found previously by high-temperature series expansions. The transition is characterized by a correlation-length exponent ν=1.0(1) and an order-parameter exponent β=0.4(1). Thus, this transition is in a different universality class from four-dimensional bond percolation, where ν=0.678(50) …
We consider the stochastic dynamics near zero-temperature of the random ferromagnetic Ising model on a Cayley tree of branching ratio $K$. We apply the Boundary Real Space Renormalization procedure introduced in our previous work (C. Monthus and T. Garel, J. Stat. Mech. P02037 (2013)) in order to derive the renormalization rule for dynamical barriers. We obtain that the probability distribution $P_n(B)$ of dynamical barrier for a subtree of $n$ generations converges for large $n$ towards some traveling-wave $P_n(B) \simeq P^*(B-nv) $, i.e. the width of the probability distribution remains fini…
The non-equilibrium dynamics of the strongly diluted random-bond Ising model in two-dimensions (2d) is investigated numerically. The persistence probability, P(t), of spins which do not flip by time t is found to decay to a non-zero, dilution-dependent, value $P(\infty)$. We find that $p(t)=P(t)-P(\infty)$ decays exponentially to zero at large times. Furthermore, the fraction of spins which never flip is a monotonically increasing function over the range of bond-dilution considered. Our findings, which are consistent with a recent result of Newman and Stein, suggest that persistence in disorde…
We revisit classical bounds of M. E. Fisher on the ferromagnetic Ising model, and show how to efficiently use them on an arbitrary given graph to rigorously upper-bound the partition function, magnetizations, and correlations. The results are valid on any finite graph, with arbitrary topology and arbitrary positive couplings and fields. Our results are based on high temperature expansions of the aforementioned quantities, and are expressed in terms of two related linear operators: the non-backtracking operator and the Bethe Hessian. As a by-product, we show that in a well-defined high-temperat…
