Random Field Ising Chain Research Articles

Infinite disorder renormalization fixed point for the continuum random field Ising chain

Infinite disorder renormalization fixed point for the continuum random field Ising chain

Random Ising chain in transverse and longitudinal fields: Strong disorder RG study

Motivated by the compound LiHoxY1-xF4, we consider the Ising chain with random couplings and in the presence of simultaneous random transverse and longitudinal fields, and study its low-energy properties at zero temperature by the strong disorder renormalization group approach. In the absence of longitudinal fields, the system exhibits a quantum-ordered and a quantum-disordered phase separated by a critical point of infinite disorder. When the longitudinal random field is switched on, the ordered phase vanishes and the trajectories of the renormalization group are attracted to two disordered fixed points: one is characteristic of the classical random field Ising chain, the other describes the quantum disordered phase. The two disordered phases are separated by a separatrix that starts at the infinite disorder fixed point and near which there are strong quantum fluctuations.

Rounding of a first-order quantum phase transition to a strong-coupling critical point

We investigate the effects of quenched disorder on first-order quantum phase transitions on the example of the $N$-color quantum Ashkin-Teller model. By means of a strong-disorder renormalization group, we demonstrate that quenched disorder rounds the first-order quantum phase transition to a continuous one for both weak and strong coupling between the colors. In the strong-coupling case, we find a distinct type of infinite-randomness critical point characterized by additional internal degrees of freedom. We investigate its critical properties in detail and find stronger thermodynamic singularities than in the random transverse field Ising chain. We also discuss the implications for higher spatial dimensions as well as unusual aspects of our renormalization-group scheme.

Transfer operator analysis of the parallel dynamics of disordered Ising chains

We study the synchronous stochastic dynamics of the random field and random bond Ising chain. For this model the generating functional analysis method of De Dominicis leads to a formalism with transfer operators, similar to transfer matrices in equilibrium studies, but with dynamical paths of spins and (conjugate) fields as arguments, as opposed to replicated spins. In the thermodynamic limit the macroscopic dynamics is captured by the dominant eigenspace of the transfer operator, leading to a relatively simple and transparent set of equations that are easy to solve numerically. Our results are supported excellently by numerical simulations.

Continuous approach for the random-field Ising chain

We study the random-field Ising chain in the limit of strong exchange coupling. In order to calculate the free energy we apply a continuous Langevin-type approach. This continuous model can be solved exactly, whereupon we are able to locate the crossover between an exponential and a power-law decay of the free energy with increasing coupling strength. In terms of magnetization, this crossover restricts the validity of the linear scaling. The known analytical results for the free energy are recovered in the corresponding limits. The outcomes of numerical computations for the free energy are presented, which confirm the results of the continuous approach. We also discuss the validity of the replica method which we then utilize to investigate the sample-to-sample fluctuations of the finite size free energy.

Exact Ground-State Energies of the Random-Field Ising Chain and Ladder

We derive the exact ground-state energy of the one-dimensional Ising model in random fields taking values h, 0 and -h with general probabilities. The random-field Ising model on a ladder is also analyzed by showing its equivalence to the random-field Ising chain with field values h, -h and 0 for h<J. The zero-temperature transger matrix is used to obtain the results.

Low-temperature properties of some disordered systems from the statistical properties of nearly degenerate two-level excitations

The thermal fluctuations that exist at very low temperature in disordered systems are often attributed to the existence of some two-level excitations. In this paper, we revisit this question via the explicit studies of the following 1D models (i) a particle in 1D random potentials (ii) the random field Ising chain with continuous disorder distribution. In both cases, we define precisely the ‘two-level’ excitations and their statistical properties, and we show that their contributions to various observables are in full agreement at low temperature with the the rigorous results obtained independently. The statistical properties of these two-level excitations moreover yield simple identities at order T in temperature for some generating functions of thermal cumulants. For the random-field Ising chain, in the regime where the Imry-Ma length is large, we obtain that the specific heat is dominated by small non-universal excitations, that depend on the details of the disorder distribution, whereas the magnetic susceptibility and the Edwards-Anderson order parameter are dominated by universal large excitations, whose statistical properties only depend on the variance of the initial disorder via the Imry-Ma length.

Exact Ground-State Energies of the Random-Field Ising Chain and Ladder

We derive the exact ground-state energy of the one-dimensional Ising model in random fields taking values h , 0 and - h with general probabilities. The random-field Ising model on a ladder is also analyzed by showing its equivalence to the random-field Ising chain with field values ± h and 0 for h < J . The zero-temperature transfer matrix is used to obtain the results.

Convolution of multifractals and the local magnetization in a random-field Ising chain

The local magnetization in the one-dimensional random-field Ising model is essentially the sum of two effective fields with multifractal probability measure. The probability measure of the local magnetization is thus the convolution of two multifractals. In this paper we prove relations between the multifractal properties of two measures and the multifractal properties of their convolution. The pointwise dimension at the boundary of the support of the convolution is the sum of the pointwise dimensions at the boundary of the support of the convoluted measures and the generalized box dimensions of the convolution are bounded from above by the sum of the generalized box dimensions of the convoluted measures. The generalized box dimensions of the convolution of Cantor sets with weights can be calculated analytically for certain parameter ranges and illustrate effects we also encounter in the case of the measure of the local magnetization. Returning to the study of this measure we apply the general inequalities and present numerical approximations of the D_q-spectrum. For the first time we are able to obtain results on multifractal properties of a physical quantity in the one-dimensional random-field Ising model which in principle could be measured experimentally. The numerically generated probability densities for the local magnetization show impressively the gradual transition from a monomodal to a bimodal distribution for growing random field strength h.

Ground states versus low-temperature equilibria in random field Ising chains

We discuss with the aid of random walk arguments and exact numerical computations the magnetization properties of one-dimensional random field chains. The ground state structure is explained in terms of absorbing and non-absorbing random walk excursions. At low temperatures, the magnetization profiles follow those of the ground states except at regions where a local random field fluctuation makes thermal excitations feasible. This follows also from the non-absorbing random walks, and implies that the magnetization length scale is a product of these two scales. It is not simply given by the Imry-Ma-like ground state domain size nor by the scale of the thermal excitations.

Random field Ising chains with synchronous dynamics

We solve a class of attractor neural network models with a mixture of 1D nearest-neighbour and infinite-range interactions, which are of a Hebbian-type form. Our solution is based on a combination of mean-field methods, transfer matrices and 1D random-field techniques, and is obtained for Boltzmann-type equilibrium (following sequential Glauber dynamics) and Peretto-type equilibrium (following parallel dynamics). Competition between the alignment forces mediated via short-range interactions, and those mediated via infinite-range ones, is found to generate novel phenomena, such as multiple locally stable `pure' states, first-order transitions between recall states, 2-cycles and non-recall states, and domain formation leading to extremely long relaxation times. We test our results against numerical simulations and simple benchmark cases and find excellent agreement.

Quantum Ising model in a transverse random field: A density-matrix renormalization-group analysis

The spin-1/2 quantum Ising chain in a transverse random magnetic field is studied by means of the density-matrix renormalization group. The system evolves from an ordered to a paramagnetic state as the amplitude of the random field is increased. The dependence of the magnetization on a uniform magnetic field in the z direction and the spontaneous magnetization as a function of the amplitude of the transverse random magnetic field are determined. The behavior of the spin-spin correlation function both above and at criticality is studied. The scaling laws for magnetization and correlation functions are tested against previous numerical and renormalization-group results.

Random S = ½ XY chains and the theory of quasi-one-dimensional ferroelectrics with hydrogen bonds

The paper presents a new numerical approach for studying the thermodynamical and dynamical properties of finite spin-1/2 XY chains. Special attention is given to examining the influence of disorder on the average transverse dynamical susceptibility of Ising chain in random transverse field.

Scaling behaviour of the correlation length for the two-point correlation function in the random field Ising chain

We study the general behaviour of the correlation length for the two-point correlation function of the local fields in an Ising chain with binary distributed fields. At zero field it is shown that is the same as the zero-field correlation length for the spin - spin correlation function. For the field-dominated behaviour of we find an exponent for the power-law divergence which is smaller than the exponent for the spin - spin correlation length. The entire behaviour of the correlation length can be described by a single crossover scaling function involving the new critical exponent.

Statistical properties of random [formula omitted] XY chains

A numerical approach to the study of equilibrium statistical properties of spin-12 XY chains is suggested. The influence of disorder on transverse dynamical susceptibility of spin-12 Ising chain in random transverse field.

Strongly coupled Ising chain under a weak random field

We present here an analytical method based on the transfer matrix M to study quenched disorder in one-dimensional spin systems in the limit of strong couplings and weak disorder. The procedure is formulated for the random-field Ising chain of finite length L, and its properties, represented as functions of M, satisfy a differential equation of the Fokker-Planck type. This equation describes “evolution” with L, and a central-limit theorem of a novel kind provides the equation and its solutions with universal character. We obtain analytical expressions for the moments of the magnetization of the infinite length chain and study the approach to the infinite coupling limit. We find that the random-field free energy f H and the Edwards-Anderson order parameter m 2 satisfy a simple relation. We discuss our results in connection to previous work by Luck and Nieuwenhuizen.

Metastable states of the random-field Ising chain

Using a numerical transfer-matrix method, the authors have determined the number of zero-temperature metastable states and their distribution in energy and magnetization for random-field Ising chains. Both discrete and continuous distributions of the random fields are considered. The degeneracy and structure of the metastable states and ground states for the discrete case may be understood in terms of a domain picture. Algorithms for generating exact ground-state configurations for both discrete and continuous distributions are described, and these demonstrate that the ground states of the random-field Ising chain have a hierarchical structure.

Low-temperature thermodynamics of random-field Ising chains: exact results

The authors consider the random-field ferromagnetic Ising chain, for a class of continuous symmetric probability distributions of the random magnetic fields, of a diluted power-times-exponential type. This class of distribution is 'exactly solvable', in the sense that the disorder can be integrated explicitly, at any temperature. A detailed analysis of the low-temperature thermodynamics is presented. Exact expressions are obtained for the ground-state energy, the zero-temperature entropy, and the amplitude of the specific heat, which vanishes linearly at low temperature. The diluted symmetric binary distribution, where the magnetic fields can only assume the values +or-HB or zero, can be viewed as a limiting case of the class of exactly solvable distributions. The low-temperature physics of this discrete model is investigated in detail, including the exponential fall-off of the specific heat.

Ground-state and quenched-state properties of a one-dimensional interacting lattice gas in a random potential

We determine the zero-temperature properties of a one-dimensional lattice gas of particles that interact via a nearest neighbor exclusion potential and are subject to a random external field. The model is a special limiting case of the random field Ising chain. We calculate (1) the energy and density of the ground state as well as the local energy-density correlation and (2) the pair correlation function. The latter calculation gives access to all higher order correlations. The structure factor is shown to be a squared Lorentzian. We also compare the ground state to the quenched state obtained by sequentially filling the lowest available energy levels.

Fractal measures in the random-field Ising model

The measure of the local field probability density in the discrete (+or-h) short-ranged coupled random-field Ising chain is studied with the use of a specific discrete one-dimensional map. The strange set corresponding to this measure resembles Cantor sets and its full scaling properties are computed by evaluating the generalised fractal dimensions Dq and the spectrum of singularities f( alpha ). The main results are also derived by perturbation theory valid for weak fields h.