One-dimensional Random-field Ising Model Research Articles

Experimental Realization of the 1D Random Field Ising Model.

We have measured magnetic-field-induced avalanches in a square artificial spin ice array of interacting nanomagnets. Starting from the ground state ordered configuration, we imaged the individual nanomagnet moments after each successive application of an incrementally increasing field. The statistics of the evolution of the moment configuration show good agreement with the canonical one-dimensional random field Ising model. We extract information about the microscopic structure of the arrays from our macroscopic measurements of their collective behavior, demonstrating a process that could be applied to other systems exhibiting avalanches.

Off-equilibrium response function in the one-dimensional random-field Ising model.

A thorough numerical investigation of the slow dynamics in the d=1 random-field Ising model in the limit of an infinite ferromagnetic coupling is presented in this paper. Crossovers from the preasymptotic pure regime to the asymptotic Sinai regime are investigated for the average domain size, the autocorrelation function, and staggered magnetization. By switching on an additional small random field at the time t(w) the linear off-equilibrium response function is obtained, which displays as well the crossover from the nontrivial behavior of the d=1 pure Ising model to the asymptotic behavior where it vanishes identically.

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.

Exact solution of return hysteresis loops in a one-dimensional random-field ising model at zero temperature

Minor hysteresis loops within the main loop are obtained exactly in the one-dimensional ferromagnetic random-field Ising model at zero temperature. Numerical simulations of the model show excellent agreement with the exact results.

Hysteresis in one-dimensional anti-ferromagnetic random-field Ising model at zero-temperature

We analyze hysteresis in a one-dimensional anti-ferromagnetic random field Ising model at zero-temperature. The random field is taken to have a rectangular distribution of width 2 Δ centered about the origin. A uniform applied field is varied slowly from −∞ to +∞ and back. Analytic expression for the hysteresis loop is obtained in the case Δ⩽| J|, where | J| is the strength of the nearest-neighbor interaction.

Chiral Optical Properties of a Helical Polymer Synthesized from Nearly Racemic Chiral Monomers Highly Diluted with Achiral Monomers

In polyisocyanates composed only of randomly distributed (R) and (S) units, the chiral optical properties of the polymer are far out of proportion to the enantiomeric excess of the monomers. This highly disproportionate relationship, which arises from a majority-rule effect among these enantiomeric units on the helical sense of the backbone, is now found to be unaffected, within certain limits, by the overwhelming presence of achiral units randomly distributed along the chain. This experimental result can be explained quantitatively by an analysis based on the one-dimensional random-field Ising model, which shows that dilution of the chiral units with achiral units increases the helical domain size in a manner that compensates for the dilution. In qualitative terms, since the random-field domain size is limited by the objection of the minority units to the helical sense dictated by the majority units, dilution of this objection acts to increase the domain size. As long as this domain size is not limited by the chain length or by thermal fluctuations, the achiral dilution will not reduce the optical activity of the polymer.

Critical temperature and density of spin flips in the anisotropic random-field Ising model

We present analytical results for the strongly anisotropic random-field Ising model, consisting of weakly interacting spin chains. We combine the mean-field treatment of interchain interactions with an analytical calculation of the average chain free energy (``chain mean-field'' approach). The free energy is found using a mapping on a Brownian motion model. We calculate the order parameter and give expressions for the critical random magnetic-field strength below which the ground state exhibits long-range order and for the critical temperature as a function of the random magnetic-field strength. In the limit of vanishing interchain interactions, we obtain corrections to the zero-temperature estimate by Imry and Ma [Phys. Rev. Lett. 35, 1399 (1975)] of the ground-state density of domain walls (spin flips) in the one-dimensional random-field Ising model. One of the problems to which our model has direct relevance is the lattice dimerization in disordered quasi-one-dimensional Peierls materials, such as the conjugated polymer trans-polyacetylene.

Cooperative chiral order in copolymers of chiral and achiral units

Polyisocyanates can be synthesized with chiral and achiral pendant groups distributed randomly along the chains. The overall chiral order, measured by optical activity, is strongly cooperative and depends sensitively on the concentration of chiral pendant groups. To explain this cooperative chiral order theoretically, we map the random copolymer onto the one-dimensional random-field Ising model. We show that the optical activity as a function of composition is well-described by the predictions of this theory.

One-dimensional random-field ising model: Gibbs states and structure of ground states

We consider the random Gibbs field formalism for the ferromagnetic ID dichotomous random-field Ising model as the simplest example of a quenched disordered system. We prove that for nonzero temperatures the Gibb state is unique for any realization of the external field. Then we prove that asT→0, the Gibbs state converges to a limit, a ground state, for almost all realizations of the external field. The ground state turns out to be a probability measure concentrated on an infinite set of configurations, and we give a constructive description of this measure.

Correlation functions of the one-dimensional random-field Ising model at zero temperature.

We consider the one-dimensional random field Ising model, where the spin-spin coupling, $J$, is ferromagnetic and the external field is chosen to be $+h$ with probability $p$ and $-h$ with probability $1-p$. At zero temperature, we calculate an exact expression for the correlation length of the quenched average of the correlation function $\langle s_0 s_n \rangle - \langle s_0 \rangle \langle s_n \rangle$ in the case that $2J/h$ is not an integer. The result is a discontinuous function of $2J/h$. When $p = {1 \over 2}$, we also place a bound on the correlation length of the quenched average of the correlation function $\langle s_0 s_n \rangle$.

One-dimensional random field Ising model: Residual entropy, magnetization, and the “perestroyka” of the ground state

The discontinuous behaviour of magnetization and entropy for zero temperature can be understood as the consequence of flips of microscopic spin clusters. The residual entropy feels very sensitively whether the ration of constant and random part of the magnetic field is rational or irrational, since in the former case an additional mechanism for degeneracy of the ground state appears.

Fluctuations of the Lyapunov exponent in disordered systems

The fluctuations of the maximum Lyapunov exponent of a product of random matrices are studied analytically and numerically. It is shown that they may be expressed as a sum of two terms, one related to the order of matrices within the product, the other to fluctuations of the number of matrices of a given type. This result is then applied to the one-dimensional random-field Ising model and the discrete Schrodinger equation with a random potential.

Multifractal properties of a class of non-natural measures as an eigenvalue problem.

Invariant measures generated by the backward iteration of one-dimensional maps are studied. The method yields the generalized dimensions as eigenvalues. For a particular backward iteration, earlier results concerning the natural measure are recovered. As an application, the case of the one-dimensional random-field Ising model is discussed.

Multifractal properties in the one-dimensional random-field Ising model.

The Ising chain in a random field is investigated in the region of parameter settings where the measure related to the free-energy distribution is a fractal. The multifractal spectrum is calculated by means of a perturbation expansion in terms of the strength of the random field by exploiting the connection with the problem of a non-natural measure of a one-dimensional map. It is shown that the multifractal spectrum and the distribution of the free-energy fluctuations of finite chains are connected.

Calculation of a characteristic fractal dimension in the one-dimensional random field ising model

By exploiting the connection with the problem of a repeller in a one dimensional map a new method is applied to calculate a fractal dimension characterising the local field. It is determined analytically in powers of the strength of the random field and also by an iteration procedure.

One-dimensional random field Ising model and discrete stochastic mappings

Previous results relating the one-dimensional random field Ising model to a discrete stochastic mapping are generalized to a two-valued correlated random (Markovian) field and to the case of zero temperature. The fractal dimension of the support of the invariant measure is calculated in a simple approximation and its dependence on the physical parameters is discussed.

Exactly soluble random field Ising models in one dimension

The authors solve exactly the one-dimensional random field Ising model for two classes of magnetic field distributions: symmetric exponential (model I) and non-symmetric exponential (model II). For both models, expressions for the free-energy at all finite temperatures are presented. The low-temperature region is examined in more detail; they obtain the zero-temperature energy and entropy in closed form; it is shown that the free energy of both models has an expansion in integer powers of temperature. Model I has a non-vanishing zero-point entropy for all values of the parameters as soon as randomness is diluted. In model II the zero-point entropy is zero except for a discrete sequence of values of one parameter. In some cases the zero-temperature magnetisation is positive whereas the average magnetic field is negative; the magnetisation may also change sign as a function of temperature.