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arXiv:hep-th/9304158 · Physical review. B, Condensed matter

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

Edward Farhi, Sam Gutmann

Published Oct 1, 1993• 9 citations
arXivHTMLPDF

Abstract

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 〈${\mathit{s}}_{0}$${\mathit{s}}_{\mathit{n}}$〉-〈${\mathit{s}}_{0}$〉〈${\mathit{s}}_{\mathit{n}}$〉 in the case that 2J/h is not an integer. The result is a discontinuous function of 2J/h. When p=1/2, we also place a bound on the correlation length of the quenched average of the correlation function 〈${\mathit{s}}_{0}$${\mathit{s}}_{\mathit{n}}$〉.

DOI
10.1103/physrevb.48.9508

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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 〈{\mathit{s}}_{0}$${\mathit{s}}_{\mathit{n}}〉-〈s0{\mathit{s}}_{0}s0​〉〈sn{\mathit{s}}_{\mathit{n}}sn​〉 in the case that 2J/h is not an integer. The result is a discontinuous function of 2J/h. When p=1/2, we also place a bound on the correlation length of the quenched average of the correlation function 〈{\mathit{s}}_{0}$${\mathit{s}}_{\mathit{n}}〉.
The two-point connected correlation function, or wavevector-dependent susceptibility, of the ferromagnetic Ising chain in a random field is calculated exactly at any temperature, for a two-parameter family of diluted symmetric exponential distributions of the magnetic fields. Thermodynamic properties of this model have been derived in a previous work by the authors. Besides the correlation function itself, the solution provides exact results for the (usual) susceptibility, the correlation length and the Edwards-Anderson parameter. The low-temperature regime is examined in full detail: the authors obtain in closed form the limit value at zero temperature of various quantities, and the first correction to this limit, which behaves linearly with temperature. The correlation length is discontinuous at zero temperature. They also derive the scaling form of the correlation function for small p, where the dilution p is the probability for a spin to be subjected to a non-zero random field. Even in this limit, the correlation function is more complicated than the simple Lorentzian predicted, e.g., by mean-field theory.
Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).
An exact expression for the spin-spin correlation function is derived for the zero-temperature random-field Ising model defined on a Bethe lattice of arbitrary coordination number.The correlation length describing dynamic spinspin correlations and separated from the intrinsic topological length scale of the Bethe lattice is shown to diverge as a power law at the critical point.The critical exponents governing the behaviour of the correlation length are consistent with the mean-field values found for a hypercubic lattice with dimension greater than the upper critical dimension.
The spin-spin correlation function of a one-dimensional random Ising model decays in a manner inversely proportional to the square of the partition function of the part of the system between the spins. At low temperatures and strong coupling there are logarithmic corrections to the scaling power-law dependence of the correlation length on the magnetic field. The correlation of the boundary spins as a function of the system size is evaluated analytically and numerically. At low temperatures and strong coupling this function has a crossover between different behaviors and its Fourier transform oscillates with wave vector.
The spin-spin correlation functions of the random one-dimensional Ising model are calculated exactly and found to have a probability distribution. It is then possible to define two characteristic lengths, one to describe the behaviour of the mean and the other of the standard deviation of the probability distribution. The existence of these two characteristic lengths then suggests that the thermodynamic behaviour of the system is considerably more complex than for the corresponding nonrandom system.
Studies a correlation function which is given by the canonical average of a product of one or more spin variables, for the random-bond Ising model, in which the exchange integrals are +J(J>O) and -J with probabilities p and 1-p, respectively. The authors show that an upper bound to the configurational average of the correlation functions, calculated in the thermodynamic limit in the zero external field limit, is the product of the same correlation function for the corresponding ferromagnetic Ising model at the temperature under consideration and the same quantity at the temperature T 1 which is determined by the condition T 1 =2J/ mod k B 1n(p/(1-p)) mod , where k B is the Boltzmann constant. The results are given for the random-bond Ising model of an arbitrary spin S, and also for the diluted random-bond Ising models of an arbitrary spin S, with the pair interaction and with general interaction, and for the diluted and undiluted random-bond n-vector model.