Random Bond Ising Chain Research Articles

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.

Initial static susceptibilities of nonuniform and random Ising chains

Within the conventional framework of standard linear response theory we have derived exact results for the initial static susceptibilities of nonuniform spin- 1 2 Ising chains. The results obtained permit one to study regularly alternating-bond and random-bond Ising chains. The influence of several types of nonuniformity and disorder on the temperature dependence of the initial longitudinal and transverse static susceptibilities is discussed.

The exact susceptibility of the transverse Ising chain and Ising-type chains with arbitrary spin

The zero-field susceptibility of the transverse Ising chain with arbitrary spin is exactly expressed in terms of the eigenvector (for the maximum eigenvalue) of the transfer matrix corresponding to its Ising interaction. The explicit expression of the susceptibility can be obtained at least for S ⩽ 7/2. The numerical calculations for large spin values are also given. The zero-temperature limit of the susceptibility is independent on spin value. The calculation is generalized to Hamiltonians with Ising-type interactions, such as thespin- S transverse Ising chain with parallel magnetic field H z , the S = 1 BEG model, mixed spin and mixed bond chains with periodic structures such as ferrimagnet, and random bond Ising chains. Other quantities such as the specific heat are also calculated for arbitrary spin value.

The susceptibility of anisotropic Heisenberg alternating- and random-bond chains by a density matrix product approximation

The magnetic susceptibility of an anisotropic Heisenberg chain with alternating exchange bonds is calculated analytically within the framework of a product approximation in which the density matrix of the system is replaced by the sum of all the products of the density matrices of clusters containing a few spins. Both the dimer- and trimer-product approximations are used to study the effect of cluster size. In the isotropic limit the results are compared with those of numerical calculations for finite uniform chains. Further, a general analytic expression for the susceptibility of the anisotropic Heisenberg chain with arbitrary exchange interactions is obtained in the dimer-product approximation. In the special case of a random distribution of two types of exchange interactions the susceptibility is evaluated and compared to that for the regular alternating-bond chain. In the anisotropic limit, the present method correctly reproduces the exact results for both the alternating- and random-bond Ising chains.

Exact initial transverse susceptibility for a random-bond Ising chain

The exact initial transverse susceptibility, χ ⊥, of a random-bond Ising chain has been derived by a method based on the use of a statistical identity of the type introduced by Callen and by Suzuki for spin correlation functions. Besides being applicable to general random-bond Ising chains, among which the uniform and alternating-bond Ising chains are special examples, the method is particularly simple in comparison to the exact derivation of χ ⊥ for the uniform Ising chain that already exists. Numerical results, especially for chains having two types of exchange bonds, are presented and discussed.

Random-bond Ising chain in a transverse magnetic field: A finite-size scaling analysis

We investigate the zero-temperature quantum phase transition of the random bond Ising chain in a transverse magnetic field. Its critical properties are identical to those of the McCoy-Wu model, which is a classical Ising model in two dimensions with layered disorder. The latter is studied via Monte Carlo simulations and transfer matrix calculations and the critical exponents are determined with a finite-size scaling analysis. The magnetization and susceptibility obey conventional rather than activated scaling. We observe that the order parameter-- and correlation function--probability distribution show a nontrivial scaling near the critical point which implies a hierarchy of critical exponents associated with the critical behavior of the generalized correlation lengths.

Zeros on the temperature axis of spin correlations in a random bond Ising chain

The authors study the correlation between two spins diametrically opposite in a random Ising loop of 2r spins with random nearest-neighbour couplings. As a function of the inverse temperature beta , this correlation undergoes random sign changes with a density rho r( beta ). They determine this density for an arbitrary distribution P(K) of the coupling constants for (i) r to infinity and (ii) the scaling limit beta to infinity , r to infinity at r/ beta fixed. For r to infinity , the total number of zeros on the beta -axis grows asymptotically as 1/2 1/21 zeta (3)-1/4 pi 21/2 log r. As an application, the density of zeros is calculated for the spin-spin correlation in a double infinite Ising chain with random bonds, having, with a probability p, an infinite transverse coupling between each pair of corresponding sites. A connection is made with predictions from spin glass theory.

Remanence, irreversibility and non-exponential relaxation in the one-dimensional spin glass

The Glauber kinetic equations have been solved numerically for the relaxation of the magnetisation in a one-dimensional random bond Ising chain with periodic boundary conditions. The chain relaxed from the completely orientated state. Calculations have been performed for the short time relaxation of the magnetisation using the Fernandez-Medina picture. These are in agreement with the author's numerical data. Irreversible behaviour similar to that occurring in laboratory spin glasses has been observed. Finally in the long time regime they perform a lower bound calculation for the magnetisation which yields non-exponential relaxation of the form M(t) approximately exp(-(t/ tau )12/). The numerical data give the same non-exponential relaxation (with a different tau ) provided that a natural prefactor of 1/3 is introduced. Hence they find M(t) approximately 1/3exp(-(t/ tau ')12/).

Low-temperature properties of frustrated Ising chains

Recently published methods used to examine the ground-state properties of the random-bond Ising chain in a uniform magnetic field are extended to describe the system at low temperatures. The entropy spikes broaden asymmetrically, and the excitation energy and specific heat become asymmetric double-peaked functions of the magnetic field with zero values between the peaks at B=2J/r for integer r. The magnetisation steps at B=2J/r are smoothed out asymmetrically. Both the ground-state and finite-temperature properties of a simple periodic frustrated Ising chain are investigated using the transfer-matrix method. The ground-state properties are similar to those of the random bond chain; the ground-state entropy, for instance, is finite for J<or=B<or=2J, with spikes at J and 2J.

Low temperature behaviour of a random-bond ising chain

We discuss the random bond Ising chain (or Landau-Blume model) within the functional equation method. We present a higher-order perturbation theory which yields exact low-temperature results even if the magnetic field is near a critical value where at T = 0 one obtains discontinuities in magnetization and entropy.

Simple frustrated systems: chains, strips and squares

The ground-state properties of several systems showing the Toulouse frustration effect are investigated analytically. First, the energy and entropy of the random-bond Ising chain in a uniform field are obtained for all concentrations of antiferromagnetic bonds using the transfer-matrix method. The susceptibility has discontinuities for an infinite number of critical values of the field, where the entropy shows spikes in addition to discontinuities. These effects are related to the physics of frustration. The second system studied consists of frustrated strips, which it is argued are the proper one-dimensional limit of spin glasses. Two kinds of strips are considered and the results are compared with recent numerical works on two-dimensional spin glasses and with the exact results for random (3*3) squares.

Exact results on random bond Ising Chains in magnetic fields

In this contribution we present exact results on the random bond Ising Chain in a magnetic field. The original problem is reduced to the solution of a functional equation for a certain probability distribution, which can be used to evaluate thermodynamics and correlation functions. We give a sufficiently accurate solution for low temperatures, which yields the complete low-temperature behaviour. Comparison is made with different Monte-Carlo-calculations performed on this system.

Random-bond Ising chain in a magnetic field at low temperatures

The low-temperature behavior ($T\ensuremath{\ll}h, J$) of a random-bond Ising chain in a magnetic field is considered [$H=\ensuremath{-}(\frac{1}{2})J\ensuremath{\Sigma}{T}_{i}{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{i+1}\ensuremath{-}h\ensuremath{\Sigma}{\ensuremath{\sigma}}_{i}, {\ensuremath{\sigma}}_{i}=\ifmmode\pm\else\textpm\fi{}1$, ${{T}_{i}}$ is a fixed random sequence of numbers +1 and -1 with concentrations ${c}_{1}$ and ${c}_{2}=1\ensuremath{-}{c}_{1}$, respectively]. The ground-state energy ${E}_{0}$, magnetization ${\ensuremath{\mu}}_{0}$ and zero-point entropy ${S}_{0}$ are calculated exactly. It is shown that ${\ensuremath{\mu}}_{0}$ and ${S}_{0}$ are discontinuous functions of magnetic field having jumps at $h=\frac{J}{n}$, $n=1, 2, \dots{}$.

Properties of a random bond Ising chain in a magnetic field

The Ising chain with random bonds in a magnetic field $\mathcal{H}=\ensuremath{-}\ensuremath{\Sigma}{i}^{}{J}_{i}{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{i+1}\ensuremath{-}h\ensuremath{\Sigma}{i}^{}{\ensuremath{\sigma}}_{i}$, where ${J}_{i}=\ifmmode\pm\else\textpm\fi{}1$ at random, and $\ensuremath{\Sigma}{i}^{}{J}_{i}=0$, represents a model of a magnetic glass, or of heteropolymer melting. Calculations of the thermodynamic properties of the chain as a function of field strength and temperature have been performed by Monte Carlo techniques. These results are compared with perturbation calculations for small and large values of $\frac{h}{T}$. The Monte Carlo results show, in agreement with the perturbation calculations, that the field-induced magnetization is generally smaller for the random bond model than for a chain of noninteracting spins. As $T\ensuremath{\rightarrow}0$ the magnetization approaches the result for noninteracting spins.