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Journal of Physics A Mathematical and General

Limit distribution of an infinite-range random Ising model

D. Wagner

Published Oct 1, 1982• 1 citation
Publisher

Abstract

The limit distribution of a random Ising model (site disorder) with an exchange interaction of infinite range is calculated exactly. It is shown that randomness changes the analytic properties of the limit distribution at the critical point considerably. At criticality this system cannot be described by a Hamiltonian which is a polynomial in the spin variables.

DOI
10.1088/0305-4470/15/10/033

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The limit distribution of a random Ising model (site disorder) with an exchange interaction of infinite range is calculated exactly. It is shown that randomness changes the analytic properties of the limit distribution at the critical point considerably. At criticality this system cannot be described by a Hamiltonian which is a polynomial in the spin variables.
The author considers an Ising chain where the random interaction matrix has the symmetrised form J=PV+VP. The matrix elements of V and P are V ij =exp(- gamma mod i-j mod ) and P ij =P i delta j,i-1 +P i+1 delta j,i+1 , where i, j indicate sites on the chain. The quantities P i are independent random variables with the same probability distribution. Their mean and variance are fixed to be (P i )=J 0 gamma /4; (P i 2 )-(P i ) 2 =J 2 gamma /4, where gamma -1 is the range of the interaction. The author then obtains Sigma j (J ij )=2J 0 , Sigma j (J ij 2 )=J 2 in the limit gamma to 0, as in a model Hamiltonian that describes a spin-glass transition. The author evaluates exactly in this limit the density of eigenvalues of the random matrix PV. An integral equation for the distribution probability of new local random variables is derived by using the continued fraction method and the asymptotic solution in the limit gamma to 0 is obtained analytically. The largest eigenvalue of PV is found to be J 0 independent of J. However the largest eigenvalue of J=PV+VP is found to be mu M =J 0 +(J 0 2 +NJ 2 ) 1 2/, when gamma =0 and N is the total number of sites. An independent evaluation of the ground state energy shows that E 0 <-J square root NN, hence E 0 is not extensive. These results show that the scaling of the variance of the random interactions as J 2 N -1 gives highly unphysical results for the present model.
Exact analytic results are presented for the critical properties of the dilute Ising model on a Bethe lattice, where a fraction p of the sites are occupied at random by spins. It is found that the critical behaviour is the same as that of the pure system (p=1) except for transitions which occur by varying p at zero temperature, where the critical behaviour is different and corresponds to the percolation problem. For T to 0 and p to p c , the percolation concentration, the zero-field magnetization shows crossover behaviour of the form expected from scaling theories. Outside the critical region, a rapidly convergent numerical method is developed and used to calculate the spontaneous magnetization.
Extensive Monte Carlo simulations are performed on a two-dimensional random field Ising model. The purpose of the present work is to study the disorder-induced changes in the properties of disordered spin systems. The time evolution of the domain growth, the order parameter, and the spin-spin correlation functions are studied in the nonequilibrium regime. The dynamical evolution of the order parameter and the domain growth shows a power law scaling with disorder-dependent exponents. It is observed that for weak random fields, the two-dimensional random field Ising model possesses long-range order. Except for weak disorder, exchange interaction never wins over pinning interaction to establish long-range order in the system.
We study with exact diagonalization techniques the Heisenberg model for a system of SU(2) spins with S=1/2S=1/2S=1/2 and random infinite-range exchange interactions. We calculate the critical temperature Tg{T}_{g}Tg​ for the spin-glass to paramagnetic transition. We obtain ${T}_{g}\ensuremath{\approx}0.13,$ in good agreement with previous quantum Monte Carlo and analytical estimates. We provide a detailed picture for the different kind of excitations which intervene in the dynamical response ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega},T)$ at T=0T=0T=0 and analyze their evolution as T increases. We also calculate the specific heat Cv(T).{C}_{v}(T).Cv​(T). We find that it displays a smooth maximum at ${T}_{M}\ensuremath{\approx}0.25,$ in good qualitative agreement with experiments. We argue that the fact that TM>Tg{T}_{M}>{T}_{g}TM​>Tg​ is due to a quantum disorder effect.
Using exact diagonalization techniques we study the dynamical response of the anisotropic disordered Heisenberg model for systems of S=1/2 spins with infinite range random exchange interactions at temperature T=0. The model can be considered as a generalization, to the quantum case, of the well known Sherrington-Kirkpatrick classical spin-glass model. We also compute and study the behavior of the Edwards Anderson order parameter and energy per spin as the anisotropy evolves from the Ising to the Heisenberg limits.