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

Abstract

We derive the exact ground-state energies of the random-field Ising chain and ladder. The random fields consist of three possible values for the chain model and bimodal fields for the ladder model. The results are piecewise linear as a function of the random-field strength. The zero-temperature transfer matrix method is used to obtain these results. §1. IntroductionThe one-dimensional random Ising model is one of the simplest examples of systems with quenched disorder. In the long history of its intensive studies, a number of results have been published which give exact solutions in the ground state. 1)-5) Mattis and Paul 6) have analyzed the random-bond Ising model on the ladder geometry at finite temperature. Their result does not have a closed analytical form because the probability distribution of eigenvalue involves devil's staircase. Kadowaki et al. 5) calculated explicitly the exact ground-state energies of the ±J random bond model and the site-random model on strips of various widths. To obtain these exact results, they used the transfer matrix method at zero-temperature. Being motivated by the method proposed in Ref. 5), we apply their method to the Ising chain in three possible random fields and Ising ladder in ±h random fields. 7) Th solutions are both piecewise linear as a function of the random-field strength. In the next section, we consider the Ising chain in random fields taking ±h and 0 and in §3 we show the exact ground-state energy of the Ising ladder in ±h random fields. §2. Random-field Ising chain Recursion relationsFirst, we consider the Ising chain in three possible random fields ±h and 0. The random field is chosen to be +h with probability p 1 , 0 with p 2 and −h with p 3 and these probabilities satisfy p 1 + p 2 + p 3 = 1. Considering the transfer matrix at zero-temperature, we obtain the recursion relations, 5)where x n is the ground-state energy with the n'th spin state being +1, J is the ferromagnetic coupling constant and h n+1 is the random field at site n + 1. The energy difference between the edge spin state being +1 and −1 is represented by a n . The function f (x) is piecewise linear as +J for x > J, −J for x < −J and x for −J < x < J. As shown below, using the recursive calculation of the above two Downloaded from https://academic.oup.

Low-energy excitations in the three-dimensional random-field Ising model

“…Similar to the more prominent spin glasses (SGs) [2,3,4], there are still many open questions concerning the low-temperature properties of the RFIM. During the last few years, the RFIM and the related diluted antiferromagnet in a field have attracted growing attention [5,6,7,8,9,10,11,12,13,14,15,16], in particular within simulation studies at finite [17,18,19,20,21,22,23], and zero temperature [24,25,26,27,28,29,30,31,32,33,34,35,36].…”