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We derive the exact ground-state energy of the one-dimensional Ising model in random fields taking values h, 0 and Àh with general probabilities. The random-field Ising model on a ladder is also analyzed by showing its equivalence to the random-field Ising chain with field values AEh and 0 for h < J. The zero-temperature transfer matrix is used to obtain the results.

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.

We calculate several correlation functions and distribution functions of dynamical variables for the gauge glass and the Villain model using the spin wave approximation and the gauge transformation. The results show that the spin wave approximation gives the exact solutions on the Nishimori line in the phase diagram. This implies that vortices play no role in the thermodynamic behavior of the system as long as some correlation and distribution functions are concerned. These results apply to any dimensions including the two-dimensional case.