The Energy-Energy Correlation Function of the Random Bond Ising Model in Two Dimensions

The energy-energy correlation function of the two-dimensional Ising model with weakly fluctuating random bonds is evaluated in the large scale limit. Two correlation lengths exist in contrast to one correlation length in the pure 2D Ising model: one is finite whereas the other is divergent at the critical points. The corresponding exponent of the divergent correlation length is νe = 1/2 in contrast to the pure system where νe = 1. The calculation is based on a previously developed effective field theory for the energy density fluctuations. PACS numbers: 05.50.+q, 64.60.Cn, 64.60.Fr Typeset using REVTEX 1 The two-dimensional Ising model [1] with random bonds is an interesting example for the competition of thermal fluctuations and quenched disorder. According to the Harris criterion [2] neither of these two effects dominates over the other near the critical point. This picture is supported by a number of investigations for weak disorder, treated in perturbation theory with respect to the variance of the bond distribution, which came to the conclusion that the thermodynamic properties are not dramatically changed by quenched disorder [3–6]. This is a consequence of the fact that the fixed point of the pure Ising model is stable against disorder (at least in one-loop order). The effect of disorder on the power law of the spin-spin correlation 〈SRS0〉 ∼ R of the pure system is controversal. While some authors claim there is no effect on the exponent 1/4 [4–6] others find a change for the correlation function to exp{−const.[log(logR)]2} [3]. On a naive basis one expects a reduction of the correlation near the critical point due to the additional fluctuations of the random bonds. Numerical simulations [7,8] do not indicate a significant change of the correlation function. Apart from the perturbative approaches it was found more recently in a non-perturbative treatment that disorder modifies the phase diagram of the two-dimensional Ising model [9,10]. In particular, disorder creates a new phase between the ferroand the paramagnetic phase. However, the width of this phase is small and vanishes for any finite order in perturbation theory. For the same reason it is rather unlikely that this this phase can be observe in a numerical simulation. Nevertheless, there is a rigorous proof for its existence [11]. This phase is characterized by a new order parameter. It was not possible in previous studies to understand its role in terms of the thermodynamic functions. However, it was related to the inverse correlation length of the average spin-spin correlation function by Braak [12]. In the following it will be shown that it also plays the role of an inverse correlation length of the average energy-energy correlation function. The Ising model describes a system of discrete spins Sr = ±1 on a lattice. In this article a square lattice will be considered. The spins on nearest neighbor sites of the lattice are coupled with the energy