Random layered Ising model and Griffiths-McCoy singularities

Our next major project was to generalize the translationally invariant interaction (1) to a non translationally invariant problem where not just a half plane boundary was present but to a case where

1.

The interaction energies E^v were allowed to vary from row to row but translational invariance in the horizontal direction was preserved

2.

The interaction energies E^v(j) between the rows were chosen as independent random variables with a probability distribution P(E^v).

This was the first time that such a random impurity problem had ever been studied for a system with a phase transition and the entire computation was a new invention [7]. In particular we made the first use in physics of Furstenburg's theory [8] of strong limit theorems for matrices. We felt that the computation was a starting success because we found that for any probability distribution, no matter how small the variance, there was a temperature scale, depending on the variance, where there was new physics that is not present in the translationally invariant model. For example the divergence in the specific heat at T_c decreases from the logarithm of Onsager to an infinitely differentiable essential singularity. Moreover the average over the distribution P(E^v) of the correlation functions of the boundary could be computed [9] and it was seen that there was an entire temperature range surrounding T_c where the boundary susceptibility was infinite. Thus the entire picture of critical exponents which had been invented several years before to describe critical phenomena in pure systems was not sufficient to describe these random systems [10]. We were very excited.

But then something happened which I found very strange. Instead of attempting to further explore the physics we had found, arguments were given as to why our effect could not possibly be relevant to real systems. This has lead to arguments which continue to this day.

We were, and are, of the opinion that the effects seen in the layered Ising model are caused by the fact that the zeroes of the partition function do not pinch the real temperature axis at just one point but rather pinch in an entire line segment. An closely related effect was simultaneously discovered by Griffiths [11] and it was, and is, our contention that in this line segment there is a new phase of the system. But this line segment is not revealed by approximate computations and for decades it was claimed that our new phase was limited to layered systems; a claim which some continue to make to this day.

However, fortunately for us, there is an alternative interpretation of the Ising model in terms of a one dimensional quantum spin chain in which our layered classical two dimensional system becomes a randomly impure quantum chain [12]. In this interpretation there is no way to argue away the existence of our new phase and finally in 1995, a quarter century after we first found the effect, D. Fisher [13], in an astounding paper, was able to craft a theory of the physics of rare events based on an exact renormalization group computation which not only reproduced our the results of our layered model on the boundary but extended the computations to bulk quantities which in 1969 we had been unable to compute. With this computation I think the at the existence of what are now called Griffiths-McCoy singularities is now accepted but it has taken a quarter century for it to happen