W-T identities and a candidate “droplet” Lagrangean for the Ising Spin Glass

In search for a microscopic description of “droplet-like” properties for the Ising spin glass (single component order parameter, zero modes i.e. correlation functions vanishing at infinity) we reconsider the two-packet model of Bray and Moore, which is effectively Replica-Symmetric and enjoys zero modes but only up to one-loop. We show how an appropriate change in the limits of the basic parameters of the model (packet size and replica number) allows for a derivation of Ward–Takahashi (WT) identities, thus ensuring the existence of zero modes to all orders and opening the way for a Lagrangean formulation of a “droplet-like” field theory for the Ising spin glass. Spin glass theory has presented so far the schizophrenic aspect of two conflicting approaches difficult to reconcile. The so-called mean-field like approach developed around Parisi [1]-[3] solution of the Sherrington-Kirkpatrick [4] meanfield model uses standard field theory (mean-field, loops, renormalisation group, WT identities, ...) on a Lagrangean built with replicated fields. It is a “microscopic” theory for the spin glass, in the same sense as the φ theory for the ferromagnet. The alternative droplet-like theory of Fisher and Huse [5] and Bray and Moore [6], despite abundant results, does not appeal to such a Lagrangean field theory starting point and is far removed from “microscopic” description. So there is a strong motivation to search better into the replicated φ Lagrangean and examine whether the characteristic features for a “droplet” like theory could fit in. If one were to put up one, it would have the features of a theory with an effective Replica-Symmetric order parameter (a ferromagnetic in disguise). It would also possess zero modes to allow for an algebraic decrease at large distances of its correlations. But such features were indeed present, years before the birth of droplet theory, in an ansatz proposed by Bray and Moore [7] (BM), were replica symmetry was broken into “two-packets”, and restored at the very end. From their results, calculated at one-loop, it could be checked that both