ISING SPIN-GLASS - RECENT PROGRESS IN THE FIELD-THEORY APPROACH

A loop expansion around Parisi's replica symmetry breaking mean field theory is constructed in zero field. We obtain the equation of state (and associated Parisi's solution) below the upper critical dimension d(u) = 6, and, in particular, explicit corrections in epsilonlnt and (epsilonlnt)2 with t = (T(c) - T)/T(c) and epsilon = 6 - d. This allows us to verify that standard scaling is satisfied with beta = 1 + epsilon/2 + 0 (epsilon2) . We have also investigated the transverse (replicon) correlation function, particularly at zero overlap, Near d(u), G00(p) is similar to t/p4 (for p is similar to 0) is the most obvious obstacle to a meaningful theory in d = 3. If we make the twofold assumption that (i) scaling applies, and (ii) the replicon propagator is dominated by the spectrum of the (small) transverse masses, we obtain the softer behavior G00(p) is similar to (1/p2-eta). (t/p1/nu)beta and a prediction for a soft replicon spectrum in t(gamma)k2gamma/beta instead of tk2 at d(u). We have checked t(gamma) to one loop and work is in progress to check k2gamma/beta to the same order. Taking the above divergence in p-(d+2-eta)/2 as the leading divergence defines the lower critical dimension d(l) by d(l) = 2 - eta (d(l)). Known values of eta (at d = 6, from the epsilon expansion at T(c), or from numerical work at d = 4,3) are compatible with d(l) is similar to 2.5 +/- .3.