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 εlnt and (εlnt) 2 with t=(T c −T)/T c and ε=6−d. This allows us to verify that standard scaling is satisfied with β=1+ε/2+0(ε 2 ). We have also investigated the transverse (replicon) correlation function, particularly at zero overlap. Near d u , G 00 (p) ~ t/P 4 (for p~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 G 00 (p) ~ (1/p 2–η ). (t/p 1 /v) β and a prediction for a soft replicon spectrum in t γ k 2γ/β γ/β instead of tk 2 at d u . We have checked t γ to one loop and work is in progress to check k 2γ/β to the same order. Taking the above divergence in p −(d+2−η)/2 as the leading divergence defines the lower critical dimension d ℓ by d ℓ=2−η (d ℓ). Known values of η (at d=6, from the ε expansion at T c , or from numerical work at d=4, 3) are compatible with d ℓ ~ 2.5±.3.