Critical point scaling of Ising spin glasses in a magnetic field

Critical point scaling in a field H applies for the limits t→0 (where t=T/Tc-1) and H→0 but with the ratio R=t/H^2/Δ finite. Δ is a critical exponent of the zero-field transition. We study the replicon correlation length ξ and from it the crossover scaling function $f(R)$ defined via 1/(ξH^4/(d+2-η))~f(R). We have calculated analytically $f(R)$ for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension $d=3$, we have determined the exponents and the critical scaling function $f(R)$ within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for $f(R)$ in $d=3$ in reasonable agreement with those of the Monte Carlo simulations at the values of R for which they can be compared. If there were a de Almeida-Thouless (AT) line for d≤6, it would appear as a zero of the function $f(R)$ at some negative value of R, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for d≤6, which we review.