Lower critical dimension of Ising spin glasses: a numerical study

Abstract

A transfer matrix method is used to study the variation with length scale L of the distribution PL(Jeff) of effective couplings in Ising spin glasses at zero temperature. For a gaussian initial distribution the authors find in two dimensions J(L) identical to ( mod Jeff mod ) varies as L-1 nu / for 2<or=L<or=12, with nu =3.4+or-0.1, implying a zero-temperature phase transition with correlation length exponent nu . In three dimensions the effective couplings initially increase with length scale, J(L) varies as L-1 nu / for 2<or=L<or=4, with -1/ nu approximately=0.2, suggesting a phase transition at finite temperature. The results are described surprisingly well by the zero-temperature version of the Migdal-Kadanoff renormalisation group scheme.