The two-dimensional ± J Ising spin glass: a model at its lower critical dimension

The ± J Ising spin glass [probabilities p and (1−p) associated with ferromagnetic and antiferromagnetic couplings, respectively] is studied, at zero temperature, on a hierarchical lattice that approaches the square lattice. It is shown that, for a wide range of values of p [between (1−pc) and pc, where pc represents the critical value of p separating the paramagnetic and ferromagnetic phases at zero temperature (pc 0.89)], the renormalized probability distribution for the coupling constants {Jij} approaches a fixed shape, after a few renormalization steps. An analytical expression for the fixed probability distribution, in terms of six delta-function peaks with a large predominance in the weights associated with the ± J peaks, is proposed. Such a result indicates that the present model is at its lower critical dimension, for the whole range from (1−pc) to pc, characterized by a stiffness exponent y=0.