Spin glass stiffness in zero dimensions

A unique analytical result for the Migdal–Kadanoff hierarchical lattice is obtained. The scaling of the defect energy for a zero-dimensional spin glass is derived for a bond distribution that is continuous at the origin. The value of the ‘stiffness’ exponent in zero dimensions, y0 = −1, corresponds to the value also found in one dimension. This result complements and completes earlier findings for yd at d > 0. PACS numbers: 75.10.Nr, 05.50.+q, 05.40.−a A quantity of fundamental importance for the modelling of amorphous magnetic materials through spin glasses [1] is the ‘stiffness’ exponent y [2, 3]. The stiffness of a spin configuration describes the typical rise in magnetic energy E due to an induced defect interface of size L. In a glassy system, the potential energy function resembles a high-dimensional mountain landscape over its variables [4]. Any defect-induced displacement of size L in such a landscape may move a system equally likely up or down in energy E. Averaging over many incarnations of such a system then results in a typical energy scale 〈| E|〉 ∼ L (L → ∞). (1) The importance of this exponent for small excitations in disordered spin systems has been discussed in many contexts [1–3, 5–8]. In particular, it signifies a renormalized coupling strength (across any hypothetical interface) between regions in space separated by a distance L [3]: if yd > 0, regions in space are strongly coupled at low temperature and spin glass ordering ensues, i.e. Tg > 0. Reference [9] provided a description of yd as a continuous function of dimension d using a fit to the data obtained in [10] for d = 2, 3, . . . , 6.That fit became credible in that it reproduced the exactly known result in d = 1, y1 = −1, to within less than 1%. Hence, it validated the values for yd found in [10, 11] and produced a number of predictions such as that dl = 5/2 may be the lower critical dimension (the dimension in which yd = 0) for Ising spin glasses, in accordance with an earlier calculation invoking replica symmetry breaking [12]. In a quest for understanding universality in spin glasses, there has been considerable interest recently in the behaviour of yd even for d < dl, where any spin glass ordering is unstable. Presumably, for divergent energy scales [8] in equation (1), i.e. for yd > 0, 0305-4470/06/3410641+06$30.00 © 2006 IOP Publishing Ltd Printed in the UK 10641