Order parameter for non-mean-field spin glasses

Abstract

We propose a novel renormalization group (RG) method for non mean-field models of spin glasses, which leads to the emergence of a novel order parameter. Unlike previous approaches where the RG procedure is based on a priori notions on the system, our analysis follows a minimality principle, where no a priori assumption is made. We apply our approach to a spin-glass model built on a hierarchical lattice. In the RG decimation procedure, a novel order parameter spontaneously emerges from the system symmetries, and self-similarity features of the RG transformation only. This order parameter is the projection of the spin configurations on the ground state of the system. Kadanoff's majority rule for ferromagnetic systems is replaced by a more complex scheme, which involves such novel order parameter. The ground state thus acts as a pattern which translates spin configurations from one length scale to another. The rescaling RG procedure is based on a minimal, information-theory approach and, combined with the decimation, it yields a complete RG transformation. Below the upper critical dimension, the predictions for the critical exponent $\nu$, which describes the critical divergence of the correlation length, are in excellent agreement with numerical simulations from both this and previous studies. Overall, this study opens new avenues in the understanding of the critical ordering of realistic spin glasses, and it can be applied to spin-glass models on a cubic lattice and nearest-neighbor couplings which directly model spin-glass materials, such as AuFe, CuMn and other magnetic alloys.