Real-space renormalization-group methods for hierarchical spin glasses

Abstract

We focus on two real-space renormalization-group (RG) methods recently proposed for a hierarchical model of a spin glass: A sample-by-sample method, in which the RG transformation is performed separately on each disorder sample, and an ensemble RG (ERG) method [M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi. Ensemble renormalization group for disordered systems. Phys. Rev. B, 87(13):134201, 2013] in which the transformation is based on an average over samples. Above the upper critical dimension, the sample-by-sample method yields the correct mean-field value for the critical exponent ν related to the divergence of the correlation length, while it does not predict the correct qualitative behavior of ν below the upper critical dimension. On the other hand, the ERG procedure has been claimed to predict the correct behavior of ν both above and below the upper critical dimension. Here, we straighten out the reasons for the discrepancy between the two methods above, by demonstrating that the ERG method predicts a marginally stable critical fixed point, thus implying a prediction for the critical exponent ν given by 2 1/ν 1. This prediction disagrees, on a qualitative and quantitative level, both with the mean-field value of ν above the critical dimension, and with numerical estimates of ν below the upper critical dimension. Therefore, our results show that finding a real-space RG method for spin glasses which yields the correct prediction for universal quantities below the upper critical dimension is still an open problem, for which our analysis may provide some general guidance for future studies.

Reply to comment on ‘Real-space renormalization-group methods for hierarchical spin glasses’

“…which is, in general, different from one, and which corresponds to the value in Fig. 5a of [3] which is reached by…”

“…where σ t 1 1 and σ t 2 1 are the standard deviations of the spin couplings at the first hierarchical level of the hierarchical Edwards-Anderson model (HEA), at the t-th ensemble renormalization group (ERG) step, and at inverse temperatures β 1 and β 2 , respectively. However, in [3] we have shown that…”

“…In this Section we will address the criticisms raised in [1] concerning the conclusions of [3] and finite-size effects in the ERG. We recall that the ERG method is composed of the following steps [1]: 1) A HEA with n hierarchical levels (a) and a HEA n − 1 levels (b) are considered.…”

“…Angelini et al write that "the relation that links the exponent ν to the eigenvalue with the largest norm λ max of the linearized matrix M, 2 [4)] is just introduced to iterate the procedure, given the finite size of the analyzed systems, and it is thus the step responsible for finite-size effects in the ERG results. The prediction 2 1/ν = 1 in [ [3]] is thus completely dominated by finite-size effects". In response to this statement, we observe the following: the fact that λ max = λ 1 = 1 is a feature specific to the ERG transformation, which is not present in other real-space RG approaches for the hierarchical model [5].…”

“…It follows that the value of ν obtained in [2] from (A) cannot describe the exponent ν, which is characteristic of a long-wavelength, critical regime where the correlation length diverges. As the ERG transformation is further iterated and flows away from (A) to (B), the finite length scales of (A) are integrated out, and the RG iteration enters the longwavelength, critical regime (B), which has been used in [3] to correctly obtain ν. Importantly, we observe that the fact that (B) involves finite-size effects is a feature specific to the ERG, and gives no rationale for extracting ν from regime (A), which is unrelated to the critical one, nor to apply Eq.…”