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

TL;DR

This paper defends the validity of a previous method for estimating the critical exponent nu in hierarchical spin glasses, countering criticisms about the mathematical relations used and the regimes for analysis.

Contribution

The authors clarify and reaffirm that their ERG-based approach correctly predicts the critical exponent nu, addressing and refuting specific criticisms in the comment.

Findings

The relation between the largest eigenvalue and nu remains valid for ERG.

The critical exponent nu is correctly estimated from the asymptotic regime.

The ERG method predicts 2^{1/nu}=1 for the critical exponent.

Abstract

In their comment, Angelini et al. object to the conclusion of [J. Phys. A: Math. Theor., 52:445002, 2019] (1), where we show that in [Phys. Rev. B, 87:134201, 2013] the exponent $\nu$ has been obtained by applying a mathematical relation in a regime where this relation is not valid. We observe that the criticism above on the mathematical validity of such relation has not been addressed in the comment. Our criticism thus remains valid, and disproves the conclusions of the comment. This constitutes the main point of this reply. We also provide a point-by-point response and discussion of Angelini et al.'s claims. First, Angelini et al. claim that the prediction $2^{1/\nu}=1$ of [1] is incorrect, because it results from the relation $\lambda_{\rm max}=2^{1/\nu}$ between the largest eigenvalue of the linearized renormalization-group (RG) transformation and $\nu$, which cannot be applied to the ensemble renormalization group (ERG) method, because for the ERG $\lambda_{\rm max} =1 $. However, the feature $\lambda_{\rm max}=1$ is specific to the ERG transformation and it does not give any grounds for questioning the validity of the general relation $\lambda_{\rm max}=2^{1/\nu}$ specifically for the ERG transformation. Second, Angelini et al. claim that $\nu$ should be extracted from an early RG regime (A), as opposed to the asymptotic regime (B) used to estimate $\nu$ in [1] and that (B) is dominated by finite-size effects. Still, (A) is a small-wavelength, non-critical regime, which cannot characterize the critical exponent $\nu$ related to the divergence of the correlation length. Also, the fact that (B) involves finite-size effects is a feature specific to the ERG, and gives no rationale for extracting $\nu$ from (A). Finally, we refute the remaining claims made by Angelini et al., and thus stand by our assertion that the ERG method yields a prediction given by $2^{1/\nu}=1$.