Reply to comment on 'Real-space renormalization-group methods for hierarchical spin glasses'
aa r X i v : . [ c ond - m a t . d i s - nn ] F e b Reply to comment on ‘Real-spacerenormalization-group methods for hierarchicalspin glasses’
Michele Castellana
Laboratoire Physico-Chimie Curie, Institut Curie, PSL Research University,CNRS UMR 168, Paris, FranceSorbonne Universit´es, UPMC Univ. Paris 06, Paris, FranceE-mail: [email protected]
Abstract.
In their comment, Angelini et al. object to the conclusion of [
J. Phys.A: Math. Theor. , 52(44):445002, 2019], where we show that in [
Phys. Rev. B ,87(13):134201, 2013] the exponent ν has been obtained by applying a mathematicalrelation in a regime where this relation is not valid. We observe that the criticismabove on the mathematical validity of such relation has not been addressed in thecomment. Our criticism thus remains valid, and disproves the conclusions of thecomment. This constitutes the main point of this reply.In addition, we provide a point-by-point response and discussion of Angelini etal.’s claims. First, Angelini et al. claim that the prediction 2 /ν = 1 of [ J. Phys.A λ max = 2 /ν between thelargest eigenvalue of the linearized renormalization-group (RG) transformation and ν , which cannot be applied to the ensemble renormalization group (ERG) method,given that for the ERG λ max = 1. However, the feature λ max = 1 is specificto the ERG transformation, and it does not give any grounds for questioning thevalidity of the general relation λ max = 2 /ν specifically for the ERG transformation.Second, Angelini et al. claim that ν should be extracted from an early RG regime(A), as opposed to the asymptotic regime (B) used to estimate ν in [ J. Phys. A ν 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 rationalefor extracting ν from (A).Finally, we refute the remaining claims made by Angelini et al. As a result ofour analysis, we stand by our assertion that the ERG method yields a predictiongiven by 2 /ν = 1. In [1], Angelini et al. write that the estimate of ν made in [2] relies on therelation ∆ t, ≡ β σ t − β σ t β − β ∝ t/ν , (1)where σ t and σ t are the standard deviations of the spin couplings at the firsthierarchical level of the hierarchical Edwards-Anderson model (HEA), at the t -thensemble renormalization group (ERG) step, and at inverse temperatures β and β ,respectively. However, in [3] we have shown that∆ t, = X l ( λ l ) t v l R 1 ( v l L · σ ) , (2)where v L and v R are the left and right eigenvectors of the matrix that linearizesthe renormalization-group (RG) transformation at the critical fixed point, and σ = ( σ , · · · , σ n ) the standard deviations of the spin couplings on hierarchical levels1 , · · · , n at the beginning of the RG iteration. If t is large, i.e., large enough that theeigenvalue with the largest norm dominates the sum in Eq. (2), then the relation (2)reduces to ∆ t, = [ v
1R 1 ( v · σ )] ( λ ) t (large t ) , (3)where λ and v , R are the (marginally) relevant eigenvalue and the correspondingleft and right eigenvectors, respectively. As a result, for large t , Eq. (1) is correct,with λ = 2 /ν . On the other hand, Eq. (2) demonstrates that Eq. (1) does not holdif t is small, i.e., small enough that multiple eigenvalues contribute to Eq. (2), see“[h]owever, we observe that the exponential dependence ... for t ≤
3, is incorrect”in [3]. The fact that the exponential form (1) can hold for large t only has beenshown also in [4]—see “[i]f λ is the eigenvalue ... then for large n ” on page 110 andEqs. (4.33), (4.37) and (4.38) therein.Importantly, our criticism above on the mathematical grounds and validity ofEq. (1) for small t , which was raised in [3], has not been addressed in [1]. As a result,this criticism remains valid, and refutes the conclusions of [1]. This constitutes themain point of this reply.In addition, we comment on the use of the exponential form (3) in [2]. Weobserve that Eq. (3) involves a proportionality factor v
1R 1 ( v · σ ) which is, ingeneral, different from one, and which corresponds to the value in Fig. 5a of [3]which is reached by P l ( λ l ) t v l R 1 ( v l L · σ ) for large t . On the other hand, the actualrelation which has been used in [2] to estimate ν is∆ t, = 2 tν , (4)see “[t]o extract critical exponents ... in our case” and “[t]he procedure used is thesame as that in the FM case” [2]. The relation (4) differs from Eq. (3) in the pref-actor in the right-hand side which, in general, is different from one in Eq. (3), andequals one in Eq. (4). If Eq. (3) were erroneously applied to the small- t regime,where it is not valid, then for t = 0 its left-hand side would equal one, while itsright-hand side would differ from one, thus resulting in a contradiction. To circum-vent this inconsistency when improperly applying Eq. (4) to the small- t regime, itsprefactor v
1R 1 ( v · σ ) needs to be manually altered, and set to one.Overall, the analysis above reflects the fact that, while Eq. (3) has been formallyderived from the RG equations [3], Eq. (4) lacks a mathematical justification. Withreference to such a lack of mathematical grounds, in Section 3 we will show how theimproper use of the exponential form (4) for small t in [2] leads to an arbitrarinessin the estimate of ν .In addition to the discussion above, in what follows we will present a point-by-point response and discussion of the remaining claims made in [1].
1. Finite-size effects
In this Section we will address the criticisms raised in [1] concerning the conclusionsof [3] and finite-size effects in the ERG. We recall that the ERG method is composedof the following steps [1]:1) A HEA with n hierarchical levels (a) and a HEA n − i -th hierarchical level of (a) and (b) are normally distributed,with zero mean and standard deviation σ i and σ ′ i , respectively.2) A set of observables for (a) is computed.3) The standard deviations σ ′ = ( σ ′ , · · · , σ ′ n − ) of the couplings of (b) aredetermined by matching a set of observables of (b) with a corresponding setof observables of (a).4) Two (b) models are joined and a n -level HEA is built, where couplings at the n -th level are drawn from the distribution of n -th level couplings of (a).Angelini et al. write that “the relation that links the exponent ν to the eigenvaluewith the largest norm λ max of the linearized matrix M , 2 /ν = λ max , cannot beapplied to the matrix M [...] as done in [[3]]. This is because in this case theeigenvalue with the largest norm is always λ max = λ = 1, the presence of thiseigenvalue λ is due to step [4)] of the ERG method. [...] [S]tep [4)] is just introducedto iterate the procedure, given the finite size of the analyzed systems, and it is thusthe step responsible for finite-size effects in the ERG results. The prediction 2 /ν = 1in [[3]] is thus completely dominated by finite-size effects”. In response to thisstatement, we observe the following: the fact that λ max = λ = 1 is a feature specificto the ERG transformation, which is not present in other real-space RG approachesfor the hierarchical model [5]. Along with its relation to finite-size effects, this featuredoes not give any grounds for questioning the validity of the relation 2 /ν = λ max specifically for the ERG transformation, i.e., on an arbitrary, case-by-case basis: Infact, the relation above can be formally derived, and is demonstrated to be valid nomatter what the value of λ max [4].In addition, we recall that the flow which follows from the ERG method [2]is characterized by an early, small- t regime t . t ∗ (A) where ∆ t, grows, andby an asymptotic, large- t regime t & t ∗ (B) where ∆ t, levels off, and reaches aplateau—see Fig. 6 of [2] and Fig. 5 of [3]. Angelini et al. claim that using Eq.(1) in regime (B) corresponds to considering the RG flow in an unphysical regiondominated by finite-size effects, and that ν should be extracted from regime (A) [1].In this regard, we observe the following. Instead of describing a long-wavelengthregime dominated by relevant eigenvalues only, which is characteristic of systemsat their critical point [4, 6, 7], the ERG flow in (A) involves irrelevant eigenvalues,and is characterized by finite length scales [7]—including the system size. It followsthat the value of ν obtained in [2] from (A) cannot describe the exponent ν , whichis characteristic of a long-wavelength, critical regime where the correlation lengthdiverges.As the ERG transformation is further iterated and flows away from (A) to (B), thefinite length scales of (A) are integrated out, and the RG iteration enters the long-wavelength, 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 featurespecific to the ERG, and gives no rationale for extracting ν from regime (A), whichis unrelated to the critical one, nor to apply Eq. (3) to such regime.
2. Large- n behavior Angelini et al. claim that the plateau in (B) disappears for large n , see “[i]n [[2]] itwas also shown that t ∗ grows if larger systems are used [...], going to t ∗ = ∞ in the n → ∞ limit” [1]. However, the fact that the number of RG iterations t ∗ needed totransition from (A) to (B) is an increasing function of n is simply due to the factthat, the larger n , the more the largest norm of irrelevant eigenvalues gets close toone—see for example Figs. 3 and 4 in [3]. In fact, the larger n , the larger the numberof iterations t needed for the irrelevant eigenvalues ( λ l ) t to go to zero in Eq. (2). Nomatter how large n and t ∗ , regime (A) is characterized by irrelevant eigenvalues andfinite characteristic lengths, thus it does not correspond to an asymptotic critical,long-wavelength region such as (B)—see the discussion in Section 1. As a result,the critical exponent ν determined from (A) does not represent the critical exponentrelated to the divergence of the correlation length, no matter how large n .As far as the large- n limit is concerned, Angelini et al. also claim acorrespondence between extracting ν from regime (A), and the large- n limit of theERG transformation, see “fitting data with eq. [(1)] for t < t ∗ [...] takes contributionnot only from λ , but also from λ i with i = 2 , , n = 4. This is exactly thecorrect thing to do, because, in the large n limit, where step [4)] of ERG is no moreneeded, the matrix M will not have the λ = 1 eigenvalue, but only the others[,irrelevant] eigenvalues λ i with 2 ≤ i ≤ n ” [1].However, we observe that the argument above is not valid, for two reasons.First, while the argument claims that the large- n limit is characterized byirrelevant eigenvalues only, regime (A) is characterized by both marginally relevantand irrelevant eigenvalues: This substantial difference invalidates the claimedcorrespondence between extracting ν from (A) and the large- n limit. Second, in theabsence of step 4), see Section 1, the large- n limit is not characterized by irrelevanteigenvalues as claimed by Angelini et al. In fact, as depicted in Fig. 1 of [2], step 4)is required in order to produce an ensemble of rescaled systems which have the samesize as the original one [1, 2]: this is a fundamental property, which is closely relatedto the self-similarity of the RG transformation at the critical point. As a result, inthe absence of step 4), and no matter what the value of n , the ERG transformationwould map n standard deviations σ , · · · , σ n of the coupling distributions of model(a) onto n − σ ′ , · · · , σ ′ n − of (b)—see steps 1), 2) and 3) inSection 1. It follows that the matrix M which linearizes the ERG transformationwould be non-square, and the concept itself of eigenvalues λ , · · · , λ n in Angelini etal.’s argument above would be ill-defined.
3. Exponential fit and value of ν In response to Angelini et al.’s statement “we showed how well an exponential fit ofthe form [(1)] fits the data obtained for the ferromagnetic model with n = 13”, inFig. 1 we show the data for the ERG applied to the ferromagnetic version of theHEA, from Fig. 3 of [2]. The quantity plotted in red dots, ∆ F t, , is analogous to ∆ t, for the HEA, where ‘F’ stands for ‘ferromagnetic’ and the standard deviations arereplaced by the values of the ferromagnetic couplings, and this quantity is plottedas a function of t in an interval which belongs to regime (A). In addition, the solidand dashed lines i) and ii) have been obtained by fitting log ∆ F t, vs. t with a linearfunction for t ≤ t ≥
30, respectively. Shown in semi-logarithmic scale, thestraight lines i) and ii) have different slopes, thus demonstrating that the pointslog ∆ F t, vs. t have a slope which depends on t . This analysis shows that, in regime(A), Eq. (4) does not hold, i.e., the ERG data does not have the exponential form∆ F t, = 2 t/ν , thus confirming the analysis at the beginning of this reply. Given thatthe slope is not constant and that the fitting region used in [2] to determine ν isarbitrary, the predicted value of ν from [1, 2] is also affected by arbitrariness.Also, Angelini et al. write that the fit above has been done “in a much broaderregime 0 < t ≤ < t ∗ ” compared to the corresponding fit for the HEA. In thisregard we observe that, despite the fact that the interval 0 < t ≤ < t ∗ mayappear to be broad, such interval still belongs to the pre-asymptotic, non-criticalregime (A)—see the discussion in Section 1.Finally, in response to Angelini et al.’s statement that the ERG method for the F t, t ∆ F t, ∆ F t, t Fit i)∆ F t, t Fit ii)
Figure 1.
Difference between the ensemble renormalization group (ERG) flowsof couplings at two different temperatures, ∆ F t, , for the ferromagnetic version ofthe hierarchical Edwards-Anderson model (red dots) from [2], in semi-logarithmicscale. The lines represent the fit of log ∆ F t, vs. t with a straight line, where fit i)is made for t ≤ t ≥
30 (blue dashed line). ferromagnetic model “giv[es] a value of ν that is in perfect agreement with the exactresult” [1], we observe that there is a ∼
7% discrepancy between the ERG prediction ν = 2 . ν = 1 . ... from [8].As a result of the analysis provided in this reply, we stand by our assertion thatthe ERG method yields a prediction for ν given by 2 /ν = 1 [3]. Acknowledgments
We thank A. Barra for valuable discussions. [1] M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi. Comment on ‘Real-space renormalization-group methods for hierarchical spin glasses’.
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