Multifractal Scalings across the Many-Body Localization Transition
Nicolas Macé Fabien Alet Nicolas Laflorencie











    Trapsrev4-1nicolas.mace@irsamc.ups-tlse.frLaboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, Francefabien.alet@irsamc.ups-tlse.frLaboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, Francenicolas.laflorencie@irsamc.ups-tlse.frLaboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, France
        
        In contrast with Anderson localization where a genuine localization is observed in real space, the many-body localization (MBL) problem is much less understood in the Hilbert space, support of the eigenstates. In this work, using exact diagonalization techniques we address the ergodicity properties in the underlying N-dimensional complex networks spanned by various computational bases for up to L=24 spin-1/2 particles (i.e. Hilbert space of size N≃ 2.7 10^6). We report fully ergodic eigenstates in the delocalized phase (irrespective of the computational basis), while the MBL regime features a generically (basis-dependent) multifractal behavior, delocalized but non-ergodic. The MBL transition is signaled by a non-universal jump of the multifractal dimensions.
        
        
        
        Multifractal Scalings across the Many-Body Localization Transition
            Nicolas Laflorencie
            December 30, 2023
        ==================================================================
        
        
         Introduction— 
        Anderson localization (AL) <cit.> is a fundamental phenomenon where transport is hindered by disorder in quantum systems composed of free particles.
        When interactions between the particles is added, localization tends to be prohibited and thermalization is favored.
         The eigenstate thermalization hypothesis (ETH) <cit.>, which provides a theoretical understanding of how thermal equilibrium is encoded at the level of individual many-body eigenfunctions, applies generically to most interacting systems (with the important exception of integrable models). 
         It is thus quite remarkable that disordered quantum interacting systems can still show absence of transport and thermalization in the form of many-body localization (MBL) <cit.>, a topic that has attracted considerable interest recently (see reviews Ref. nandkishore_many-body_2015,abanin_recent_2017,alet_many_2018,abanin_ergodicity_2018). 
         Besides arrested transport, MBL features several striking properties, e.g.  anomalously small (area-law) entanglement <cit.>, emergent integrability and Poisson spectral statistics <cit.>, very slow (logarithmic) entanglement spreading <cit.>, some of which expose the differences between MBL and AL. 
         Numerical simulations of 1d lattice models <cit.>, analytical rigorous proofs <cit.>, and effective localized-bits ( l-bits) models <cit.> have been instrumental in understanding these properties of the MBL phase. 
         
        
        
        However, several important questions remain open regarding the MBL-ETH transition and the nature of eigenstates. This contrasts with the much-better understood case of AL where the spatial extension of  single-particle orbitals is known to display  multifractality at criticality on finite-dimensional lattices, a behavior which characterizes the transition and its underlying field-theoretical description <cit.>. In the MBL context on the other hand, the behavior of  many-body eigenstates in the Hilbert space is not firmly established <cit.>. We aim at addressing three open issues:  (i) the possible existence of an intermediate non-ergodic delocalized phase in the ETH phase <cit.>; (ii) the wavefunctions properties in the MBL regime sometimes described as Fock-space localized <cit.>, while perturbative <cit.> and numerical <cit.> results indicate multifractality; (iii) a consistent description of the critical regime. 
        
        
        
        
        
        The central quantities of interest in our work are the participation entropies (PE) S_q, derived from the q^ th moments of a wavefunction |Ψ⟩ expressed in a given basis:
        
        |Ψ⟩=∑_α=1^ Nψ_α|α⟩   and   S_q=1/1-qln(∑_α=1^ N |ψ_α|^2q).
        
        In the Shannon limit S_1=-∑_α |ψ_α|^2ln |ψ_α|^2, while the case q=2 recovers the usual inverse participation (IPR) <cit.> with S_2=-ln( IPR). 
        Let us first describe the possible leading asymptotic behaviors for S_q with the support size N. 
        For a perfectly delocalized state S_q=ln N.
        In contrast, if a state is localized on a finite set, one gets S_q=constant, as observed for AL. In an intermediate situation, wavefunctions are extended but non-ergodic, with S_q=D_qln N (D_q< 1 are q-dependent multifractal dimensions).
        
        
        In this Letter, we inspect  many-body eigenstates in the Hilbert space by studying D_q across the ETH-MBL transition for model  Eq. (<ref>). 
        Building on large-scale numerics for the most relevant basis sets {|α⟩}, we provide a complete scaling analysis describing both phases and the critical point, which captures finite-size effects. We conclude for full ergodicity (D_q=1) for ETH, generic multifractality in the entire MBL regime, and a non-universal jump of D_q at the transition, as sketched in Fig. <ref> <cit.>.
        
        
         MBL as a complex network Anderson problem—
        We focus on the random-field XXZ S=1/2 chain model
        
         H=∑_i=1^L[Δ S_i^z S_i+1^z-h_iS_i^z+1/2(S_i^+ S_i+1^-+S_i^- S_i+1^+)],
        
        with periodic boundary conditions and h_i randomly drawn from a uniform distribution [-h,h]. Eq. (<ref>) is equivalent <cit.> 
         to interacting spinless fermions in a random potential.
        This system has been intensively studied <cit.> and its phase diagram is well-known for the case Δ=1 with a critical disorder estimated to beh_c=3.7(2) in the middle of the many-body spectrum {E} such that ϵ=(E-E_ min)/(E_ max-E_ min)=0.5. This Hamiltonian can be recast as a single particle Anderson problem of the general form
        
         H=∑_αμ_α|α⟩⟨α| +∑_⟨αβ⟩t_αβ|α⟩⟨β|,
        
        in a given basis {|α⟩}.
        Of course, the localization properties measured by the PE depend crucially on the choice of {|α⟩}. We focus on two bases: spin configurations {|α⟩}_S and Fock basis {|α⟩}_F, which we argue are the most relevant for the model Eq. (<ref>): (i) both bases diagonalize H in specific limits where localization is well-understood (the non-interacting limit Δ=0 for {|α⟩}_F, the limit h→∞ for {|α⟩}_S) and are thus used as a starting point for the  l-bits construction or efficient numerical simulations of MBL;  (ii) they implement the U(1) conservation rule (particle number or magnetization conservation) of the model; 
        
        (iii) H is  sparse in both bases: an ingredient which eases  numerical diagonalization and may favor ergodicity breaking.
        
        a) Spin configuration basis.
        The basis {|α⟩}_S uses the local projection of S_i^z, i.e.|α⟩=| -0.15cm ↑↓↑…⟩. We restrict the study to the zero magnetization sector ∑_i S_i^z =0 (half-filling for fermions) of dimension N=LL/2≃ 2^L/√(L). In the spin configuration basis, Eq. (<ref>) becomes a hopping problem Eq. (<ref>) with disordered on-site energies μ_α=⟨α|∑_iΔ S_i^z S_i+1^z-h_iS_i^z|α⟩, and constant hopping terms t_αβ=1/2 (allowing tunnelling between neighboring states ⟨αβ⟩ connected by spin-flip terms of Eq. (<ref>)). As the site-dependent connectivity z≈ L/2 grows faster with system size L than the average on-site disorder strength σ_μ
        ∼ h√(L), general arguments would prohibit genuine AL in this complex network (note however the strong correlations of the potential between neighboring sites).
        
        
        b) Fock basis.{|α⟩}_F are many-body states built from non-interacting localized orbitals which diagonalize the free-fermion part H-Δ∑_i S_i^z S_i+1^z of Eq. (<ref>).
         On-site potentials μ_α are the sum of the non-interacting orbital energies corrected by the Hartree-Fock term, and off-diagonal hoppings t_αβ are built from the interaction terms <cit.>.  Viewing MBL as an Anderson problem on graphs defined by these Fock states has been promoted in very early works <cit.>. Nevertheless, the hopping problem expressed in {|α⟩}_F is qualitatively different than from in {|α⟩}_S. While diagonal terms have similar behaviors, there is a much larger number of non-zero matrix elements z_αβ between Fock states which is constant over the graph: z_αβ=1/4[L/2(L/2-1)]^2+L^2/4 (in the ∑_i S_i^z=0 sector). Moreover, hoppings t_αβ are not constant but random, both in sign and magnitude <cit.>.
        
        
        
         0.15cm
         Participation entropies: large scale numerics— Eigenstates of Eq. (<ref>) are extracted from full and shift-invert subset <cit.> diagonalization, focusing on high-energy states (ϵ=0.5) on various system sizes, ranging from L=8 spins (with N=70) up to L=24 (N = 2 705 432). Disorder average is performed over many independent samples, typically tens of thousand for the smallest sizes L≤ 16, and several hundreds for the largest samples 18≤ L≤ 24 <cit.>. We fix Δ=1.
        
        
        We first discuss the scaling with Hilbert space size N of the disorder-average PE S_q defined by Eq. (<ref>) and shown in Fig. <ref> for both Fock and configuration spaces for a few representative values of the disorder.
        In the ETH phase (Fig. <ref> a,b), low disorder data follow a purely ergodic scaling of the form S_q=ln N+b_q with b_q<0 for both Fock and configuration spaces. Upon increasing h a curvature develops, indicating that an asymptotic scaling regime might be eventually reached for larger N, as exemplified by h=2.6 and h=3.2 data. In the MBL regime (Fig. <ref> (c,d) with q=1,2) at h=5 both bases exhibit a delocalized behavior S_q=D_qln N+b_q with a q- and basis-dependent multifractal dimension D_q<1 and a  correction b_q>0. 
        
        Very interestingly, b_q changes sign (negative for ETH and positive for MBL) as seen in Fig. <ref> where a crossing point in S_q/ln N appears in the vicinity of the critical disorder strength h_c∼ 3.8. Crossings are equally observed for Fock and configuration spaces in Fig. <ref> (a,d,e). This effect is also clearly visible from the distributions P(S_q/ln N) shown in Figs. <ref> (b,c) where we observe strikingly distinct behaviours on both sides of the transition, with qualitatively different finite size effects and opposite skewness. As S_q/ln N is restricted to the interval [0,1], the distributions at small disorder slowly converge (from below) towards unity with system size. Conversely, in the MBL regime, rare events tails extend above the average while distributions shrink with system size (with a variance vanishing as a power-law <cit.>), thus leading to positive finite size corrections. As argued below, the negative constant correction b_q in the ETH phase can be physically related to a non ergodicity volume Λ_q=exp(-b_q). In contrast,  the positive value of b_q in the MBL regime seems to be rooted in rare events showing up in positive skewness distributions.
         0.15cm
         Scaling analysis— We turn to a finite size scaling analysis of PE across the MBL transition. Fig. <ref> shows results for q=2 in the two basis where very good collapses can be obtained using the following ansatz <cit.>, inspired by a recent investigation <cit.> of the Anderson problem on random graphs:
        
        
            S_q( N)-S_q,c( N) =
               G_ vol( N/Λ_q)    if  h< h_c 
             G_ lin(ln N/ξ_q)          if  h> h_c.
        
         Note that we first subtracted S_q,c( N) the critical PE at h_c=3.8 <cit.>. In the ETH regime (top panels), a volumic scaling G_ vol( N/Λ) yields an excellent description of the crossover with Λ interpreted as a non-ergodicity volume <cit.>. When the Hilbert space dimension N≫Λ, full ergodicity is recovered with S_q=ln N-(1-D_q,c)lnΛ,  D_q,c being the multifractal dimension at the transition, while the critical behavior S_q→ S_q,c  is recovered below the non-ergodicity volume  N≪Λ. This volumic scaling  is found to be universal as it occurs for both Fock and spin basis, with a non-ergodicity volume diverging very fast as the transition is approached: lnΛ∼ (h_c-h)^-a (insets (a) in Fig. <ref>), with a≈ 0.45 <cit.>. An analogous volume law with a= 0.5 was reported <cit.> for the Anderson problem on random regular graphs, in agreement with analytical predictions in infinite dimensions <cit.>.
        
        
        Contrasting with ETH, data in the MBL regime rather follow a linear scaling function G_ lin(lnN/ξ) with a universal form
        
        
        S_q, MBL-S_q,c=-D_q,cln N/ξ,
        
        again yielding a very good data collapse (bottom panels of Fig. <ref>).
        The length scale ξ, extracted from the best collapse, is shown in panels (b) of Fig. <ref> as a function of the distance to criticality h-h_c, where again h_c=3.8 has been fixed. In Eq. (<ref>), ξ lies in the range (1,+∞) which guarantees S_q, MBL to remain positive in the limit ln N≫ξ where the leading term follows S_q=D_q,c(1-1/ξ)ln N. In the other limit ln N≪ξ, one retrieves the critical scaling. Close to criticality, we observe a divergence of the length scale ξ∝ (h-h_c)^-a', with a'≈ 0.76.
        
        
        
        It is crucial to make a distinction between random graphs which display  AL at large disorder (see for instance Refs. <cit.>), and the present many-body problem where instead, multifractality is present in the entire MBL phase ∀ h≥ h_c. 
        
        We clearly observe <cit.>D_q, MBL(h≫ h_c) ∝ 1/h, thus vanishing only in the infinite disorder limit, in agreement with recent results <cit.>. This behavior can be accounted for by simple strong disorder arguments <cit.>. Numerical data further suggest that this form persists almost down to the critical point such that D_q, MBL(h) ≃ D_q,ch_c/h.
        
        
         0.15cm
         Discussion— Finite-size effects might affect the estimates of MBL critical point and exponents <cit.>. However our scaling analysis carefully accounts for them (Fig. 4) and strongly supports the following scenario across the ETH-MBL transition for the standard model Eq. (<ref>). (i) Eigenstates are generically ergodic, with no multifractality over the entire ETH phase (D_q=1) thus removing the possibility for an intermediate non-ergodic state with D_q<1 in the model Eq. (<ref>). (ii) The entire MBL regime harbors delocalized non-ergodic eigenstates with disorder and basis dependent multifractal dimensions, only vanishing in the infinite disorder limit. (iii) The ETH-MBL transition point belongs to the same non-ergodic regime, with a jump of the multifractal dimension D_q<1 at the transition, in agreement with recent predictions for a non-thermal critical point <cit.>. ETH-MBL criticality displays two radically different scalings across the transition for eigenstates extension through the Hilbert space: a volumic law G_ vol( N/Λ) for ETH with an emergent non-ergodicity volume Λ, diverging at the transition, and a linear scaling G_ lin(ln N/ξ) in the MBL regime. This is clearly compatible with a second-order critical point, but with an unsymmetrical nature, as also recently highlighted in Refs. <cit.>. To which extent ξ is related to a localization length in real space remains to be understood. 
        We do not see signs of a putative second MBL phase <cit.> in our multifractal analysis which is not directly comparable with the (real-space) MBL multifractality proposed by a renormalization group study <cit.>.
        
        
        Our results on multifractality of the MBL phase have the following consequences on other approaches for the MBL problem: (i) they give the correct scaling behavior for variational computational approaches to eigenstates based on the two basis at hand, (ii) put constraints for putative theories of the critical point (in the same spirit as the multifractal spectrum does for AL <cit.>), (iii) provide a new phenomenology, clearly different from the Anderson problem on random graphs or Cayley trees <cit.>, thus challenging possible analogies.
        
        
        Other perspectives are also opened up by the present work, such as connecting with the anomalous slow dynamics observed <cit.> in a large part of the ETH phase, e.g. 
        by considering multifractal properties of operators <cit.>. We note that while rare-regions effects <cit.> can induce such subdiffusive transport, much less dramatic consequences are expected for eigenstates (reflecting "infinite-time" physics) <cit.>.
        Another extension will consist in testing for universal behaviour by considering other models with a MBL transition or disturbing the computational basis <cit.>. Finally, our results pave the way for a better control of decimation approaches in configuration <cit.> or Fock space <cit.>, as well as for a recently proposed Hilbert space percolation approach <cit.> to the MBL transition.
        
         
        We are deeply grateful to Gabriel Lemarié for enlightening explanations about the scaling analysis performed in Ref. <cit.>. We also thank D. J. Luitz, Y. Bar Lev, and I. Khaymovich for stimulating discussions.
        This work benefited from the support of the project THERMOLOC ANR-16-CE30-0023-02 of the French National Research Agency (ANR) and by the French Programme Investissements d'Avenir under the program ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT. We acknowledge PRACE for awarding access to HLRS's Hazel Hen computer based in Stuttgart, Germany under grant number 2016153659, as well as the use of HPC resources from CALMIP (grants 2017-P0677 and 2018-P0677) and GENCI (grant x2018050225).
        
        
        
        
        
        
        
        
        
         2cm
        Supplemental material 
        
        
        § ANDERSON PROBLEM IN THE FOCK SPACE
        (i) Fermionic language—
        Using the Jordan-Wigner transformation, one can rewrite the spin Hamiltonian 
        
         H=∑_i=1^L[Δ S_i^z S_i+1^z-h_iS_i^z+1/2(S_i^+ S_i+1^-+S_i^- S_i+1^+)],
        
        as interacting fermions on a chain of L sites
        
         H=∑_i=1^L1/2(f_i^† f_i+1^+h.c.+2Δ n_i n_i+1-μ_i n_i) +C,
        
        where the random chemical potential μ_i=2h_i, and C is an irrelevant constant. 
        The non-interacting part (Δ=0)
         H_0=∑_i=1^L(f^†_if_i+1^+f^†_i+1f_i^  -μ_i n _i)
        
        is diagonalized by b_k=∑_i=1^Lϕ_k(i)f_i (ϕ_k being the single particle orbitals, Anderson localized for any finite disorder in 1D), yielding
        
        
        
        
        
        H_0=∑_k=1^Lϵ_k b^†_kb_k^.
        
        In this non-interacting basis, the interaction term V=Δ∑_in_i n_i+1 reads
        
        V=∑_jklmV_jklmb^†_jb^_kb_l^†b_m^,  with   V_jklm=Δ∑_i=1^Lϕ_j^*(i)ϕ_k(i)ϕ_l^*(i+1)ϕ_m(i+1).
        (ii) Many-body problem—
        When the system is filled with a finite density of fermions ν=N_f/L, many-body states are expressed in the the Fock basis {α}
        |α⟩=∏_k=1^L(b_k^†)^n_k|0⟩   (n_k=0,1),   with  ∑_k=1^Ln_k=N_f.
        
        We restrict the discussion to half-filling ν=1/2, putting N_f=L/2 particles on L sites yields an Hilbert space of dimension N=L!/(L/2!)^2. 
        The original many-body problem Eq. (<ref>) can be rewritten in this Fock basis as follows
         H=∑_α=1^ Nμ_α|α⟩⟨α|+∑_α≠βt_αβ|α⟩⟨β|+h.c.,   with   t_αβ=⟨α|V|β⟩   and  μ_α=∑_k=1^Ln_kϵ_k+t_αα.
        
        The hopping between sites α and β is therefore  governed by the interaction term Eq. (<ref>) which can be decomposed in two distinct contributions, 
        one-fermion terms: V_jjlmn_j b_l^†b_m^ with L^2/4 non-zero matrix elements, and two-fermion terms:
        V_jklm b^†_jb^_kb_l^†b_m^ having L^4/64-L^3/16+L^2/16 non-vanishing terms.
        
        
        Clearly, off-diagonal hoppings t_α,β are random numbers, built from products of localized orbitals in Eq. (<ref>).
        
        
        
        
        
        
        § ALTERNATIVE SCALINGS
        
        We show in Fig. <ref> that neither a linear scaling G_ lin(ln N/ξ) can describe the ETH phase, see Fig <ref> (a), nor a volumic one G_ vol( N/Λ) for the MBL regime, see Fig <ref> (b). Indeed, we observe systematic deviations as it is impossible to achieve a good collapse of the data, in contrast with Fig. 4 of the main text. Similar scaling violations observed in Fig. <ref> for S_2^ Fock occur in the configuration space.
        
        
        
        
        
        
        § CRITICAL FIELD DETERMINATION & COLLAPSE QUALITY
        
        Here we address the sensitivity of the scaling analysis with respect the critical disorder value. In our scaling analysis, we used h_c=3.8, first guided by the crossing plot (Fig. 3 in the main text). If for instance one takes h_c=4.4 as suggested for instance by the numerical linked cluster expansion result <cit.>, one gets a very poor collapse, as shown below in Fig. <ref>.
        
        
        
        One can go further, by making quantitative the quality of the data collapse, which strongly depends on the critical field h_c.
        Such a determination is non-trivial: while the most commonly accepted value is h_c = 3.7(2)<cit.>, some works put forward a significantly larger critical disorder <cit.>. Here, we estimate h_c = 3.8(1), using a method based on the optimization of Spearman's rank correlation coefficient.
        
        Given a 2D cloud of data points (x_i, y_i), Spearman's coefficient R quantifies how close the y_i are from being a monotonic function of the x_i. In other words, R quantifies how well the data points align on the graph of an (unspecified) monotonic function.
        R = 1 indicates a perfectly monotonic relashionship between y_i and x_i, while R = 0 indicates its complete absence.
        In the ETH region, finding the non-ergodicity volumes Λ(h) such that S_q - S_q,c is best described by the volumic scaling function 𝒢_vol(𝒩/Λ) amounts to maximizing Spearman's coefficient R.
        The residual 1-R quantifies the quality of the collapse.
        Similarly, in the MBL region, one can maximize Spearman's R to find the lengths ξ(h) such that S_q - S_q,c is best described by the linear scaling function 𝒢_lin(log𝒩/ξ).
        This method both automatically determines the collapse and robustly estimates its quality, without any assumption on the form of the scaling function 𝒢_vol or 𝒢_lin.
        Fig. <ref> shows the value of the residual for different candidate critical field values h_c.
        The best collapse is acheived when the candidate critical field is the closest to the transition point.
        The sharp residual dip observed on Fig. <ref> robustly points to a transition at h_c = 3.8(1).
        
        
        
        
        
        § DISTRIBUTION OF S_2 IN THE MBL REGIME
        
        The behavior of rare events tails in the MBL regime is an interesting issue for which we provide a more detailed study  at h=5, shown below in Fig. <ref>, where we confirm that the finite size effects shrink with increasing system sizes. More precisely, the standard deviation σ decays with ln N as a power-law with an exponent ≈ 0.45, as visible in the inset of Fig. <ref>. 
        
        
        
        
        
        § STRONG DISORDER LIMIT
        
        For very high disorder one can use a strong disorder decimation scheme for the spin S=1/2 Heisenberg chain
         H=∑_i=1^L(JS⃗_i·S⃗_i+1-h_iS_i^z).
        
        The idea is to gradually eliminate sites having the strongest fields.
        These sites will be frozen parallel (or anti-parallel) to the field direction for the ground-state (excited state). 
        If |h_i|=h_ max, the surrounding sites (i-1, i+1) will be perturbatively  coupled by J_(i-1, i+1)^ eff=J^2/2h_ max≪ J.
        The fields are also weakly renormalized, but we can neglect this small effect for the present argumentation.
        The decimation of sites can then be continued up to a cutoff Γ≤ |h_i|, as illustrated in Fig. <ref>.
        
        
        
        Taking an initial random field distribution uniform in [-h,h] [Fig. <ref> (c)] with h≫ J, if we decimate all the fields |h_i|≥Γ, a fraction ρ_ fr(Γ,h) of sites will be frozen, while ρ_ act=1-ρ_ fr remains active, with ρ_ act(Γ,h)=Γ/h.
        Assuming an initial spin chain with L sites, we end up with an active backbone containing  L_ act=LΓ/h active spins, as shown in Fig. <ref> (b). Only active sites contribute to the participation entropies such that the relevant portion of the spin configuration space will be N_ act∝N^Γ/h, thus giving to leading order
        
        
        S_q^ Conf∝Γ/hln N.
        
        
        Ignoring the small couplings J^ eff, the decimated chain acts as a collection of isolated active clusters of various lengths ℓ_ act^i, and the average size of such active clusters is ℓ = ρ_ fr^-1=(1-Γ/h)^-1. Including this length scale in the participation entropy at strong disorder yields
        
        
        S_q^ Conf∝ (1-1/ℓ)ln N.
        
        Moreover, we know from the scaling arguments S_q, MBL-S_q,c=-D_q,cln N/ξ (main text) that in the MBL regime,
        
        
        S_q^ Conf = D_q,c(1-1/ξ)ln N,
        
        suggesting that at strong disorder, the length scale ξ coincides with the average size of  active clusters: ξ = ℓ.
        
        
        The 1/h prediction of this strong disorder argument is well corroborated by the numerics in Fig. <ref>. It further confirms that eigenstates cannot be Anderson localized in the spin configuration space, and are generically non-ergodic with a finite multifractal dimension. 
        
        
        This naive approach is not expected to work when approaching criticality, for instance it would predict a power-law divergence of ξ with an exponent a'=1. We nevertheless find that the prefactor of the 1/h decay is such that S_q^ Conf≈ D_q,ch_c/hln N in a fair range of large h, a formula which is used to plot the straight lines in Fig. <ref>.
