The ensemble renormalization group (ERG) is one of the few real-space renormalization-group methods claimed to reproduce the correct critical exponents of strongly disordered systems, but its application to the hierarchical Edwards-Anderson (HEA) spin glass is the object of an unresolved published controversy. All published ERG results for the disordered HEA rest on hierarchy depth n≤4n \le 4, where exhaustive enumeration of spin configurations is feasible; at finite depth the size-restoring step of the ERG iteration forces an eigenvalue λ=1\lambda = 1 into the linearized transformation and produces a plateau in the renormalization-group flow at a value depending on the starting temperature. Angelini, Parisi and Ricci-Tersenghi contend these are finite-depth artifacts that vanish as n→∞n \to \infty, so the correlation-length exponent ν\nu must be fitted in the pre-plateau window t<t∗t < t^{*}; Castellana contends that this window is a non-critical regime and that the ERG prediction is the trivially wrong value 21/ν=12^{1/\nu} = 1. The open problem is to settle this: extend the ERG for the disordered HEA to depths substantially beyond n=4n = 4, determine how t∗t^{*}, the plateau structure, and the eigenvalue spectrum scale with depth, and establish whether the extracted ν\nu converges with depth to the independently known spin-glass values below the upper critical dimension or the marginal stability persists.

Background

Spin glasses are magnetic systems with quenched (fixed in place rather than thermally fluctuating) random interactions whose low-temperature phase is a frozen, disordered state. The Edwards-Anderson (EA) model places Ising spins Si=±1S_i = \pm 1 on a lattice with random nearest-neighbor couplings, and its paramagnet-to-spin-glass transition is characterized by the critical exponent ν\nu governing the divergence of the correlation length, ξ∼∣T−Tc∣−ν\xi \sim |T - T_c|^{-\nu}. Below the upper critical dimension dU=6d_U = 6 (the dimension above which mean-field theory becomes exact), no controlled analytic method computes ν\nu for the EA model: perturbative replica field theory — field theory in which the quenched disorder is averaged by introducing many fictitious copies ("replicas") of the system — is uncontrolled at strong coupling. This motivates real-space renormalization-group (RG) methods — iterative decimation-and-rescaling transformations acting directly on the lattice degrees of freedom — which are non-perturbative by construction.

A controlled testbed is the hierarchical Edwards-Anderson (HEA) model, a spin glass on the Dyson hierarchical lattice introduced by Dyson (1969): a system of 2k+12^{k+1} spins is two copies of a 2k2^k-spin system joined by random Gaussian couplings whose strength decays with the hierarchical level kk, controlled by an exponent ζ\zeta. Tuning ζ\zeta makes the model mimic a short-range spin glass in any chosen effective dimension dd; the region 1/2<ζ<2/31/2 < \zeta < 2/3 corresponds to dimensions above dUd_U and ζ=5/6\zeta = 5/6 approximately corresponds to three dimensions. Because the lattice is a tree and is mapped to itself by block decimation, real-space RG transformations can be implemented with unusual precision.

The ensemble renormalization group (ERG) of Angelini, Parisi and Ricci-Tersenghi (2013) renormalizes the probability distributions of the couplings rather than individual disorder samples: (1) ensemble-averaged observables (normalized spin-glass correlations at each level) are computed for an ensemble of nn-level systems; (2) the coupling variances of (n−1)(n-1)-level systems are fixed by matching those observables; (3) the original size is restored by re-joining two renormalized systems with fresh couplings drawn from the top-level distribution. This size-restoring step (sometimes called the reconstruction step) is needed only because the system is finite. In the disordered model, thermal averages were computed exactly by exhaustive enumeration of all spin configurations, limiting the depth to n=4n = 4; the ferromagnetic version, where exact recursions exist, was benchmarked up to n=13n = 13, where the extracted ν\nu is 2.076(6) against the exactly known 1.9487 (finite-depth errors of about 7% at n=13n = 13 and 43% at n=4n = 4, where ν=2.79(12)\nu = 2.79(12)).

Following Wilson's classic real-space RG relation, the eigenvalue of largest norm λmax⁡\lambda_{\max} of the matrix linearizing an RG transformation at a critical fixed point (a parameter set left invariant by the transformation, marking a continuous phase transition) obeys λmax⁡=21/ν\lambda_{\max} = 2^{1/\nu}; equivalently, the difference of two RG trajectories at nearby inverse temperatures β1\beta_1, β2\beta_2 grows with the RG step tt as

β1σ1t(β1)−β2σ1t(β2)β1−β2∝2t/ν\frac{\beta_1\sigma_1^{t}(\beta_1)-\beta_2\sigma_1^{t}(\beta_2)}{\beta_1-\beta_2}\propto 2^{t/\nu}

where σ1t\sigma_1^t is the renormalized coupling standard deviation at the lowest hierarchical level. In the ERG on the HEA the flow does not reach the usual fixed points: after t∗t^{*} steps it plateaus at a value σFP\sigma_{FP} that depends on the starting temperature. Angelini et al. attribute the plateau and a spurious eigenvalue λ=1\lambda = 1 (which the size-restoring step forces into the linearization matrix as a constant top-level variance) to finite-depth artifacts, and extracted ν\nu from fits of the relation above restricted to the pre-plateau window t<t∗t < t^{*}, finding agreement with Monte Carlo simulations.

Castellana (2019) re-analyzed the same method and reached the opposite conclusion: diagonalizing the linearized ERG at the critical fixed point for depths n=2,3,4n = 2, 3, 4 yields a marginally stable fixed point with largest eigenvalue exactly 1, so the ERG prediction is 21/ν=12^{1/\nu} = 1 — trivially wrong both above and below dUd_U — and the pre-plateau window used by Angelini et al. is a small-wavelength, non-critical regime in which the fitting relation is invalid. Angelini, Parisi and Ricci-Tersenghi replied in a Comment (2020) that the eigenvalue 1 and the plateau are artifacts of the size-restoring step, unnecessary at infinite size; that t∗t^{*} grows with depth, as verified for the ferromagnetic model up to n=13n = 13; and that in the n→∞n \to \infty limit the spurious eigenvalue would be absent. Castellana's Reply (2020) reasserts that the pre-plateau regime is non-critical at any depth, that the eigenvalue-1 structure is intrinsic to the ERG transformation, and that without the size-restoring step the linearization matrix is not even square, so the Comment's claimed large-depth eigenvalue structure is ill-defined. The exchange ended with both positions unreconciled.

Crucially, every load-bearing claim about the disordered HEA rests on depth n=4n = 4: no scalable implementation of the ERG for the spin glass (for example, replacing exhaustive enumeration by Monte Carlo or stochastic-approximation estimates of the ensemble observables, as Castellana did only at fixed small depth) has been reported, so the depth-dependence of t∗t^{*}, of the plateau, of the eigenvalue spectrum, and of the fitted ν\nu is unknown for the model the controversy is actually about. Castellana (2019) also states the broader problem this serves: finding a real-space RG method that yields correct universal quantities for spin glasses below the upper critical dimension.

Problem Statement

For the disordered hierarchical Edwards-Anderson (HEA) spin glass — Ising spins with random Gaussian couplings on the Dyson hierarchical lattice, with coupling strength decaying with the hierarchical level — determine the large-depth behavior of the ensemble renormalization group (ERG) and thereby settle which of the two published readings of the method is correct.

All published ERG results for the HEA rest on hierarchy depths n≤4n \le 4 (exhaustive enumeration of spin configurations in the original work; stochastic approximation of the matching step in the re-analysis, still at n≤4n \le 4). At finite depth the size-restoring step of the ERG forces an eigenvalue λ=1\lambda = 1 into the linearized transformation and produces a plateau in the RG flow at a value σFP\sigma_{FP} depending on the starting temperature; Angelini, Parisi and Ricci-Tersenghi attribute both to finite-depth artifacts that vanish as n→∞n \to \infty (with t∗→∞t^{*} \to \infty), while Castellana maintains that the pre-plateau window t<t∗t < t^{*} is a non-critical regime and that the ERG prediction is 21/ν=12^{1/\nu} = 1.

Determine, for the disordered HEA at hierarchy depths substantially beyond n=4n = 4 (the ferromagnetic benchmark indicates depths of order ten or more are needed for percent-level extrapolation):

  1. measure how the plateau onset t∗t^{*}, the plateau properties (height, σFP\sigma_{FP} and its starting-temperature dependence), and the eigenvalue spectrum of the linearized ERG transformation — in particular the fate of the λ=1\lambda = 1 mode and of the remaining eigenvalues — scale with depth, across a range of the coupling-decay exponent ζ\zeta (which sets the effective dimension of the hierarchical lattice) spanning effective dimensions both below and above the upper critical dimension dU=6d_U = 6;

  2. extract the correlation-length critical exponent ν\nu as a function of depth using the pre-plateau protocol of Angelini et al., fitting

β1σ1t(β1)−β2σ1t(β2)β1−β2∝2t/ν\frac{\beta_1\sigma_1^{t}(\beta_1)-\beta_2\sigma_1^{t}(\beta_2)}{\beta_1-\beta_2}\propto 2^{t/\nu}

for t<t∗t < t^{*}, and establish whether ν(n)\nu(n) converges as nn grows, and to what value;

  1. compare the large-depth limit of the extracted ν\nu, and of the eigenvalue-based prediction, with independently established Monte Carlo values of ν\nu for the corresponding long-range spin glasses, and state which reading of the ERG the results support: convergence to the accepted spin-glass exponents with vanishing size-restoring artifacts (vindicating the method if properly used), or persistence of the marginal stability and of the prediction 21/ν=12^{1/\nu} = 1 (confirming the criticism).

These three components jointly serve one objective — establishing whether, and in what sense, the ERG yields the correct critical exponent for the disordered hierarchical spin glass in the infinite-hierarchy limit. Equivalently, characterize the n→∞n \to \infty ERG transformation on the HEA: the fate of the λ=1\lambda = 1 mode, the existence and criticality of the fixed-point structure, and the validity range of the fitting relation in the window t<t∗t < t^{*}.

References

  1. Maria Chiara Angelini, Giorgio Parisi, and Federico Ricci-Tersenghi, 'Comment on Real-space renormalization-group methods for hierarchical spin glasses', J. Phys. A: Math. Theor. 53, 418001 (2020), DOI: 10.1088/1751-8121/ab8414, arXiv:1911.02323, https://arxiv.org/abs/1911.02323
  2. Michele Castellana, 'Reply to comment on Real-space renormalization-group methods for hierarchical spin glasses', J. Phys. A: Math. Theor. 53, 418002 (2020), DOI: 10.1088/1751-8121/ab8671, arXiv:2102.06263, https://arxiv.org/abs/2102.06263
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