Hessian spectrum and low-energy mode localization in the replica-symmetry-broken phase of the fully-connected vector spin glass in a random field

The fully-connected m-component vector spin glass in a Gaussian random field at zero temperature undergoes, for m>=3, a transition at a critical field from a paramagnetic phase with a single energy minimum to a replica-symmetry-broken spin-glass phase with many quasi-degenerate marginal minima. The Hessian spectrum at the minima was solved by the cavity method in the paramagnetic phase (pseudo-gap $\rho(\lambda)\sim\lambda^{m-1}$, quasi-localized edge modes) and at criticality (square-root edge, delocalization of the softest modes, with logarithmic corrections at $m=3$), but the analysis cannot be extended into the spin-glass phase, where replica symmetry breaking invalidates the single-minimum cavity construction. The open problem is to complete this theory: derive the replica-symmetry-broken cavity-field distribution $P(H)$ and, from it, the spectral density with prefactors and the eigenvector localization laws in the spin-glass phase (testing the conjectured square-root edge and the Bray-Moore essential singularity $P(H)\sim e^{-a/H}$); settle the $m=3$ logarithmic corrections and classify, realize, or exclude the non-square-root critical spectra $\rho(\lambda)\sim\lambda^{1/(m-1)}$ predicted for $2<m<3$; quantify the finite-size crossover $N^*\sim\epsilon^{-3}$ that masks asymptotic scaling near criticality; and determine whether the whole pseudo-gap/quasi-localization/delocalization scenario is generic deformed-Wigner (Rosenzweig-Porter) random-matrix physics or requires spin-glass-specific ingredients.

Background

Real structural glasses host, besides phonons, low-frequency nonphononic excitations that are quasi-localized and follow a universal quartic law $D(\omega)\sim\omega^4$; whether mean-field glass models can reproduce this was long unclear, because the best-solved mean-field models (spherical p-spin glasses, perceptrons) have gapped or quadratically vanishing Hessian spectra with fully delocalized eigenvectors. Numerical studies of the three-dimensional Heisenberg spin glass in a magnetic field (Baity-Jesi, Martin-Mayor, Parisi and Perez-Gaviro, 2015) and analytic studies of the XY spin glass on the Bethe lattice (the infinite random regular graph, i.e. a locally tree-like graph of fixed finite connectivity) found gapless spectra with quasi-localized $\omega^4$ modes, motivating the analysis of fully-connected models with continuous spins.

The source paper (Franz, Nicoletti, Parisi and Ricci-Tersenghi, SciPost Phys. 12, 016 (2022)) studied the fully-connected $m$-component vector spin glass at zero temperature in a Gaussian random field, with Hamiltonian

$\mathcal{H}=-\sum_{i<j}J_{ij}\,S_i\cdot S_j-\sum_i b_i\cdot S_i,$

where each spin is a unit vector with $|S_i|=1$, the couplings are Gaussian with variance $\overline{J_{ij}^2}=1/N$, and the field components $b_i^\alpha$ are independent Gaussians of variance $\Delta^2$. For $m\ge3$ the model has a zero-temperature transition at the critical field $\Delta_c=1/\sqrt{m(m-2)}$ between a paramagnetic phase at high field (a single isolated minimum) and a spin-glass phase at low field. Replica symmetry breaking (RSB) is the Parisi solution structure that replaces the single replica-symmetric (RS) order parameter by a hierarchy of overlap matrices; it signals many quasi-degenerate minima that are marginal, i.e. strongly sensitive to any perturbation.

The paper analyzed the Hessian matrix of the Hamiltonian at the energy minima: its eigenvalues $\lambda=\omega^2$ are squared excitation frequencies, and eigenvector localization is measured by the inverse participation ratio (IPR), $\sum_i|v_i|^4$, which stays finite for a mode concentrated on finitely many sites and vanishes as $1/N$ for extended modes. In the paramagnetic phase the Hessian is statistically a Rosenzweig-Porter, or deformed Wigner, random matrix: a Wigner matrix (drawn from the GOE, the Gaussian Orthogonal Ensemble of real symmetric random matrices with independent entries up to symmetry) plus independent random diagonal entries whose distribution is that of the cavity fields. The cavity method (the message-passing construction, exact for such fully-connected models, that compares the minimum of the N-spin system with the system in which one spin is removed) gives the cavity-field density $P_m(H)\sim H^{m-1}$ at small field $H$. The established results are: in the paramagnetic phase ($\Delta>\Delta_c$) the spectrum is gapless with a pseudo-gap $\rho(\lambda)\sim\lambda^{m-1}$ (i.e. $D(\omega)\sim\omega^{2m-1}$, so $\omega^5$ for Heisenberg spins $m=3$), the smallest eigenvalue scales as $\lambda_1\sim\epsilon\,N^{-1/m}$ with $\epsilon=1-A$ the distance from criticality ($A=(1-1/m)\langle H^{-2}\rangle$, and $\epsilon$ coincides with the replicon eigenvalue, the stability eigenvalue of the RS solution), and the lowest eigenvectors are quasi-localized on small-field sites with finite IPR; at criticality ($\Delta=\Delta_c$) the edge becomes square-root, $\rho(\lambda)\sim\sqrt{\lambda}$ for $m\ge4$ with a logarithmic correction $\rho(\lambda)\sim\sqrt{\lambda/|\log\lambda|}$ at $m=3$, $\lambda_1\sim N^{-2/3}$, and the lowest eigenvectors delocalize, $\mathrm{IPR}_1\sim N^{-2/3}$: the paramagnet-to-spin-glass transition is a delocalization transition of the softest modes. Near criticality the spectrum crosses over at $\lambda^*\sim\epsilon^2$ between the pseudo-gap and the square-root behavior, producing a crossover size $N^*\sim\epsilon^{-3}$ (for $m=3$) below which simulations display pre-asymptotic $N^{-2/3}$-like scaling. The spectral edge laws are consistent with rigorous results of Lee and Schnelli for deformed Wigner matrices with power-law diagonal distributions.

Inside the spin-glass phase ($\Delta<\Delta_c$), however, the RS cavity analysis fails: the assumption that the minimum deforms continuously when a spin is added or removed breaks down because of the many quasi-degenerate, marginal minima and the level crossings between them. The paper argues only heuristically that the spectrum keeps a square-root edge there, assuming a divergent spin-glass susceptibility $\chi_{SG}=\langle 1/\lambda^2\rangle$ throughout the phase and a cavity-field density vanishing faster than $H^{m-1}$, as predicted for RSB metastable states by the classic Bray-Moore analysis (an essential singularity $P(H)\sim e^{-a/H}$). No explicit $P(H)$, no spectral prefactors, and no eigenvector localization thresholds are derived inside the RSB phase; the leading logarithmic correction $\rho(\lambda)\sim\sqrt{\lambda/|\log\lambda|}$ of the $m=3$ critical spectrum is derived, but the logarithmic factors in the smallest-eigenvalue scaling, in the inverse participation ratio and in the crossover scale are neglected or only estimated there, and the formally consistent non-square-root critical spectra $\rho(\lambda)\sim\lambda^{1/(m-1)}$ for noninteger $2<m<3$ (with no known physical realization) are left unclassified; the finite-size crossovers are quantified only as scaling estimates; and the authors explicitly leave open whether the whole pseudo-gap/quasi-localization/delocalization scenario follows from generic deformed-Wigner (Rosenzweig-Porter) random-matrix structure or requires additional physical ingredients such as the entropic features of continuous spins or the marginality of many nearby minima.

Problem Statement

Complete the analytic theory of the Hessian spectrum and low-energy eigenmode localization of the fully-connected $m$-component vector spin glass in a Gaussian random field at zero temperature (Hamiltonian $\mathcal{H}=-\sum_{i<j}J_{ij}S_i\cdot S_j-\sum_i b_i\cdot S_i$ with $|S_i|=1$, $\overline{J_{ij}^2}=1/N$, Gaussian field components of variance $\Delta^2$), extending it from the replica-symmetric paramagnetic phase and the critical point - where it is established - into the replica-symmetry-broken spin-glass phase ($\Delta<\Delta_c=1/\sqrt{m(m-2)}$) and to the boundary cases in $m$. The common objective is a closed, explicit description of the excitation spectrum and mode localization over the entire zero-temperature phase diagram for all $m\ge3$, at the same standard of explicitness as the existing paramagnetic solution, together with the identification of the mechanism that generates it. Concretely, the unresolved questions raised by the source analysis are:

1. RSB-phase cavity statistics and spectral laws. Derive the cavity-field distribution $P(H)$ for the minima dominating the spin-glass phase, accounting for replica symmetry breaking - the many quasi-degenerate marginal minima and the level crossings that invalidate the single-minimum replica-symmetric cavity construction. From it, derive the Hessian spectral density near the lower edge with explicit prefactors - in particular decide whether the conjectured square-root behavior $\rho(\lambda)\sim\sqrt{\lambda}$ holds throughout the spin-glass phase for all $m$, and whether the Bray-Moore essential singularity $P(H)\sim e^{-a/H}$ is the correct small-field law - and derive the localization law of the lowest eigenvectors there (whether they are extended, and the scaling of the inverse participation ratio), including how spectral prefactors and localization thresholds are modified by replica-symmetry-breaking effects.

2. The $m\le3$ boundary. Settle the logarithmic corrections at $m=3$ - firm up the leading critical-spectrum correction $\rho(\lambda)\sim\sqrt{\lambda/|\log\lambda|}$ and determine the logarithmic factors neglected in the smallest-eigenvalue scaling and in the inverse participation ratio - and classify the critical behavior for $2<m<3$: determine whether the formally consistent non-square-root critical spectra $\rho(\lambda)\sim\lambda^{1/(m-1)}$ can occur, and whether any physically realizable glass model with continuous bounded degrees of freedom realizes them, or establish that they cannot.

3. Finite-size control near criticality. Quantify systematically the finite-$N$ corrections to the asymptotic spectral and localization laws near $\Delta_c$ - the crossover at $\lambda^*\sim\epsilon^2$ with crossover size $N^*\sim\epsilon^{-3}$ ($\epsilon=1-A$, the replicon-type distance from criticality) - so that the asymptotic scaling can be extracted from simulations of accessible sizes without confusing the pre-asymptotic regime ($\lambda_1$ scaling as at criticality, $N^{-2/3}$-like) with the asymptotic one ($\lambda_1\sim\epsilon N^{-1/m}$).

4. Mechanism. Determine to what extent the pseudo-gap, the quasi-localization of edge modes, and the delocalization transition at $\Delta_c$ follow from the deformed-Wigner (Rosenzweig-Porter) random-matrix structure of the Hessian together with the field statistics alone, and what, if anything, requires the specific entropic and marginality properties of vector spin glasses.

Items 1-3 are the missing pieces of one phase diagram, and item 4 is the mechanistic explanation of that picture. All four questions are mandatory: the task is resolved only when each is answered or explicitly excluded, so a partial result that settles only part of this picture does not resolve it. The $2<m<3$ range of item 2 lies within this scope, as one of the boundary cases in $m$ referred to above, even though the physical phase diagram is defined for $m\ge3$.

References

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