Glassy and marginally stable phases of the ferromagnetic random-field XY model on sparse random graphs: RSB structure, the vanishing RSB window at large connectivity, critical behavior, and clock-discretization corrections

The ferromagnetic XY model with quenched, randomly oriented fields on random regular graphs was solved at the replica-symmetric (RS) level by Lupo, Parisi and Ricci-Tersenghi (J. Phys. A 52, 284001 (2019)) using belief propagation with a $Q$-state clock discretization (spin angles restricted to $Q$ equispaced values). That work found a tiny low-temperature region where the RS solution is unstable — evidence for a replica-symmetry-broken (RSB) spin-glass phase in a purely ferromagnetic vector model, which is provably impossible for Ising spins and absent in the fully connected limit — together with a marginally stable ferromagnetic phase (stability parameter $\lambda_{BP}$ compatible with zero throughout, due to global-rotation zero modes surviving the O(2)-symmetric random field) and power-law clock-discretization corrections with exponent $\alpha\in[0.8,1.0]$ in the regime where the ordering effect of the ferromagnetic couplings prevails. Five linked questions — the first declared open in the paper itself, the remaining four describing limitations of its replica-symmetric analysis — remain open: (i) whether the zero-temperature RSB field window of width $\Delta H_c(C)$ vanishes identically beyond some finite connectivity (numerically undetectable for $C\gtrsim 8$) or persists, possibly exponentially or algebraically small, for all finite $C$; (ii) the construction of the RSB solution itself (order-parameter function, one-step versus full RSB); (iii) critical exponents and universality class of the paramagnetic-RSB and ferromagnetic-RSB transitions; (iv) finite-size behavior of the marginal global-rotation modes and of the glassy ordering; and (v) an analytic theory of the discretization exponent $\alpha$. All five questions form one coherent open objective: completing the characterization of the glassiness and marginal stability of this model.

Background

The random-field XY (RFXY) model endows two-component unit spins, written as angles $\theta_i\in[0,2\pi)$, with ferromagnetic nearest-neighbor couplings and quenched random fields: $\mathcal{H}=-J\sum_{(i,j)\in E}\cos(\theta_i-\theta_j)-\sum_i H_i\cos(\theta_i-\varphi_i),$ with $J>0$, field moduli $H_i$, and independently drawn field directions $\varphi_i$ (the canonical choice being uniform on $[0,2\pi)$, which leaves the field distribution invariant under global spin rotations — an O(2)-symmetric distribution). For scalar spins, the random-field Ising model (RFIM) provably admits no replica-symmetry-broken (RSB) spin-glass phase at thermal equilibrium on any topology: positive couplings force connected correlation functions to be nonnegative, so the spin-glass susceptibility is bounded by the ferromagnetic one. This argument fails for vector spins, because small transverse fluctuations about an equilibrium configuration can generate negative effective correlations even with purely ferromagnetic couplings. In the fully connected (Sherrington–Kirkpatrick-type) ferromagnetic RFXY no RSB phase appears either. Whether graph sparsity changes this conclusion was the question addressed by Lupo, Parisi and Ricci-Tersenghi (2019).

They studied the model on random regular graphs (RRGs) — sparse random graphs in which every vertex has exactly $C$ neighbors and short loops are rare, so the graph is locally tree-like and the Bethe approximation becomes exact in the thermodynamic limit. Equilibrium statistics are then given by belief propagation (BP), the iterative solution of self-consistency equations for cavity messages $\eta_{i\to j}(\theta_i)$, the probability density of a spin angle in the graph with one neighbor removed; the disorder-averaged solution is obtained by the population dynamics algorithm, a Monte Carlo representation of the distribution of messages. Because messages are functions on a circle, they are computed via a $Q$-state clock discretization (restricting angles to $Q$ equispaced values, an approximation to the continuous model whose error must be controlled as $Q\to\infty$; the same authors had previously established that this approximation converges exponentially fast in $Q$ when quenched disorder dominates, and developed the corresponding methodology for the XY spin glass in a field on sparse graphs, where the transverse (Gabay–Toulouse) and longitudinal (de Almeida–Thouless) instability lines of vector spin glasses in a field can be compared). The stability of the replica-symmetric (RS) solution is monitored by $\lambda_{BP}$, the logarithmically averaged growth rate of the norm of perturbations of the BP messages around the fixed point, averaged over iterations and over the message population: $\lambda_{BP}<0$ indicates stability of the RS solution, $\lambda_{BP}=0$ identifies the critical (marginal) line, and $\lambda_{BP}>0$ signals instability — conventionally interpreted as the onset of replica symmetry breaking, the Parisi mechanism in which a glass phase is described by an order parameter that is a probability distribution over overlaps (or, on sparse graphs, a distribution over messages), with one-step (1RSB) or full hierarchical (fullRSB) structure.

The 2019 work found: (a) a tiny RS-unstable ("RSB") region at very low temperatures for small connectivity — at $T=0$ on $C=3$ RRGs it spans a field window of width $\Delta H_c\approx 0.21$ (in units where $J=1/(C-1)$, chosen so that the $C\to\infty$ limit stays finite), bounded by critical fields $H_c^{-}=0.367(3)$ and $H_c^{+}=0.578(3)$; the window shrinks with growing $C$ and becomes numerically undetectable for $C\gtrsim 8$, while the fully connected limit has no RSB region at all ($H_c^{\pm}\to 1/2$); (b) the entire ferromagnetic phase is marginally stable: $\lambda_{BP}$ is slightly negative at any finite $Q$ and its $Q\to\infty$ extrapolation is compatible with zero throughout the phase, because the O(2)-symmetric field distribution allows a coherent global rotation of all spins at no extensive energy cost — a Goldstone-like zero mode (a continuous-symmetry mode that costs no energy) surviving the random field, and lost as soon as a directional bias is switched on; (c) the $Q\to\infty$ approach of the fitted slope and intercept of $\lambda_{BP}(H)$ over the field intervals covering the $T=0$ RSB window follows a power law $Q^{-\alpha}$ with $\alpha$ reliably inside $[0.8,1.0]$ — the regime where the ordering effect of the ferromagnetic couplings prevails — in contrast with the exponential-in-$Q$ convergence that prevails when quenched disorder dominates; (d) a $2\times 2$ correlation-matrix analysis showing that the transverse second moments are the most critical at the RS instability. Of the five unresolved issues above, the first is stated as open in the paper itself — whether the RSB field window at $T=0$ is identically zero beyond some finite connectivity or only becomes too small to detect — and the remaining four describe limitations of the analysis stated across the paper.

Problem Statement

Complete the characterization of the glassy and marginally stable phases of the ferromagnetic random-field XY model on random $C$-regular graphs — the model with Hamiltonian $\mathcal{H}=-J\sum_{(i,j)\in E}\cos(\theta_i-\theta_j)-\sum_i H_i\cos(\theta_i-\varphi_i)$, $J=1/(C-1)$, field directions $\varphi_i$ drawn from an O(2)-symmetric distribution (uniform, or Gaussian components) — beyond the replica-symmetric level at which Lupo, Parisi and Ricci-Tersenghi (2019) solved it. The shared objective binding the following five linked questions is to determine the actual low-temperature phase structure of this model: what the replica-symmetry-breaking (RSB) instability develops into, over which region of the ($H$, $T$, $C$) parameter space it exists, and how it is characterized.

(i) Connectivity dependence of the RSB window. At zero temperature the RS-unstable field window has width $\Delta H_c(C)$ that is numerically undetectable for $C\gtrsim 8$ while being present for small $C$ (e.g. $\Delta H_c\approx 0.21$ at $C=3$). Determine whether $\Delta H_c(C)$ vanishes identically for all sufficiently large but finite $C$, or whether a nonzero RSB window — perhaps exponentially small, or power-law small — persists for every finite $C$, closing only in the fully connected limit where $H_c^{\pm}\to 1/2$ and no RSB region exists. A rigorous argument or controlled large-$C$ expansion settling either alternative answers this question; the source paper states this sub-question explicitly as open.

(ii) Structure of the RSB phase. Inside the RS-unstable region, construct the RSB solution: compute the RSB order parameter (the distribution over cavity messages for continuous-angle spins) and determine whether the structure is one-step RSB, finite-step RSB, or full RSB, including the fate of the ferromagnetic ordering (mixed ferromagnetic–glassy order) within it.

(iii) Critical behavior of the RS-instability transitions. Determine the critical exponents and universality class of the two transitions at which the RS solution loses stability — the paramagnetic–RSB transition and the ferromagnetic–RSB transition — going beyond the correlation-length divergence diagnostics already computed in the source.

(iv) Finite-size behavior. Characterize how the marginal global-rotation modes and the glassy ordering manifest at accessible finite $N$: the finite-size scaling of the RS instability and of the marginal ($\lambda_{BP}=0$) modes, and the dynamics of the slow global rotations whose energetic cost vanishes in the thermodynamic limit.

(v) Analytic theory of the discretization corrections. Derive the slow $Q$-state clock discretization convergence observed in the regime where the ordering effect of the ferromagnetic couplings prevails: over the field intervals covering the $T=0$ RSB window, the slope $m$ and intercept $q$ of the piecewise-linear fit to the stability parameter $\lambda_{BP}(H)$ approach their continuous-limit values as $m(Q)\sim m(\infty)+a/Q^{\alpha}$ and $q(Q)\sim q(\infty)+b/Q^{\alpha}$ with $\alpha$ reliably inside $[0.8,1.0]$ (the source notes that, since its range of $Q$ values does not span many orders of magnitude, an exponential-in-$Q$ approach with a large characteristic scale is also compatible with the data). Identify the mechanism that sets the functional form and the exponent of this convergence, in contrast to the exponential-in-$Q$ convergence that prevails when quenched disorder dominates.

The five sub-questions are scientifically linked as facets of one objective — establishing whether, where, and in what form a genuinely ferromagnetic vector-spin model with a quenched random field develops a spin-glass (RSB) phase on locally tree-like sparse graphs, and why its ferromagnetic phase is marginal.

References

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