The 2004 numerical study of Cavagna, Giardina and Parisi (Phys. Rev. Lett. 92, 120603) found that the non-paramagnetic stationary points of the Thouless–Anderson–Palmer (TAP) free energy of the Sherrington–Kirkpatrick (SK) spin glass come in minimum–saddle pairs that coalesce into marginally unstable states as the system size grows, and that the annealed complexity of these states agrees with Bray and Moore's 1980 supersymmetry-breaking calculation, indicating that the Becchi–Rouet–Stora–Tyutin (BRST) supersymmetry of the TAP counting problem is spontaneously broken — but only at the annealed level and at finite size. The two central questions raised there remain open: (i) whether the supersymmetry breaking survives the quenched disorder average and the thermodynamic limit, i.e. what is the quenched complexity of the marginal TAP states and whether it vanishes at the Parisi equilibrium free energy as required for consistency with equilibrium thermodynamics; and (ii) what dynamical role, if any, the marginal states play in the out-of-equilibrium aging dynamics of the SK model. A September 2026 preprint (de Chirico and Leuzzi, arXiv:2609.04006) computes the quenched complexity for the first time within a two-group replica ansatz (two replica groups describing the coalesced minimum–saddle pair) with one step of replica symmetry breaking, but explicitly leaves the full-RSB limit and its central conjecture open; the dynamical question is sharpened, not settled, by simulations showing strong ergodicity breaking (two-time correlations that fail to decay to zero even at asymptotically large time separations) in SK aging.

Background

The Sherrington–Kirkpatrick (SK) model is the mean-field model of a spin glass: NN Ising spins σi=±1\sigma_i=\pm1 interact through independent Gaussian couplings of mean zero and variance 1/N1/N. Below its transition temperature TcT_c its equilibrium thermodynamics is described by Parisi's full replica symmetry breaking (full RSB) solution, in which the Gibbs measure splits into a nested hierarchy of pure states encoded in a continuous order-parameter function. Thouless, Anderson and Palmer (TAP) described the individual states through a mean-field free energy FTAP(m)F_{\mathrm{TAP}}(m) of the local magnetizations mim_i; its stationary points, solutions of the TAP equations, are the candidate pure and metastable states. Below TcT_c their number is expected to grow exponentially with NN, and its density per spin is the complexity (also called configurational entropy), Σ(f)=N−1ln⁡N(f)\Sigma(f)=N^{-1}\ln\mathcal{N}(f), where N(f)\mathcal{N}(f) is the number of TAP stationary points of free-energy density ff. The physical, quenched complexity averages the logarithm of N(f)\mathcal{N}(f) over the disorder (an average denoted by an overbar, ln⁡N(f)‾\overline{\ln\mathcal{N}(f)}); the annealed approximation averages N(f)\mathcal{N}(f) itself and is exact only where the annealed saddle is stable. Counting TAP stationary points is a Kac–Rice-type problem whose replica representation carries a supersymmetry: the Hessian determinant in the counting integral can be represented by Grassmann ('fermion') variables, and the combined theory of magnetizations ('bosons') and Grassmann variables is invariant under a Becchi–Rouet–Stora–Tyutin (BRST) supersymmetry, the same algebraic structure first identified in gauge-field quantization. When this supersymmetry is unbroken, its Ward identities force the dominant stationary points to be minima. Bray and Moore's 1980 calculation of the SK complexity spontaneously broke this supersymmetry and obtained a nonzero annealed complexity on a finite interval of free-energy densities; supersymmetric recomputations in 2003 (Cavagna, Giardina, Parisi and Mézard; Annibale, Cavagna, Giardina and Parisi) found a much smaller complexity, stable only at the ground-state free energy — if the supersymmetry were unbroken, the complexity of metastable states would vanish. The 2004 numerical study of Cavagna, Giardina and Parisi settled the annealed-level part of this dispute: solving the full TAP equations at T=0.2TcT=0.2T_c for N≤80N\leq80, they found (i) all non-paramagnetic stationary points are minima or index-one (one unstable direction) saddles, organized in minimum–saddle pairs whose free-energy difference scales as Δf∼N−1.26\Delta f\sim N^{-1.26} and whose smallest Hessian eigenvalue scales as ∣λmin⁡∣∼N−1.40|\lambda_{\min}|\sim N^{-1.40}, so the pairs plausibly coalesce into marginally unstable (soft-mode) states as N→∞N\to\infty; and (ii) the annealed complexity is compatible with the Bray–Moore supersymmetry-breaking curve, supporting spontaneous BRST supersymmetry breaking in the SK model. Physical consistency adds a sharp constraint: no state can lie below the equilibrium free-energy density f0f_0 of the Parisi full-RSB solution, yet the annealed Bray–Moore curve is unstable below a free energy f∗f^{\ast} and its zero lies below f0f_0 — an unphysical remnant that a correct quenched calculation must remove. A second, dynamical frame is the theory of aging: the out-of-equilibrium dynamics of mean-field spin glasses is described asymptotically by the Cugliandolo–Kurchan solution under the weak ergodicity breaking hypothesis (the system ages forever but asymptotically loses all memory of its initial state). Very long simulations of SK aging have since shown that this hypothesis itself fails in SK: two-time correlations remain bounded away from zero at arbitrarily large time separations, a confinement now called strong ergodicity breaking. In p-spin-like models (pure spin models with pp-body interactions, such as the spherical pp-spin glass), solved by one step of replica symmetry breaking (one-step RSB), the long-time dynamics is dominated by marginally stable threshold states; the 2004 paper observed the contrasting fact that in SK the dynamics asymptotically reproduces the static results 'with no role played by the metastable states', while explicitly deferring the question of what the marginally unstable TAP states actually do to the dynamics.

Problem Statement

Two linked questions about the marginal TAP states of the SK spin glass, raised by the 2004 numerical study of Cavagna, Giardina and Parisi, remain open after the intervening literature. (1) Quenched complexity in the thermodynamic limit: compute the quenched complexity Σ(f)=lim⁡N→∞N−1 ln⁡N(f)‾\Sigma(f)=\lim_{N\to\infty}N^{-1}\,\overline{\ln\mathcal{N}(f)} (the overbar denotes the average over the disorder) of the dominant marginal (BRST-supersymmetry-breaking) TAP states of the SK model as a function of the free-energy density ff and temperature T<TcT<T_c, including the T→0T\to0 limit; this entails determining (a) whether the spontaneous breaking of the BRST supersymmetry survives the quenched disorder average and the thermodynamic limit, and (b) the fate of the complexity at low free energy — in particular whether, in the full-RSB limit, Σ(f)\Sigma(f) vanishes exactly at the Parisi equilibrium free-energy density f0f_0, so that the counting of TAP states is reconciled with Parisi's equilibrium thermodynamics (no states below f0f_0, no unphysical annealed remnant below f∗f^{\ast}). (2) Dynamical role of the marginal states: determine whether and how the marginally unstable (coalesced) TAP states influence the out-of-equilibrium, long-time (aging) dynamics of the SK model — in particular, determine whether the asymptotic dynamics after a quench is confined to, or attracted by, marginal TAP states, how their soft modes enter the aging spectrum, and how this relates to the strong ergodicity breaking observed in long-time SK aging simulations. The two questions share a common objective: the population and stability of the marginal TAP states at the quenched level in the thermodynamic limit — the full quenched complexity Σ(f,T)\Sigma(f,T) for all T<TcT<T_c, including the T→0T\to0 limit, of which the verdicts (a) and (b) are part — and their physical function in the dynamics, comprising the three aspects of question (2): confinement or attraction of the asymptotic dynamics, soft modes in the aging spectrum, and the relation to strong ergodicity breaking.

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