Finite-size ($1/N$) corrections to the number of TAP solutions in the Sherrington–Kirkpatrick spin glass: the vanishing-prefactor problem of the Bray–Moore complexity

In the Sherrington–Kirkpatrick (SK) spin glass, the exponentially many solutions of the Thouless–Anderson–Palmer (TAP) equations are counted by the complexity $\Sigma(f)$, computed at leading order by the Bray–Moore (BM) saddle point. This saddle point breaks the BRST supersymmetry (SUSY) of the counting action, and as a consequence the prefactor of the exponential contribution $\exp[N\Sigma]$ vanishes at every order in the $1/N$ expansion (Kurchan's all-orders argument, extended by Parisi and Rizzo to every free energy), because the TAP Hessian has an isolated eigenvalue that is exactly zero on the BM saddle and of order $1/\sqrt{N}$ at finite $N$. The true number of TAP solutions — counted with the modulus of the Hessian determinant — is nevertheless expected to possess non-zero finite-size corrections, and how to compute them has been open since the 2004 Parisi–Rizzo paper. The subsequent literature lineage: Rizzo (2005) bypassed the vanishing prefactor by overlap-biased counting; Belius, Concetti and Genovese (2024) rigorously controlled the annealed determinant only at exponential scale; Kent-Dobias (2026) computes annealed complexities with non-vanishing prefactors in a four-index superspace yet states explicitly that the zero-prefactor tension in the SK model remains open; de Chirico and Leuzzi (2026) computed the leading quenched complexity of marginal states, closing the leading-order version of the replicated extension but not the sub-exponential structure. The surviving open core is the sub-exponential (finite-$N$) structure of the annealed true TAP complexity of the SK model; the quenched (replicated) structure is a related open direction but is not part of the present problem.

Background

The Sherrington–Kirkpatrick (SK) model is the mean-field Ising spin glass: $N$ spins $\sigma_i=\pm1$ interact through independent Gaussian random couplings $J_{ij}$ of variance $1/N$. Below its transition temperature the model's free-energy landscape contains exponentially many metastable states. These are described by the Thouless–Anderson–Palmer (TAP) approach: one introduces local magnetizations $m_i$ and a mean-field free-energy function $F_{\mathrm{TAP}}(m)$ (containing the random coupling term, the local entropy, and an Onsager reaction term — the standard mean-field correction that removes each spin's contribution to its own local field — depending on the self-overlap $q=\frac{1}{N}\sum_i m_i^2$), whose stationary points $\partial_i F_{\mathrm{TAP}}(m)=0$ — the TAP equations — encode the metastable states. The complexity, or configurational entropy, $\Sigma(f)=N^{-1}\log\mathcal{N}(f)$, is the logarithm of the number $\mathcal{N}(f)$ of TAP solutions of free-energy density $f$, normalized by $N$; it controls the statistics of the landscape, the barriers, and the slow dynamics of spin glasses. Counting the solutions uses an integral representation (the Kac–Rice idea): besides the magnetizations and bosonic Lagrange multiplier fields $x_i$ enforcing the TAP equations, one introduces anticommuting Grassmann (fermionic) fields $\bar{\psi}_i,\psi_i$ whose Gaussian integral reproduces the determinant of the Hessian $\partial_i\partial_j F_{\mathrm{TAP}}$. The resulting action is invariant under a Becchi–Rouet–Stora–Tyutin (BRST) transformation, a 'supersymmetry' (SUSY) mixing bosonic and fermionic variables, of the same type as the Parisi–Sourlas supersymmetry of disordered systems. A crucial technical point is the modulus: the Grassmann representation gives the determinant without its absolute value, so the SUSY machinery naturally computes a sign-weighted count, which by the Morse theorem (the alternating sum of stationary points by index equals the Euler–Poincaré characteristic of the magnetization space, here $1$) must equal $1$ — while the physical complexity requires the count with the modulus. Bray and Moore (BM, 1980) evaluated the annealed count (the disorder average of the number of solutions, not of its logarithm) at a saddle point that breaks this SUSY; the SUSY-preserving alternative saddle is unstable in models whose low-temperature equilibrium solution is the full replica-symmetry-breaking (FRSB) one (the replica method computes disorder averages of logarithms by introducing identical copies, 'replicas', of the system and trading the average of the logarithm for a logarithm of an average over the replicas; in FRSB the symmetry between replicas is broken with a continuum of overlap values between copies) such as the SK model (Crisanti, Leuzzi, Parisi and Rizzo, 2004), leaving the SUSY-breaking BM saddle as the only viable description. Kurchan (1991) proved a general consequence of breaking the SUSY at the saddle: the prefactor multiplying $\exp[N\Sigma]$ vanishes identically at every order in the $1/N$ expansion. Parisi and Rizzo (2004) extended this vanishing to solutions at any free energy, gave a simple argument for it (the fermionic block of the Hessian of the macroscopic action is singular), and proved rigorously that the isolated eigenvalue that Aspelmeier, Bray and Moore (2004) had found numerically in the TAP Hessian spectrum — an eigenvalue detached below the continuous band, generated by the rank-one projector term of the Hessian — is exactly zero on the BM saddle, as a direct consequence of SUSY breaking. At finite $N$ this isolated eigenvalue is of order $1/\sqrt{N}$, and each marginal solution (one with a vanishing Hessian eigenvalue) splits into a close pair of a minimum and an index-one saddle (the index of a stationary point being the number of negative eigenvalues of its Hessian), which reconciles the exponential complexity with the Morse theorem. The unsettled question is the sub-exponential structure of the true count: writing the number of TAP solutions with the modulus as $\mathcal{N}_N(f)\sim\exp[N\Sigma_{\mathrm{BM}}(f)]\,C_N(f)$ with $\log C_N(f)=c_0(f)+c_1(f)/N+\cdots$, the standard expansion yields $C_N\equiv0$ order by order, yet the true complexity is expected to have non-zero finite-size corrections. Computing these $1/N$ corrections — or establishing that they are non-perturbative — is the open problem. The sub-exponential structure of the quenched (replicated) count is a related open direction but is not part of the present problem; there the average is taken of the logarithm of the number of solutions and replica symmetry must be broken in the space of the counted states, via the two-group ansatz of Bray and Moore in which the replicas divide into two groups — physically representing a minimum and a rank-one saddle that coalesce into a single marginal state — a leading-order version of which has only recently been obtained.

Problem Statement

In the Sherrington–Kirkpatrick spin glass below the transition temperature (zero field), write the number of solutions of the TAP equations at free-energy density $f$, counted with the modulus of the Hessian determinant, as $\mathcal{N}_N(f)\sim\exp[N\Sigma_{\mathrm{BM}}(f)]\,C_N(f)$, where $\Sigma_{\mathrm{BM}}(f)$ is the Bray–Moore complexity and the sub-exponential prefactor is defined as the ratio $C_N(f):=\mathcal{N}_N(f)\,e^{-N\Sigma_{\mathrm{BM}}(f)}$. Compute the sub-exponential, finite-size structure of this true count: determine whether $\log C_N(f)$ admits a non-zero perturbative $1/N$ expansion, $\log C_N(f)=c_0(f)+c_1(f)/N+\cdots$ — and if so compute its leading term $c_0(f)$ and the scaling form of the corrections — or establish that the corrections are non-perturbative and determine the asymptotic form of $C_N(f)$ in that case. The known obstruction is that the standard SUSY/BRST saddle-point expansion around the Bray–Moore saddle gives a prefactor that vanishes identically at every order in $1/N$, because the fermionic block of the Hessian of the macroscopic action is singular there and the TAP Hessian has an isolated eigenvalue that is exactly zero in the $N\to\infty$ limit and of order $1/\sqrt{N}$ at finite $N$. In the related quenched (replicated) setting — in which the average is taken of the logarithm of the number of solutions, with replica symmetry broken in the space of the counted states (the replicas dividing into two groups that represent a minimum and a rank-one saddle coalescing into a marginal state) — the source paper announced the extension of its results as work in progress, and the leading-order complexity of marginal states (states with a vanishing Hessian eigenvalue) is now known at the one-step-replica-symmetry-breaking two-group level; the sub-exponential structure of the quenched count is a related open direction but is not part of the present problem. The determination of the leading finite-size correction(s) to the true number of TAP solutions must respect the Morse-theorem constraint that the sign-weighted count equals $1$.

References

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