Quenched TAP complexity of the Sherrington-Kirkpatrick model: the supersymmetry-breaking saddle, the Hessian modulus, and level-wise Legendre relations

The complexity - the exponential growth rate of the number of metastable states with their free energy - of the Sherrington-Kirkpatrick (SK) spin glass can be attacked with a supersymmetric field-theoretic method based on the Becchi-Rouet-Stora-Tyutin (BRST) symmetry. Annibale, Cavagna, Giardina and Parisi proved with this method that the BRST-supersymmetric quenched complexity of the Thouless-Anderson-Palmer (TAP) states equals the Legendre transform of the static free energy with respect to the largest replica-symmetry-breaking point, while explicitly acknowledging unresolved issues: whether a quenched saddle-point solution of the full variational equations that breaks the supersymmetry exists and what its form is; the unknown regime of validity of dropping the modulus of the Hessian determinant (legitimate only when stable minima dominate over unstable saddles); and the unproved conjecture that the complexity of clusters at an intermediate replica-symmetry-breaking level is the Legendre transform of the free energy with respect to the corresponding breaking point. Later work established that the supersymmetric quenched solution is replicon-unstable at every free energy except the equilibrium one, that typical SK stationary points are marginally stable index-one saddles whose count is described by the Bray-Moore solution obeying modified Ward identities, and - for annealed SK and for quenched spherical models with full replica symmetry breaking - how to count stationary points with the determinant modulus retained. A self-consistent, stable, index-resolved quenched full-RSB computation for the SK model that breaks the supersymmetry and resolves the vanishing-prefactor and zero-mode tension, together with a proof or refutation of the level-wise Legendre conjecture, remains open: a recent quenched computation (De Chirico and Leuzzi, 2026) treats only the restricted two-group, one-step case, leaving the full-RSB, most-general construction unbuilt, and the recent supersymmetric formalism (2026) explicitly marks the SK zero-prefactor tension as an open problem and defers the full-RSB quenched calculation to future work.

Background

A spin glass is a disordered magnet whose low-temperature behavior is dominated by a huge number of long-lived metastable states rather than by a single ordered ground state. The Sherrington-Kirkpatrick (SK) model is the fully connected, exactly solvable representative of this class: $N$ Ising spins $s_i=\pm 1$ coupled by Gaussian random interactions of variance $1/N$. Its equilibrium is described by Parisi's full replica symmetry breaking (FRSB) solution, in which the statistics of overlaps between spin configurations is encoded in an order-parameter function $q(x)$ on $x\in[0,1]$; the points $x_i$ at which new overlap values appear are the replica-symmetry-breaking points (breaking points), and the largest breaking point plays a distinguished role in the relations below.

A concrete handle on metastability is the Thouless-Anderson-Palmer (TAP) free energy $F^{TAP}(m)$, a mean-field free energy written as a function of the continuous local magnetizations $m_i\in(-1,1)$; its stationary points $\partial_i F^{TAP}=0$ are the TAP states, and the physical metastable states are its local minima. The complexity $\Sigma(\beta,f)$ is the logarithm, per spin, of the number of TAP states with free-energy density $f$ at inverse temperature $\beta$: that number grows as $\exp[N\Sigma(\beta,f)]$ as $N\to\infty$. Counting stationary points is a Kac-Rice problem - an integral over configuration space of a delta function imposing $\nabla F^{TAP}=0$ multiplied by the determinant of the Hessian - and the modulus of that determinant is the crux of the difficulty: $|\det \partial_i\partial_j F^{TAP}|$ has no direct field-theoretic representation, while the signed determinant does, via Grassmann (anticommuting) variables. The index of a stationary point is its number of negative Hessian eigenvalues, so minima have index zero and saddles have positive index.

Bray and Moore computed the SK complexity in 1980, finding an annealed solution (averaging the raw count rather than the logarithm of the count); Bray, Moore and Young extended the quenched calculation in 1984 and uncovered a rescaling relation between the TAP and static breaking points. The supersymmetric method reorganizes the counting: the Becchi-Rouet-Stora-Tyutin (BRST) supersymmetry, a symmetry between the commuting fields that implement the gradient constraint and the anticommuting (Grassmann) fields that represent the determinant, ties the three objects $F^{TAP}$, its gradient and its Hessian together and imposes Ward identities that select a distinguished saddle point of the resulting replica field theory. For systems with one step of replica symmetry breaking, Monasson's method of coupled replicas had shown that the complexity is the Legendre transform of the static free energy with respect to the breaking point $x$.

Annibale, Cavagna, Giardina and Parisi (the source paper) used the BRST supersymmetry to perform the quenched computation of the SK TAP complexity, proving that the supersymmetric complexity is exactly the Legendre transform of the static free energy $F(\beta;x)$ with respect to the largest breaking point: $\Sigma(\beta,f)=\beta x f-\beta x F(\beta;x),\qquad f=F(\beta;x)+x\,\frac{\partial F(\beta;x)}{\partial x}.$ The derivation is restricted to the BRST saddle: it drops the modulus of the Hessian determinant, which is justified only if stable minima dominate over unstable saddles; and its physical admissibility is diagnosed by the replicon eigenvalue, a fluctuation mode of the replica field theory associated with variations of the overlap structure, whose non-negativity (the Plefka criterion) is required for the solution to be internally consistent. The paper explicitly leaves open: (i) which saddle is physically correct at the annealed level, given that a non-BRST annealed solution due to Bray and Moore competes with the BRST one; (ii) whether a quenched non-BRST solution of the full variational equations exists and what its form is, noting that the previous quenched calculation of Bray, Moore and Young had turned out to be effectively supersymmetric; (iii) the regimes in which dropping the determinant modulus is valid; (iv) the extension of the Legendre-transform relation to the intermediate breaking points, conjectured by Montanari and Ricci-Tersenghi; and (v) the efficacy of the BRST method in generic glassy systems beyond the SK model.

Problem Statement

Determine the physically correct quenched complexity of the TAP states of the Sherrington-Kirkpatrick model and its exact relation to equilibrium thermodynamics, completing the program left open by the supersymmetric (BRST) computation of Annibale, Cavagna, Giardina and Parisi. In the supersymmetric formulation of the SK TAP complexity at inverse temperature $\beta$ below the critical temperature, the problem has three linked components:

1. The BRST-breaking quenched saddle. Construct the self-consistent solution of the full quenched variational (saddle-point) equations of the TAP-complexity functional at the full replica symmetry breaking level in which the BRST supersymmetry is broken - the most general scheme in which the order parameters are independent rather than tied by the standard Ward identities (the equations written down, but not solved, by Crisanti, Leuzzi, Parisi and Rizzo) - and determine the resulting complexity $\Sigma(\beta,f)$ as a function of the free-energy density $f$, its support, and the lower band edge. The candidate expected from cavity arguments is the Bray-Moore solution, which satisfies modified Ward identities; the answer must state which saddle is the physical one and why.

2. The Hessian modulus and the nature of the counted states. Establish when dropping the modulus of the Hessian determinant is valid - equivalently, determine the index composition (minima versus saddles of each index) and the marginal-stability properties of the stationary points being counted - and reconcile the result with the vanishing subexponential prefactor and the isolated zero Hessian eigenvalue that BRST breaking implies at typical stationary points. The counting must be consistent with what the TAP equations actually enumerate in the thermodynamic limit.

3. The level-wise Legendre relation. Prove or refute, within the supersymmetric formalism, the conjecture of Montanari and Ricci-Tersenghi that in systems with more than one step of replica symmetry breaking the complexity of clusters of states at level $i$ is the Legendre transform of the static free energy with respect to the breaking point $x_i$, of which the proven relation at the largest breaking point (where clusters are just states) is the boundary case.

These three components are one scientific objective: a physically consistent, stable, index-resolved quenched theory of the number and nature of the metastable TAP states of the SK model, together with the exact dictionary between that counting and the static free energy. They are not independent: whether the physical saddle breaks the BRST supersymmetry, and how the determinant modulus, zero modes and marginal stability are thereby related, is itself part of what must be determined, and the Legendre-transform dictionary holds exactly on the supersymmetric branch. The result recovers the known equilibrium free energy at the lower band edge and reduces consistently to the annealed results in the appropriate limit.

References

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