Geometric organization of the metastable TAP states in the spherical p-spin model: ultrametricity, threshold coalescence, and the constrained-complexity versus three-replica-potential discrepancy

In the spherical p-spin spin glass, a paradigmatic exactly solvable mean-field glass model, the low-temperature phase contains exponentially many metastable states, all of vanishing thermodynamic weight, so the standard overlap distribution is trivial and the organization of states must be resolved entropically. Cavagna, Giardina and Parisi (1997) probed this hidden structure with the constrained complexity (the exponential growth rate of the number of TAP solutions at given free-energy density and overlap from a reference state) and found a continuous overlap distribution, an anomalous regime in which the constrained complexity grows with overlap, and a coalescent, channel-like organization of the marginal threshold states. They posed as the main open issue a unified frame describing this full geometric organization, with two concrete unresolved questions: whether the states possess an ultrametric organization as the Sherrington-Kirkpatrick pure states do, and a discrepancy between the ending points of the dominant-state energy spectrum computed from the constrained complexity and from the three-replica potential — the spectrum from the potential terminates at a smaller overlap (a greater distance) than the nearest TAP solutions counted by the constrained complexity, so the potential misses the states closest to the reference state, and it is not known whether the potential fails to see those states because the constrained second replica does not thermalize in their vicinity, or whether the counted solutions are not true minima of the TAP free energy. Later literature has re-derived the constrained complexity as the (annealed-exact) two-point complexity within the modern Kac-Rice landscape program and computed the quenched three-point complexity of stationary points, but the ultrametricity question and the endpoint discrepancy remain unsettled, and no unified geometric description of the state organization exists.

Background

Mean-field spin glasses are disordered systems in which infinitely many degrees of freedom interact through random couplings. In the Sherrington-Kirkpatrick (SK) model, the low-temperature phase is described by Parisi's replica symmetry breaking: the Gibbs measure splits into many pure states, the mutual overlaps $q_{ab}=N^{-1}\sum_i s_i^a s_i^b$ (normalized scalar products between spin configurations of two replicas $a,b$) have a nontrivial distribution $P(q)$, and the pure states are organized ultrametrically — among any three states the two smallest of the three mutual overlaps coincide, so states sit at the leaves of a hierarchical tree.

The spherical p-spin model is a different mean-field spin glass: continuous spins constrained to a sphere, $\sum_i s_i^2=N$, with a Hamiltonian that is a random homogeneous polynomial of degree $p\ge 3$ in the spins, $H(\sigma)=-\sum_{i_1<\cdots<i_p}J_{i_1\ldots i_p}\sigma_{i_1}\cdots\sigma_{i_p}$ with independent Gaussian couplings. On cooling it has two transitions: a dynamic (mode-coupling) transition at $T_d$, where relaxation gets trapped, and a static thermodynamic transition at $T_s<T_d$ described by one step of replica symmetry breaking (1RSB). Its metastable states are described by the Thouless-Anderson-Palmer (TAP) approach: mean-field self-consistency equations whose solutions are the stationary points of a mean-field TAP free-energy functional, candidate minima of which represent the metastable glass states. The number of TAP solutions with free-energy density $f$ grows as $\exp(N\Sigma(f))$, where $\Sigma(f)$ is the complexity (configurational entropy). A crucial difference from the SK model: in the window $T_s<T<T_d$ every state, the equilibrium ones included, has vanishing thermodynamic weight and is exponentially numerous in $N$, so no state can be discarded and $P(q)$ carries no information; the state statistics must be resolved entropically, keeping the energy density as an explicit additional variable.

Cavagna, Giardina and Parisi (CGP97; J. Phys. A 30, 7021 (1997)) investigated this hidden structure with two tools. The first, introduced in their companion paper (J. Phys. A 30, 4449 (1997)), is the three-replica potential $V_3$: within the real-replica method (studying real copies of the system with constrained overlaps), a first replica is thermalized in an equilibrium state, a second is forced to thermalize at overlap $q_{12}$ with the first, and $V_3$ is the free-energy cost of a third replica as a function of its distances $q_{13},q_{23}$ from the first two. The second tool is the constrained complexity $\Sigma_c(f,q)$: the exponential growth rate of the number of TAP solutions of free-energy density $f$ lying at overlap $q$ from a fixed reference (equilibrium) state. Their main findings for the $p=3$ model: (i) a continuous spectrum of states at all overlaps from $0$ up to a maximum $q_{\mathrm{last}}$; for sub-threshold states $q_{\mathrm{last}}$ is below the self-overlap (well-separated states), while threshold states — the marginal states at the threshold energy, which trap the asymptotic dynamics and where the Hessian develops zero modes — have $\Sigma_c$ vanishing continuously at their self-overlap, i.e. they coalesce along their marginal directions into connected channels winding through phase space, with no sharp separation; (ii) an anomalous regime in which $\Sigma_c$ increases with $q$: states are more crowded at small distances than random placement would predict, the two coinciding only in the $p\to\infty$ random-energy limit; (iii) from the spectrum of dominant states (minimizing $\phi_c(f,q)=f-T\Sigma_c(f,q)$), the states nearest to any given state always have higher energy density, and the number of dominant states drops from exponential in $N$ to order $N$ beyond an overlap $q_{\mathrm{low}}$.

In their conclusions and Appendix A they state the open problems: the pursuit of a unifying frame synthesizing all these results into one description of the state organization; the absence of any evidence for or against an ultrametric organization of the states, to be investigated through the correlations among triplets of states (the analogue of the Mézard-Parisi-Sourlas-Toulouse-Virasoro analysis for the SK model); and a discrepancy in the ending points of the dominant-state energy spectrum, which from the constrained complexity ends at the overlap of the nearest TAP solutions counted by $\Sigma_c$, while the spectrum extracted from $V_3$ ends earlier in overlap: the potential ceases to see states around the reference replica at a distance strictly greater than that of the nearest TAP solutions counted by $\Sigma_c$, so the states closest to the reference replica are missed — with two possible explanations left open. A modern reformulation exists: what CGP97 computed is the annealed two-point complexity, and the prescribed triplet test is the three-point complexity; both are now computed via the Kac-Rice method (a formula expressing the expected number of stationary points of a random Gaussian field as an integral over the joint statistics of its gradient and Hessian), in a quenched treatment (averaging the complexity over the disorder before taking the exponential growth rate).

Problem Statement

For the spherical p-spin spin glass at $p=3$, in the temperature window $T_s<T<T_d$ between the static and dynamic transitions, determine the full geometric organization of its TAP states — equilibrium and metastable states alike, all exponentially numerous and of vanishing thermodynamic weight. As posed by Cavagna, Giardina and Parisi, the objective comprises three linked components:

(1) Unified framework. Construct a frame in which the observed structures — the continuous overlap distribution of states, the anomalous regime of the constrained complexity $\Sigma_c$ (its growth with overlap $q$), the coalescent channel-like organization of threshold states, and the role of the energy density as an additional organizing variable — are jointly described as one geometry of states: a single overlap- and energy-resolved description of the TAP states from which all of these structures follow.

(2) Ultrametricity. Decide whether the states possess an ultrametric organization, as the SK pure states do. The question concerns the mutual organization of the full set of TAP states, not only the equilibrium Gibbs measure.

(3) Endpoint discrepancy. The energy spectrum of dominant states around a reference state, computed from the constrained complexity (minimizing $\phi_c(f,q)=f-T\Sigma_c(f,q)$), ends at the overlap of the nearest TAP solutions counted by $\Sigma_c$; the spectrum extracted from the three-replica potential $V_3$ ends earlier in overlap: the potential ceases to see states around the reference replica at a distance strictly greater than that of the nearest TAP solutions counted by $\Sigma_c$, so the states closest to the reference replica are missed. Establish which alternative holds: (a) the potential fails to see the last states because the second replica does not thermalize in their vicinity; or (b) the last solutions counted by $\Sigma_c$ are not minima of the TAP free energy.

The three components are linked: (2) and (3), together with the continuous overlap distribution, the anomalous regime, and threshold coalescence, fall within the unified geometric description of (1).

References

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