Extension of the rigorous TAP variational principle beyond the constrained-overlap spherical 2-spin (vector) Sherrington–Kirkpatrick model

Belius, Fröber and Ko (arXiv:2304.04031, 2023) proved that the constrained-overlap multiple-replica spherical Sherrington–Kirkpatrick (SK) model — the spherical vector-spin spin glass whose free energy is the Franz–Parisi potential — admits a rigorous Thouless–Anderson–Palmer (TAP) variational principle: at all inverse temperatures and external fields its limiting free energy is the supremum of an explicit TAP functional over magnetizations satisfying a spectral Plefka condition, and the formula confirms that the model is replica symmetric. This was only the second spin glass model (after the one-replica spherical SK model) with a mathematically rigorous TAP computation of the free energy at all temperatures and external fields. The authors state the hope that the TAP approach be extended to a wider class of spin glass models, note that replica symmetry cannot presently be deduced from the model's positive-temperature Parisi formula, and leave full smoothness of a limiting spectral functional as a technical refinement. The open problem is to establish a rigorous TAP variational principle, with an explicit (e.g. spectral Plefka-type) characterization of the admissible magnetizations, for constrained-overlap multiple spherical spin glasses beyond 2-spin interactions — primarily the mixed/pure p-spin spherical vector models whose Parisi formulas are already known. No resolution is known in the literature to date: existing extensions of TAP reach only high temperature, pure models without external field, TAP equations rather than variational principles, or different model classes.

Background

Mean-field spin glasses are disordered systems whose random Hamiltonians are built from independent Gaussian couplings. The prototype is the Sherrington–Kirkpatrick (SK) model, whose Hamiltonian $H_N(\sigma)=\sqrt{N}\sum_{i,j=1}^N J_{ij}\sigma_i\sigma_j$ with i.i.d. standard Gaussian $J_{ij}$ is a random quadratic form; in the spherical variant the spin vector $\sigma\in\mathbb{R}^N$ is constrained to the unit sphere $S^{N-1}$. The central exact-solvability result of the field is the Parisi formula: the thermodynamic limit of the normalized free energy (the logarithm of the partition function) is given by an explicit variational minimization, proved rigorously for Ising and spherical mixed p-spin models (mixed p-spin models allow Hamiltonians that are general mixtures of degree-$p$ polynomials of the spins; the SK case is $p=2$).

Multi-replica variants probe correlations between independent copies ("replicas") of the system. The overlap between two replicas is their normalized inner product $Q_{kl}=N^{-1}\sum_i\sigma^k_i\sigma^l_i$. In the constrained-overlap multiple-replica model, also called the vector spin model, one fixes a positive definite constraint matrix $Q\in\mathbb{R}^{n\times n}$ with unit diagonal and computes the free energy restricted to configurations whose overlap matrix is approximately $Q$ (if $Q$ is not positive definite the constrained free energy is $-\infty$). In physics this quantity is the Franz–Parisi potential, introduced to describe metastable states of one-replica spin glasses. For the spherical versions of these models the limiting free energy is known: Panchenko and Talagrand proved an upper bound via the Guerra interpolation scheme, and Ko established the matching lower bound — a Parisi-type minimization over matrix-valued paths — using the Aizenman–Sims–Starr comparison scheme (a cavity-type recentering identity for the free energy) together with synchronization properties of constrained vector models from Panchenko's work on mixed p-spin models with vector spins; Ko also proved the Crisanti–Sommers form (the Legendre-dual variational formula familiar from spherical models) for vector spins. Auffinger and Zhou analyzed the spherical mixed vector p-spin model at zero temperature and showed that the zero-temperature Parisi minimizer is replica symmetric for 2-spin interactions. ("Replica symmetric" means the Parisi minimizer is the trivial one with no replica-symmetry breaking; equivalently, the asymptotic overlap distribution of the Gibbs measure is concentrated at the diagonal value.)

The Thouless–Anderson–Palmer (TAP) approach proposes an alternative route to the free energy: rather than minimizing over Parisi order parameters, one maximizes an explicit functional of the magnetization $m$ (the vector of coordinate-wise averages of the spins under the Gibbs measure). The TAP free energy contains a nontrivial "Onsager" reaction term correcting for the response of the spin environment to the imposed magnetization; for 2-spin interactions it is quadratic. Plefka identified a convergence condition for the TAP iteration in the Ising SK model, which in the spherical vector setting becomes the spectral condition $\|\beta^{1/2}(Q-mm^{T})\beta^{1/2}\|_2\le 1/\sqrt{2}$, where $\beta=\mathrm{diag}(\beta_1,\dots,\beta_n)$ collects the replicas' inverse temperatures and $\|\cdot\|_2$ is the spectral norm; geometrically it says that the "effective temperature after recentering" lies in the high-temperature region. Rigorous TAP computations of the free energy at all temperatures and external fields are rare: Belius and Kistler proved the TAP–Plefka variational principle for the one-replica spherical SK model, Subag computed the free energy of pure p-spin spherical models without external field from the TAP approach, and Chen, Panchenko and Subag developed the "generalized TAP free energy" framework for Ising mixed p-spin models.

In this landscape, Belius, Fröber and Ko considered the constrained-overlap model with $n$ replicas on the sphere sharing one SK disorder, with replicas possibly at different inverse temperatures $\beta_k$ and external fields $h^k$, and the constrained free energy $F_N^\epsilon(\beta,h,Q)=\frac{1}{N}\log\int_{Q^\epsilon}\exp\Big(\sum_{k=1}^n\beta_kH_N(\sigma^k)+N\sum_{k=1}^n h^k\cdot\sigma^k\Big)\,d\sigma,$ where $Q^\epsilon=\{\sigma:\|\sigma\sigma^T-Q\|_\infty\le\epsilon\}$ restricts the overlaps to $Q$ within tolerance $\epsilon$ in sup-norm. Their main theorem states that, for positive definite $Q$, in probability, $\lim_{\epsilon\to0}\lim_{N\to\infty}\Big|F_N^\epsilon(\beta,h,Q)-\sup_{m\in\mathrm{Plef}_N(Q,\beta)}\frac{\mathcal{F}_{\mathrm{TAP}}(m)}{N}\Big|=0,$ with the TAP functional $\mathcal{F}_{\mathrm{TAP}}(m)=\frac{N}{2}\log\lvert Q-mm^{T}\rvert+\sum_{k=1}^n\beta_kH_N(m^k)+N\sum_{k=1}^n h^k\cdot m^k+\frac{N}{2}\beta^{T}(Q-mm^{T})^{\odot 2}\beta,$ where $|\cdot|$ is the determinant, $(\cdot)^{\odot 2}$ the entrywise (Hadamard) square, and $\mathrm{Plef}_N(Q,\beta)$ the set of magnetizations with $0\le mm^{T}<Q$ in the strict Loewner order (i.e. $Q-mm^{T}$ positive definite) satisfying the spectral Plefka condition above. They also evaluate the supremum explicitly when the external fields are collinear, obtaining a closed-form ground-state energy, and they show that any maximizer of a modified TAP functional used in the upper-bound analysis must satisfy the Plefka condition. Because the variational problem involves a single matrix, the formula confirms that the model is replica symmetric — a fact that is natural but, as the authors note, not presently deducible from the model's positive-temperature Parisi formula. Beyond this 2-spin case the TAP machinery has not been extended; the authors state the hope that "a further improvement of the present TAP approach can be extended to a wider class of spin glass models."

Problem Statement

Extend the rigorous TAP variational principle of Belius, Fröber and Ko beyond the constrained-overlap multiple spherical model with 2-spin (SK) interactions, per the source's stated hope that 'a further improvement of the present TAP approach can be extended to a wider class of spin glass models'. The common scientific objective is a rigorous TAP-type variational theory of the limiting constrained free energy for constrained-overlap (vector spin) spherical models with non-2-spin interactions; the natural primary target is the spherical mixed or pure p-spin vector model (p≥3 or mixed couplings), the family for which Parisi-type formulas are already known (Ko 2019, 2020; Panchenko 2018).

A qualifying solution must, for at least one genuinely wider model class of this kind:

1. Exhibit an explicit TAP-type functional of the multi-replica magnetization $m$ (with the correct generalized Onsager reaction term) and prove that the limiting constrained free energy at all inverse temperatures and external fields equals the supremum of this functional over an explicitly characterized admissible set.
2. Prove (or refute) that maximizers of the modified TAP functional used in the upper-bound argument, where the admissibility criterion is not imposed a priori, must satisfy the proposed criterion — in the 2-spin case this is the spectral Plefka condition $\|\beta^{1/2}(Q-mm^{T})\beta^{1/2}\|_2\le 1/\sqrt{2}$.

Two companion questions from the source's outlook are optional and secondary to the extension; resolving the extension alone qualifies: (a) whether replica symmetry of the constrained multiple 2-spin model can be deduced directly from the Parisi-type formula at positive temperature, which the source declares presently infeasible; and (b) the technical refinement of proving that the limiting spectral functional $\mathcal{F}_K$ is smooth on the closed domain $[0,\infty)$, removing the approximation arguments currently used for temperature vectors with zero components. Positive results for non-spherical geometries or Ising-type vector spins are legitimate answers within the same objective.

References

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