Dimension dependence of the constrained overlap correlation function in the Ising spin glass: reconciling the Franz-Parisi-Virasoro interface theory with the high-dimensional mean-field expansion

In the low-temperature ordered phase of the $D$-dimensional Edwards-Anderson Ising spin glass at zero magnetic field, two analytic frameworks make conflicting predictions for the large-distance decay of the constrained connected overlap correlation function $C(x|Q)=\langle q(x)q(0)\rangle_Q-Q^2$, the connected correlation of the local overlap between two independent copies (replicas) of the same sample whose total overlap is held fixed at $Q$. Dimensional analysis of the Franz-Parisi-Virasoro (FPV) interface theory - the computation underlying the prediction $5/2$ for the lower critical dimension of the spin glass phase - gives $C(x|Q)\propto x^{-\alpha}$ with $\alpha=\frac{4}{5}\left(D-\frac{5}{2}\right)$, independent of $Q$ below the Edwards-Anderson overlap $q_{EA}$. The loop expansion around the Sherrington-Kirkpatrick mean field, evaluated at the replica-symmetry-broken saddle point, instead gives in high dimensions $\alpha(0)=D-4$ at $Q=0$ and $\alpha(Q)=D-3$ for $0<Q<q_{EA}$. The two predictions cross at $D=10$ (for $Q=0$) and at $D=5$ (for $Q\neq 0$), and the source paper (Maiorano and Parisi, PNAS 2018) states it is unclear whether something special happens at these dimensions. The later literature leaves the question open: measurements of the exponent exist only at $D=3$ (consistent with the FPV value) and $D=4$ (intermediate between the two predictions), the finite-dimensional corrections to the mean-field propagators have never been computed without approximations, and the most recent work (2026) solves the analogous fixed-overlap decay problem only for hierarchical proxy models, explicitly leaving the short-range case open.

Background

A spin glass is a magnet with frozen random interactions. The minimal lattice model is the Edwards-Anderson (EA) Ising spin glass, with Hamiltonian $H=-\sum_{\langle i,j\rangle}J_{ij}\sigma_i\sigma_j$, where the spins $\sigma_i=\pm 1$ sit on a $D$-dimensional cubic lattice and the nearest-neighbor couplings $J_{ij}=\pm 1$ are drawn once per sample and kept fixed. In zero magnetic field the model develops a spin glass phase below a critical temperature $T_c$ (which exists for $D\ge 3$). Because a sample has many nearly degenerate equilibrium configurations with no simple symmetry relation between them, the natural order parameter is built from two independent equilibrium copies (replicas) $\sigma$ and $\tau$ of the same sample: the local overlap $q(x)=\sigma_x\tau_x$ and its spatial average $Q$. The distribution of $Q$ over equilibrium states is nontrivial in the ordered phase; its edges $\pm q_{EA}$, where $q_{EA}$ is the Edwards-Anderson parameter (the self-overlap of a pure thermodynamic state), mark the maximal typical agreement between two states. Replica symmetry breaking (RSB) is Parisi's mean-field theory of this phase: it is exact for the infinite-range Sherrington-Kirkpatrick (SK) model and conjectured to describe finite-dimensional EA models. In RSB the ordered phase is marginal - the analogue of a continuous symmetry broken without a spectral gap - and supports an infinite family of soft fluctuation modes whose correlations decay algebraically; the most divergent of these soft modes is called the replicon.

A sharp probe of these soft modes is the constrained connected correlation function. One equilibrates the two-replica system restricted to total overlap $Q$ (with $|Q|<q_{EA}$, inside the support of the overlap distribution) and measures

$$C(x|Q)=\langle q(x)q(0)\rangle_Q-Q^2,$$

where $\langle\cdot\rangle_Q$ is the constrained equilibrium average and $Q^2$ subtracts the disconnected part. Its large-distance decay defines an exponent $\alpha$ through $C(x|Q)\propto x^{-\alpha}$, and equivalently its Fourier transform behaves as $\tilde C(k|Q)\propto k^{-(D-\alpha)}$ at small momentum $k$. Two routes predict $\alpha$. First, the interface theory of Franz, Parisi and Virasoro (FPV): computing, within a mean-field-based effective theory, the free-energy cost of an interface between two regions forced to different overlaps yields a non-analytic effective functional of the overlap profile, of the form $\Delta F\propto\sum_i|q_i-q_{i+1}|^{5/2}$; the cost of an overlap difference over a region of linear size $L$ then scales as $L^{D-5/2}$, which vanishes at $D=5/2$. This is the FPV prediction for the lower critical dimension $D_{lc}$ (the dimension below which the ordered phase disappears), and it is strongly supported by numerical work. Dimensional analysis of the same functional yields

$$\alpha=\frac{4}{5}\left(D-\frac{5}{2}\right),$$

independent of $Q$ for $|Q|<q_{EA}$ (giving $\alpha=2/5$ in $D=3$). Second, the loop expansion around the mean field: De Dominicis, Kondor and Temesvari computed the Gaussian propagators of replica field theory at the RSB saddle point of the SK model in the ordered phase, identifying the constrained correlation function with specific propagator components (their review chapter is the reference used by the source paper). The far-infrared behavior of these components is $1/k^4$ in the zero-overlap sector and $1/k^3$ at generic nonzero overlap, corresponding in real space to $\alpha(0)=D-4$ and $\alpha(Q)=D-3$ for $0<Q<q_{EA}$. These replicon-sector divergences are so strong (up to $1/k^4$) that the loop expansion of the equation of state is controlled only for $D>8$; the tree-level propagator spectrum is nonetheless the mean-field benchmark, and the transition itself has upper critical dimension $6$.

The two predictions disagree at every dimension except two crossing points: FPV crosses the mean-field $\alpha(0)=D-4$ at $D=10$ and $\alpha(Q)=D-3$ at $D=5$. In $D=3$ the FPV prediction $\alpha=2/5$ is close to the measured value $0.38\pm 0.02$ (later simulations place it in the range $0.35$-$0.4$), while the mean-field values are $0$ and $-1$. In $D=4$ the zero-overlap (replicon) exponent, measured through non-equilibrium dynamics and connected to the equilibrium exponent by the statics-dynamics dictionary (the established mapping between the decay of correlations in an out-of-equilibrium system whose coherence length grows in time and the equilibrium correlations of a system of that size), is $\approx 1.0$, temperature-independent, below the FPV value $6/5$ and far above the mean-field value $0$. At large effective dimension (EA models on diluted hypercubes - the EA model on a randomly diluted high-dimensional hypercube graph, whose built-in graph distance gives a tunable effective spatial dimension - studied at $T=0.7T_c$ through non-equilibrium dynamics up to effective dimension $22$) the $1/k^4$ law of the mean-field expansion is verified in the zero-overlap sector. The FPV prediction thus appears favored near the lower critical dimension and the mean-field spectrum at high dimension, and no measurement or derivation exists in the intermediate range $D=5$ to $D=12$ where the crossings sit. Maiorano and Parisi (2018), after numerically confirming the FPV interface-barrier scaling in $D=3,4$, posed the reconciliation question that defines this problem.

Problem Statement

For the $D$-dimensional Edwards-Anderson Ising spin glass at zero magnetic field in its low-temperature replica-symmetry-broken phase, determine the exponent $\alpha(Q,D)$ governing the large-distance decay of the constrained connected overlap correlation function $C(x|Q)=\langle q(x)q(0)\rangle_Q-Q^2$ for $|Q|<q_{EA}$ (equivalently, the small-$k$ behavior of its Fourier transform $\tilde C(k|Q)\propto k^{-(D-\alpha)}$), as a function of the spatial dimension $D$ and the imposed overlap $Q$.

Two analytic routes currently give conflicting predictions: the Franz-Parisi-Virasoro interface theory, by dimensional analysis of the non-analytic overlap-interface functional (the same computation that yields the lower critical dimension $D_{lc}=5/2$), predicts $\alpha=\frac{4}{5}\left(D-\frac{5}{2}\right)$, independent of $Q$; the loop expansion around the Sherrington-Kirkpatrick mean field, evaluated at the RSB saddle point, predicts $\alpha(0)=D-4$ in the zero-overlap sector and $\alpha(Q)=D-3$ for $0<Q<q_{EA}$ in high dimensions. The predictions cross at $D=10$ for $Q=0$ and at $D=5$ for $Q\neq 0$.

The problem is to reconcile them: establish the regime structure of $\alpha(Q,D)$ - which prediction, if either, is correct in which range of dimensions and where, if anywhere, the two cross over into one another - whether the crossing dimensions $D=10$ and $D=5$ are physically special (for example, new crossover dimensions of the theory) or merely artifacts of extrapolating two results outside their regimes of validity, and what the true $\alpha(Q,D)$ is.

References

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