Link-site overlap coefficient near the lower critical dimension and the $q^2$-dependence of overlap correlations in bulk-perturbed Ising spin-glass ground states

Marinari and Parisi (Phys. Rev. Lett. 86, 3887 (2001)) computed pairs of ground states of the three-dimensional Edwards-Anderson Ising spin glass with Gaussian couplings, before and after a bulk perturbation that repels the second ground state from the first, and found that the link overlap conditioned on the squared site overlap obeys $q_l(q^2)\approx 1-A(1-q^2)$ with $A=1-q_l(0)$ extrapolating to 0.24-0.33 in the thermodynamic limit, vanishing conditional fluctuations, and an overlap-overlap correlation function decaying with exponent $\phi\approx 0.4$ at small $q$; they conjectured that $A\to 0$ as the spatial dimension approaches the lower critical dimension of the spin glass. The later literature shows that the qualitative three-dimensional claims are now well supported: overlap equivalence (vanishing conditional variance and a one-to-one $q_l$-$q^2$ relation) is established at low temperature, and a 2026 extrapolation of equilibrium finite-temperature simulations to $T=0$ gives $1-q_l(0)\approx 0.18$-$0.19$, confirming a nonzero, space-filling excitation-interface density. However, the dimensional dependence of $A$ -- in particular the conjectured vanishing at the lower critical dimension $d_l\approx 2.5$ -- has never been measured, the decay exponent $\phi(q^2)$ has not been characterized at $T=0$ beyond the small-$q$ window, and the assumed $\varepsilon$-independence of $q_l(q^2)$ has not been directly tested at $T=0$. The problem is to settle these surviving questions, whose common objective is to establish how the replica-symmetry-breaking-like overlap geometry of the perturbed ground state degenerates toward the droplet-like geometry as spin-glass ordering itself disappears.

Background

The Edwards-Anderson (EA) Ising spin glass places Ising spins $\sigma_i=\pm 1$ on a lattice with quenched (frozen) random nearest-neighbor couplings $J_{ij}$; here the couplings are independent Gaussian variables of zero mean and unit variance on simple cubic lattices of $N=L^3$ sites with periodic boundary conditions (Edwards and Anderson 1975). Which theory describes the ordered spin glass phase of this model in three spatial dimensions is a long-standing open question. In the droplet/scaling picture (Fisher and Huse 1988), the ordered phase consists essentially of a single pair of pure states related by the global spin-flip symmetry; large-scale excitations are droplets of flipped spins whose energy grows as $L^\theta$ with stiffness exponent $\theta$ ($\theta\approx 0.2$ in 3D, $\theta<0$ in 2D), so excitations costing energy of order 1 cannot be system spanning. In the replica-symmetry-breaking (RSB) picture inherited from the exactly solvable infinite-range Sherrington-Kirkpatrick (SK) model (Sherrington and Kirkpatrick 1975; Mezard, Parisi and Virasoro 1987), the phase is a mosaic of many pairs of pure states, and there exist system-spanning excitations of order-1 energy whose interfaces are space filling, i.e. have positive surface-to-volume ratio. Intermediate TNT (trivial/nontrivial) scenarios posit order-1 excitations with fractal, non-space-filling interfaces.

Two overlaps between spin configurations $\sigma$ and $\tau$ diagnose the phase: the site overlap $q=N^{-1}\sum_i\sigma_i\tau_i$ and the link overlap $q_l=(DN)^{-1}\sum_{i,\mu}\sigma_i\sigma_{i+\hat\mu}\tau_i\tau_{i+\hat\mu}$, the fraction of nearest-neighbor bonds on which the two configurations agree. Unlike $q$, $q_l$ is unchanged when one configuration is globally flipped, and it is sensitive only to the interface between regions where the two configurations differ. In the SK model two independent equilibrium configurations satisfy $q_l=q^2$ exactly. In finite dimensions the relation between $q_l$ and $q^2$ is nontrivial and diagnostic: the droplet picture predicts a trivial (delta-function) link-overlap distribution, while RSB predicts a nontrivial one-to-one functional relation $q_l(q^2)$.

Because equilibrating large samples far below the critical temperature is extremely hard, one can instead sample the $T=0$ landscape directly: compute the ground state $\sigma^0$ of $H_0(\sigma)=-\sum_{\langle ij\rangle}J_{ij}\sigma_i\sigma_j$, and then the ground state $\tau^0$ of the bulk-perturbed Hamiltonian

$$H_1(\tau)=H_0(\tau)+\frac{\varepsilon}{ND}\sum_{i,\mu}\sigma^0_i\sigma^0_{i+\hat\mu}\tau_i\tau_{i+\hat\mu}=H_0(\tau)+\varepsilon\,q_l(\sigma^0,\tau),$$

which penalizes agreement with the original ground state -- it raises the energy of $\sigma^0$ itself by exactly $\varepsilon$ -- and thereby forces an excitation whose energy above the ground state is of order 1 (the perturbation is $O(1)$ while $H_0$ is $O(N)$). The repulsive sign above is the one of Palassini and Young's original construction and is required for the perturbation to have any effect; the arXiv preprint of Marinari and Parisi prints the perturbation with the opposite (attractive) sign, an apparent typographical slip, since the same preprint describes the term as a repulsion from the true ground state and its nontrivial overlap distributions are impossible with the attractive sign (with $\varepsilon>0$ and the attractive sign, $\sigma^0$ would remain the ground state of $H_1$). This construction was introduced by Palassini and Young (2000) and in a related form by Krzakala and Martin (2000); Franz and Parisi (2000) supplied its RSB-theoretic rationale: a finite repulsive coupling induces a nontrivial overlap distribution $P(q^2)$ between the two ground states, computable in closed form for the ultrametric overlap trees of RSB theory (ultrametric meaning that the pure states are organized hierarchically by their mutual overlaps), and although $P(q^2)$ changes with $\varepsilon$, the conditional relation $q_l(q^2)$ is expected to become $\varepsilon$-independent as $L\to\infty$.

Marinari and Parisi (2001) applied this method to $L=4$-$14$ in 3D with a genetic-algorithm ground-state heuristic and found: (i) $q_l(q^2)=1-A(1-q^2)$ for $q^2$ not too close to 1, with $A(L)=1-q_l(0)$ extrapolating to 0.24-0.33 depending on the fit form; (ii) the conditional variance of $q_l$ at fixed $q^2$ vanishes as $L\to\infty$; (iii) the conditioned overlap-overlap correlation function $C(x,L|q^2)=\langle q(x)q(0)\rangle_{(q^2)}$ (the correlation of the local overlap field between the two ground states, restricted to pairs whose squared site overlap is $q^2$) decays at small $q$ as $x^{-\phi}$ with $\phi\approx 0.4$, consistent with RSB and incompatible with droplet-like scaling. They concluded that the surface-to-volume ratio of the induced excitations tends to a nonzero limit, i.e. space-filling interfaces with fractal dimension $d_s=d$, in tension with Palassini and Young's contemporaneous conclusion $d-d_s=0.42(2)$. The coefficient $A$ interpolates between the SK value $A=1$ and the 3D value $A\approx 0.2$-$0.3$, which led the authors to conjecture that $A\to 0$ as the spatial dimension approaches the lower critical dimension $d_l$ of the spin glass, i.e. the dimension below which no spin glass phase survives, equivalently where the stiffness exponent $\theta(d)$ vanishes (current estimates place $d_l\approx 2.5$; in 2D, below it, the ground state is unique up to global flip and the droplet picture holds). The statement that $q_l$ at fixed $q^2$ has vanishing fluctuations and becomes a one-to-one function of $q^2$ was later isolated as the key structural property, called overlap equivalence, by Contucci, Giardina, Giberti and Vernia (2006). The source paper explicitly left open: the systematic error of the extrapolations of $A$; the assumed $\varepsilon$-independence of $q_l(q^2)$; the use of the scaling ansatz for $C$ at distance $x=1$; the $q^2$-dependence of $\phi$; the description of the genetic algorithm; and the conjecture $A\to 0$ near $d_l$.

Problem Statement

The problem is to complete the characterization, initiated by Marinari and Parisi (2001), of the overlap structure of the ground state of the Edwards-Anderson Ising spin glass with Gaussian couplings probed by the bulk repulsion perturbation $H_1$, along three linked fronts that the later literature has left open.

(1) Behavior of the link-site overlap coefficient near the lower critical dimension (central component). The relevant thermodynamic-limit coefficient is

$$A(d)=\lim_{L\to\infty}\bigl(1-q_l^{(d)}(0)\bigr),$$

where $q_l^{(d)}(0)$ is the link overlap between the ground state of $H_0$ and the ground state of $H_1$ in dimension $d$, at fixed perturbation strength $\varepsilon$, conditioned on squared site overlap $q^2=0$ (equivalently, the slope of the relation $q_l(q^2)=1-A(1-q^2)$ valid for $q^2$ away from 1); the $\varepsilon$-independence of this limit is itself the content of component (3). Known points are $A=1$ in the Sherrington-Kirkpatrick limit and $A\approx 0.18$-$0.33$ in $d=3$ (estimator-dependent). Decide whether $A(d)\to 0$ as $d$ approaches the lower critical dimension $d_l$ (where the stiffness exponent $\theta(d_l)=0$; $d_l\approx 2.5$ by current estimates) from above, as the source authors conjectured. Since $d_l$ is not an integer lattice dimension, the limit $d\to d_l^+$ must be realized through an effective-dimension construction, or by a controlled theoretical computation, whose faithfulness to $d_l$ is justified.

(2) $q^2$-dependence of the overlap-correlation decay exponent at $T=0$. Characterize the decay exponent $\phi(q^2)$ of the conditioned overlap-overlap correlation function $C(x,L|q^2)$ of the two ground states beyond the small-$q$ window ($q\le 0.1$) examined in the source: determine whether $\phi$ depends on $q^2$, and in particular whether it takes distinct values at $q=0$ and $q\neq 0$ (with a possible rise near $q^2=q_{EA}^2$, the squared self-overlap of the ordered phase), as mean-field theory suggests.

(3) Direct $\varepsilon$-independence test at $T=0$. Verify directly, at several perturbation strengths $\varepsilon$ and fixed $q^2$, that $q_l(q^2)$ computed from bulk-perturbed ground states is $\varepsilon$-independent in the $L\to\infty$ limit -- an assumption made in the source and since supported only indirectly, via overlap equivalence and finite-temperature comparison. The three components share one scientific objective: to establish, with controlled statistical and systematic errors, how the RSB-like overlap geometry of the bulk-perturbed ground state degenerates toward the droplet-like geometry ($A=0$, trivial link overlap) as spin-glass order itself weakens and disappears at the lower critical dimension. The question comprises the conjectured limit $A(d)\to 0$ as $d\to d_l^+$ together with components (2) and (3), each determined with quantified errors. Resolving the broader RSB-versus-droplet/TNT/metastate controversy (the metastate, a probability measure over asymptotic Gibbs states, being the framework in terms of which one rival scenario is formulated) is not required; the measured quantities are to be interpreted within that controversy.

References

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