Thermodynamic-limit structure of the overlap distribution in the three-dimensional Edwards-Anderson spin glass: the small-$x$ moment relations and the continuous part of $P(q)$

Marinari, Parisi, and Ruiz-Lorenzo (1998) studied the three-dimensional Edwards-Anderson (EA) Ising spin glass with Gaussian couplings in equilibrium down to $T=0.7$ (about $0.74\,T_c$) on lattices up to $L=16$, finding the overlap distribution $P(q)$ consistent with a replica-symmetry-breaking (RSB) phase: a delta peak at $q_{\mathrm{EA}}\simeq0.7$ carrying weight about 0.52 plus a nontrivial continuous part, and mean-field RSB moment relations for the sample-to-sample fluctuations of the cumulative overlap that hold at large cumulative overlap but fail at small values. They explicitly left open whether these small-$x$ violations vanish in the thermodynamic limit and whether the finite-size broadening mechanism they proposed is the explanation; their analysis of the delta weight and position and of the continuous part of $P(q)$, together with their remark that the same broadening argument applies to the tails of the distribution, called for a much more accurate determination of the quantitative shape of the continuous part of $P(q)$. Later work resolved two of the acknowledged limitations: the corrections-to-scaling exponent is now measured as $\omega$ of about 1.0-1.12 (not the one-loop value 3), and equilibrium simulations have reached $L=32$ below $T_c$ and $L=40$ near $T_c$ for the $\pm J$ realization. The surviving open core, which is the problem posed here, is to establish at thermodynamic-limit level whether the RSB moment relations for the cumulative overlap hold down to small $x$, and what the exact structure of $P(q)$ is (delta weight and position, continuous-part functional form and small-$q$ behavior) - in other words, whether the low-temperature phase of the three-dimensional EA spin glass is RSB-like (many pure states) or droplet-like (a single pair of pure states).

Background

A spin glass is a magnet with quenched (frozen) random interactions. Its minimal lattice model is the Edwards-Anderson (EA) model: Ising spins $\sigma_i=\pm1$ on a simple cubic lattice of linear size $L$ with Hamiltonian $H=-\sum_{\langle ij\rangle}J_{ij}\sigma_i\sigma_j$, where the nearest-neighbor couplings $J_{ij}$ are independent quenched random variables. The source study used Gaussian couplings of zero mean and unit variance (the alternative, cheaper "bimodal" or $\pm J$ realization takes $J_{ij}=\pm1$; both are believed to lie in one universality class, though this has been questioned). In three dimensions the model orders at a spin-glass transition at $T_c\simeq0.95$ for Gaussian couplings ($T_c\simeq1.102$ for $\pm J$).

The order parameter is the overlap $q=V^{-1}\sum_i\sigma_i^{(1)}\sigma_i^{(2)}$ between two independent equilibrium replicas (system copies sharing the same disorder). For a given disorder realization $J$ its distribution is $P_J(q)$; the disorder average is $P(q)$. Two competing theories describe the low-temperature phase. The replica-symmetry-breaking (RSB) picture, inherited from the exactly solved fully connected Sherrington-Kirkpatrick (SK) mean-field model, holds that phase space splits into infinitely many thermodynamic pure states organized ultrametrically (any three equilibrium states have mutual overlaps of which the two smallest coincide); $P(q)$ is then nontrivial - a delta peak at $\pm q_{\mathrm{EA}}$ (the Edwards-Anderson order parameter) plus a continuous part - and physical quantities keep sample-to-sample fluctuations in the thermodynamic limit (non-self-averaging). The droplet/scaling picture instead predicts a single pair of pure states (related by global spin flip), $P(q)=\frac12[\delta(q-q_{\mathrm{EA}})+\delta(q+q_{\mathrm{EA}})]$ with no continuous part, and vanishing sample fluctuations as $L\to\infty$.

The source study simulated the Gaussian model with simulated tempering and parallel tempering (Monte Carlo schemes in which the temperature itself is updated at run time, allowing the system to cross the free-energy barriers of the glassy phase) on $L=4\ldots16$ lattices down to $T=0.7$. It established a transition at $T_c=0.95(4)$ with $\nu=2.00(15)$, and read $P(q)$ as a delta component at $q_{\mathrm{EA}}\simeq0.7$ of weight $\simeq0.52(4)$ plus a continuous part carrying $\simeq0.48$ of the weight. It also tested the mean-field RSB predictions for the sample statistics of the cumulative overlap $x_J(q_c)=\int_0^{q_c}P_J(q)\,dq$ (the cumulative probability, within sample $J$, that the overlap is smaller than $q_c$, the single-sample overlap distribution being folded onto one sign of $q$ so that $x_J\to1$ as $q_c\to1$): the distribution $\hat\Pi_{q_c}$ of $x_J(q_c)$ over samples should have moments $x_n(q_c)=\int dr\,r^n\hat\Pi_{q_c}(r)$ obeying the moment relations $x_2=\frac13 x_1+\frac23 x_1^2,\qquad x_3=\frac{x_1}{15}\left(2+8x_1+5x_1^2\right),$ derived for the RSB mean-field solution by Mezard, Parisi, Sourlas, Toulouse, and Virasoro. Equivalent symmetric-convention relations follow from the Ghirlanda-Guerra identities (a hierarchy of exact constraints on the joint moments of the overlaps of several replicas) and stochastic stability (invariance of the Gibbs state under small random perturbations of the couplings), which are rigorous theorems for SK-like mean-field models (as is ultrametricity, proven by Panchenko), but not for finite-dimensional models. The 1998 data satisfied the relations for $x_1\gtrsim0.5$ but violated them at small $x_1$; the authors proposed a finite-size mechanism (each sample's delta peaks are broadened to a width $\Delta$, injecting spurious weight at small overlap, with $\Delta\approx0.15$ at $L=16$) but could not decide whether the discrepancy vanishes in the thermodynamic limit.

The study also acknowledged three further gaps: the corrections-to-scaling exponent $\omega$ (governing the approach of finite-size quantities such as Binder cumulant crossings - the Binder cumulant is a dimensionless ratio of moments of an observable commonly used to locate transitions - to their asymptotic scaling form; one-loop field theory gives $\omega=6-D=3$ in $D=3$, modified by renormalization) could not be estimated accurately; $L=24$ could not be thermalized at $0.7T_c$ with the available 25-Gflop computer; and the quantitative shape of the continuous part of $P(q)$ and its tails required larger sizes and better statistics. Later work determined $\omega$ of about 1.0-1.12 (from $\pm J$ simulations up to $L=40$), pushed equilibrium simulations to $L=32$ at $0.64T_c$ ($\pm J$) and to $L=16$ at $0.44T_c$ with rigorous equilibration tests (Gaussian), retested the moment relations at $L=32$ with deviations shrinking with size, and produced quantile analyses, metastate constructions, and boundary-condition studies that variously support an RSB-like picture or warn that accessible sizes remain droplet-compatible. No study has delivered the missing thermodynamic-limit statement: whether the small-$x$ deviations from the moment relations vanish and what the exact weights, functional form, and small-$q$ behavior of $P(q)$ are as $L\to\infty$.

Problem Statement

The source question (Marinari, Parisi, Ruiz-Lorenzo, 1998, Gaussian-coupling realization) asks for thermodynamic-limit statements about the equilibrium spin-glass phase of the three-dimensional Edwards-Anderson Ising model; two of its acknowledged limitations (the unknown corrections-to-scaling exponent $\omega$, and the inability to thermalize $L=24$ at $0.7T_c$) have been substantially resolved by later work ($\omega=1.12(10)$ for the $\pm J$ model; equilibrium data up to $L=32$ below $T_c$ and $L=40$ near $T_c$). The surviving open core is the following.

Define, for a disorder realization $J$ with single-sample overlap distribution $P_J(q)$ folded onto one sign of $q$ (so that $x_J\to1$ as $q_c\to1$), the cumulative overlap $x_J(q_c)=\int_0^{q_c}P_J(q)\,dq$, its sample-to-sample distribution $\hat\Pi_{q_c}$, and the moments $x_n(q_c)=\int dr\,r^n\hat\Pi_{q_c}(r)$. The mean-field RSB theory predicts $x_2=\frac13 x_1+\frac23 x_1^2$ and $x_3=\frac{x_1}{15}(2+8x_1+5x_1^2)$ (in the symmetric convention $X_J(q)=\int_{-q}^{q}P_J(q')\,dq'$ used by later numerical studies, the corresponding Ghirlanda-Guerra/stochastic-stability relations read $X_2=\frac13 X_1+\frac23 X_1^2$ and $X_3=\frac{1}{15}\left(2X_T+2X_1+6X_1^2+5X_1^3\right)$, where $X_T$ is the three-replica cumulative probability; under ultrametricity $X_T=X_1^2$, giving $X_3=\frac{1}{15}\left(2X_1+8X_1^2+5X_1^3\right)$).

(1) Small-$x$ moment relations. Establish whether, in the thermodynamic limit ($L\to\infty$ at fixed $T<T_c$), these moment relations (and the related quantile/stochastic-stability constraints) hold at all $x_1>0$ - that is, whether the observed small-$x_1$ violations are finite-size effects that vanish as the delta peaks of $P_J$ sharpen, quantifying the scaling of the finite-size deviations and of the peak broadening $\Delta(L)$ with $L$ - or whether they persist, refuting the RSB reading of the sample statistics.

(2) Quantitative structure of $P(q)$. Determine the thermodynamic-limit overlap distribution in the spin-glass phase: the position $q_{\mathrm{EA}}(T)$ and weight of the delta component, and the functional form, total weight, and small-$q$ behavior of the continuous part between $q=0$ and $q_{\mathrm{EA}}$ (equivalently, the large-$L$ behavior of the single-sample statistics such as the fraction of centrally peaked samples).

Both parts serve one objective: decide, at the level of defensible thermodynamic-limit statements, whether the low-temperature phase of the 3D EA spin glass is RSB-like (many pure states, so that $P(q)$ has a nontrivial continuous part) or droplet-like (a single pair of pure states, so that $P(q)$ is trivial). In the droplet-like branch, however, the continuous-part structure of part (2) is fixed (zero weight, so that $P(q)$ reduces to the pair of delta peaks at $\pm q_{\mathrm{EA}}$ and is fixed by the delta position $q_{\mathrm{EA}}(T)$ and its weight). The research object is the three-dimensional EA spin glass in either the Gaussian-coupling realization of the source study or the $\pm J$ realization, which are believed to lie in one universality class; the relation between the two must be argued, not re-derived (the still-open Gaussian-model residuals - no dedicated large-$L$ determination of $\omega$ for Gaussian couplings, and equilibrium data below $T_c$ currently limited to $L\le16$ for Gaussian couplings - are context that need not be closed).

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