Thermodynamic-limit structure of the boundary-condition-induced ground-state rearrangement in the three-dimensional Edwards-Anderson Ising spin glass

The three-dimensional Edwards-Anderson Ising spin glass with Gaussian couplings is the canonical model of a disordered frustrated magnet, and the structure of its low-temperature phase — many thermodynamic states as in replica symmetry breaking (RSB), or a single spin-reversed pair as in the droplet picture — remains unresolved. Marinari and Parisi (Phys. Rev. B 62, 11677 (2000)) attacked the question by computing pairs of ground states of the same disorder realization under periodic and anti-periodic boundary conditions (linear sizes up to $L=14$), measuring the induced rearrangement through the link overlap, window overlaps and their correlation functions, and extrapolating a positive thermodynamic-limit value $\lim(1-q_l)\approx 0.245$ with decay exponents $\delta\approx 0.32$ and $\lambda=1.5$ — concluding that the boundary-condition-induced interface is space filling, as RSB predicts. The twenty-five-year controversy that followed: competing ground-state studies found non-space-filling excitations; exact ground-state computations, large-scale equilibrium simulations and a 2026 equilibration study at $T=0.2\approx 0.21\,T_c$ support space filling; droplet-side analyses of interface fractal dimensions, crossover-length arguments and metastate theory contest it. The surviving open problem: determine the thermodynamic-limit structure of the boundary-condition-induced ground-state rearrangement (space filling versus fractal, with an unknown crossover length), whether the zero-temperature-first order of limits equals the equilibrium order, and which excitation the boundary change selects and why. The question is open: the most recent study (2026) still frames the choice among the droplet, TNT, chaotic-pairs and RSB scenarios as undecided.

Background

A spin glass is a magnet in which randomly placed frustrated interactions prevent any spin configuration from satisfying all of them. The standard lattice model is the Edwards-Anderson (EA) Hamiltonian $H=-\sum_{\langle ij\rangle}J_{ij}\sigma_i\sigma_j$ with Ising spins $\sigma_i=\pm 1$ on a simple cubic lattice and nearest-neighbor couplings $J_{ij}$ drawn independently from a Gaussian distribution of zero mean; the couplings are quenched, i.e. fixed during thermal equilibration. In three dimensions the model orders below a critical temperature $T_c\approx 0.95$ (in units set by the coupling variance) into a phase whose internal structure is the subject of a forty-year debate.

Two frameworks make opposite predictions. In the droplet model of Fisher and Huse (and earlier scaling treatments), the ordered phase consists of a single pair of pure thermodynamic states related by a global spin flip, and the lowest excitations are clusters ('droplets') whose energy grows as $L^\theta$ with the stiffness exponent $\theta\approx 0.24$ in 3D; an interface forced into the system is a fractal surface of dimension below the space dimension. In replica symmetry breaking (RSB), Parisi's mean-field solution of the infinite-range model extended to finite dimensions, the phase contains many thermodynamic pure states, excitations of order-one energy with system-spanning extent exist, and interfaces between distant states are space filling, so two independent equilibrium replicas differ on a positive fraction of bonds. Intermediate 'TNT' (trivial-nontrivial) scenarios combine a nontrivial spin-overlap distribution with non-space-filling excitation surfaces. Chaotic-pairs scenarios, also close to the droplet picture, keep a single spin-reversed pair of pure states but allow the pair selected by the boundary conditions to change chaotically with volume.

Ground-state computation gives exact access to the $T=0$ limit: with Gaussian couplings the ground state (global energy minimum) of a finite box is unique, so the structure of the phase must be probed by perturbing the system and observing how the ground state responds. Changing the boundary conditions in one lattice direction from periodic (P) to anti-periodic (AP) is equivalent, by a gauge transformation that flips the sign of the couplings crossing one plane, to imposing an interface. The pair of ground states $\sigma$ (P) and $\tau$ (AP) for the same disorder defines the local spin overlap $q(i)=\sigma(i)\tau(i)$ and the gauge-invariant link overlap $q_l$, the average of $q(i)q(j)$ over the nearest-neighbor bonds $\langle ij\rangle$ of the lattice (with the bonds crossing the anti-periodic plane entering with reversed sign); equivalently, $q_l$ is twice the fraction of bonds on which the two ground states agree minus one, so that $1-q_l$ is twice the interface-bond density. If the induced rearrangement is droplet-like, $1-q_l$ vanishes as a power of $L$; if it is RSB-like, $q_l$ stays below 1 in the thermodynamic limit. Following Palassini and Young, who studied the induced change inside a fixed block far from the boundary and concluded in favor of droplet-like triviality, Marinari and Parisi computed pairs of ground states for 16000 samples at $L=4$ down to 670 samples at $L=14$ with a modified genetic algorithm (a stochastic optimizer related to methods later used for spin-glass ground states), and measured the average link overlap, the probability that a cube of side $M$ intersects the interface ('hole' probability), the window-overlap distribution (the spin overlap restricted to interior cubes of side $M$), and the overlap and link-overlap correlation functions. Their conclusions: the link overlaps extrapolate to $\lim_{L\to\infty}(1-q_l)=0.245\pm 0.015$; the window overlap scales as $q^2_{M,L}\simeq L^{-\delta}f(M/L)$ with $\delta=0.32$; the overlap correlation function decays with exponent $\delta\simeq 0.32$; the link-overlap correlation function requires $\lambda=1.5$ rather than a droplet-expected value near $0.3$; and the hole probability indicates a space-filling interface. They concluded that the RSB picture correctly describes 3D spin glasses.

Three methodological assumptions are acknowledged in the source. First, ground-state studies take $T\to 0$ before $L\to\infty$, whereas equilibrium thermodynamics takes $L\to\infty$ first; the assumed commutation of the two limits is stated as a tentative hypothesis, not a result. Second, the infinite-$L$ extrapolations depend on the chosen fit form (polynomial in $1/L$ versus power law with subleading corrections give 0.20 to 0.25 for $\lim(1-q_l)$). Third, the genetic algorithm's correctness guarantees are deferred to a companion publication, and the paper gives no independent audit of ground-state optimality. Finally, the paper offers no microscopic derivation of which excited state the boundary-condition change selects. The metastate framework of Newman and Stein — a probability measure over the coupling-dependent ground states of large volumes, describing how boundary conditions select them — has been used to analyze how boundary conditions select the ground state (Read 2014; Billoire et al. 2017); the source treats the interface-energy structure as an uncontrolled assumption.

Problem Statement

For the three-dimensional Edwards-Anderson Ising spin glass with Gaussian couplings of zero mean on an $L^3$ simple-cubic lattice, let $\sigma$ and $\tau$ denote the ground states of the same disorder realization under periodic boundary conditions and under boundary conditions that are anti-periodic in one lattice direction (periodic in the other two). Let $q_l(L)$ be the gauge-invariant link overlap between them: $q_l(L)=N_b^{-1}\sum_{\langle ij\rangle}\sigma_i\tau_i\,\sigma_j\tau_j$, the average over the $N_b$ nearest-neighbor bonds of the lattice, with the bonds crossing the anti-periodic plane entering with reversed sign; equivalently, twice the fraction of bonds on which the two ground states agree minus one, so that $1-q_l(L)$ equals twice the interface-bond density (the source's convention, under which the extrapolated values quoted below are stated). The problem is to determine, for this experiment and in the thermodynamic limit $L\to\infty$:

(1) Interface structure. Does $1-\langle q_l\rangle$ (disorder average) converge to a positive constant — the source's extrapolation gives $0.245\pm 0.015$ with a systematic range of 0.20 to 0.25 depending on the fit form, i.e. a space-filling rearrangement of the ground state as replica symmetry breaking predicts — or does it vanish as a power of $L$, corresponding to a fractal interface of dimension $d_f<3$ as the droplet and TNT pictures predict, possibly only beyond a crossover length not accessible at current lattice sizes? Correspondingly, are the source's scaling exponents — $\delta\approx 0.32$ for the window overlap $q^2_{M,L}\simeq L^{-\delta}f(M/L)$ and for the decay of the overlap correlation function, and $\lambda=1.5$ in the scaling of the link-overlap correlation function — asymptotic values, or pre-asymptotic effective exponents? The fit-form dependence of the infinite-$L$ extrapolation (polynomial in $1/L$ versus power law with subleading corrections) is itself part of the problem: is the resulting conclusion insensitive to the assumed fit form?

(2) Order of limits. Does the ground-state approach, which sends $T\to 0$ before $L\to\infty$, compute the same quantities as the equilibrium limit that sends $L\to\infty$ before $T\to 0$? Establish the commutation, or exhibit how the two orders differ for these observables.

(3) Selection mechanism. Identify which low-energy excited state the boundary-condition change selects, and determine whether boundary-condition perturbations generate excitations typical of the low-temperature equilibrium ensemble, justifying the determination.

Items (1)-(3) are jointly posed: together they determine the structure of the low-temperature phase of the three-dimensional Ising spin glass as probed by this experiment (many thermodynamic states with a nontrivial link overlap versus a single spin-reversed pair with $q_l\to 1$), and the zero-temperature method carries the conclusion only if items (2)-(3) hold.

References

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