Controlling systematic errors and extending the applicability of the off-equilibrium finite-size method for critical behavior analyses

The off-equilibrium finite-size method of Lulli, Parisi, and Pelissetto (Phys. Rev. E 93, 032126 (2016)) extracts critical temperatures and exponents from Monte Carlo data taken long before equilibration: time is reparametrized in terms of the renormalization-group invariant ratio $R_\xi=\xi/L$, so that finite-size crossings of Binder cumulants locate the critical point without thermalization. Applied to the three-dimensional $\pm J$ Ising spin glass ($8\le L\le 64$), it yielded $\beta_c=0.910(4)$, $\nu=2.47(10)$, $\eta=-0.39(1)$, $\omega=1.3(2)$, and $z=6.80(15)$, but with acknowledged limitations: neglected subleading scaling corrections and approximated nonlinear scaling fields produce cut-dependent systematic effects comparable to the statistical errors and estimated subjectively; the Monte Carlo cost at fixed $R_\xi$ scales as $L^{z}$ with $z\approx 7$, capping sizes at $L\le 64$; the error on $z$ is deliberately conservative while earlier off-equilibrium estimates disagree; and the polynomial scaling-function parametrizations are justified only within a narrow temperature interval. The problem is to develop the method into a quantitatively controlled and more widely applicable instrument: quantify and incorporate the neglected corrections so that systematic-error budgets are determined rather than guessed, mitigate the $L^{z}$ sampling cost, determine $z$ for the three-dimensional $\pm J$ Ising spin glass with controlled systematics, and validate more flexible functional forms for wider temperature ranges and other models. The method has been transferred to the Gardner transition of hard-sphere glasses (Li et al., PNAS 2021) without controlling the correction or cost limitations, and no known work has quantified the method's systematic errors, extended its reach beyond $L\le 64$, or systematically assessed $z$.

Background

A spin glass is a magnetic system with quenched (frozen) random interactions in which frustration, the impossibility of simultaneously satisfying all local interaction constraints, produces a rugged free-energy landscape and extremely slow dynamics. A standard theoretical model is the three-dimensional Edwards-Anderson Ising spin glass with bimodal couplings: Ising spins $\sigma_x=\pm 1$ on a cubic lattice with nearest-neighbor couplings $J_{xy}=\pm 1$ chosen independently with equal probability (the $\pm J$ model). Locating its paramagnetic-to-glass transition and measuring the associated critical exponents is a benchmark problem in the statistical mechanics of disordered systems.

The standard tool is Monte Carlo simulation combined with finite-size scaling (FSS): because a finite lattice of linear size $L$ cannot sustain an infinite correlation length, critical singularities appear as power laws in $L$. A renormalization-group (RG) invariant quantity $U$, such as a Binder cumulant (a dimensionless ratio of moments of the spin-glass order parameter, computed from the overlap between two independent replicas evolving with the same disorder realization), becomes a universal function of $(\beta-\beta_c)L^{1/\nu}$, where $\beta=1/T$ is the inverse temperature, $\beta_c$ its critical value, and $\nu$ the correlation-length exponent; curves of $U$ versus $\beta$ for different $L$ cross at $\beta_c$. Two obstacles limit precision. First, scaling corrections: the leading behavior is corrected by a term $u_\omega(\beta)L^{-\omega}$, where $\omega$ is the leading correction-to-scaling exponent and $u_\omega(\beta)$ the associated nonlinear scaling field, the (generally nontrivial, analytic but non-linear) dependence of the correction amplitude on temperature. Second, critical slowing down: the Monte Carlo time needed to equilibrate grows as $L^{z}$ with a dynamic exponent $z\approx 7$ for the three-dimensional Ising spin glass, so equilibrium simulations, even using parallel tempering (an algorithm that simulates many temperatures together and swaps configurations to tunnel over free-energy barriers), are restricted to sizes of order $L\approx 30$–$40$ and require costly sample-by-sample equilibration checks.

Previous off-equilibrium methods sidestepped equilibration by quenching the system from a random initial configuration and analyzing the subsequent aging dynamics, but they generally assumed $L$ so large that finite-size effects are negligible, an assumption that fails for disordered systems where such $L$ are unreachable. Lulli, Parisi, and Pelissetto (Phys. Rev. E 93, 032126 (2016)) instead built a genuine off-equilibrium finite-size method: since for a quench from a disordered start the correlation length $\xi$ grows with Monte Carlo time $t$, the dynamic scaling variable $tL^{-z}$ can be traded for the RG-invariant ratio $R_\xi=\xi/L$, and at fixed $R_\xi$ the quantity $U$ behaves exactly as in equilibrium, with crossings locating $\beta_c$ and slopes giving $\nu$, yet no equilibration is needed. They fitted the five Binder cumulants simultaneously with the ansatz

$U(t,L,\beta)=P_1(R_\xi)+P_2(R_\xi)(\beta-\beta_c)L^{1/\nu}+P_3(R_\xi)L^{-\omega}$

(with $P_1$, $P_2$, $P_3$ polynomials of degree 6, 3, and 3, and 78 free parameters in total), and the susceptibility separately with an analogous ansatz, in simulations of the $\pm J$ model with $8\le L\le 64$, up to a few million disorder realizations per size and temperature, all stopped well before equilibrium ($R_\xi\le 0.5$ versus the equilibrium value $0.652(3)$). They obtained $\beta_c=0.910(4)$ (i.e., $T_c=1.099(5)$), $\nu=2.47(10)$, $\eta=-0.39(1)$, $\omega=1.3(2)$, and $z=6.80(15)$, competitive with the best equilibrium estimates of Hasenbusch, Pelissetto, and Vicari (Phys. Rev. B 78, 214205 (2008)) and the Janus Collaboration (Phys. Rev. B 88, 224416 (2013)) at a fraction of the computational cost.

The demonstration, however, carried acknowledged limitations. Varying the data-selection cuts ($L\ge L_{\min}$, $\xi\ge\xi_{\min}$, $R_\xi\ge R_{\xi,\min}$) produced scatter somewhat larger than the statistical errors, with $\omega$ dropping by more than one error bar when $\xi_{\min}$ increased from 3 to 4, so the quoted systematic error was an explicitly subjective inflation of the statistical one. The nonlinear scaling fields were approximated as $u_\beta(\beta)=\beta-\beta_c$ and $u_\omega(\beta)=1$, and the polynomial ansatz with a first-order expansion in the scaling variable was justified only by the narrow temperature interval (five $\beta$ values between 0.880 and 0.910). Because data must be collected at fixed $R_\xi$ for every $L$, and the relevant observables do not self-average (their sample-to-sample fluctuations do not decrease with system size) in disordered systems, the Monte Carlo time at fixed statistical error scales as $L^{z}$ even off equilibrium, which capped the study at $L\le 64$. Finally, the estimate $z=6.80(15)$ carried a deliberately conservative, not systematically derived, error, while earlier off-equilibrium estimates of $z$ disagreed among themselves, plausibly because subleading corrections were neglected in those analyses.

Problem Statement

Develop the off-equilibrium finite-size method into a quantitatively controlled and more widely applicable instrument for determining critical parameters, overcoming the limitations acknowledged in its founding demonstration on the three-dimensional $\pm J$ Ising spin glass. The common objective uniting the requested results is that the method's systematic errors be quantified by the analysis rather than subjectively estimated, and that its demonstrated reach (system sizes, temperature ranges, model classes) be extended, with the three-dimensional $\pm J$ Ising spin glass as the benchmark system. Concretely, the source work leaves four linked gaps:

1. (Correction control.) Neglected subleading scaling corrections beyond the leading $L^{-\omega}$ term, and the approximations $u_\beta(\beta)=\beta-\beta_c$ and $u_\omega(\beta)=1$ for the nonlinear scaling fields, produce systematic effects comparable to the statistical errors at accessible sizes: fitted $\omega$ and $\beta_c$ drift by more than one error bar under variation of the data-selection cuts $L_{\min}$, $\xi_{\min}$, $R_{\xi,\min}$, and the quoted systematic error is an explicitly subjective inflation. Quantify and incorporate these corrections so that the error budget of $\beta_c$, $\nu$, $\eta$, $\omega$, and $z$ is determined by the analysis rather than chosen.

2. (Sampling cost.) Because data must be collected at fixed $R_\xi$ for every $L$, and the relevant observables do not self-average in disordered systems, the Monte Carlo time at fixed statistical error scales as $L^{z}$ with $z\approx 7$ for the Ising spin glass; this is why the original study stopped at $L=64$. Reduce this cost so that substantially larger system sizes, or the same sizes at quantitatively smaller cost, become accessible.

3. (Dynamic exponent.) The estimate $z=6.80(15)$ carries a deliberately conservative, not systematically derived, error, and earlier off-equilibrium estimates of $z$ disagree, plausibly because subleading corrections were neglected in those analyses. Determine $z$ for the three-dimensional $\pm J$ Ising spin glass with a controlled assessment of how the fit parametrization, the polynomial degrees, the selection cuts, and the treatment of time correlations affect the result.

4. (Functional forms.) The polynomial ansätze for the scaling functions and the first-order expansion in the scaling variable are justified only within the narrow temperature interval simulated. Develop and validate more flexible functional forms with controlled convergence for wider temperature ranges and for models with stronger nonlinearities of the scaling fields.

Strands 1, 3, and 4 are the systematic-error-control program on one and the same fit framework, benchmarked by the $\pm J$ Ising spin glass; strand 2 is the practical-reach axis of the same method. All four strands are jointly required for a complete solution: strand 3 is an independent deliverable that requires a best controlled estimate of $z$ itself, beyond the determination of its error budget already implied by strand 1.

References

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