Nontrivial maturation metastate-average state in a one-dimensional long-range Ising spin glass: Above and below the upper critical range

S. Jensen, N. Read, and A. P. Young

Phys. Rev. E 104, 034105 – Published 7 September 2021

Abstract

Understanding the low-temperature pure state structure of spin glasses remains an open problem in the field of statistical mechanics of disordered systems. Here we study Monte Carlo dynamics, performing simulations of the growth of correlations following a quench from infinite temperature to a temperature well below the spin-glass transition temperature $T_{c}$ for a one-dimensional Ising spin-glass model with diluted long-range interactions. In this model, the probability $P_{i j}$ that an edge ${i, j}$ has nonvanishing interaction falls as a power law with chord distance, $P_{i j} \propto 1 / R_{i j}^{2 σ}$ , and we study a range of values of $σ$ with $1 / 2 < σ < 1$ . We consider a correlation function $C_{4} (r, t)$ . A dynamic correlation length that shows power-law growth with time $ξ (t) \propto t^{1 / z}$ can be identified in the data and, for large time $t, C_{4} (r, t)$ decays as a power law $r^{- α_{d}}$ with distance $r$ when $r ≪ ξ (t)$ . The calculation can be interpreted in terms of the maturation metastate averaged Gibbs state, or MMAS, and the decay exponent $α_{d}$ differentiates between a trivial MMAS ( $α_{d} = 0$ ), as expected in the droplet picture of spin glasses, and a nontrivial MMAS ( $α_{d} \neq 0$ ), as in the replica-symmetry-breaking (RSB) or chaotic pairs pictures. We find nonzero $α_{d}$ even in the regime $σ > 2 / 3$ which corresponds to short-range systems below six dimensions. For $σ < 2 / 3$ , the decay exponent $α_{d}$ follows the RSB prediction for the decay exponent $α_{s} = 3 - 4 σ$ of the static metastate, consistent with a conjectured statics-dynamics relation, while it approaches $α_{d} = 1 - σ$ in the regime $2 / 3 < σ < 1$ ; however, it deviates from both lines in the vicinity of $σ = 2 / 3$ .

Received 25 June 2021
Accepted 13 August 2021

DOI:https://doi.org/10.1103/PhysRevE.104.034105

Physics Subject Headings (PhySH)

Classical statistical mechanics

Disordered systems Spin glasses

Monte Carlo methods Stochastic processes & statistics

Condensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

S. Jensen ¹, N. Read^2,3, and A. P. Young⁴

¹Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
²Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA
³Department of Applied Physics, Yale University, P.O. Box 208284, New Haven, Connecticut 06520-8284, USA
⁴Physics Department, University of California, Santa Cruz, Santa Cruz, California 95064, USA

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Vol. 104, Iss. 3 — September 2021

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Images

Figure 1
$C_{4} (r, t)$ as a function of distance along the lattice $r$ for multiple lattice sizes $N$ for the largest times reached in the simulations for a few representative interaction parameters (a) $σ = 0.625$ , (b) $σ = 0.720$ , and (c) $σ = 0.896$ . We see that finite-size effects are well controlled already at $N = 2^{14}$ for the largest $σ$ values but we require lattice sizes $N = 2^{26}$ for $σ = 0.625$ .
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Figure 2
$C_{4} (r, t)$ as a function of distance along the lattice $r$ for a given lattice size (with controlled finite-size effects) for varying times. Panel (a) shows interaction $σ = 0.625$ for lattice size $N = 2^{26}$ up to time $t = 2^{14}$ . Panel (b) shows results at $σ = 2 / 3$ for lattice size $N = 2^{24}$ up to time $t = 2^{17}$ . Panels (c)–(f) are for interactions in the regime $σ > 2 / 3$ , which is above the upper critical range. We also show the short-distance power-law behavior for each interaction $σ$ (dashed lines) and the large-distance power-law behavior (dash-dotted lines) for the largest available value of $t$ .
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Figure 3
Data collapse results for multiple interaction parameters from (a) $σ = 0.625$ , below the upper critical range, to (f) $σ = 0.840$ , well above the upper critical range. The collapse is performed with the ansatz $C_{4} (r, t) = \frac{1}{r^{α_{d}}} f (\frac{r}{ξ (t)})$ for the listed lattice sizes where $ξ (t) \propto t^{1 / z}$ . The best-fit values used in the collapse are determined as discussed in the Appendix.
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Figure 4
Exponent $α_{d}$ (dark-blue squares) describing the decay of $C_{4} (r, t)$ as a function of interaction power-law exponent $σ$ . $α_{d} > 0$ for all interactions studied, suggesting a nontrivial metastate at low $T$ for $σ < 1$ . We clearly see the predicted behavior $α_{d} = 3 - 4 σ$ (light-blue dashed line) for $σ < 2 / 3$ . The dependence of $α_{d}$ on $σ$ approaches $α_{d} = 1 - σ$ (light-green dotted line) for $σ > 2 / 3$ . We also show the bound $α_{d} \leq 2 - 2 σ$ [24] (purple dash-dotted line) which is satisfied by the simulation results.
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Figure 5
Collapse quality $S$ heatmap for $σ = 0.685$ . The quality is considered good for $S \approx 1$ though, as discussed in the main text, it can be much smaller due to the highly correlated nature of the data. The quality shown was evaluated with data for $t_{min} = 2^{13}$ to $t_{max} = 2^{16}$ for $N = 2^{24}$ and lattice distances $r_{min} = 2^{6}$ .
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Figure 6
Histograms for the best-fit collapse exponent $α_{d}$ with $N_{b} = 1000$ bootstrap samples with interaction exponents (a) $σ = 0.625$ , (b) $σ = 0.667$ , (c) $σ = 0.685$ , (d) $σ = 0.720$ , (e) $σ = 0.784$ , and (f) $σ = 0.840$ . Each panel shows the bootstrap histogram for varying $r_{min}$ . We see that the best estimate value for $α_{d}$ varies significantly for small $r_{min}$ , e.g., for $σ = 0.625$ we observe variations up to $r_{min} \approx 2^{4}$ . The best-fit collapse value of $α_{d} = 0.501$ (used in Fig. 3) and the error $δ α_{d} = 0.009$ are determined with $r_{min} = 2^{7}$ . The values for each interaction exponent are given in Table 1.
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