Evidence of many thermodynamic states of the three-dimensional Ising spin glass

Open Access

Evidence of many thermodynamic states of the three-dimensional Ising spin glass

Wenlong Wang, Mats Wallin, and Jack Lidmar

Phys. Rev. Research 2, 043241 – Published 16 November 2020

Abstract

We present a large-scale simulation of the three-dimensional Ising spin glass with Gaussian disorder to low temperatures and large sizes using optimized population annealing Monte Carlo. Our primary focus is investigating the number of pure states regarding a controversial statistic, characterizing the fraction of centrally peaked disorder instances, of the overlap function order parameter. We observe that this statistic is subtly and sensitively influenced by the slight fluctuations of the integrated central weight of the disorder-averaged overlap function, making the asymptotic growth behavior very difficult to identify. Modified statistics effectively reducing this correlation are studied, and essentially monotonic growth trends are obtained. The effect of temperature is also studied, finding a larger growth rate at a higher temperature. Our state-of-the-art simulation and variance reduction data analysis suggest that the many pure states picture is most likely and coherent.

Received 25 May 2020
Revised 4 September 2020
Accepted 15 October 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.043241

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Classical statistical mechanics Frustrated magnetism

Spin glasses

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Wenlong Wang ^1,2,*, Mats Wallin ¹, and Jack Lidmar ¹

¹Department of Physics, Royal Institute of Technology, Stockholm, SE-106 91, Sweden
²College of Physics, Sichuan University, Chengdu 610065, China

^*wenlongcmp@scu.edu.cn

Article Text

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Issue

Vol. 2, Iss. 4 — November - December 2020

Subject Areas

Statistical Physics

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Images

Figure 1
(a) The disorder-averaged overlap function for different system sizes $L$ at typical low temperatures $T = 2 / 3$ (left) and $T = 0.42$ (right), respectively. (b) The central weight $I (0.2)$ is approximately independent of the system size $L$ for both temperatures, in agreement with the many-state picture. Note that the maximum size is $L = 16$ , compared with $L = 12$ of [25].
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Figure 2
The statistics $Δ (0.2, 1)$ and $Δ (0.4, 1)$ as a function of $L$ at two temperatures $T = 2 / 3$ and $T = 0.42$ (a). While $Δ$ appears to have a growing trend, it is not fully regular. We find that $Δ$ is subtly influenced by the central weight; cf., Fig. 1. To illustrate this correlation, two windows of the same size but of very different weights are studied at $T = 0.42$ , and the Window B with a larger weight also has consistently larger $Δ$ (b). This correlation can be effectively reduced by looking at the statistic $Δ / I$ as shown in panel (c). Similarly, the modified $Δ$ as shown in panel (d) has a much cleaner growth trend and is in fact remarkably monotonic. Here $〈 I 〉$ is the average of $I$ over the sizes. See the discussion for more details.
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Figure 3
Different from the large-size regime, the effect of the temperature on $Δ$ for small sizes is complex because the rapidly increasing correlation and decreasing weight $I$ compete with each other as the temperature is decreased. The top panel (a) shows the Pearson correlation coefficient, and the bottom panel (b) shows $Δ$ and $I$ for $L = 4$ . Here a local maximum of $Δ$ is observed. As $T$ decreases, $Δ$ first increases due to the rapidly growing correlation or peak sharpening between $0.3 ≲ T < 0.8$ . When typical peaks are sufficiently narrow and tall $T ≲ 0.3$ , the weight effect takes over and $Δ$ decreases with $I$ , converging to 0 as $T \to 0$ . The figure also illustrates clearly the fluctuation of $Δ$ is systematically larger than that of the central weight due to its discrete nature and thus amplifies fluctuation.
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Figure 4
(a, b) A more traditional statistic $Δ - I$ to decorrelate $Δ$ and $I$ ; the left (a) and right (b) panels are for $q_{0} = 0.2$ and 0.4, respectively. The statistic shows a growing behavior as the modified $Δ$ , in agreement with the many-state picture. By contrast, the statistic should converge to 0 for a two-state picture. (c, d) The same as above but for a refined statistic $Δ - λ I$ using the control variate analysis with a size-averaged parameter $λ$ . The data appear more regular as the error bar is further suppressed, showing again a growing behavior.
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Figure 5
The left column of top panels (a) shows the scatter plot of $Δ_{J}$ vs $I_{J}$ for $q_{0} = 0.2$ at $T = 2 / 3$ . Note the empty top left and bottom right regions in the scatter plot, suggesting sample-wise correlation. To further illustrate the scatter plot, the histograms of $I_{J}$ for the $Δ_{J} = 0$ and 1 classes are shown in the second and third columns, respectively. Similar correlations are found for $q_{0} = 0.4$ (b), as well as the lower temperature $T = 0.42$ in the corresponding panels (c, d). Other system sizes show very similar features.
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