Interface free-energy exponent in the one-dimensional Ising spin glass with long-range interactions in both the droplet and broken replica symmetry regions

T. Aspelmeier, Wenlong Wang, M. A. Moore, and Helmut G. Katzgraber

Phys. Rev. E 94, 022116 – Published 11 August 2016

Abstract

The one-dimensional Ising spin-glass model with power-law long-range interactions is a useful proxy model for studying spin glasses in higher space dimensions and for finding the dimension at which the spin-glass state changes from having broken replica symmetry to that of droplet behavior. To this end we have calculated the exponent that describes the difference in free energy between periodic and antiperiodic boundary conditions. Numerical work is done to support some of the assumptions made in the calculations and to determine the behavior of the interface free-energy exponent of the power law of the interactions. Our numerical results for the interface free-energy exponent are badly affected by finite-size problems.

Received 13 June 2016

DOI:https://doi.org/10.1103/PhysRevE.94.022116

Physics Subject Headings (PhySH)

Interfaces Spin glasses

Ising model

Statistical Physics & Thermodynamics

Authors & Affiliations

T. Aspelmeier^1,2,3, Wenlong Wang⁴, M. A. Moore⁵, and Helmut G. Katzgraber^4,6

¹Felix Bernstein Institute for Mathematical Statistics in the Biosciences, Georg August University of Göttingen, 37077 Göttingen, Germany
²Institute for Mathematical Stochastics, University of Göttingen, 37073 Göttingen, Germany
³Statistical Inverse Problems in Biophysics, Max Planck Institute for Biophysical Chemistry, 37077 Göttingen, Germany
⁴Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA
⁵School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
⁶Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA

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Vol. 94, Iss. 2 — August 2016

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Images

Figure 1
Estimates of the exponent $μ$ (sample-to-sample variation of the ground-state energy) as a function of $σ$ , i.e., $δ E \sim L^{μ}$ . Here KY denotes results obtained on samples up to $L = 256$ by Katzgraber and Young [19]. The expectation for $μ$ is that $μ = 1 / 6$ for $σ < 1 / 2$ [28] and $1 / 2$ for all $σ > 1 / 2$ , as the system is now self-averaging [29] (dashed lines). Finite-size effects make the transition between these two values for $μ$ spread over a large range of $σ$ . Notice that the results for $μ$ in the SK model region $σ = 0.1$ are moving closer to the theoretical prediction of $1 / 6$ as $L$ increases.
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Figure 2
Spin overlap distributions for three values of $σ$ as a function of $L$ at $T = 0.2 T_{c}$ . The distributions include $P^{π, π} (q), P^{\bar{π}, \bar{π}} (q)$ , and $P^{\bar{π}, \bar{π}} (q)$ . The diagonal distributions have substantial peaks close to $\pm 1$ , with decreasing $q_{EA}$ as $L$ increases, while the off-diagonal distributions $P^{π, \bar{π}} (q)$ peak only at $q = 0$ , becoming increasingly localized towards the center as $L$ increases for the system sizes studied. Note that in the third panel, $P^{π, \bar{π}} (q)$ appears to saturate to a non- $δ$ function. In all panels the systems sizes increase from bottom to top as seen from the center of the distribution for the cases where there is a central peak; otherwise, it is as seen from the peaks at large values of $| q |$ .
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Figure 3
Spin overlap distributions for $L = 400$ for various values of $σ$ at $T = 0.2 T_{c}$ . The diagonal distributions are those with peaks close to $\pm 1$ , while the off-diagonal distributions $P^{π, \bar{π}} (q)$ have peaks only at $q = 0$ and become increasingly localized towards the center as $σ$ decreases. For the distributions with a peak at the center, the values of $σ$ increase with decreasing peak height. For the distributions with large support for $| q |$ large the values of $σ$ increase for increasing peak height.
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Figure 4
Parisi overlap $P^{π, \bar{π}} (0)$ as a function of $\sqrt{L}$ for three representative values of $σ$ at $T = 0.2 T_{c}$ . Note that $P^{π, \bar{π}} (0)$ grows approximately linearly in $\sqrt{L}$ in the RSB regime, but seems to level off in the droplet or scaling regime $(σ > 2 / 3)$ .
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Figure 5
Estimates of the exponent $θ$ as a function of $σ$ . Here KY denotes results obtained on samples up to $L = 256$ in Ref. [19] by Katzgraber and Young. The dashed line denotes the droplet regime prediction for $θ = 1 - σ$ . We expect this to apply for $2 > σ \geq 2 / 3$ . When $σ < 2 / 3$ we predict that $θ = 1 / 6$ and the horizontal dashed line shows this prediction. Notice that the result for $θ$ in the SK model region $(σ = 0.1)$ is moving closer to the theoretical prediction of $1 / 6$ as $L$ increases, albeit very slowly.
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