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arXiv:cond-mat/9210005 · Europhysics Letters (EPL)

Multicanonical Study of the 3D Ising Spin Glass

Bernd A. Berg, T. Çelik, Ulrich H. E. Hansmann

Published Apr 1, 1993• 38 citations
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Abstract

We simulated the Edwards-Anderson Ising spin glass model in three dimensions via the recently proposed multicanonical ensemble. Physical quantities such as energy density, specific heat and entropy are evaluated at all temperatures. We studied their finite-size scaling, as well as the zero-temperature limit to explore the ground-state properties. Our estimate of the ground-state entropy per spin is s 0 = 0.046 ÷ 0.002.

DOI
10.1209/0295-5075/22/1/012

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We performed numerical simulations of 2D and 3D Edwards-Anderson spin glass models by using the recently developed multicanonical ensemble. Our ergodicity times increase with the lattice size approximately as V3V^3V3. The energy, entropy and other physical quantities are easily calculable at all temperatures from a single simulation. Their finite size scalings and the zero temperature limits are also explored.
We study zero-temperature properties of the three-dimensional Edwards-Anderson Ising spin glass on finite lattices up to size 123{12}^{3}123. Using multicanonical sampling we generate large numbers of ground-state configurations in thermal equilibrium. Finite-size scaling fits of the data are carried out for two hypothetical scenarios: Parisi mean-field theory versus a droplet scaling ansatz. With a zero-temperature scaling exponent y=0.72\ifmmode\pm\else\textpm\fi{}0.12 the data are well described by the droplet scaling ansatz. Alternatively, a description in terms of the Parisi mean-field behavior is still possible. The two scenarios give significantly different predictions on lattices of size \ensuremath{\ge}123{12}^{3}123.
Large numbers of ground states of three-dimensional (3D) Edwards-Anderson Ising spin glasses are calculated for sizes up to 103{10}^{3}103 using a combination of a genetic algorithm and cluster-exact approximation. A detailed analysis shows that true ground states are obtained. The ground-state stiffness (or domain wall) energy \ensuremath{\Delta} is calculated. A $|\ensuremath{\Delta}|\ensuremath{\sim}{L}^{{\ensuremath{\Theta}}_{S}}$ behavior with ${\ensuremath{\Theta}}_{S}=0.19(2)$ is found that strongly indicates that the 3D model has an equilibrium spin-glass-paramagnet transition for nonzero Tc.{T}_{c}.Tc​.
We report a high-precision finite-size scaling study of the critical behavior of the three-dimensional Ising Edwards-Anderson model (the Ising spin glass). We have thermalized lattices up to L=40L=40L=40 using the Janus dedicated computer. Our analysis takes into account leading-order corrections to scaling. We obtain Tc=1.1019(29){T}_{\mathrm{c}}=1.1019(29)Tc​=1.1019(29) for the critical temperature, $\ensuremath{\nu}=2.562(42)$ for the thermal exponent, $\ensuremath{\eta}=\ensuremath{-}0.3900(36)$ for the anomalous dimension, and $\ensuremath{\omega}=1.12(10)$ for the exponent of the leading corrections to scaling. Standard (hyper)scaling relations yield $\ensuremath{\alpha}=\ensuremath{-}5.69(13)$, $\ensuremath{\beta}=0.782(10)$, and $\ensuremath{\gamma}=6.13(11)$. We also compute several universal quantities at Tc{T}_{\mathrm{c}}Tc​.
We study the three-dimensional Edwards-Anderson model with binary interactions by Monte Carlo simulations. Direct evidence of finite-size scaling is provided, and the universal finite-size scaling functions are determined. Monte Carlo data are extrapolated to infinite volume with an iterative procedure for correlation lengths up to $\ensuremath{\xi}\ensuremath{\approx}140$. The infinite-volume data are consistent with a conventional power-law singularity at finite temperature Tc{T}_{c}Tc​. Taking into account corrections to scaling, we find ${T}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.156\ifmmode\pm\else\textpm\fi{}0.015$, $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.8\ifmmode\pm\else\textpm\fi{}0.2$, and $\ensuremath{\eta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}0.26\ifmmode\pm\else\textpm\fi{}0.04$. The data are also consistent with an exponential singularity at finite Tc{T}_{c}Tc​, but not with an exponential singularity at zero temperature.