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arXiv:1212.4023 · Journal of Statistical Mechanics Theory and Experiment

Numerical study of the overlap Lee–Yang singularities in the three-dimensional Edwards–Anderson model

R. A. Baños, Jose Miguel Gil-Narvion, Jorge Monforte-Garcia, Juan Jesus Ruiz-Lorenzo, David Yllanes

Published Feb 18, 2013• 2 citations
arXivHTMLPDF

Abstract

We have characterized numerically, using the Janus computer, the Lee–Yang complex singularities related to the overlap in the 3 D Ising spin-glass with binary couplings over a wide range of temperatures (both in the critical and in the spin-glass phase). Studying the behavior of the zeros at the critical point, we have obtained an accurate measurement of the anomalous dimension in very good agreement with the values quoted in the literature. In addition, by studying the density of the zeros we have been able to characterize the phase transition and to investigate the Edwards–Anderson order parameter in the spin-glass phase, finding agreement with the values obtained using more conventional techniques.

DOI
10.1088/1742-5468/2013/02/p02031

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We have characterized numerically, using the Janus computer, the Lee–Yang complex singularities related to the overlap in the 3 D Ising spin-glass with binary couplings over a wide range of temperatures (both in the critical and in the spin-glass phase). Studying the behavior of the zeros at the critical point, we have obtained an accurate measurement of the anomalous dimension in very good agreement with the values quoted in the literature. In addition, by studying the density of the zeros we have been able to characterize the phase transition and to investigate the Edwards–Anderson order parameter in the spin-glass phase, finding agreement with the values obtained using more conventional techniques.
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 characterize numerically the properties of the phase transition of the three-dimensional Ising spin glass with Gaussian couplings and of the low-temperature phase. We compute critical exponents on large lattices. We study in detail the overlap probability distribution and the equilibrium overlap-overlap correlation functions. We find a clear agreement with off-equilibrium results from previous work. These results strongly support the existence of a continuous spontaneous replica symmetry breaking in three-dimensional spin glasses.
Using the decay of the out equilibrium spin-spin correlation function we compute the equilibrium Edward-Anderson order parameter in the three dimensional binary Ising spin glass in the spin glass phase. We have checked that the Edward-Anderson order parameter computed from out of equilibrium numerical simulations follows with good precision the critical law as determined in experiments and in numerical studies at equilibrium. We have also studied the dependence of the order parameter with the lattice size. Finally we present a large time study of the scaling of the off-equilibrium fluctuation-dissipation relations.
We introduce a novel method for numerical spin glass investigations: Simulations of two replica at fixed temperature, weighted to achieve a broad distribution of the Parisi overlap parameter qqq (multioverlap). We demonstrate the feasibility of the approach by studying the 3D Edwards-Anderson Ising ( ${J}_{\mathrm{ik}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ifmmode\pm\else\textpm\fi{}1$) spin glass in the broken phase ( $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$). This makes it possible to obtain reliable results about spin glass tunneling barriers. In addition, our results indicate a nontrivial scaling behavior of the canonical qqq distributions not only at the freezing point but also deep in the broken phase.
We compute the spin-glass order parameter, qEA, in the low-temperature phase of the D=3 Edwards-Anderson model. Our approach focuses on the connected spatial correlation function, regarded as a function of the spin overlap q. We investigate by means of Finite Size Scaling the phase transition suffered by this correlation function precisely at q=qEA. Thanks to the Janus dedicated computer, we include in the analysis equilibrium data for L=32 lattices, down to temperatures T=0.64Tc. We compute the exponent that rules finite size effects, which turns out to be compatible with the replicon exponent.
The Edwards and Anderson model (1975) for spin glasses is investigated by a real space rescaling transformation on an S= 1 / 2 Ising Hamiltonian with nearest-neighbour-only interactions in three dimensions. A transition to a spin glass ordered phase is found at about half the mean transition temperature, in contrast to earlier work on a two-dimensional model where no transition was found. The results suggest that the lower critical dimensionality, d c , for spin glass order is slightly greater than 2, although possibly d c =2 exactly. The specific heat is calculated and found to exhibit a broad maximum above T c , with no discernable singularity at the transition since the exponent alpha is large and negative.