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arXiv:cond-mat/9712320 · Physical Review Letters

Multioverlap Simulations of the 3D Edwards-Anderson Ising Spin Glass

Bernd A. Berg, Wolfhard Janke E

Published May 25, 1998• 82 citations
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Abstract

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 $q$ (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 $q$ distributions not only at the freezing point but also deep in the broken phase.

DOI
10.1103/physrevlett.80.4771

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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 ( Jik=\ifmmode±\else\textpm\fi1{J}_{\mathrm{ik}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ifmmode\pm\else\textpm\fi{}1Jik​=\ifmmode±\else\textpm\fi1) spin glass in the broken phase ( \ensuremathβ=1\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1\ensuremathβ=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 report large-scale simulations of the three-dimensional Edwards-Anderson Ising spin glass system using the recently introduced multi-overlap Monte Carlo algorithm. In this approach the temperature is fixed and two replica are coupled through a weight factor such that a broad distribution of the Parisi overlap parameter q is achieved. Canonical expectation values for the entire q-range (multi-overlap) follow by reweighting. We present an analysis of the performance of the algorithm and in particular discuss results on spin glass free-energy barriers which are hard to obtain with conventional algorithms. In addition we discuss the non-trivial scaling behavior of the canonical q-distributions in the broken phase.
Abstract We report large‐scale simulations of the three‐dimensional Edwards‐Anderson Ising spin glass system using the recently introduced multi‐overlap Monte Carlo algorithm. In this approach the temperature is fixed and two replica are coupled through a weight factor such that a broad distribution of the Parisi overlap parameter q is achieved. Canonical expectation values for the entire q ‐range (multi‐overlap) follow by reweighting. We present an analysis of the performance of the algorithm and in particular discuss results on spin glass free‐energy barriers which are hard to obtain with conventional algorithms. In addition we discuss the non‐trivial scaling behavior of the canonical q ‐distributions in the broken phase.
For the Edwards-Anderson Ising spin-glass model in three and four dimensions (3d(3d(3d and 4d)4d)4d) we have performed high statistics Monte Carlo calculations of those free-energy barriers FBq{F}_{B}^{q}FBq​ which are visible in the probability density PJ(q){P}_{\mathcal{J}}(q)PJ​(q) of the Parisi overlap parameter q. The calculations rely on the recently introduced multioverlap algorithm. In both dimensions, within the limits of lattice sizes investigated, these barriers are found to be non-self-averaging and the same is true for the autocorrelation times of our algorithm. Further, we present evidence that barriers hidden in q dominate the canonical autocorrelation times.
We have simulated Edwards-Anderson (EA) as well as Sherrington-Kirkpatrick systems of L3{L}^{3}L3 spins. After averaging over large sets of EA system samples of $3\ensuremath{\le}L\ensuremath{\le}10$, we obtain accurate numbers for distributions p(q)p(q)p(q) of the overlap parameter qqq at very low-temperature TTT. We find $p(0)/T\ensuremath{\rightarrow}0.233(4)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with the droplet scenario of spin glasses. We also study the number of mismatched links---between replica pairs---that come with large scale excitations. Contributions from small scale excitations are discarded. We thus obtain for the fractal dimension of outer surfaces of $q\ensuremath{\sim}0$ excitations in the EA model ${d}_{s}\ensuremath{\rightarrow}2.59(3)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with ${d}_{s}\ensuremath{\rightarrow}3$ as $T\ensuremath{\rightarrow}0$ that is predicted by mean-field theory for the macroscopic limit.
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 present results of recent high-statistics Monte Carlo simulations of the Edwards-Anderson Ising spin-glass model in three and four dimensions. The study is based on a non-Boltzmann sampling technique, the multi-overlap algorithm which is specifically tailored for sampling rare-event states. We thus concentrate on those properties which are difficult to obtain with standard canonical Boltzmann sampling such as the free-energy barriers F^q_B in the probability density P_J(q) of the Parisi overlap parameter q and the behaviour of the tails of the disorder averaged density P(q) = [P_J(q)]_av.