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arXiv:cond-mat/0503486 · Physical Review B

Reduction of two-dimensional dilute Ising spin glasses

Stefan Boettcher, Alexander K. Hartmann

Published Jul 14, 2005• 5 citations
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Abstract

The recently proposed reduction method is applied to the Edwards-Anderson model on bond-diluted square lattices. In combination with a graph-theoretical matching algorithm, this allows us to calculate numerically exact ground states of large systems. Low-temperature domain-wall excitations are studied to determine the stiffness exponent ${y}_{2}$. A value of ${y}_{2}=\ensuremath{-}0.281(3)$ is found, consistent with previous results obtained on undiluted lattices. This comparison demonstrates the validity of the reduction method for bond-diluted spin systems and provides strong support for similar studies proclaiming accurate results for stiffness exponents in dimensions $d=3,\dots{},7$.

DOI
10.1103/physrevb.72.014429

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The recently proposed reduction method is applied to the Edwards-Anderson model on bond-diluted square lattices. In combination with a graph-theoretical matching algorithm, this allows us to calculate numerically exact ground states of large systems. Low-temperature domain-wall excitations are studied to determine the stiffness exponent y2{y}_{2}y2​. A value of y2=\ensuremath−0.281(3){y}_{2}=\ensuremath{-}0.281(3)y2​=\ensuremath−0.281(3) is found, consistent with previous results obtained on undiluted lattices. This comparison demonstrates the validity of the reduction method for bond-diluted spin systems and provides strong support for similar studies proclaiming accurate results for stiffness exponents in dimensions d=3,…,7d=3,\dots{},7d=3,…,7.
Recently, a method has been proposed to obtain accurate predictions for low-temperature properties of lattice spin glasses that is practical even above the upper critical dimension, dc=6{d}_{c}=6dc​=6. This method is based on the observation that bond-dilution enables the numerical treatment of larger lattices, and that the subsequent combination of such data at various bond densities into a finite-size scaling ansatz produces more robust scaling behavior. In the present study we test the potential of such a procedure, in particular, to obtain the stiffness exponent for the hierarchical Migdal-Kadanoff lattice. Critical exponents for this model are known with great accuracy and any simulations can be executed to very large lattice sizes at almost any bond density, effecting an insightful comparison that highlights the advantages---as well as the weaknesses---of this method. These insights are applied to the Edwards-Anderson model in d=3d=3d=3 with Gaussian bonds.
We present a collection of simulations of the Edwards–Anderson lattice spin glass at T=0 to elucidate the nature of low-energy excitations over a range of dimensions that reach from physically realizable systems to the mean-field limit. Using heuristic methods, we sample the ground states of instances to determine their energies while eliciting excitations through manipulating boundary conditions. We exploit the universality of the phase diagram of bond-diluted lattices to make such a study in higher dimensions computationally feasible. As a result, we obtain a variety of accurate exponents for domain wall stiffness and finite-size corrections, which allow us to examine their dimensional behavior and their connection with predictions from mean-field theory. We also provide an experimentally testable prediction for the thermal-to-percolative crossover exponent in dilute lattice Ising spin glasses.
Extensive computations of ground-state energies of the Edwards-Anderson spin glass on bond-diluted, hypercubic lattices are conducted in dimensions d =3, ..., 7. Results are presented for bond densities exactly at the percolation threshold, p = p c , and deep within the glassy regime, p > p c , where finding ground states is one of the hardest combinatorial optimization problems. Finite-size corrections of the form 1/ N ω are shown to be consistent throughout with the prediction ω=1− y / d , where y refers to the "stiffness" exponent that controls the formation of domain wall excitations at low temperatures. At p = p c , an extrapolation for d → ∞ appears to match our mean-field results for these corrections. In the glassy phase, however, ω does not approach its anticipated mean-field value of 2/3, obtained from simulations of the Sherrington-Kirkpatrick spin glass on an N -clique graph. Instead, the value of ω reached at the upper critical dimension matches another type of mean-field spin glass models, namely those on sparse random networks of regular degree called Bethe lattices.
A new approach to exploring low-temperature excitations in finite-dimensional lattice spin glasses is proposed. By focusing on bond-diluted lattices just above the percolation threshold, large system sizes L can be obtained which lead to enhanced scaling regimes and more accurate exponents. Furthermore, this method in principle remains practical for any dimension, yielding exponents that so far have been elusive. This approach is demonstrated by determining the stiffness exponent for dimensions d = 3, d = 6 (the upper critical dimension), and d = 7. Key is the application of an exact reduction algorithm, which eliminates a large fraction of spins, so that the reduced lattices never exceed ∼ 10 3 variables for sizes as large as L = 30 in d = 3, L = 9 in d = 6, or L = 8 in d = 7. Finite-size scaling analysis gives y 3 = 0.24(1) for d = 3, significantly improving on previous work. The results for d = 6 and d = 7, y 6 = 1.1(1) and y 7 = 1.24(5), are entirely new and are compared with mean-field predictions made for d ⩾ 6.
The recently proposed reduction method for diluted spin glasses is investigated in depth. In particular, the Edwards-Anderson model with $\ifmmode\pm\else\textpm\fi{}J$ and Gaussian bond disorder on hypercubic lattices in d=2d=2d=2, 3, and 4 is studied for a range of bond dilutions. The results demonstrate the effectiveness of using bond dilution to elucidate low-temperature properties of Ising spin glasses and provide a starting point to enhance the methods used in reduction. Based on that, a greedy heuristic called ``dominant bond reduction'' is introduced and explored.
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​.