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arXiv:0911.2583 · The European Physical Journal B

Numerical results for spin glass ground states on Bethe lattices: Gaussian bonds

Stefan Boettcher

Published Mar 18, 2010• 8 citations
arXivHTMLPDF
DOI
10.1140/epjb/e2010-00095-4

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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.
We consider the effect of perturbing a single bond on ground states of nearest-neighbor Ising spin glasses, with a Gaussian distribution of the coupling constants, across various two- and three-dimensional lattices and regular random graphs. Our results reveal that the ground states are strikingly fragile with respect to such changes. Altering the strength of only a single bond beyond a critical threshold value leads to a new ground state that differs from the original one by a droplet of flipped spins whose boundary and volume diverge with the system size-an effect that is reminiscent of the more familiar phenomenon of disorder chaos. These elementary fractal-boundary zero-energy droplets and their composites feature robust characteristics and provide the lowest-energy macroscopic spin-glass excitations. Remarkably, within numerical accuracy, the size of such droplets conforms to a universal power-law distribution with exponents that depend on the spatial dimension of the system. Furthermore, the critical coupling strengths adhere to a stretched exponential distribution that is predominantly determined by the local coordination number.
The standard short-range two-dimensional Ising spin glass is numerically well accessible, in particular, because there are polynomial-time ground-state algorithms. On the other hand, in contrast to higher dimensional spin glasses, it does not exhibit a rich behavior, i.e., no ordered phase at finite temperature. Here, we investigate whether long-range correlated bonds change this behavior. This would still keep the model numerically well accessible while exhibiting a more interesting behavior. The bonds are drawn from a Gaussian distribution with a two-point correlation for bonds at distance r that decays as (1+r^{2})^{-a/2}, a≥0. We study numerically with exact algorithms the ground-state and domain-wall excitations. Our results indicate that the inclusion of bond correlations still does not lead to a spin-glass order at any finite temperature. A further analysis reveals that bond correlations have a strong effect at local length scales, inducing ferro- and antiferromagnetic domains into the system. The length scale of ferro- and antiferromagnetic order diverges exponentially as the correlation exponent approaches a critical value, a→a_{crit}=0. Thus, our results suggest that the system becomes a ferro- or antiferromagnet only in the limit a→0.