Ground states of the Sherrington–Kirkpatrick spin glass with Levy bonds

Abstract

Ground states of Ising spin glasses on fully connected graphs are studied for a broadly distributed bond family. In particular, bonds J distributed according to a Levy distribution P(J) ∝ 1/|J|^1+α, |J| > 1, are investigated for a range of powers α. The results are compared with those for the Sherrington–Kirkpatrick (SK) model, where bonds are Gaussian distributed. In particular, we determine the variation of the ground-state energy densities with α, their finite-size corrections, measure their fluctuations, and analyze the local field distribution. We find that the energies themselves at infinite system size attain universally the Parisi-energy of the SK as long as the second moment of P(J) exists (α > 2). They compare favorably with recent one-step replica symmetry breaking predictions well below α = 2. At and just below α = 2, the simulations deviate significantly from theoretical expectations. The finite-size investigation reveals that the corrections exponent ω decays from the putative SK value ω _SK  = 2/3 already well above α = 2, at which point it reaches a minimum. This result is justified with a speculative calculation of a random energy model with Levy bonds. The exponent ρ that describes the variations of the ground-state energy fluctuations with system size decays monotonically from its SK value for decreasing α and appears to vanish at α = 1.

Keywords:

• spin glass
• optimization
• Sherrington
• disordered systems
• glass
• numerical simulation

Acknowledgements

I am delighted to dedicate this work to David Sherrington in honor of his 70th birthday. I am deeply indebted to David for his inspiration and encouragement infusing much of my research, and for his relentless support at many turns of my career. Many thanks to J.-P. Bouchaud for his suggestions that initiated this project. I am very grateful to K. Janzen and A. Engel for many clarifying discussion and for providing me with some of their data, and to the Fulbright Kommission for supporting my stay at Oldenburg University. This work has been supported also by grants 0312510 and 0812204 from the Division of Materials Research at the National Science Foundation and by the Emory University Research Council.

Notes

Notes

1. A copy of the SK code is available at http://www.physics.emory.edu/faculty/boettcher/Research/EO_demo/

2. It is well-known 25,46 that is the Fourier transform of a symmetric distribution Q(J), similar to P(J) in Equation (1) with Q(J) ∼P(J) ∼|J|^−1−α for |J| → ∞. For the sum of n such random variables, , it is easy to show that . Hence, for the respective standard deviations σ _n ∼σ₁/n ^1−1/α, so that the mean ⟨J⟩ = lim _n→∞ S _n is approached with error ∼1/n ^1−1/α. Arguably, see Figure 2, for α → 1⁺, energies are dominated by a few large bonds such that their distribution also follows the power-law in Equation (1), implying the slow decrease in Equation (5).