[Paper Review] Numerical study of ground state energy fluctuations in spin glasses

Claudio Giberti, Cecilia Vernia|ArXiv.org|Jul 8, 2008

Theoretical and Computational Physics29 references3 citations

TL;DR

This paper presents a stochastic annealing algorithm to study ground state energy fluctuations in spin glasses, focusing on the Sherrington-Kirkpatrick (SK) and Edwards-Anderson (EA) models. It confirms power-law scaling of energy fluctuations with exponents ρ ≈ 0.501 for EA (d=2) and ω ≈ 1.25 for d=2, and finds that the ground state energy distribution approaches Gaussianity in the thermodynamic limit, contrasting with the Gumbel-like behavior seen in finite-size SK systems.

ABSTRACT

Using a stochastic algorithm introduced in a previous paper, we study the finite size volume corrections and the fluctuations of the ground state energy in the Sherrington-Kirkpatrick and the Edwards-Anderson models at zero temperature. The algorithm is based on a suitable annealing procedure coupled with a balanced greedy-reluctant strategy that drives the systems towards the deepest minimum of the energy function.

Motivation & Objective

• To investigate the finite-size scaling of ground state energy fluctuations in spin glass models with quenched disorder.
• To validate a stochastic annealing algorithm for computing ground states in complex spin glass systems.
• To determine the scaling exponents ρ and ω for the standard deviation and mean energy density, respectively.
• To analyze the limiting shape of the ground state energy probability distribution p_N(e_N) in the thermodynamic limit.

Proposed method

• A stochastic annealing algorithm with a balanced greedy-reluctant strategy is employed to locate the global energy minimum in spin glass Hamiltonians.
• The algorithm is applied to both the mean-field Sherrington-Kirkpatrick (SK) and short-range Edwards-Anderson (EA) models.
• Energy density fluctuations are computed via disorder averaging over many independent realizations of the quenched couplings J.
• Finite-size scaling is performed using power-law fits: ε_N = e_∞ + bN^(-ω) and σ_N = aN^(-ρ), with N being the system size.
• The distribution p_N(e_N) is rescaled as x_N = (e_N - ε_N)/σ_N to study its limiting form in the large-N limit.
• Statistical moments—skewness and kurtosis—are computed to assess convergence to Gaussianity.

Experimental results

Research questions

• RQ1 How do the standard deviation σ_N and mean energy density ε_N scale with system size N in the Edwards-Anderson model?
• RQ2 Does the probability distribution p_N(e_N) of the ground state energy density converge to a Gaussian in the thermodynamic limit for the EA model?
• RQ3 What are the finite-size correction exponents ω and ρ for the mean energy and its fluctuations in the EA model (d=2 and d=3)?
• RQ4 How does the shape of p_N(e_N) evolve with system size, and does it transition from Gumbel-like to Gaussian behavior as predicted by extreme value theory?
• RQ5 Is the stochastic annealing algorithm effective in accurately computing ground state energies across different spin glass models?

Key findings

• For the Edwards-Anderson model in two dimensions, the standard deviation of the ground state energy scales as σ_N ∝ N^(-0.501), with a best-fit exponent ρ = 0.501 ± 6×10⁻³.
• In three dimensions, the finite-size correction to the mean energy density follows ε_N = -1.698 + 2.077N^(-1), consistent with literature estimates of ω ≈ 0.967.
• The skewness and kurtosis of the ground state energy distribution decay with system size, supporting convergence to a Gaussian distribution in the thermodynamic limit.
• For small system sizes, the energy distribution in the EA model (d=2) resembles a Gumbel distribution (m=6), but shifts toward Gaussianity as L increases.
• The limiting distribution for the SK model remains Gumbel-like, while the EA model exhibits a transition to Gaussianity, indicating weaker correlations in the energy contributions.
• The stochastic annealing algorithm produces results consistent with prior studies, validating its use for ground state energy computation in spin glasses.