[Paper Review] How to evaluate ground-state landscapes of spin glasses thermodynamical correctly

E.P. Nakhmedov|arXiv (Cornell University)|Oct 19, 2012

Theoretical and Computational Physics2 references7 citations

TL;DR

This paper presents a thermodynamically correct method to evaluate ground-state landscapes in three-dimensional Edwards-Anderson ±J spin glasses by combining cluster-exact approximation with a ballistic search and T=0 Monte Carlo simulations to ensure all ground states are sampled with equal probability. The key result is that the overlap distribution P(|q|) collapses to a delta function in the thermodynamic limit, indicating a single dominant cluster of ground states, contradicting mean-field predictions of multiple pure states.

ABSTRACT

Ground states of three-dimensional $\\pm J$ Ising spin glasses are calculated for sizes up to 143 using a combination of a genetic algorithm and cluster-exact approximation. For each realization several independent ground states are obtained. Then, by applying ballistic search and T=0 Monte-Carlo simulations, it is ensured that each ground state appears with the same probability. Consequently, the results represent the true T=0 thermodynamic behavior. The distribution P(|q|) of overlaps is evaluated. For increasing size the width of P(|q|) and the fraction of the distribution below $q_0\\equiv 0.5$ converge to zero. This indicates that for the infinite system P(|q|) is a delta function, in contrast to previous results. Thus, the ground-state behavior is dominated by few large clusters of similar ground states.

Motivation & Objective

• To resolve the long-standing debate on whether spin glasses exhibit multiple pure states or only two (global spin-flip related) in the thermodynamic limit.
• To correct the bias in prior ground-state sampling methods, where different algorithms assign unequal statistical weights to degenerate ground states, violating thermodynamic principles.
• To provide a reliable, unbiased sampling technique that ensures each ground state contributes equally to observables, enabling accurate evaluation of the true T=0 thermodynamic behavior.
• To investigate the structure of the ground-state landscape using overlap distributions and cluster statistics in finite-size systems, with extrapolation to infinite size.

Proposed method

• Combines cluster-exact approximation (CEA) with a genetic algorithm to compute true ground states for system sizes up to L=14.
• Applies a ballistic search and T=0 Monte Carlo simulations to ensure all ground states are generated with equal probability, achieving correct thermodynamic sampling.
• Uses a multi-step procedure: first, find all ground states via CEA-genetic algorithm; second, apply T=0 MC sweeps to explore the full ground-state manifold within each cluster.
• Imposes a constraint on bond configurations (ΣJij=0) to reduce finite-size fluctuations and improve statistical convergence.
• Evaluates the overlap distribution P(|q|) between independent ground states, where q = (1/N)Σσiσ′i, and analyzes its width and fraction below q0=0.5.
• Performs finite-size extrapolations of σ²(|q|) and X₀.₅(L) (fraction of overlaps below 0.5) to infer the infinite-system behavior.

Experimental results

Research questions

• RQ1 Does the ground-state landscape of 3D spin glasses support multiple pure states, as predicted by mean-field theory, or only two (global spin-flip related) states, as suggested by droplet theory?
• RQ2 Can standard ground-state algorithms like genetic CEA provide a thermodynamically correct sampling of degenerate ground states, or do they introduce statistical biases?
• RQ3 What is the behavior of the overlap distribution P(|q|) in the thermodynamic limit, and does it converge to a delta function as predicted by droplet theory?
• RQ4 How does the distribution of ground-state clusters evolve with system size, and does one dominant cluster emerge in the large-L limit?
• RQ5 To what extent is the ground-state space ultrametric, as indicated by the distribution of overlap differences δq = q₂ - q₁ in triplets of states?

Key findings

• The variance σ²(|q|) of the overlap distribution decreases with system size L and extrapolates to σ²∞ = -0.01(1), indicating the width of P(|q|) vanishes in the thermodynamic limit, implying P(|q|) is a delta function.
• The fraction of overlaps below q₀=0.5, denoted X₀.₅(L), extrapolates to X∞ = -0.01(2), meaning that for infinite systems, no pairs of ground states have small overlap, confirming dominance of a single cluster.
• The distribution P(δq) of overlap differences in triplets of ground states becomes sharply peaked at δq=0 with increasing L, supporting the ultrametric structure expected in droplet theory.
• The average overlap ⟨q⟩ remains high across system sizes, and the distribution P(|q|) becomes increasingly narrow, indicating strong similarity among ground states.
• The probability P(nc=1) that a realization has only one ground-state cluster decreases with L, but the system remains dominated by a single large cluster, not by many small ones.
• Finite-size extrapolations of σ²(|q|) and X₀.₅(L) are consistent with a thermodynamic limit where P(|q|) collapses to a delta function at q=1, indicating a single pure state (up to global spin flip).