Complex Replica Zeros of $\pm J$ Ising Spin Glass at Zero Temperature

Abstract

Zeros of the $n$th moment of the partition function $[Z^n]$ are investigated in a vanishing temperature limit $\beta \to \infty$, $n\to 0$ keeping $y=\beta n\sim O(1)$. In this limit, the moment parameterized by $y$ characterizes the distribution of the ground-state energy. We numerically investigate the zeros for $\pm J$ Ising spin glass models with several ladder and tree systems, which can be carried out with a feasible computational cost by a symbolic operation based on the Bethe--Peierls method. For several tree systems we find that the zeros tend to approach the real axis of $y$ in the thermodynamic limit implying that the moment cannot be described by a single analytic function of $y$ as the system size tends to infinity, which may be associated with breaking of the replica symmetry. However, examination of the analytical properties of the moment function and assessment of the spin-glass susceptibility indicate that the breaking of analyticity is relevant to neither one-step or full replica symmetry breaking.

27 pages, 13 figures. Added references, some comments, and corrections to minor errors

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• Distribution of partition function zeros of the $\pm J$ model on the Bethe lattice
• Thermodynamic Construction of an One-Step Replica-Symmetry-Breaking Solution in Finite Connectivity Spin Glasses
• On analyticity with respect to the replica number in random energy models I: an exact expression of the moment of the partition function
• On analyticity with respect to the replica number in random energy models II: zeros on the complex plane
• Zero-Temperature Complex Replica Zeros of the $\pm J$ Ising Spin Glass on Mean-Field Systems and Beyond