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

TL;DR

This paper studies the zeros of the moments of the partition function in zero-temperature spin glasses, revealing complex behaviors related to replica symmetry breaking through numerical and analytical methods.

Contribution

It introduces a numerical approach to analyze the complex zeros of the partition function moments in spin glasses, linking zero distributions to replica symmetry breaking.

Findings

Zeros approach the real axis in certain systems, indicating non-analytic behavior.

The non-analyticity is not directly linked to known replica symmetry breaking mechanisms.

The method provides insights into the ground-state energy distribution in spin glasses.

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.