Zero-Temperature Complex Replica Zeros of the $\pm J$ Ising Spin Glass on Mean-Field Systems and Beyond

TL;DR

This paper investigates the zeros of the partition function's moments in complex replica number space for $m{J}$-distributed Ising spin glasses at zero temperature, revealing analyticity breaking and phase transition indications.

Contribution

It introduces a numerical method to analyze complex replica zeros in spin glasses on Cayley trees and hierarchical lattices, highlighting differences between mean-field and finite-dimensional systems.

Findings

Zeros approach the real axis indicating analyticity breaking.

Presence of zero-temperature phase transition in hierarchical lattices.

Zeros spread widely in the complex plane for hierarchical lattices.

Abstract

Zeros of the moment of the partition function $[Z^n]_{\bm{J}}$ with respect to complex $n$ are investigated in the zero temperature limit $\beta \to \infty$, $n\to 0$ keeping $y=\beta n \approx O(1)$. We numerically investigate the zeros of the $\pm J$ Ising spin glass models on several Cayley trees and hierarchical lattices and compare those results. In both lattices, the calculations are carried out with feasible computational costs by using recursion relations originated from the structures of those lattices. The results for Cayley trees show that a sequence of the zeros approaches the real axis of $y$ implying that a certain type of analyticity breaking actually occurs, although it is irrelevant for any known replica symmetry breaking. The result of hierarchical lattices also shows the presence of analyticity breaking, even in the two dimensional case in which there is no finite-temperature spin-glass transition, which implies the existence of the zero-temperature phase transition in the system. A notable tendency of hierarchical lattices is that the zeros spread in a wide region of the complex $y$ plane in comparison with the case of Cayley trees, which may reflect the difference between the mean-field and finite-dimensional systems.