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

摘要

Zeros of the moment of the partition function $[Z^n]_{\bm{J}}$ with respectto complex $n$ are investigated in the zero temperature limit $\beta \to\infty$, $n\to 0$ keeping $y=\beta n \approx O(1)$. We numerically investigatethe zeros of the $\pm J$ Ising spin glass models on several Cayley trees andhierarchical lattices and compare those results. In both lattices, thecalculations are carried out with feasible computational costs by usingrecursion relations originated from the structures of those lattices. Theresults for Cayley trees show that a sequence of the zeros approaches the realaxis of $y$ implying that a certain type of analyticity breaking actuallyoccurs, although it is irrelevant for any known replica symmetry breaking. Theresult of hierarchical lattices also shows the presence of analyticitybreaking, even in the two dimensional case in which there is nofinite-temperature spin-glass transition, which implies the existence of thezero-temperature phase transition in the system. A notable tendency ofhierarchical 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 thedifference between the mean-field and finite-dimensional systems.