Existence and Absence of Spin-Glass Phases on Migdal--Kadanoff Hierarchical Lattices: A Complete Classification

Abstract

We consider the Ising spin-glass model on Migdal--Kadanoff (MK) hierarchical lattices, characterized by the fractal dimension $d=1+\log s/\log b$, where $s$ and $b$ denote the number of parallel branches and the number of bonds in series within each branch, respectively. For independent Gaussian couplings, we rigorously establish a necessary and sufficient condition for the existence of finite-temperature spin-glass order for arbitrary integers $b,s\ge 2$. We prove that a finite-temperature spin-glass phase exists whenever \[ \frac{b}{\sqrt{s}} \left( \frac{b\sqrt{s(s-1)}}{bs-1} \right)^{s-1}<1. \] In particular, this condition holds for all three-dimensional MK hierarchical lattices ($d=3$, or equivalently $s=b^2$), regardless of the choice of $b$. Conversely, we prove the absence of finite-temperature spin-glass order for fractal dimensions $d<1+\log 17/\log 6\approx 2.58$, as well as in the large-$b$ limit for any fixed $d<3$. In the intermediate regime $1+\log 17/\log 6\le d<3$, we provide a finite-step procedure to determine the presence or absence of spin-glass order, which depends on $b$ and $s$. Our results provide a complete characterization of finite-temperature spin-glass order on MK hierarchical lattices.

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