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≥2. We prove that a finite-temperature spin-glass phase exists whenever b/√s ( (b√(s(s−1)))/(bs−1) )s−1<1. In particular, this condition holds for all three-dimensional MK hierarchical lattices (d=3, or equivalently s=b2), regardless of the choice of b. Conversely, we prove the absence of finite-temperature spin-glass order for fractal dimensions d<1+log17/log6≈2.58, as well as in the large-b limit for any fixed d<3. In the intermediate regime 1+log17/log6≤ 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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BibTeX
@misc{okuyama2026existence,
title = {{Existence and Absence of Spin-Glass Phases on Migdal--Kadanoff Hierarchical Lattices: A Complete Classification}},
author = {Manaka Okuyama and Masayuki Ohzeki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32291},
eprint = {2610.32291},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32291},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Existence and Absence of Spin-Glass Phases on Migdal--Kadanoff Hierarchical Lattices: A Complete Classification
AU - Manaka Okuyama
AU - Masayuki Ohzeki
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32291
UR - https://arxiv.org/abs/2610.32291
AB - 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≥2. We prove that a finite-temperature spin-glass phase exists whenever b/√s ( (b√(s(s−1)))/(bs−1) )^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+log17/log6≈2.58, as well as in the large-b limit for any fixed d<3. In the intermediate regime 1+log17/log6≤ 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.
ER -