Lower-Critical Spin-Glass Dimension from 23 Sequenced Hierarchical Models

TL;DR

This paper precisely calculates the lower-critical dimension for the Ising spin-glass phase as approximately 2.52 using 23 hierarchical lattice models, providing insights into phase transition behavior in fractional dimensions.

Contribution

It introduces a nearly exact numerical method to determine the lower-critical dimension for spin-glass phases using a family of hierarchical lattices.

Findings

Lower-critical dimension found to be approximately 2.52

Phase transition temperature and critical exponents calculated across dimensions

Provides a highly accurate fit with R^2 = 0.999999

Abstract

The lower-critical dimension for the existence of the Ising spin-glass phase is calculated, numerically exactly, as $d_L = 2.520$ for a family of hierarchical lattices, from an essentially exact (correlation coefficent $R^2 = 0.999999$) near-linear fit to 23 different diminishing fractional dimensions. To obtain this result, the phase transition temperature between the disordered and spin-glass phases, the corresponding critical exponent $y_T$, and the runaway exponent $y_R$ of the spin-glass phase are calculated for consecutive hierarchical lattices as dimension is lowered.