Reduction of Spin Glasses applied to the Migdal-Kadanoff Hierarchical Lattice

TL;DR

This paper introduces a reduction method for finding ground states of spin glasses on hierarchical lattices, enabling precise calculations of critical exponents and properties in low-dimensional systems.

Contribution

A novel reduction procedure for spin glasses on hierarchical lattices that allows exact ground state calculations and scaling analysis.

Findings

Determined stiffness exponents y_3=0.25546(3) in 3D and y_4=0.76382(4) in 4D.

Achieved complete lattice reduction up to size 2^{100} in 3D and 2^{35} in 4D.

Provided exact ground state energies, entropies, and overlap distribution approximations.

Abstract

A reduction procedure to obtain ground states of spin glasses on sparse graphs is developed and tested on the hierarchical lattice associated with the Migdal-Kadanoff approximation for low-dimensional lattices. While more generally applicable, these rules here lead to a complete reduction of the lattice. The stiffness exponent governing the scaling of the defect energy $\Delta E$ with system size $L$, $\sigma(\Delta E)\sim L^y$, is obtained as $y_3=0.25546(3)$ by reducing the equivalent of lattices up to $L=2^{100}$ in $d=3$, and as $y_4=0.76382(4)$ for up to $L=2^{35}$ in $d=4$. The reduction rules allow the exact determination of the ground state energy, entropy, and also provide an approximation to the overlap distribution. With these methods, some well-know and some new features of diluted hierarchical lattices are calculated.