Majority Rule at Low Temperatures on the Square and Triangular Lattices

TL;DR

This paper investigates the behavior of majority rule renormalization group transformations on square and triangular lattices at low temperatures, revealing non-existence in some cases and locality in others.

Contribution

It proves the non-existence of the transformation for square lattices at low temperatures and establishes the locality of the transformation at zero temperature for triangular lattices.

Findings

Transformation not defined for square lattice at low temperatures.

Zero temperature transformation on triangular lattice is local.

Renormalized Hamiltonian on triangular lattice has 14 interaction types.

Abstract

We consider the majority rule renormalization group transformation applied to nearest neighbor Ising models. For the square lattice with 2 by 2 blocks we prove that if the temperature is sufficiently low, then the transformation is not defined. We use the methods of van Enter, Fernandez and Sokal, who proved the renormalized measure is not Gibbsian for 7 by 7 blocks if the temperature is too low. For the triangular lattice we prove that a zero temperature majority rule transformation may be defined. The resulting renormalized Hamiltonian is local with 14 different types of interactions.