Decimations for One- and Two-dimensional Ising and Rotator Models II: Continuous versus Discrete Symmetries

TL;DR

This paper investigates how decimated Gibbs measures with continuous symmetries can become non-Gibbsian due to spin-flop transitions, despite their discrete counterparts exhibiting phase transitions.

Contribution

It reveals the mechanism by which continuous symmetry measures can lose Gibbsian properties through discrete symmetry-breaking transitions under decimation.

Findings

Decimated measures with continuous symmetry can be non-Gibbsian.

Spin-flop transitions cause loss of Gibbsianness in certain configurations.

Discrete symmetry-breaking occurs despite continuous symmetry preservation.

Abstract

We show how decimated Gibbs measures which have an unbroken continuous symmetry due to the Mermin-Wagner theorem, although their discrete equivalents have a phase transition, still can become non-Gibbsian. The mechanism rests on the occurrence of a spin-flop transition with a broken discrete symmetry, once the model is constrained by the decimated spins in a suitably chosen "bad" configuration.