Two-cycles in spin systems: multi-state Ising-type ferromagnets

TL;DR

This paper derives Hamiltonians for multi-state spin-glass systems with Ising symmetry, analyzes the effects of different updating schemes, and investigates the emergence of two-cycles, supported by analytical solutions and numerical simulations.

Contribution

It introduces exact Hamiltonians for multi-state spin systems and compares their behavior under sequential and synchronous updates, revealing new physical phenomena.

Findings

Two-cycles occur under synchronous updating in certain models.

Differences between Q-Ising and Blume-Emery-Griffiths ferromagnets are identified.

Numerical simulations confirm analytical results.

Abstract

Hamiltonians for general multi-state spin-glass systems with Ising symmetry are derived for both sequential and synchronous updating of the spins. The possibly different behaviour caused by the way of updating is studied in detail for the (anti)-ferromagnetic version of the models, which can be solved analytically without any approximation, both thermodynamically via a free-energy calculation and dynamically using the generating functional approach. Phase diagrams are discussed and the appearance of two-cycles in the case of synchronous updating is examined. A comparative study is made for the Q-Ising and the Blume-Emery-Griffiths ferromagnets and some interesting physical differences are found. Numerical simulations confirm the results obtained.