Ising Model on periodic and quasi-periodic chains in presence of magnetic field: some exact results

TL;DR

This paper derives exact partition functions for Ising models on periodic and Fibonacci quasiperiodic chains under magnetic fields, revealing recurrence relations and asymptotic behaviors in large systems.

Contribution

It introduces a general method for exact partition function calculation on both periodic and quasiperiodic chains, including recurrence relations for Fibonacci chains.

Findings

Exact partition functions for periodic chains with magnetic field.

Recurrence relations among Fibonacci chain partition functions.

Asymptotic relation for free energy in large Fibonacci systems.

Abstract

We present a general procedure for calculating the exact partition function of an Ising model on a periodic chain in presence of magnetic field considering both open and closed boundary conditions. Using same procedure on a quasiperiodic (Fibonacci) chain we have established a recurrence relation among partition functions of different Fibonacci generations from n-th to (n+6)-th. In the large N limit we find $(2\tau + 1){F_{n+1}}={F_{n+2}}$; where $\tau$ is the golden mean and $F_n$ stands for free energy/spin for the n-th generation. We have also studied chemical potential in both cases.