Exact solution of a generalized ANNNI model on a Cayley tree

TL;DR

This paper analytically solves the recurrence equations of a generalized ANNNI model on a Cayley tree, providing exact critical temperatures, phase counts, and the partition function, advancing understanding beyond previous numerical studies.

Contribution

It offers the first exact analytical solutions for the recurrence equations of this generalized ANNNI model on a Cayley tree, including critical temperatures and phase enumeration.

Findings

Derived exact critical temperatures and phase boundaries.

Determined the number of phases present in the model.

Calculated the partition function analytically.

Abstract

We consider the Ising model on a Cayley tree of order two with nearest neighbor interactions and competing next nearest neighbor interactions restricted to spins belonging to the same branch of the tree. This model was studied by Vannimenus and found a new modulated phase, in addition to the paramagnetic, ferromagnetic, antiferromagnetic phases and a (+ + - -) periodic phase. Vannimenus's results based on the recurrence equations (relating the partition function of an $n-$ generation tree to the partition function of its subsystems containing $(n-1)$ generations) and most results are obtained numerically. In this paper we analytically study the recurrence equations and obtain some exact results: critical temperatures and curves, number of several phases, partition function.