An exact solution on the ferromagnetic Face-Cubic spin model on a Bethe lattice

TL;DR

This paper provides an exact analytical solution for the ferromagnetic Face-Cubic spin model on a Bethe lattice, revealing detailed phase diagrams and critical points for various component numbers.

Contribution

It introduces an exact recursive relation approach to analyze the Face-Cubic spin model on Bethe lattices, including phase diagrams and critical point characterization.

Findings

Phase diagrams with multiple tricritical points for Q ≤ 2.

Identification of a triple point and a tricritical point for Q > 2.

Exact calculation of thermodynamic properties using recursive relations.

Abstract

The lattice spin model with $Q$--component discrete spin variables restricted to have orientations orthogonal to the faces of $Q$-dimensional hypercube is considered on the Bethe lattice, the recursive graph which contains no cycles. The partition function of the model with dipole--dipole and quadrupole--quadrupole interaction for arbitrary planar graph is presented in terms of double graph expansions. The latter is calculated exactly in case of trees. The system of two recurrent relations which allows to calculate all thermodynamic characteristics of the model is obtained. The correspondence between thermodynamic phases and different types of fixed points of the RR is established. Using the technique of simple iterations the plots of the zero field magnetization and quadrupolar moment are obtained. Analyzing the regions of stability of different types of fixed points of the system of recurrent relations the phase diagrams of the model are plotted. For $Q \leq 2$ the phase diagram of the model is found to have three tricritical points, whereas for $Q> 2$ there are one triple and one tricritical points.