SPIN ONE ISING MODEL WITH COMPETING INTERACTIONS ON THE BETHE LATTICE

Abstract

The phase diagram of the spin one Ising model with competing interactions between first and second nearest neighbors generations, single ion anisotropy and quadrupolar interactions is studied. Modulated phases, multiphase point and reentrances are observed and discussed. For high competing single ion anisotropy the paramagnetic phase reaches a finite region of the ground state. We study the phase diagram of the spin one Ising model on the Bethe lattice of general coordination number z = q+ l (q is the connectivity) with competing first and second nearest neighbors interactions, quadrupolar interaction, single-ion anisotropy and under an external magnetic field. A mean field analysis of a similar model has been considered by [I]. They studied the phase diagram for a particular plane in the parameter space finding several modulated phases with reentrant behavior and the existence of a Lifshitz point. The present work provides the analysis of the phase diagram in the entire parameters space. In our approach [2] the partition function, the local magnetization, the non-critical density and the pair correlation function, involving sites deep in the interior of the lattice, are obtained exactly as a function of the fixed point solutions (attractors) of a set of coupled recursion relations of appropriated effective fields. The reduced model Hamiltonian describing the system is where p is the inverse of temperature, n labels the generation shells of the Bethe lattice and Si = &I, 0 are the spin variables. The first and third sums are over all pairs of nearest neighbors spins while the second sum is restricted to the pairs of second nearest neighbors spins belonging to the same branch. K1, K2, K, D are B are the coupling constant of the first, second nearest neighbors interactions, quadrupolar interaction, single-ion anisotropy and external magnetic field respectively. Following [2] we define appropriate effective fields obeying the following discrete mapping relating three consecutive generations an+l = B D + K ~ + K + F ~ ( ~ , , ~ , ) + G ~ ~ bn+i = B D K I + K + F I ( c ~ , ~ ~ ) + G , ~ cn+i = B D K I + K + F I ( & , ~ ~ ) + G ~ ~ dn+l = -B D + KI + K + F-I (cn, dn) + Gi-1 fn+l = B D + F o ( a n , b n ) + ~ L l gn+l = -B D + FO (cn, dn) + G L 1