Lattice models on Cayley tree with competing interactions

The Ising models with few competitive interactions play a significant role in studying the behavior of spin glasses. In this paper we review some generally known results about models on Cayley tree and present new ones. It is known that a phase diagram of models with competitive interbranch neighbors on Cayley tree of second order can contain ferromagnetic, paramagnetic, antiferromagnetic phases and paramodulated phase with period 2 only. We investigate the Uncle-Nephew Ising model on Cayley tree of third order with competitive interbranch neighbors and show that it has very rich phase diagram with many modulated phases, consisting of periodic (commensurate) and aperiodic (incommensurate) domains