The Heisenberg Model on the Cayley Tree

Abstract

The ferromagnetic Heisenberg model on the finite most symmetrical Cayley tree is studied in the limit of large size. The spectrum and wavefunctions of the single-magnon states are obtained rigorously. They have many unusual properties. The magnon .density excited as a whole at infinitesimal temperatures is logarithmically divergent when the mag­ non is approximated as boson. Nevertheless, the possibility of some spontaneous magnetiza­ tion at the central part of the tree remains. § 1. Introduction Recently, Matsuda!) and Eggarte~' have independently pointed out that the self-consistent Bethe-Peierls treatment 3' of the ferroma'gnetic Ising model on the Cayley tree, which had been named the Bethe-lattice by Kurata, Kikuchi and Watari 4' and had been believed to be exact, 5~ does not correspond to the correct thermodynamic limit. This situation is owing to the existence of a huge number of surface atoms on a Cayley tree. The Cayley tree is one-dimensional because it contains no closed loops and· at the same time rather infinite-dimensional because the ratio of the number of the surface atoms to .the total does not tend to zero but remains finite in the thermodynamic limit. Thus there coexist low- and high­ dimensional characters in this pseudolattice and it is interesting to see which charac­ ter reflects on what physical properties. In this paper, the ferromagnetic Heisenberg model on the finite most sym­ metrical Cayley tree is studied in the limit of large size. The high symmetry of this model enables us to obtain the spectrum and the wavefunctions of the single­ magnon states rigorously. This spectrum is unusual as partly pointed out by Rubin and Zwanzig6' for the vibratiom~l frequency spectrum of the rooted Cayley tree. Even in the limit of infinitely large system size, there remain the finite portion of modes whose wavefunctions are not zero only on the finite number of atoms. It is shown that the total number of magnons excited per atom at an infinitesimal temperature is logarithmically divergent just like in the two-dimensional case and then the ferromagnetic .ground state is unstable as a result of such excitations, if magnon is approximated as boson. Nevertheless, in the central part of the system,