The spherical hierarchical model

Abstract

The spherical version of Dyson's hierarchical model is analyzed. A particular case which is designed to simulate the long-range Ising problem is dealt with in detail. A phase transition is found with critical temperature $$\beta _c = \tfrac{1}{2}(2^\alpha - 2)(4 - 2^\alpha )^{ - 1} $$ wherenth neighbor spins interact with a strength ofn−α. Critical exponents are calculated for this particular case and are found to be identical with the critical exponents of the long-range spherical Ising model.