Privman-Fisher hypothesis on finite systems: Verification in the case of the spherical model of ferromagnetism

Abstract

The Privman-Fisher hypothesis on the singular part of the free-energy density of a finite system, near the bulk critical point T=$T_c$, is examined in the context of the spherical model of ferromagnetism. A d-dimensional hypercubical lattice (of size $N_1$a×$N_2$a×...×$N_d$a) is considered and, subject to periodic boundary conditions, explicit expressions are derived for the free energy, the specific heat, and the magnetic susceptibility of the system at temperatures close to $T_c$. The relevant scaling functions governing the critical behavior of the system are obtained and, with the use of the asymptotic properties of these functions, various predictions of the Privman-Fisher hypothesis are verified. By implication, the passage of the given system towards standard bulk behavior, as $N_j$→∞, is also elucidated.