Thermodynamic geometry of Ising ferromagnet in an external magnetic field: A self-consistent field theory calculation

Abstract

We present a complete geometrical description for the ferromagnetic Ising model in the pair approximation as introduced by Balcerzak (2003) using self-consistent field theory. A metric is defined in a two-dimensional phase space of magnetization (M) and nearest-neighbour correlation function (C). Based on the metric elements an expression for the thermodynamic Ricci scalar (R) is derived in terms of the lattice coordination number q. We study R as the temperature (T), magnetic field (h) and exchange energy coupling (J) are varied and show that there are T and h dependent critical properties for q=6. By direct comparison, we demonstrate that the special case q=2 provides a consistent behaviour with the already known exact formula in Janyszek and Mrugała work (1989) for the one-dimensional Ising model.