On the failure of effective-field theory in predicting a spurious spontaneous ordering and phase transition of Ising nanoparticles, nanoislands, nanotubes and nanowires

Abstract

The present work clarifies a failure of the effective-field theory in predicting a false spontaneous long-range order and phase transition of Ising nanoparticles, nanoislands, nanotubes and nanowires with either zero- or one-dimensional magnetic dimensionality. It is conjectured that the standard formulation of the effective-field theory due to Honmura and Kaneyoshi generally predicts for the Ising spin systems a spurious spontaneous long-range order with nonzero critical temperature regardless of their magnetic dimensionality whenever at least one Ising spin has coordination number greater than two. The failure of the effective-field theory is exemplified on a few paradigmatic exactly solved examples of zero- and one-dimensional Ising nanosystems: star, cube, decorated hexagon, star of David, branched chain, sawtooth chain, two-leg and hexagonal ladders. The presented exact solutions illustrate eligibility of a few rigorous analytical methods for exact treatment of the Ising nanosystems: exact enumeration, graph-theoretical approach, transfer-matrix method and decoration-iteration transformation. The paper also provides a substantial survey of the scientific literature, in which the effective-field theory led to a false prediction of the spontaneous long-range ordering and phase transition.