  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.
