  The Ising model describes collective behaviors such as phase transitions and
critical phenomena in various physical, biological, economical, and social
systems. It is well-known that spontaneous phase transition at finite
temperature does not exist in the Ising model with short-range interactions in
one dimension. Yet, little is known about whether this forbidden phase
transition can be approached arbitrarily closely -- at fixed finite
temperature. To describe such asymptoticity, here I introduce the notion of
marginal phase transition (MPT) and use symmetry analysis of the transfer
matrix to reveal the existence of spontaneous MPT at fixed finite temperature
$T_0$ in one class of one-dimensional Ising models on decorated two-leg
ladders, in which $T_0$ is determined solely by on-rung interactions and
decorations, while the crossover width $2\delta T$ is independently,
exponentially reduced ($\delta T = 0$ means a genuine phase transition) by
on-leg interactions and decorations. These findings establish a simple ideal
paradigm for realizing an infinite number of one-dimensional Ising systems with
spontaneous MPT, which would be characterized in routine lab measurements as a
genuine first-order phase transition with large latent heat thanks to the
ultra-narrow $\delta T$ (say less than one nano-kelvin), paving a way to push
the limit in our understanding of phase transitions and the dynamical actions
of frustration arbitrarily close to the forbidden regime.
