  The laws that govern the nature often set certain limits or bans. Approaching
those limits is a common paradigm of breakthroughs in human knowledge and
technological application, such as time slowing to a stop as one approaches the
speed of light, electrons flowing freely with no resistance as one approaches
the absolute zero, and an event horizon arising as one approaches a black hole.
A well-known ban concerning the statistical mechanics of cooperative phenomena,
such as ice melting into water, is the nonexistence of phase transition at
finite temperature in the Ising model with short-range interactions in one
dimension. Yet, little is known about whether this forbidden transition could
be approached arbitrarily closely at fixed finite temperature. To explore such
asymptoticity, the notion of marginal phase transition (MPT) was introduced
recently, and both the spontaneous MPT and the field-induced MPT were
discovered respectively in ladder and single-chain Ising models decorated with
geometric frustration. Here I push the limit down to the minimal single-chain
Ising model -- without geometric frustration -- in a magnetic field; then, a
hidden frustration is revealed to drive the first-order MPT with large latent
heat. Furthermore, as one approaches the zero field, MPT appears continuous
with no latent heat, displays abrupt change in sublattice magnetization and a
kink-like minimum in overall magnetization with temperature, and vanishes when
the field drops below a threshold. These findings deepen our understanding of
cooperative phenomena via the doubly forbidden and open new avenues to the
design and engineering of the Ising systems with the different types of MPT for
exploring exotic phenomena and low-dimensional device applications.
