Approaching the forbidden phase transition in the Ising model, III: Hidden frustration, continuous marginal phase transition, and the minimal model

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.