The Ising model, with short-range interactions between constituents, is a basic mathematical model in statistical mechanics. It has been widely used to describe collective phenomena such as order-disorder phase transitions in various physical, biological, economical, and social systems. However, it was proven that spontaneous phase transitions do not exist in the one-dimensional Ising models. Besides low dimensionality, frustration is the other well-known suppressor of phase transitions. Here I show that surprisingly, a strongly frustrated one-dimensional two-leg ladder Ising model can exhibit a marginal finite-temperature phase transition. It features a large latent heat, a sharp peak in specific heat, and unconventional order parameters, which classify the transition as involving an entropy-favored intermediate-temperature ordered state and further unveil a crossover to an exotic normal state in which frustration effectively decouples the two strongly interacted legs in a counterintuitive non-mean-field way. These exact results expose a mathematical structure that has not appeared before in phase-transition problems, and shed new light on our understanding of phase transitions and the dynamical actions of frustration. Applications of this model and its mechanisms to various systems with extensions to consider higher dimensions, quantum characters, or external fields, etc. are anticipated and briefly discussed---with insights into the puzzling phenomena of strange strong frustration and intermediate-temperature orders such as the Bozin-Billinge orbital-degeneracy-lifting recently discovered in real materials.
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