  One-dimensional systems---ranging from travelling light to circuit cables and
from DNA to superstrings---are ubiquitous and critically important to the human
knowledge of the universe. However, our engagement with one-dimensional systems
in the research and education of spontaneous phase transitions, the phenomena
wherein materials can change rapidly between different phases (e.g., gas,
liquid, solid, etc.) on their own, has not been largely exercised, since it was
proven that one-dimensional systems do not contain phase transitions in the
textbook Ising model almost 100 years ago [1] and its quantum counterpart, the
Heisenberg model, over 50 years ago [2]. Recently, a spontaneous marginal phase
transition (MPT) was discovered in a one-dimensional Ising model containing
strong geometrical frustration [3]. Here, by exploring the symmetry of the new
mathematical structure underlying the MPT, I report the finding and
classification of an infinite number of MPT cases---with highly tunable
intriguing behaviors like phase reentrance, the dome shape of transition
temperature, pairing, and gauge freedom. These discoveries reveal the
possibility of building the MPT-based one-dimensional Ising Machine that can be
used to simulate the complex phenomena of phase competition in strongly
correlated systems and provide insights with its unambiguous exact solutions.
They also form a rich playground for exploring unconventional phase transitions
in one-dimensional Heisenberg models.
