Insights gained from solvable models into a variety of phase transitions, including emergent assemblies plus isoelectronic series of atomic ions

Three solvable models are set out in some detail in reviewing different types of phase transitions. Two of these relate directly to emergent critical phenomena, viz. melting and magnetic transitions in heavy rare-earth metals, and secondly, via the $3d$ Ising model, to critical behaviour in an insulating ferromagnet such as CrBr$_3$. The final `transition', however, concerns ionization of an electron in an isoelectronic series with $N$ electrons as the atomic number $Z$ is reduced below that of the neutral atom. These solvable models are, throughout, brought into contact either with experiment, or with very precise numerical modelling on real materials.

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