Abstract
This chapter further supports the case for the Ising spin as the Drosophila of statistical mechanics, that is the system that can be used to model virtually every interesting thermodynamic phenomenon and to test every theoretical method. Exact solutions are extremely rare in many-body physics. Here, we consider three of them: (1) the one-dimensional Ising model, (2) the one-dimensional Ising model in a transverse field, the simplest quantum spin system, and (3) the two-dimensional Ising model in zero field. The results are paradigms for a host of more complex systems and situations that cannot be solved exactly but which can be understood qualitatively and even quantitatively on the basis of simulations and asymptotic methods such as series expansions and the renormalization group.
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