Thomas Bothner

The Ising model from 1920 to 2019

"Of all the systems in statistical mechanics on which exact calculations have been performed, the two-dimensional Ising model is not only the most thoroughly investigated; it is also the richest and most profound." These are the opening lines in Barry McCoy's and Tai Tsun Wu's classical 1973 monograph and since then several new features of the model have been discovered. In this (semi)-review lecture we will first familiarize ourselves with a few classical aspects of the model: the one-dimensional version, Peierls's argument for the spontaneous magnetization, Onsager's, Kaufmann's and Yang's remarkable exact computations and the analysis of spin-spin correlation functions due to Cheng and Wu. After that we shall discuss the massive scaling limit of the same correlation functions and highlight the appearance of Painlevé-III and Painlevé-VI functions as discovered in the works of Barouch, McCoy, Tracy, Wu, Jimbo and Miwa in 1976-1980. Our lecture will conclude with the speaker's recent introduction (arXiv:1710.04295, arXiv:1808.02606) of Hamiltonian action integral methods in the scaling analysis of the model. This talk is intended for a broad math and physics audience in particular including students.