In the 1930s Rudolf Peierls argued that the one-dimensional electrons interacting with phonons undergo an instability, leading to the formation of a periodic structure known as an electronic crystal. Peierls's instability stands in a short list of major phenomena of condensed matter physics.
From a mathematical perspective, a comprehensive solution to the Peierls problem was given in papers by Igor Krichever and co-authored by Natasha Kirova, Sergei Brazovski, and Igor Dzyaloshinsky In the early 80’s. It was found that electronic crystals are periodic solutions of soliton equations, falling within the framework of Krichever-Novikov's theory of finite-gap potentials.
The Peierls phenomenon also emerges as a limiting case of models of interacting fermions, such as Gross-Neveu models with a large rank symmetry group when the rank of the group tends to infinity. These models are solvable by the Bethe Ansatz for finite rank groups. The talk presents the result of a recent paper co-authored by Konstantin Zarembo, Valdemar Melin, and Yoko Sekiguchi, where Krichever’s finite-gaps solutions of soliton equations were obtained as a singular large rank limit of the Bethe Ansatz solution of models with Lie group symmetry.