Norm of Bethe Wave Function as a Determinant

TL;DR

This paper discusses the determinant representation of the norm of Bethe wave functions in integrable models, highlighting historical developments and its significance in algebraic combinatorics.

Contribution

It reviews the historical proof that the norm squared equals a determinant of the linearized Bethe equations, emphasizing its importance and applications.

Findings

Norm of Bethe wave function equals a determinant of the linearized system

Historical proof published in 1982 in Communications in Mathematical Physics

Applications in algebraic combinatorics and vertex models

Abstract

This is a historical note. Bethe Ansatz solvable models are considered, for example XXZ Heisenberg anti-ferromagnet and Bose gas with delta interaction. Periodic boundary conditions lead to Bethe equation. The square of the norm of Bethe wave function is equal to a determinant of linearized system of Bethe equations (determinant of matrix of second derivatives of Yang action). The proof was first published in Communications in Mathematical Physics, vol 86, page 391 in l982. Also domain wall boundary conditions for 6 vertex model were discovered in the same paper [see Appendix D]. These play an important role for algebraic combinatorics: alternating sign matrices, domino tiling and plane partition. Many publications are devoted to six vertex model with domain wall boundary conditions.