The Hubbard chain: Lieb-Wu equations and norm of the eigenfunctions

TL;DR

This paper establishes a determinant formula for the norm of Hubbard model eigenfunctions, linking it to the linearization of Lieb-Wu equations and their Hessian, revealing non-degeneracy conditions.

Contribution

It introduces a novel determinant expression for the eigenfunction norm based on the linearized Lieb-Wu equations and an associated action functional.

Findings

Norm squared proportional to determinant of a linearized matrix

Non-degeneracy of Lieb-Wu solutions when wave function is non-zero

Expression of norm in terms of Hessian determinant of an action

Abstract

We argue that the square of the norm of the Hubbard wave function is proportional to the determinant of a matrix, which is obtained by linearization of the Lieb-Wu equations around a solution. This means that in the vicinity of a solution the Lieb-Wu equations are non-degenerate, if the corresponding wave function is non-zero. We further derive an action that generates the Lieb-Wu equations and express our determinant formula for the square of the norm in terms of the Hessian determinant of this action.