Existence of Solutions to the Bethe Ansatz Equations for the 1D Hubbard Model: Finite Lattice and Thermodynamic Limit

TL;DR

This paper proves the existence of solutions to the Bethe Ansatz equations for the 1D Hubbard model on finite lattices and in the thermodynamic limit, confirming the validity of the known solutions and their properties.

Contribution

It provides a rigorous proof of the existence and continuity of solutions to the Bethe Ansatz equations for the 1D Hubbard model, including in the thermodynamic limit.

Findings

Existence of real, ordered solutions for finite lattices.

Continuity of solutions from small to large interaction U.

Convergence of solutions to known distribution in the thermodynamic limit.

Abstract

In this work, we present a proof of the existence of real and ordered solutions to the generalized Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any positive U to the limit of large interaction is also shown. This continuity property, when combined with the proof that the wavefunction obtained with the generalized Bethe Ansatz is normalizable, is relevant to the question of whether or not the solution gives us the ground state of the finite system, as suggested by Lieb and Wu. Lastly, for the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the well known solution of the 1D Hubbard model.