Bethe-ansatz studies of energy level crossings in the one-dimensional Hubbard model

TL;DR

This paper uses the Bethe ansatz to analyze energy level crossings in the one-dimensional Hubbard model, explicitly constructing degenerate eigenstates and revealing the role of dynamical symmetries and conserved operators in spectral degeneracies.

Contribution

It provides explicit construction of degenerate eigenstates at level crossings and highlights the importance of dynamical symmetries and conserved quantities in the Hubbard model.

Findings

Confirmed all level crossings via Bethe ansatz solutions.

Identified that some crossings are due to dynamical symmetries, not U-independent symmetries.

Discovered a twofold permanent degeneracy in Bethe ansatz wavefunctions.

Abstract

Motivated by Heilmann and Lieb's work, we discuss energy level crossings for the one-dimensional Hubbard model through the Bethe ansatz, constructing explicitly the degenerate eigenstates at the crossing points. After showing the existence of solutions for the Lieb--Wu equations with one-down spin, we solve them numerically and construct Bethe ansatz eigenstates. We thus verify all the level crossings in the spectral flows observed by the numerical diagonalization method with one down-spin. For each of the solutions we obtain its energy spectral flow along the interaction parameter U. Then, we observe that some of the energy level crossings can not be explained in terms of U-independent symmetries. Dynamical symmetries of the Hubbard model are fundamental for identifying each of the spectral lines at the level crossing points. We show that the Bethe ansatz eigenstates which degenerate at the points have distinct sets of eigenvalues of the higher conserved operators. We also show a twofold permanent degeneracy in terms of the Bethe ansatz wavefunction.