Completeness of Bethe ansatz for 1D Hubbard model with AB-flux through combinatorial formulas and exact enumeration of eigenstates

For the one-dimensional Hubbard model with Aharonov-Bohm-type magnetic flux, we study the relation between its symmetry and the number of Bethe states. First we show the existence of solutions for Lieb-Wu equations with an arbitrary number of up-spins and one down-spin, and exactly count the number of the Bethe states. The results are consistent with Takahashi's string hypothesis if the system has the so(4) symmetry. With the Aharonov-Bohm-type magnetic flux, however, the number of Bethe states increases and the standard string hypothesis does not hold. In fact, the so(4) symmetry reduces to the direct sum of charge-u(1) and spin-sl(2) symmetry through the change of AB-flux strength. Next, extending Kirillov's approach, we derive two combinatorial formulas from the relation among the characters of so(4)- or (u(1)\oplus sl(2))-modules. One formula reproduces Essler-Korepin-Schoutens' combinatorial formula for counting the number of Bethe states in the so(4)-case. From the exact analysis of the Lieb-Wu equations, we find that another formula corresponds to the spin-sl(2) case.