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

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 Takahashiʹs 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 Kirillovʹs approach [J. Sov. Math. 30 (1985) 2298, J. Sov. Math. 36 (1987) 115], we derive two combinatorial formulas from the relation among the characters of so(4)- or (u(1)⊕sl(2))-modules. One formula reproduces Essler–Korepin–Schoutensʹ 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.