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

TL;DR

This paper investigates the completeness of Bethe ansatz solutions for the 1D Hubbard model with AB-flux, deriving combinatorial formulas and exact enumeration methods that reveal how flux affects symmetry and state counting.

Contribution

It introduces new combinatorial formulas and exact enumeration techniques for Bethe states in the Hubbard model with AB-flux, extending previous results to flux-induced symmetry reductions.

Findings

Number of Bethe states increases with AB-flux.

Standard string hypothesis does not hold under flux.

Derived combinatorial formulas match known and new state counts.

Abstract

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.