The Bethe ansatz in the ХХХ model of Heisenberg for the 4-inverted spins

Abstract

The Bethe ansatz approach will be applied in quantum integrable systems, namely to the Heisenberg XXX spin
chain model, in dimension d = 3 , with four inverted spins. The application of this approach will be implemented taking into account the boundary conditions and the periodicity condition. Parametrizing the system of Bethe equations, which will provide an algebraic form of writing, and get the desired solution to the problem. As a consequence of a number of trivial calculations and the introduction of additional restrictive conditions, solutions of the system of Bethe XXX-chain equations in the form of corresponding spectral parameters were presented.Using the spectral parameter, we expressed the full momentum of this state
and its energy. Comparing the energy at the bound and unbound States, it is shown that at the unbound state the system has the greatest energy. The energy values for a finite number of spins in the spin chain are also obtained.