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Physica Scripta

On the Bethe Ansatz approach to the quantum inverse problem for non-ultralocal integrable systems

Asesh Roy Chowdhury, Shibani Sen

Published Mar 1, 1991• 0 citations
Publisher

Abstract

We have reformulated the Bethe Ansatz approach to a Derivative Nonlinear Schrödinger equation, known to be a non-ultralocal system. In this present version we have shown that an approach exactly similar to the usual NLS case is possible. Furthermore the Yang-Baxter equation and factorization is shown to be valid. Afterwards the R (μ, v ) matrix has been constructed from the Bethe Ansatz result (which is impossible otherwise, as the Faddeev approach is not applicable) and the commutation rules for the scattering data is deduced. These are then utilised to construct the algebraic Bethe ansatz eigen states. Next the periodicity condition is invoked and the eigenvalues are determined via some algebraic equations which are also seen to be reproduced via the algebraic approach. Lastly we consider the possibility of exceptional states in the algebraic Bethe states.

DOI
10.1088/0031-8949/43/3/001

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We have reformulated the Bethe Ansatz approach to a Derivative Nonlinear Schrödinger equation, known to be a non-ultralocal system. In this present version we have shown that an approach exactly similar to the usual NLS case is possible. Furthermore the Yang-Baxter equation and factorization is shown to be valid. Afterwards the R (μ, v ) matrix has been constructed from the Bethe Ansatz result (which is impossible otherwise, as the Faddeev approach is not applicable) and the commutation rules for the scattering data is deduced. These are then utilised to construct the algebraic Bethe ansatz eigen states. Next the periodicity condition is invoked and the eigenvalues are determined via some algebraic equations which are also seen to be reproduced via the algebraic approach. Lastly we consider the possibility of exceptional states in the algebraic Bethe states.
We have formulated the quantum inverse problem for an alternative version of the nonlinear Schr\"odinger equation (NLS), which is a variant of the derivative NLS (DNLS) equation. While the DNLS equation is nonultralocal and cannot be formulated as a problem of the quantum inverse scattering method, our equation is ultralocal, and hence it is possible to construct the quantum R matrix and the commutation rules for the scattering data. Next we have constructed the algebraic Bethe ansatz, from which the eigenvalue equation for the excited states is derived. This eigenvalue equation admits stringlike configurations, and the density of states corresponding to real and complex eigenvalues is explicitly derived in the thermodynamic limit.
The authors continue their analysis of the extended derivative nonlinear Schrodinger equation with the help of the Bethe-ansatz technique. In the previous analysis by the QISM approach it was not possible to consider the reduction of the extended system to the derivative NLSE, as the theory then becomes non-ultralocal. Therefore, they have applied the approach of the Bethe ansatz to both the extended DNLSE and the usual DNLSE, and show that it is possible to construct multiparticle quantum states in both the cases.
The generalisation of the quantum inverse scattering method is used to derive the Bethe ansatz equation of a system for a multicomponent nonlinear Schrodinger models with n species (or colours) of bosons or fermions. The matrix periodic boundary conditions for bosons and fermions are obtained and the relationship between them discussed. The eigenstates for the infinite conserved quantities of the system are constructed. Lastly, the author generalises these results to the non-linear Schrodinger model with the mirror-image delta potentials proposed by Schulz (1987).
A lattice regularized Lax operator for the non-ultralocal modified Korteweg de Vries (mKdV) equation is proposed at the quantum level with the basic operators satisfying a q-deformed braided algebra. From the associated quantum R and Z-matrices the exact integrability of the model is proved through the braided quantum Yang-Baxter equation which is a suitably generalized equation for the non-ultralocal models. Using the algebraic Bethe ansatz the eigenvalue problem of the quantum mKdV model is exactly solved and its connection with the spin-1/2 XXZ chain is established, facilitating the investigation of the corresponding conformal properties.