We study the exact solution of the Gaudin model with Dzyaloshinsky–Moriya and Kaplan–Shekhtman–Entin–Wohlman–Aharony interactions. The energy and Bethe ansatz equations of the Gaudin model can be obtained via the off-diagonal Bethe ansatz method. Based on the off-diagonal Bethe ansatz solutions, we construct the Bethe states of the inhomogeneous XXX Heisenberg spin chain with the generic open boundaries. By taking a quasi-classical limit, we give explicit closed-form expression of the Bethe states of the Gaudin model. From the numerical simulations for the small-size system, it is shown that some Bethe roots go to infinity when the Gaudin model recovers the U(1) symmetry. Furthermore, it is found that the contribution of those Bethe roots to the Bethe states is a nonzero constant. This fact enables us to recover the Bethe states of the Gaudin model with the U(1) symmetry. These results provide a basis for the further study of the thermodynamic limit, correlation functions, and quantum dynamics of the Gaudin model.We study the exact solution of the Gaudin model with Dzyaloshinsky–Moriya and Kaplan–Shekhtman–Entin–Wohlman–Aharony interactions. The energy and Bethe ansatz equations of the Gaudin model can be obtained via the off-diagonal Bethe ansatz method. Based on the off-diagonal Bethe ansatz solutions, we construct the Bethe states of the inhomogeneous XXX Heisenberg spin chain with the generic open boundaries. By taking a quasi-classical limit, we give explicit closed-form expression of the Bethe states of the Gaudin model. From the numerical simulations for the small-size system, it is shown that some Bethe roots go to infinity when the Gaudin model recovers the U(1) symmetry. Furthermore, it is found that the contribution of those Bethe roots to the Bethe states is a nonzero constant. This fact enables us to recover the Bethe states of the Gaudin model with the U(1) symmetry. These results provide a basis for the further study of the thermodynamic limit, correlation functions, and quantum dynamics of the Gaudin model.

Chinese Physics BDec 1, 20204 citations

Algebraic Bethe ansatz for a discrete-state BCS pairing model

Jan von Delft, Rubik Poghossian

We show in detail how Richardson's exact solution of a discrete-state BCS (DBCS) model can be recovered as a special case of an algebraic Bethe-ansatz solution of the inhomogeneous XXX vertex model with twisted boundary conditions: by implementing the twist using Sklyanin's $\mathbf{K}$ -matrix construction and taking the quasiclassical limit, one obtains a complete set of conserved quantities ${\mathbf{H}}_{i}$ from which the DBCS Hamiltonian can be constructed as a second order polynomial. The eigenvalues and eigenstates of the ${\mathbf{H}}_{i}$ (which reduce to the Gaudin Hamiltonians in the limit of infinitely strong coupling) are exactly known in terms of a set of parameters determined by a set of on-shell Bethe ansatz equations, which reproduce Richardson's equations for these parameters. We thus clarify that the integrability of the DBCS model is a special case of the integrability of the twisted inhomogeneous XXX vertex model. Furthermore, by considering the twisted inhomogeneous XXZ model and/or choosing a generic polynomial of the ${\mathbf{H}}_{i}$ 's as Hamiltonian, more general exactly solvable models can be constructed. To make the paper accessible to readers that are not Bethe-ansatz experts, the introductory sections include a self-contained review of those of its feature which are needed here.We show in detail how Richardson's exact solution of a discrete-state BCS (DBCS) model can be recovered as a special case of an algebraic Bethe-ansatz solution of the inhomogeneous XXX vertex model with twisted boundary conditions: by implementing the twist using Sklyanin's $\mathbf{K}$-matrix construction and taking the quasiclassical limit, one obtains a complete set of conserved quantities ${\mathbf{H}}_{i}$ from which the DBCS Hamiltonian can be constructed as a second order polynomial. The eigenvalues and eigenstates of the ${\mathbf{H}}_{i}$ (which reduce to the Gaudin Hamiltonians in the limit of infinitely strong coupling) are exactly known in terms of a set of parameters determined by a set of on-shell Bethe ansatz equations, which reproduce Richardson's equations for these parameters. We thus clarify that the integrability of the DBCS model is a special case of the integrability of the twisted inhomogeneous XXX vertex model. Furthermore, by considering the twisted inhomogeneous XXZ model and/or choosing a generic polynomial of the ${\mathbf{H}}_{i}$'s as Hamiltonian, more general exactly solvable models can be constructed. To make the paper accessible to readers that are not Bethe-ansatz experts, the introductory sections include a self-contained review of those of its feature which are needed here.

arXiv:cond-mat/0106405Physical review. B, Condensed matterOct 4, 200262 citations

Relaxation of the general solution for Richardson-Gaudin models

Razvan Teodorescu, I. L. Aleǐner

Bulletin of the American Physical SocietyMar 23, 20050 citations

\mathbf{K}

-matrix construction and taking the quasiclassical limit, one obtains a complete set of conserved quantities

{\mathbf{H}}_{i}

from which the DBCS Hamiltonian can be constructed as a second order polynomial. The eigenvalues and eigenstates of the

{\mathbf{H}}_{i}

(which reduce to the Gaudin Hamiltonians in the limit of infinitely strong coupling) are exactly known in terms of a set of parameters determined by a set of on-shell Bethe ansatz equations, which reproduce Richardson's equations for these parameters. We thus clarify that the integrability of the DBCS model is a special case of the integrability of the twisted inhomogeneous XXX vertex model. Furthermore, by considering the twisted inhomogeneous XXZ model and/or choosing a generic polynomial of the

{\mathbf{H}}_{i}

's as Hamiltonian, more general exactly solvable models can be constructed. To make the paper accessible to readers that are not Bethe-ansatz experts, the introductory sections include a self-contained review of those of its feature which are needed here.