The exactly solvable BCS Hamiltonian of superconductivity is considered from several viewpoints: Richardson's ansatz, conformal field theory, integrable inhomogenous vertex models and Chern-Simons theory.

arXiv:hep-th/0111114Nov 13, 20013 citations

Classification of Exactly Solvable Two-Component Models

Kiyoshi Sogo, M. Uchinami, Yuko Akutsu +1 more

Progress of Theoretical PhysicsAug 1, 198277 citations

arXiv:math-ph/0211003 · Nuclear Physics B

Exactly solvable discrete BCS-type Hamiltonians and the six-vertex model

А. А. Овчинников

Published Apr 23, 2003• 7 citations

arXiv HTML PDF

DOI: 10.1016/s0550-3213(03)00218-9

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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.

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In the case of two-component models, all possible solutions for factorization equations of S-matrix are obtained. These solutions give a classification of solvable vertex models. It is found that Yang-Baxter relation is more general than the factorization equation. The six vertex model by Yang, Yang and Sutherland is derived not from the factorization equation but from Yang-Baxter relation. Thus it is concluded that Yang-Baxter relation lies at the basis of all the known solvable two component models. Furthermore, spin Hamiltonians corresponding to our vertex models are presented in explicit forms.In the case of two-component models, all possible solutions for factorization equations of S-matrix are obtained. These solutions give a classification of solvable vertex models. It is found that Yang-Baxter relation is more general than the factorization equation. The six vertex model by Yang, Yang and Sutherland is derived not from the factorization equation but from Yang-Baxter relation. Thus it is concluded that Yang-Baxter relation lies at the basis of all the known solvable two component models. Furthermore, spin Hamiltonians corresponding to our vertex models are presented in explicit forms.

Progress of Theoretical PhysicsAug 1, 198277 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.