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Chinese Physics

Trigonometric SU(N) Gaudin model

Junpeng Cao, Bo-Yu Hou, Yue Rui-Hong

Published Sep 25, 2001• 1 citation
Publisher

Abstract

In this paper, we obtain the eigenstates and the eigenvalues of the Hamiltonians of the trigonometric SU(N) Gaudin model based on the quasi-classical limit of the trigonometric SU(N) chain with the periodic boundary condition. By using the quantum inverse scattering method, we also obtain the eigenvalues of the generating function of the trigonometric SU(N) Gaudin model.

DOI
10.1088/1009-1963/10/10/308

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In this paper, we obtain the eigenstates and the eigenvalues of the Hamiltonians of the trigonometric SU(N) Gaudin model based on the quasi-classical limit of the trigonometric SU(N) chain with the periodic boundary condition. By using the quantum inverse scattering method, we also obtain the eigenvalues of the generating function of the trigonometric SU(N) Gaudin model.
In this paper we deal with the trigonometric Gaudin model, generalized using a nontrivial triangular reflection matrix (corresponding to non-periodic boundary conditions in the case of anisotropic XXZ Heisenberg spin-chain). In order to obtain the generating function of the Gaudin Hamiltonians with boundary terms we follow an approach based on Sklyanin’s derivation in the periodic case. Once we have the generating function, we obtain the corresponding Gaudin Hamiltonians with boundary terms by taking its residues at the poles. As the main result, we find the generic form of the Bethe vectors such that the off-shell action of the generating function becomes exceedingly compact and simple. In this way—by obtaining Bethe equations and the spectrum of the generating function—we fully implement the algebraic Bethe ansatz for the generalized trigonometric Gaudin model.
Richardson–Gaudin models are a class of quantum integrable models connected to many physical systems, including pairing Hamiltonians from the theory of superconductivity. They can be obtained in the quasi-classical limit of the Quantum Inverse Scattering Method, which is based on an R-matrix and a Lax operator satisfying the Yang-Baxter equation. They can also be obtained from the Boundary Quantum Inverse Scattering Method, which relies on solutions of the reflection equations known as K-matrices. In this thesis we study these latter models systematically, explore the connections betweenthem and investigate the interpretation of the “boundary”.First of all, we consider Richardson–Gaudin models obtained from the spin-1/2 su(2) Boundary Quantum Inverse Scattering Method with diagonal K-matrices. We prove that the trigonometric boundary construction is equivalent to its rational limit, through a change of variables, rescaling, and a basis transformation. Moreover, we prove that the twisted-periodic and boundary constructions are equivalent in the trigonometric case, but not in the rational limit. Thus, including the “boundary” does not lead to a new model in this case.Next, we investigate Richardson–Gaudin models obtained from the spin-1/2 su(2) Boundary Quantum Inverse Scattering Method with non-diagonal K-matrices. Here the situation is different. The conserved operators in the boundary construction are no longer equivalent to the ones in the twisted-periodic construction. Also, the rational and the trigonometric boundary constructions are not equivalent. In the rational case this allows us to construct a generalisation of the p+ip pairing Hamiltonian with external interaction terms. In the trigonometric case the expressions for the conserved operators involve several free parameters, which can be adjusted to construct a variety of Hamiltonians. This result offers opportunities for future investigations.Finally, we study the case of the q-deformed bosonic Lax operator. This case is much more challenging than the case of the spin Lax operator. It is not straightforward to define the rational and quasi-classical limits of the bosonic Lax operator. Even after making modifications to the Lax operator for these limits to be well defined, it turns out that the limits do not commute. We state some open questions for future work.
Following Sklyanin’s proposal in the rational case, we derive the generating function of the Gaudin Hamiltonians in the trigonometric case. Our derivation is based on the quasi-classical expansion of the linear combination of the transfer matrix of the inhomogeneous XXZ Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians are obtained as the residues of the generating function. PACS: 02.20.Tw, 03.65.Fd, 05.50+q, 75.10.Jm; MSC 2010: 81R12, 82B23.