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arXiv:1405.7451 · Nuclear Physics B

On the boundaries of quantum integrability for the spin-1/2 Richardson–Gaudin system

Inna Lukyanenko, Phillip S. Isaac, Jon Links

Published Jun 25, 2014• 28 citations
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Abstract

We discuss a generalised version of Sklyanin's Boundary Quantum Inverse Scattering Method applied to the spin-1/2, trigonometric sl ( 2 ) case, for which both the twisted-periodic and boundary constructions are obtained as limiting cases. We then investigate the quasi-classical limit of this approach leading to a set of mutually commuting conserved operators which we refer to as the trigonometric, spin-1/2 Richardson–Gaudin system. We prove that the rational limit of the set of conserved operators for the trigonometric system is equivalent, through a change of variables, rescaling, and a basis transformation, to the original set of trigonometric conserved operators. Moreover, we prove that the twisted-periodic and boundary constructions are equivalent in the trigonometric case, but not in the rational limit.

DOI
10.1016/j.nuclphysb.2014.06.018

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We discuss a generalised version of Sklyanin's Boundary Quantum Inverse Scattering Method applied to the spin-1/2, trigonometric sl ( 2 ) case, for which both the twisted-periodic and boundary constructions are obtained as limiting cases. We then investigate the quasi-classical limit of this approach leading to a set of mutually commuting conserved operators which we refer to as the trigonometric, spin-1/2 Richardson–Gaudin system. We prove that the rational limit of the set of conserved operators for the trigonometric system is equivalent, through a change of variables, rescaling, and a basis transformation, to the original set of trigonometric conserved operators. Moreover, we prove that the twisted-periodic and boundary constructions are equivalent in the trigonometric case, but not in the rational limit.
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.
We extend the quantum–classical duality to the trigonometric (hyperbolic) case. The duality establishes an explicit relationship between the classical N-body trigonometric Ruijsenaars–Schneider model and the inhomogeneous twisted XXZ spin chain on N sites. Similarly to the rational version, the spin chain data fixes a certain Lagrangian submanifold in the phase space of the classical integrable system. The inhomogeneity parameters are equal to the coordinates of particles while the velocities of classical particles are proportional to the eigenvalues of the spin chain Hamiltonians (residues of the properly normalized transfer matrix). In the rational version of the duality, the action variables of the Ruijsenaars–Schneider model are equal to the twist parameters with some multiplicities defined by quantum (occupation) numbers. In contrast to the rational version, in the trigonometric case there is a splitting of the spectrum of action variables (eigenvalues of the classical Lax matrix). The limit corresponding to the classical Calogero–Sutherland system and quantum trigonometric Gaudin model is also described as well as the XX limit to free fermions.
In the derivation of the generating function of the Gaudin Hamiltonians with boundary terms, we follow the same approach used previously in the rational case, which in turn was based on Sklyanin's method in the periodic case. Our derivation is centered on the quasi-classical expansion of the linear combination of the transfer matrix of the XXZ Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield strikingly simple off-shell action of the generating function, we fully implement the algebraic Bethe ansatz, obtaining the spectrum of the generating function and the corresponding Bethe equations.
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.