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Journal of Physics A Mathematical and Theoretical

Z 2 -graded classical r -matrices and algebraic Bethe ansatz: applications to integrable models of quantum optics and nuclear physics

T. Skrypnyk

Published Aug 12, 2016• 11 citations
Publisher

Abstract

Abstract We consider quantum integrable models based on the Lie algebra gl ( n ) and non-skew-symmetric classical r -matrices associated with Z 2 -gradings of gl ( n ) of the following type: gl ( n ) = gl ( n ) 0 ¯ + gl ( n ) 1 ¯ , where gl ( n ) 0 ¯ = gl ( n 1 ) ⊕ gl ( n − n 1 ) . Among the considered models are Gaudin-type models with an external magnetic field, used in nuclear physics to produce proton–neutron Bardeen–Cooper–Schrieer-type models, n -level many-mode Jaynes–Cummings–Dicke-type models of quantum optics, matrix generalization of Bose–Hubbard dimers, etc. We diagonalize the constructed models by means of the ‘generalized’ nested Bethe ansatz.

DOI
10.1088/1751-8113/49/36/365201

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Abstract We consider quantum integrable models based on the Lie algebra gl ( n ) and non-skew-symmetric classical r -matrices associated with Z 2 -gradings of gl ( n ) of the following type: gl ( n ) = gl ( n ) 0 ¯ + gl ( n ) 1 ¯ , where gl ( n ) 0 ¯ = gl ( n 1 ) ⊕ gl ( n − n 1 ) . Among the considered models are Gaudin-type models with an external magnetic field, used in nuclear physics to produce proton–neutron Bardeen–Cooper–Schrieer-type models, n -level many-mode Jaynes–Cummings–Dicke-type models of quantum optics, matrix generalization of Bose–Hubbard dimers, etc. We diagonalize the constructed models by means of the ‘generalized’ nested Bethe ansatz.
We consider quantum integrable systems associated with reductive Lie algebra g l ( n ) and Cartan-invariant non-skew-symmetric classical r -matrices. We show that under certain restrictions on the form of classical r -matrices “nested” or “hierarchical” Bethe ansatz usually based on a chain of subalgebras g l ( n ) ⊃ g l ( n − 1 ) ⊃ . . . ⊃ g l ( 1 ) is generalized onto the other chains or “hierarchies” of subalgebras. We show that among the r -matrices satisfying such the restrictions there are “twisted” or Z p -graded non-skew-symmetric classical r -matrices. We consider in detail example of the generalized Gaudin models with and without external magnetic field associated with Z p -graded non-skew-symmetric classical r -matrices and find the spectrum of the corresponding Gaudin-type hamiltonians using nested Bethe ansatz scheme and a chain of subalgebras g l ( n ) ⊃ g l ( n − n 1 ) ⊃ g l ( n − n 1 − n 2 ) ⊃ g l ( n − ( n 1 + . . . + n p − 1 ) ) , where n 1 + n 2 + . . . + n p = n .
We consider quantum integrable systems associated with the Lie algebra gl(n) and Cartan-invariant non-dynamical non-skew-symmetric classical r-matrices. We describe the sub-class of Cartan-invariant non-skew-symmetric r-matrices for which exists the standard procedure of the nested Bethe ansatz associated with the chain of embeddings gl(n)⊃gl(n−1)⊃gl(n−2)⊃⋯⊃gl(1). We diagonalize the corresponding quantum integrable systems by its means. We illustrate the obtained results by the examples of the generalized Gaudin systems with and without external magnetic field associated with three classes of non-dynamical non-skew-symmetric classical r-matrices.
We construct quantum integrable systems associated with non-skew-symmetric gl(2)-valued classical r-matrices. We find a new explicit multiparametric family of such the non-skew-symmetric classical r-matrices. We consider two classes of examples of the corresponding integrable systems, namely generalized Gaudin systems with and without an external magnetic field. In the case of arbitrary r-matrices diagonal in a standard gl(2)-basis, we calculate the spectrum of the corresponding quantum integrable systems using the algebraic Bethe ansatz. We apply these results to a construction of integrable fermionic models and obtain a wide class of integrable Bardeen–Cooper–Schrieffer (BCS)-type fermionic Hamiltonians containing the pairing and electrostatic interaction terms. We also consider special cases when the corresponding integrable Hamiltonians contain only pairing interaction term and are exact analogs of the “reduced BCS Hamiltonian” of Richardson.
In the present paper we develop the algebraic Bethe ansatz approach to the case of non-skew-symmetric gl(2)⊗gl(2)-valued Cartan-non-invariant classical r-matrices with spectral parameters. We consider the two families of these r-matrices, namely, the two non-standard rational r-matrices twisted with the help of second order automorphisms and realize the algebraic Bethe ansatz method for them. We study physically important examples of the Gaudin-type and BCS-type systems associated with these r-matrices and obtain explicitly the Bethe vectors and the spectrum for the corresponding quantum hamiltonians in terms of solutions of Bethe equations.
We analyze symmetries of the integrable generalizations of Jaynes–Cummings and Dicke models associated with simple Lie algebras g and their reductive subalgebras gK [T. Skrypnyk, “Generalized n-level Jaynes-Cummings and Dicke models, classical rational r-matrices and nested Bethe ansatz,” J. Phys. A: Math. Theor. 41, 475202 (2008)]. We show that their symmetry algebras contain commutative subalgebras isomorphic to the Cartan subalgebras of g, which can be added to the commutative algebras of quantum integrals generated with the help of the quantum Lax operators. We diagonalize additional commuting integrals and constructed with their help the most general integrable quantum Hamiltonian of the generalized n-level many-mode Jaynes–Cummings and Dicke-type models using nested algebraic Bethe ansatz.