arXiv:0806.1810

Quantum Invariants of the Pairing Hamiltonian

Yamaç Pehlivan

Published Jun 11, 2008• 2 citations

Abstract

Quantum invariants of the orbit dependent pairing problem are identified in the limit where the orbits become degenerate. These quantum invariants are simultaneously diagonalized with the help of the Bethe ansatz method and a symmetry in their spectra relating the eigenvalues corresponding to different number of pairs is discussed. These quantum invariants are analogous to the well known rational Gaudin magnet Hamiltonians which play the same role in the reduced pairing case (i.e., orbit independent pairing with non degenerate energy levels). It is pointed out that although the reduced pairing and the degenerate cases are opposite of each other, the Bethe ansatz diagonalization of the invariant operators in both cases are based on the same algebraic structure described by the rational Gaudin algebra.

DOI: 10.48550/arxiv.0806.1810

Related papers

Solutions of nuclear pairing

A. B. Balantekin, Yamaç Pehlivan

We give the exact solution of orbit dependent nuclear pairing problem between two nondegenerate energy levels using the Bethe ansatz technique. Our solution reduces to previously solved cases in the appropriate limits including Richardson's treatment of reduced pairing in terms of rational Gaudin algebra operators.

arXiv:0710.3941Physical Review CNov 14, 200718 citations

Spectra and symmetry in nuclear pairing

A. B. Balantekin, J. H. de Jesus, Yamaç Pehlivan

We apply the algebraic Bethe ansatz technique to the nuclear pairing problem with orbit dependent coupling constants and degenerate single particle energy levels. We find the exact energies and eigenstates. We show that for a given shell, there are degeneracies between the states corresponding to less than and more than half full shell. We also provide a technique to solve the equations of Bethe ansatz.

arXiv:nucl-th/0702059Physical Review CJun 4, 200716 citations

Twisted rational r-matrices and algebraic Bethe ansatz: Application to generalized Gaudin and Richardson models

T. Skrypnyk, N. Manojlović

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.

Nuclear Physics BApr 29, 20219 citations

Algebraic Bethe ansatz forQ-operators: the Heisenberg spin chain

Rouven Frassek

We diagonalize Q -operators for rational homogeneous -invariant Heisenberg spin chains using the algebraic Bethe ansatz. After deriving the fundamental commutation relations relevant for this case from the Yang–Baxter equation we demonstrate that the Q -operators act diagonally on the Bethe vectors if the Bethe equations are satisfied. In this way we provide a direct proof that the eigenvalues of the Q -operators studied here are given by Baxter's Q -functions.

Journal of Physics A Mathematical and TheoreticalJun 30, 20156 citations

“Twisted” rational r-matrices and the algebraic Bethe ansatz: Applications to generalized Gaudin models, Bose–Hubbard dimers, and Jaynes–Cummings–Dicke-type models

T. Skrypnyk

We construct quantum integrable systems associated with the Lie algebra gl ( n ) and non-skew-symmetric “shifted and twisted” rational r-matrices. The obtained models include Gaudin-type models with and without an external magnetic field, n-level ( n −1)-mode Jaynes–Cummings–Dicke-type models in the Λ-configuration, a vector generalization of Bose–Hubbard dimers, etc. We diagonalize quantum Hamiltonians of the constructed integrable models using a nested Bethe ansatz.

Theoretical and Mathematical PhysicsOct 1, 20166 citations

Algebraic Bethe ansatz for Q-operators: The Heisenberg spin chain

Rouven Frassek

We diagonalize Q-operators for rational homogeneous sl(2)-invariant Heisenberg spin chains using the algebraic Bethe ansatz. After deriving the fundamental commutation relations relevant for this case from the Yang-Baxter equation we demonstrate that the Q-operators act diagonally on the Bethe vectors if the Bethe equations are satisfied. In this way we provide a direct proof that the eigenvalues of the Q-operators studied here are given by Baxter's Q-functions.

arXiv:1504.04501Apr 17, 20151 citation