Diagonalization of the BCS Hamiltonian and the Bethe equations

Luigi Amico A, G. Falci, Rosario Fazio

We study the connection between the BCS pairing model and the inhomogeneous vertex model. The two spectral problems coincide in the quasi-classical limit of the off-shell Bethe ansatz of the disordered six-vertex model. The latter problem is transformed into an auxiliary spectral problem which corresponds to the diagonalization of the integrals of motion of the BCS model. A generating functional whose quasi-classical expansion leads to the constants of motion of the BCS model, and in particular the Hamiltonian, is identified.

J. D. Fan, Y. M. Malozovsky

D. Ya. Petrina

Luigi Amico A, G. Falci, Rosario Fazio

Abstract. We study the connection between the BCS pairing model and the inhomogeneous vertex model. The two spectral problems coincide in the quasi– classical limit of the off–shell Bethe Ansatz of the disordered six vertex model. The latter problem is transformed into an auxiliary spectral problem which corresponds to the diagonalization of the integrals of motion of the BCS model. A generating functional whose quasi classical expansion leads to the constants of motion of the BCS model and in particular the Hamiltonian, is identified. PACS: 03.65.Fd, 74.20.Fg

L. M. Garrido, P. Pascual

G. Gorohovsky, Eldad Bettelheim

We present a derivation of a previously announced result for matrix elements between exact eigenstates of the pairing Hamiltonian. Our results, which generalize the well-known Bardeen–Cooper–Schrieffer (BCS) (Bardeen et al 1957 Phys. Rev. 108 1175; 1957 Phys. Rev. 106 162) expressions for what are known as ‘coherence factors’, are derived based on the Slavnov (1989 Theor. Math. Phys. 79 502) formula for overlaps between Bethe-ansatz states, thus making use of the known connection between the exact diagonalization of the BCS Hamiltonian, due to Richardson (1963 Phys. Lett. 3 277; 1964 Nucl. Phys. A 52 221), and the algebraic Bethe ansatz. The resulting formula has a compact form after a suitable parameterization of the energy plane. Although we apply our method here to the pairing Hamiltonian, it may be adjusted to study what is termed the ‘Sutherland limit’ (Sutherland 1995 Phys. Rev. Lett. 74 816) for exactly solvable models, namely where a macroscopic number of rapidities form a large string.