Non-skew-symmetric classical r-matrices and integrable “[formula omitted]” proton–neutron BCS models

Abstract

We study integrable cases of the pairing BCS Hamiltonians containing several types of fermions and possessing non-uniform coupling constants. We prove that there exist three classes of such the integrable models associated with “Z2-graded” non-skew-symmetric classical r-matrices with spectral parameters and Lie algebras gl(2m), sp(2m) and so(2m), respectively. The proposed models are higher rank generalizations of the so-called “px+ipy” one-type fermion (m=1) BCS model. In the partial case of two types of fermions (m=2) the obtained models may be interpreted as N=Z, “px+ipy” proton–neutron integrable models. In particular, in the case of sp(4) we obtain the “px+ipy”-analogue of the famous integrable proton–neutron model of Richardson. We find the spectrum of the constructed Hamiltonians in terms of solutions of Bethe-type equations.