Antilinear Operators in Hartree-Bogolyubov Theory. I

Abstract

Hartree-Bogolyubov (HB) theory is formulated in a basis-independent way, i.e., in terms of linear and antilinear operators acting in the one-particle space. For that purpose, some basic antilinear algebra is presented. The pairing tensor and the pairing potential are shown to represent two antilinear skew-Hermitian operators. The polar factorization of the first of them (the correlation operator ${\hat{t}}_a$), i.e., ${\hat{t}}_a={(\hat{\rho }-{\hat{\rho }}^2)}^{\frac{1}{2}}{\hat{P}}_a$, shows that HB theory has only two variational (trial) operators: the density operator $\hat{\rho }$ and the antilinear pairing operator ${\hat{P}}_a$ which is defined by the properties ${{\hat{P}}_a}^{+}={{\hat{P}}_a}^{†1}=-{\hat{P}}_a$. These two operators commute. The former is the unique and very well-known variational operator of Hartree-Fock (HF) theory, and the latter represents a new variational freedom typical of HB theory. Most calculations, as for instance the Bardeen-Cooper-Schrieffer (BCS) approximation, restrict this freedom by choosing ${\hat{P}}_a$ to be the time-reversal operator. The basic dynamical (Euler-Lagrange) equations of HB theory are obtained directly by varying linear and antilinear operators. They are expressed in a compact form, using only commutators and anticommutators of the kinematical and the dynamical operators: ${\hat{A}}_a\equiv {[\hat{h},{\hat{t}}_a]}_{+}-{[{\hat{\Delta }}_a,\hat{\rho }-\frac{1}{2}]}_{+}=0, \hat{B}\equiv {[\hat{h},\hat{\rho }]}_{-}-{[{\hat{\Delta }}_a,{\hat{t}}_a]}_{-}=0,$ where ${\Delta }_a$ is the pairing potential and $\hat{h}$ is the one-particle Hamiltonian.