Poles of transition densities

It is seen immediately from Eq. (27) that the transition density matrices have imaginary axis poles at

$$z_n = \pm{i}\vert u_n/v_n\vert,$$ (28)

and, therefore, are not analytical. These poles carry over to the HFB transition energy density as well. The poles appear beyond the origin, $z_n$$\neq$0, provided all amplitudes $u_n$ are nonzero; we assume this hereafter, i.e., none of the canonical states is being blocked. We can also safely assume that all amplitudes $v_n$ are nonzero, because otherwise the corresponding states would not contribute to the density matrices at all. Of course, if there exist poles in the HFB transition energy density, they must be cancelled by the norm overlap $\langle\Phi\vert\Phi(z)\rangle$, because the Hamiltonian matrix element $\langle\Phi\vert\hat{H}\vert\Phi(z)\rangle$ is an analytical function of $z$.

However, as we discuss in the next section, whenever the transition energy density is not related to a Hamiltonian, or some approximations are involved in Hamiltonian's construction, the presence of the poles (28) requires special attention. For example, the exact HFB transition energy density,

$$E_{\mbox{\rm\scriptsize {HFB}}}(\rho_z,\chi_z,\bar{\chi}_z) ... ...) + E_{\rm field}(\rho_z) + E_{\rm pair}(\chi_z,\bar{\chi}_z),$$ (29)

is often split into the kinetic term $E_{\rm kin}(\tau_z)$ that depends on the kinetic transition density, the mean-field term $E_{\rm field}$ that depends on the particle transition density, and the pairing term $E_{\rm pair}$ that depends on the pairing transition densities. It was first realized in Ref. [30], and then discussed by several authors [31,32,37], that the poles are not cancelled separately in $E_{\rm field}$ and $E_{\rm pair}$, but only in the sum thereof, i.e., for the total HFB energy calculated for a given Hamiltonian.

As the origin of the pairing interaction is believed to be different from that of the effective interaction in the particle-hole direction, it is customary to employ different Hamiltonians to calculate $E_{\rm field}$ and $E_{\rm pair}$. This, however, leads to a non-analytical behavior of $E_{\mbox{\rm\scriptsize {HFB}}}$ due to the presence of poles in the complex $z$-plane; hence, to a priori contour-dependent projected HFB energies. We discuss this question in the next section in the more general context of the DFT energy functional.