Analytic properties

In order to proceed further, we must investigate the analytic structure of the integrand ${\cal{}E}_N(z)$ appearing in the numerator of Eq. (34):

$${\cal{}E}_N(z)= z^{-N-1} \langle\Phi\vert\Phi(z)\rangle E_{\mbox{\rm\scriptsize {DFT}}}(\rho_z,\chi_z,\bar{\chi}_z) .$$ (35)

Let us first discuss the case when the DFT energy density $E_{\mbox{\rm\scriptsize {DFT}}}(\rho,\chi,\chi^*)$ is a polynomial in local densities; hence, the DFT transition energy density $E_{\mbox{\rm\scriptsize {DFT}}}(\rho_z,\chi_z,\bar{\chi}_z)$ is a polynomial in transition densities. The case of fractional power dependence requires special attention and will be discussed in Sec. 3.6.

Within the polynomial assumption, poles of the transition densities (28) do or do not appear as poles of the integrand (35), depending on the structure of the DFT transition energy density. [For instance, quadratic ($p$=2) and cubic ($p$=3) terms are characteristic of two-body and three-body interactions, respectively.] On the one hand, each polynomial term of the order $p$ in densities (30) brings about a pole of the order $p$. On the other hand, each term in the overlap (28) produces a zero of the order $q$, where $q$ is the degeneracy factor of the HFB density matrix with the two-fold Kramers degeneracy not counted. (Note that the product in Eq. (28) contains only one term for each canonical pair.) In particular, for the spherical shell of angular momentum $j$, the degeneracy is $q=j+{\textstyle{\frac{1}{2}}}$.

When the poles of transition densities and zeros of the overlap $\langle\Phi\vert\Phi(z)\rangle$ are combined, the poles in ${\cal{}E}_N(z)$ are of the order $p-q$. For single-particle states that have only two-fold Kramers degeneracy ($q$=1), and for the terms with $p$=2, one obtains the first-order poles in ${\cal{}E}_N(z)$ with, in general, non-zero residues. Non-vanishing residues may also appear for higher-order poles corresponding to terms with $p>2$. On the other hand, for four-fold degenerate states with $q$=2, terms with $p$=2 do not produce poles in ${\cal{}E}_N(z)$, and only terms with $p>2$ may give rise to poles with non-vanishing residues. As discussed in detail in Ref. [30], for the energy density derived from a Hamiltonian, additional cancellations between terms originating from particle-hole and pairing channels occur, and the first-order poles disappear.

Figure 1: (color online) Schematic illustration of the analytic structure of the integrand (35) in the complex $z$-plane (see text). Small crossed circles denote imaginary axis poles. Three integration contours ($C0$, $C1$, $C2$), symmetric with respect to the real axis, are indicated. The poles having particle character (corresponding to canonical states lying above the chemical potential $\lambda$) are located outside the unit circle $C0$ while the hole poles lie inside $C0$. The unprojected ground state wave function corresponds to $z$=1.

In Fig. 1 we schematically illustrate the analytic structure of the integrand (35). Crossed circles on the imaginary axis represent poles of ${\cal{}E}_N(z)$. Apart from the pole at $z=0$, the integrand may have poles (28) distributed symmetrically in pairs with respect to the real axis. Poles located within the unit circle (C0 in Fig. 1) correspond to the canonical states with occupation numbers larger than 0.5, or with $u_n/v_n\le1$, i.e., with canonical energies below the Fermi energy $\lambda$. Similarly, poles outside the unit circle correspond to canonical states lying above the Fermi energy.

The unprojected HFB ground state $\vert\Phi\rangle$, located at $z$=1, is shifted along the integration contour $C$, and its overlap and DFT transition energy contribute to the integrand of Eq. (35). Standard projection formula (1) corresponds to the unit circle $C0$. Contours $C1$ and $C2$ encircle a fewer number of poles in ${\cal{}E}_N(z)$, with contour $C2$ surrounding only the single pole at the origin. Shapes of these contours are irrelevant, and only the points at which they cross the imaginary axis matter. For example, contours $C1$ and $C2$, shown in Fig. 1, are equivalent to circular contours $C1'$ and $C2'$ of Fig. 2, the latter being more practical in calculations. If the residues of the poles inside the unit circle are non-zero, the three integration contours shown in Fig. 1 may give different projected energies. Of course, contours including poles located outside the unit circle (not shown in Fig. 1) may still give different results.

Figure 2: Schematic illustration of analytic structure of the particle transition density at a fixed point in space $\bf r$. The poles (crossed circles) and zeros (dots) of $\rho _z$ are located on the imaginary-$z$ axis. The regions of real negative $\rho _z$ are shaded. Three circular integration contours $C0$, $C1'$, and $C2'$ are indicated. See text for details.