Particle-number projection and the density functional theory

In the framework of the density functional theory for superconductors, we study the restoration of the particle-number symmetry by means of the projection technique. Conceptual problems are outlined and numerical difficulties are discussed. Both are related to the fact that neither the many-body Hamiltonian nor the wave function of the system appear explicitly in the density functional theory. Similar obstacles are encountered in self-consistent theories utilizing density-dependent effective interactions.

Figure 1

Schematic illustration of the analytic structure of the integrand in Eq. (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 λ) are located outside the unit circle $C0,$ while the hole poles lie inside $C0$. The unprojected ground state wave function corresponds to $z=1$.

Figure 2

Schematic illustration of analytic structure of the particle transition density at a fixed point in space $\mathbf{r}$. 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 $C0,C{1}^{\text{'}}$, and $C{2}^{\text{'}}$ are indicated. See text for details.

Figure 3

Ordinary ($z=1$) and transition ($z=i$) densities in ${}^{18}\mathrm{O}$ calculated in HFB+SLy4 as functions of $r$. The upper part of the figure shows (in extended scale) the small region of radii where the transition density becomes negative.

Figure 4

Bottom: ordinary ($z=1$) and transition ($z=i$) densities in ${}^{26}\mathrm{O}$ calculated in HFB+SLy4 as functions of $r$. Top: the real parts (solid lines) and imaginary parts (dashed lines) of the 1/6 powers of the ordinary ($z=1$) and transition ($z={\mathit{ie}}^{\pm i\varepsilon }$) densities.

Figure 5

Modified closed contour $C{1}^{″}$ that encircles the zero of the transition density. The poles (zeros) of ${\rho }_z$ are marked by crossed circles (dots). The equivalent circular contour $C{1}^{\text{'}\text{'}\text{'}}$ lying between the zero of ${\rho }_z$ and the previous pole is also indicated. See text for more details.

Figure 6

$N=10$ projected DFT-SLy4 energies of Eq. (44) calculated in ${}^{18}\mathrm{O}$ as functions of the integration radius $r_0$. Panels (a) and (b) show, respectively, results for the terms originating from density-independent and density-dependent parts of the Skyrme force. Positions of individual neutron poles are marked by arrows. The inset shows the results near the $2s_{1/2}$ pole on an expanded scale. The result of calculations obtained with the equivalent contours passing below the branching point associated with the 2$s_{1/2}$ pole (cf. Fig. 5) is shown by a dotted line.

Figure 7

Neutron poles $z_n$ in ${}^{18}\mathrm{O}$ [Eq. (28)] as functions of quadrupole deformation β, calculated within the HFB-SIII method with volume pairing interaction.

Figure 8

Neutron (left) and proton (right) poles $z_n$ [Eq. (28)] as functions of quadrupole deformation β calculated for ${}^{32}\mathrm{Mg}$ with the SIII functional and volume pairing interaction.

Figure 9

Similar to Fig. 8, except for canonical energies $e_k$.

Figure 10

Deformation energy $E(\beta )$ as a function of quadrupole deformation β, calculated for ${}^{18}\mathrm{O}$ with the SIII force and volume pairing interaction. Results of HFB (open triangles) are compared with different variants of PAV (squares and circles) and VAP PNP (stars) (see text for details).

Figure 11

Deformation energy $E(\beta )$ as a function of quadrupole deformation β calculated for ${}^{32}\mathrm{Mg}$ with the SIII force and volume pairing interaction. Results of the PAV HFB+LN calculations (squares and triangles) are compared with the VAP PNP results (dots). The standard HFB result is shown by open triangles.

Figure 12

Contributions to the PAV energy of ${}^{32}\mathrm{Mg}$ from the three selected poles as a function of deformation β.