Particle-Number-Projected HFB

In the context of HFB theory [1], the particle-number-projected (PNP) state is given by the standard expression

$$\displaystyle \vert\Psi_N\rangle \equiv {\hat P}_N \vert\P... ...{0}^{2\pi} {\rm d}\phi\,e^{i\phi(\hat{N}-N)}\vert\Phi\rangle ,$$ (1)

where $\hat{N}$ is the PN operator, $\phi$ is the gauge angle, ${\hat P}_N$ is the projection operator for $N$ particles, and $\vert\Phi\rangle$ is the HFB wave function (generalized product state) which does not have well-defined particle number. This expression, after the integral is discretized, is most often used in practical calculations. However, it only constitutes a specific realization of a more general form [35] given by the contour integral,

$$\displaystyle \vert\Psi_N\rangle \equiv {\hat P}_N \vert\P... ...{1}{2\pi i}\oint_C {\rm d}z\,z^{\hat{N}-N-1}\vert\Phi\rangle ,$$ (2)

where C is an arbitrary closed contour encircling the origin $z=0$ of the complex plane.