Projected HFB states

The Thouless theorem (7) allows us to express the HFB state $\vert\Phi\rangle$ and shifted HFB state $\vert\Phi(z)\rangle$ as sums of components having different particle numbers,

$$\vert\Phi\rangle={\cal{}N}\sum_{k=0}^{\infty}\frac{(\hat{Z}^+)^k}{k!}\vert\rangle ,$$ (11)

$$\vert\Phi(z)\rangle={\cal{}N}\sum_{k=0}^{\infty}z^{2k}\frac{(\hat{Z}^+)^k}{k!}\vert\rangle ,$$ (12)

where $\hat{Z}^+={\textstyle{\frac{1}{2}}}\sum_{mn} Z^*_{mn} a^+_m a^+_n$ is the Thouless pair-creation operator. It then trivially follows that the shift transformation does not change any of the particle-number-projected states (2),

$$\vert\Psi_N\rangle={\cal{}N}\frac{(\hat{Z}^+)^{N/2}}{(N/2)!}\vert\rangle ,$$ (13)

but only scales the coefficients in the sum of Eq. (12).

Since the shifted states (12) are manifestly analytical in $z$, all closed contours $C$ in Eq. (2) give, by the Cauchy theorem, the same result. Among them, the integral in Eq. (1) simply corresponds to the unit circle.

The analyticity of $\vert\Phi(z)\rangle$ results in a simple and elegant representation of the projected state:

$$\vert\Psi_N\rangle \equiv {\hat P}_N \vert\Phi\rangle = \r... ...}{\mbox{\scriptsize {$z=0$}}}$}\,z^{-N-1}\vert\Phi(z)\rangle .$$ (14)

Indeed, in the sum of Eq. (12), only the term with $N=2k$ particles is multiplied by $1/z$ and thus contributes to the residue at $z=0$. This observation allowed Dietrich, Mang, and Pradal [36] to formulate the so-called method of residues for calculating all kinds of matrix elements involving the projected state $\vert\Psi_N\rangle$. For example, the average HFB energy of the projected state can be written as a ratio of two residues:

$$E_{\mbox{\rm\scriptsize {HFB}}}^N= \frac{\langle\Phi\vert\... ...riptsize {$z=0$}}}$}\,z^{-N-1}\langle\Phi\vert\Phi(z)\rangle}.$$ (15)

The invariance of the projected state with respect to the integration contour can be formulated in another way; namely, one can utilize the property that an arbitrarily shifted HFB state can be equally well used to project the particle number. Indeed, for

$$\displaystyle \vert\Psi_N(z_0)\rangle \equiv {\hat P}_N \v... ...\pi i}\oint_C {\rm d}z\,z^{\hat{N}-N-1}\vert\Phi_N(z_0)\rangle$$ (16)

we trivially have

$$\vert\Psi_N(z_0)\rangle = z_0^N \vert\Psi_N\rangle ,$$ (17)

i.e., projection from a shifted HFB state changes only the phase and normalization of the projected state. We refer to this property as shift invariance.