Shifted HFB states

Let us introduce several useful notations that will be used later. First, we call the operator appearing under the integral (2) the shift operator,

$$\hat{z}(z) = z^{\hat{N}} = e^{(\eta+i\phi)\hat{N}} ,$$ (3)

parametrized by means of a single complex number $z$, $\ln(z)$=$\eta+i\phi$. The shift operator $\hat{z}(z)$ is parametrized by the complex number $z$ and constitutes a non-unitary Bogoliubov transformation (in fact, a non-unitary single-particle basis transformation) of simple kind, i.e.,

$$\begin{array}{rcl} \hat{z} a^+_n \hat{z}^{-1} &=& z a^+_n , \\ \hat{z} a_n \hat{z}^{-1} &=& z^{-1} a_n \end{array}$$ (4)

or

$$\begin{array}{rcl} \hat{z}^{-1} a^+_n \hat{z} &=& z^{-1} a^+_n , \\ \hat{z}^{-1} a_n \hat{z} &=& z a_n . \end{array}$$ (5)

Obviously, for $z=1$, the shift operator is equal to identity.

Second, we define the shifted HFB states as

$$\vert\Phi(z)\rangle = \hat{z}(z)\vert\Phi\rangle .$$ (6)

When the HFB state $\vert\Phi\rangle$ is expressed through the Thouless theorem [1] (we assume an even number of particles for simplicity),

$$\vert\Phi\rangle={\cal{}N}\exp\left( {\textstyle{\frac{1}{2}}}\sum_{mn} Z^*_{mn} a^+_m a^+_n\right)\vert\rangle ,$$ (7)

the shifted HFB states read

$$\vert\Phi(z)\rangle={\cal{}N}\exp\left( {\textstyle{\frac{1}{2}}}z^2 \sum_{mn} Z^*_{mn} a^+_m a^+_n\right)\vert\rangle ,$$ (8)

where ${\cal{}N}$ is the normalization constant of the HFB state (7). Similarly, for the HFB state expressed in the canonical basis or for a BCS state,

$$\vert\Phi\rangle=\prod_{n>0}\left(u_n+v_n{a}_n^+{a}_{\bar{n}}^+\right)\vert\rangle ,$$ (9)

the shifted state reads

$$\vert\Phi(z)\rangle=\prod_{n>0}\left(u_n+z^2\,v_n{a}_n^+{a}_{\bar{n}}^+\right) \vert\rangle ,$$ (10)

where $u_n$ and $v_n$ are the real HFB occupation amplitudes in the canonical basis and the product $\prod_{n>0}$ involves only one state from each pair of canonical partners (see Ref. [1] for details).

We call $\hat{z}(z)$ a shift, because it moves the HFB state $\vert\Phi\rangle$= $\vert\Phi(1)\rangle$ from its original position at $z=1$ to a different point $z$ in the complex plane. Since consecutive shift transformations correspond to products of the shift parameters $z$, the parameters $\eta$ and $\phi$ in Eq. (3) are additive.