The Lipkin method

The Lipkin idea of flattening the function $E_\Phi(\bm{P})$ is implemented in the following way. Guided, by the GOA result in Eq. (9), one defines the Lipkin operator

$$\hat{K}=k\hat{\bm{P}}^2,$$ (14)

for which the projected average values $K(\bm{P})$ are defined as:

$$K(\bm{P}) = {\langle\Phi\vert\hat{K}\vert\bm{P}\rangle}/ {\langle\Phi \vert\bm{P}\rangle} ,$$ (15)

which in turn fulfill the sum rule:

$$\int {\rm {d}}^3 \bm{P} K(\bm{P}){\langle\Phi\vert\bm{P}\rangle}= \langle\Phi\vert\hat{K}\vert\Phi\rangle .$$ (16)

Then, by subtracting Eqs. (12) and (16), one obtains

$$\int {\rm {d}}^3 \bm{P} \left(E_\Phi(\bm{P})-K(\bm{P})\right... ...{P}\rangle}= \langle\Phi\vert\hat{H}-\hat{K}\vert\Phi\rangle .$$ (17)

It is clear that the flattest difference $E_\Phi(\bm{P})-K(\bm{P})$ is obtained by adjusting the constant $k$ so as to best fulfill the equation

$$K(\bm{P}) = E_\Phi(\bm{P}) - E_\Phi(\bm{0}) .$$ (18)

Note that since $K(\bm{0})=0$, the exact projected energy of the system at rest, $E_\Phi(\bm{0})$, must, by definition, appear on the right-hand side of Eq. (18).

In case that $E_\Phi(\bm{P})$ grows exactly parabolically, one obtains that

$$E_\Phi(\bm{0}) = {\langle\Phi\vert\hat{H}-\hat{K}\vert\Phi\rangle},$$ (19)

that is, the projected energy of the system at rest $E_\Phi(\bm{0})$ can be calculated without performing any projection at all.

The essence of the Lipkin method is in finding a suitable value of the correcting parameter $k$, which for each Slater determinant $\vert\Phi\rangle$ must describe the parabolic growth of the function $E_\Phi(\bm{P})$. We note here that this growth has nothing to do with the physical translational motion of the system boosted to momentum $\bm{P}$, in which case the energy must grow as $E({\bm{P}}^2)=\hbar^2\bm{P}^2/2mA$. The function $E_\Phi(\bm{P})$ simply characterizes the distribution of projected energies within the Slater determinant, that is, the degree of the symmetry breaking in the Slater determinant at rest. Therefore, the correcting parameter $k$ has no obvious relation with the true translational mass of the system. Moreover, the correcting parameter must depend on all kinds of approximations or space truncations, which are made when obtaining the Slater determinant $\vert\Phi\rangle$, that is, the Lipkin method corrects for these approximations too.

The right-hand side of Eq. (19) can be minimized by the standard self-consistent method, whereby the optimum state $\vert\Phi\rangle$ can be found. At each stage of the iterative procedure one has to determine $k$, that is, $k$ must parametrically depend on $\vert\Phi\rangle$. Note that at any given iteration of the self-consistent method, this parametric dependence must not be varied. Finally, after the iteration converges, $E_\Phi(\bm{0})$ is given by the obtained minimum value of the Lipkin projected energy (19). Obviously, the quality of the result crucially depends on the quality of the calculation of $k$, which now will be discussed.