Lipkin method of particle-number restoration to higher orders
X.B. Wang, J. Dobaczewski, M. Kortelainen, L.F. Yu, M.V. Stoitsov
Abstract:
Background:
On the mean-field level, pairing correlations are incorporated
through the Bogoliubov-Valatin transformation, whereupon the particle
degrees of freedom are replaced by quasiparticles.
This approach leads to a spontaneous breaking of the particle-number
symmetry and mixing of states with different particle numbers. In
order to restore the particle number, various methods have been
employed, which are based on projection approaches before or after
variation. Approximate variation-after-projection (VAP) schemes,
utilizing the Lipkin method, have mostly been used within the
Lipkin-Nogami prescription.
Purpose:
Without recurring to the Lipkin-Nogami prescription, and using
instead states rotated in the gauge space, we derive the Lipkin
method of particle-number restoration up to sixth order and we test
the convergence and accuracy of the obtained expansion.
Methods:
We perform self-consistent calculations using the higher-order Lipkin
method to restore the particle-number symmetry in the framework of
superfluid nuclear energy-density functional theory. We also apply
the Lipkin method to a schematic exactly solvable two-level pairing
model.
Results:
Calculations performed in open-shell tin and lead isotopes show that
the Lipkin method converges at fourth order and satisfactorily
reproduces the VAP ground-state energies and energy kernels. Near
closed shells, the higher-order Lipkin method cannot be applied because of
a non-analytic kink in the ground-state energies in function of the
particle number.
Conclusions:
In open-shell nuclei, the higher-order Lipkin method provides a good
approximation to the exact VAP energies. The method is computationally
inexpensive, making it particularly suitable, for example, for future
optimizations of the nuclear energy-density functionals and
simultaneous restoration of different symmetries.