One of the biggest successes of the mean-field (MF) and energy-density-functional (EDF) methods applied to many-body systems consists in including correlations through the mechanism of the symmetry breaking. A single broken-symmetry MF state or a one-body broken-symmetry density can, in fact, represent a large class of correlations that are physically important. Moreover, the breaking of symmetry on the MF level corresponds to the appearance or disappearance of certain correlations that in finite systems may have all features of the phase transitions.
A link between the broken-symmetry MF states and correlated symmetry-conserving states is provided by the symmetry-restoration methods [1]. The broken-symmetry states can be viewed as auxiliary objects, which facilitate obtaining the real quantum mechanical states that have all good quantum numbers. Within the EDF methods, this leads to a very fruitful idea of the projected energy being a functional of the symmetry-breaking one-body density [2,3]. However, variational equations then become quite difficult to solve, and the full symmetry-restored MF or EDF methods were to date applied only in several particular cases [2,3,4].
Numerous approximate methods to restore broken symmetries by variation before or after projection (VAP) were formulated and implemented in the past [1,5]. The main issue here is the feasibility of the method - it must be a reasonable compromise between the precision and closeness to the exact VAP on one side, and with the numerical effort to execute it on the other side. In this respect, the Lipkin method [6] has very many advantages, which are discussed in the present study. It allows for calculating an approximate VAP energy without any necessity to perform the projection at all. In the past, it has been mostly used for the particle-number restoration, within the so-called Lipkin-Nogami [7] formulation. However, the original Lipkin method was formulated for restoring the translational symmetry, and in this study this case is studied in detail.