Abstract
The mean-field approximation based on effective interactions or density functionals plays a pivotal role in the description of finite quantum many-body systems that are too large to be treated by ab initio methods. Some examples are strongly interacting medium and heavy mass atomic nuclei and mesoscopic condensed matter systems. In this approach, the linear Schrödinger equation for the exact many-body wave function is mapped onto a non-linear one-body potential problem. This approximation, not only provides computationally very simple solutions even for systems with many particles, but due to the non-linearity, it also allows for obtaining solutions that break essential symmetries of the system, often connected with phase transitions. In this way, additional correlations are subsumed in the system. However, the mean-field approach suffers from the drawback that the corresponding wave functions do not have sharp quantum numbers and, therefore, many results cannot be compared directly with experimental data. In this article, we discuss general group-theory techniques to restore the broken symmetries, and provide detailed expressions on the restoration of translational, rotational, spin, isospin, parity and gauge symmetries, where the latter corresponds to the restoration of the particle number. In order to avoid the numerical complexity of exact projection techniques, various approximation methods available in the literature are examined. Applications of the projection methods are presented for simple nuclear models, realistic calculations in relatively small configuration spaces, nuclear energy density functional (EDF) theory, as well as in other mesoscopic systems. We also discuss applications of projection techniques to quantum statistics in order to treat the averaging over restricted ensembles with fixed quantum numbers. Further, unresolved problems in the application of the symmetry restoration methods to the EDF theories are highlighted in the present work.
Highlights
Difficulties encountered in restoring symmetries with nuclear energy density functional (EDF)
We discuss the case of a many-body setting. These three simple models are exactly solvable, which allow us to analyze the problem of symmetry breaking and restoration in the quantum mechanical context and to clearly delineate the role of approximations that unavoidably have to be made in realistic situations
Both approaches give excellent reproduction of the exact results, for the half-filled shell [figure 6(b)], at or below the critical pairing strength both fail. This is so because the kink in the dependence of the exact energies on the particle number, which is a characteristic feature of a shell gap, cannot be reproduced by the quadratic (Dobaczewski and Nazarewicz 1993) or higher-order (Wang et al 2014) form of the Lipkin operator
Summary
Mean-field approaches play a central role in the description of quantum many-body problems in areas like quantum chemistry, atomic, molecular, condensed matter, and nuclear physics. This is the case, for instance, in the BCS and HFB theories, where the associated mean-field wave functions, which are the vacua of the corresponding quasiparticle operators, do not represent states with good particle number. It is common to use different (effective) interactions for each of the three contributions to the energy coming from a two-body operator, namely the direct, exchange, and pairing contributions (the most typical case is probably the use of the Slater approximation for the Coulomb exchange contribution and the neglect of the Coulomb antipairing field) In this case, a naive use of the generalized Wick’s theorem can lead to spurious contributions and specific ways to deal with this problem have to be devised.
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