Symmetry restoration in mean-field approaches

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.

Application of the variational principle to a coherent-pair condensate: The BCS case

“…Variation of the average energy Ē (11) under an infinitesimal θ has been derived in Sec. V of the arXiv manuscript [43] (but has not been published in a journal yet).…”

“…[7][8][9][10] in the BCS case and Refs. [11][12][13][14][15] in the HFB case. * Electronic address: liyuan.jia@usst.edu.cn When feasible, VAP is preferred [1,2,5] over PAV.…”

Static octupole deformations in $^{96}$Zr from angular momentum and parity projections

“…The resulting states usually break fundamental symmetries and conservation laws, such as rotational symmetry, translational invariance and particle number conservation. As a consequence, projectionafter-variation (PAV) techniques are employed to restore the broken symmetries and calculate observables with good quantum numbers [33][34][35][36][37][38][39][40][41]. PAV calculations have been applied to study the octupole deformations in Ra isotopes based on shell model wave functions [42], in Ra isotopes [43] and 20 Ne [44] based on RHB wave functions, in Ba, Ra isotopes [45,46], Zr isotopes [47], 16 O [48], actinides, superheavies [49] and other nuclei [50] based on HFB wave functions.…”

Operator overlaps in harmonic oscillator bases with different oscillator lengths

“…In this paper we apply the formalism to the common situation where one deals with two finite harmonic oscillator (HO) bases with different oscillator lengths. The present results could be of interest to carry out generator coordinate method (GCM) calculations and/or symmetry restoration [3][4][5][6] in fission -see [7,8] for a recent review and for a recent application. In fission one has to consider a set of HFB wave functions usually labeled in terms of the axial quadrupole moment |φ(q 20 ) to study fission dynamics.…”