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. Examples are strongly interacting atomic nuclei and mesoscopic condensed matter systems. In this approach, the linear Schrodinger equation for the exact many-body wave function is mapped onto a non-linear density-dependent 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. However, 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 theoretical 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. We present applications of the projection methods to simple nuclear models, realistic calculations in relatively small configuration spaces, nuclear energy density functional 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 energy density functional theories are highlighted. CONTENTSL. Other electronic systems 60 M. Other emerging directions 60 VIII. Projected statistics 61 A. Symmetry restoration at finite temperature 62 B. Thermo-field dynamics 63 IX. Summary, conclusions, and perspectives 63 X. Acknowledgments 64 A. Overlaps and matrix elements between HFB states: the generalized Wick's theorem 65

Permanent variational wave functions for bosons

“…It is a common observation in the practice of Hartree-Fock approximation that the solution often spontaneously breaks the symmetries of the Hamiltonian [34]. This could also happen with us, as we have seen in Fig.…”

Axially-deformed solution of the Skyrme-Hartree-Fock-Bogoliubov equations using the transformed harmonic oscillator basis (IV) hfbtho (v4.0): A new version of the program

“…PNP curves, corroborating the fact that the discrepancy in projected energies stems from the initial difference in HFB states. Finally, an instructive limit to consider is the case of a collapsing pairing interaction, which is a common feature of PNP models that perform variation before projection [15]. Note that the collapse of pairing happens around Q20 = 2.5 b in our calculation.…”

“…Since its first published version, the hfbtho program has implemented the particle number restoration in the canonical basis. The current version includes a new module for the simultaneous restoration of rotational, particle number, and reflection symmetry of the HFB states [1,15,16].…”

Solution of universal nonrelativistic nuclear DFT equations in the Cartesian deformed harmonic-oscillator basis. (IX) HFODD (v3.06h): a new version of the program

“…The HFB method accounts for pairing correlations through the breaking of the U(1) symmetry associated with particle-number conservation. The above computation of overlaps allows us to implement the restoration of correct proton and neutron numbers by projection after variation of a symmetry-unrestricted HFB state |Φ [65]. In version (v3.04n) of the code hfodd, projection on the total particle number A was implemented, which allows for a full particle-number-symmetry restoration in nuclei where one of the species, protons or neutrons, are unpaired in |Φ .…”