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arXiv:1808.05672 · Journal of Physics G Nuclear and Particle Physics

Convergence and efficiency of angular momentum projection

Calvin W. Johnson, C. F. Jiao

Published Nov 5, 2018• 16 citations
arXivHTMLPDF

Abstract

Abstract In many so-called ‘beyond-mean-field’ many-body methods, one creates symmetry-breaking states and then projects out states with good quantum number(s); the most important example is angular momentum. Motivated by the computational intensity of symmetry restoration, we investigate the numerical convergence of two competing methods for angular momentum projection with rotations over Euler angles, the textbook-standard projection through quadrature, and a recently introduced projection through linear algebra. We find well-defined patterns of convergence with increasing number of mesh points (for quadrature) and cut-offs (for linear algebra). Because the method of projection through linear algebra requires inverting matrices generated on a mesh of Euler angles, we discuss two methods for robustly reducing the number of required evaluations. Reviewing the literature, we find our inversion involving rotations about the z -axis is equivalent to trapezoidal ‘quadrature’ commonly used as well as Fomenko projection used for particle-number projection. The efficiency depends upon the number of angular momentum J to be projected, but in general inversion methods, including Fomenko projection/trapezoidal ‘quadrature’ dramatically improve the efficiency.

DOI
10.1088/1361-6471/aaee20

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Abstract In many so-called ‘beyond-mean-field’ many-body methods, one creates symmetry-breaking states and then projects out states with good quantum number(s); the most important example is angular momentum. Motivated by the computational intensity of symmetry restoration, we investigate the numerical convergence of two competing methods for angular momentum projection with rotations over Euler angles, the textbook-standard projection through quadrature, and a recently introduced projection through linear algebra. We find well-defined patterns of convergence with increasing number of mesh points (for quadrature) and cut-offs (for linear algebra). Because the method of projection through linear algebra requires inverting matrices generated on a mesh of Euler angles, we discuss two methods for robustly reducing the number of required evaluations. Reviewing the literature, we find our inversion involving rotations about the z -axis is equivalent to trapezoidal ‘quadrature’ commonly used as well as Fomenko projection used for particle-number projection. The efficiency depends upon the number of angular momentum J to be projected, but in general inversion methods, including Fomenko projection/trapezoidal ‘quadrature’ dramatically improve the efficiency.
The authors provide a state-of-the-art overview of projection techniques employed in beyond-mean-field many-body methods. They focus on two symmetries that are of critical importance for the description of nuclear spectroscopy, particle number, and angular momentum. The presentation covers formal aspects of symmetry restoration (group theory) and practical aspects related to numerical implementation. A number of illustrative examples are given, including cranked states in odd-AAA nuclei.
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 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. 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 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 in the present work.
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.
The conventional method of constructing wave functions of pure angular momentum for a composite system is based on the idea of coupling the angular momenta of the constituents. Instead of this synthetic method, an analytical approach is here introduced. It is pointed out that an arbitrary trial wave function for the total system in a unique way must be resolvable into orthogonal components of pure angular momentum associated with different quantum numbers. A particular component can be selected by means of certain 'projection operators', which annihilate all components except the one desired. Such a projection operator is simply a product of commuting factors, each one of which annihilates a specific eigenfunction to the angular momentum under consideration. Physically this idea is of importance, since one can now start out from a rough model wave function based essentially on qualitative arguments, for instance of the independent-particle type, and then obtain a mathematically and physically much better trial function by selecting the particular component of the original function which has the correct symmetry type desired. The projection operators are studied in some detail both as products and in expanded form. The case of degenerate subspaces of the same angular momentum quantum numbers is discussed, and the problem of constructing an orthogonal subset of functions is solved by a simple elimination procedure; the connection with the conventional seniority idea is briefly discussed.
The restoration of particle number and angular momentum symmetry using projection operators has been theoretically investigated for mean-field Hartree-Fock-Bogoliubov (HFB) states. A computer code for the projection of particle number was then implemented. To do so efficiently and avoid the sign ambiguity of the Onishi formula, the computation of overlap between quasi-particle vaccums using Pfaffians was investigated. Different algorithms for the the Pfaffian from~\cite{GBBertschRobledo},\cite{Wimmer}, was tested for performance and accuracy. For the same reason different truncations of the model space was also investigated. The particle number projector was then implemented in \textit{HOSPHE} \cite{Carlsson} for calculations of ground states pertaining to an effective Quadrupole plus Pairing Hamiltonian calculated using the SLy4 parametrization of the Skyrme interaction.