Projection on particle number and angular momentum: Example of triaxial Bogoliubov quasiparticle states

Background: Many quantal many-body methods that aim at the description of self-bound nuclear or mesoscopic electronic systems make use of auxiliary wave functions that break one or several of the symmetries of the Hamiltonian in order to include correlations associated with the geometrical arrangement of the system's constituents. Such reference states have been used already for a long time within self-consistent methods that are either based on effective valence-space Hamiltonians or energy density functionals, and they are presently also gaining popularity in the design of novel ab initio methods. A fully quantal treatment of a self-bound many-body system, however, requires the restoration of the broken symmetries through the projection of the many-body wave functions of interest onto good quantum numbers.

Purpose: The goal of this work is threefold. First, we want to give a general presentation of the formalism of the projection method starting from the underlying principles of group representation theory. Second, we want to investigate formal and practical aspects of the numerical implementation of particle-number and angular-momentum projection of Bogoliubov quasiparticle vacua, in particular with regard of obtaining accurate results at minimal computational cost. Third, we want to analyze the numerical, computational, and physical consequences of intrinsic symmetries of the symmetry-breaking states when projecting them.

Methods: Using the algebra of group representation theory, we introduce the projection method for the general symmetry group of a given Hamiltonian. For realistic examples built with either a pseudopotential-based energy density functional or a valence-space shell-model interaction, we then study the convergence and accuracy of the quadrature rules for the multidimensional integrals that have to be evaluated numerically and analyze the consequences of conserved subgroups of the broken symmetry groups.

Results: The main results of this work are also threefold. First, we give a concise, but general, presentation of the projection method that applies to the most important potentially broken symmetries whose restoration is relevant for nuclear spectroscopy. Second, we demonstrate how to achieve high accuracy of the discretizations used to evaluate the multidimensional integrals appearing in the calculation of particle-number and angular-momentum projected matrix elements while limiting the order of the employed quadrature rules. Third, for the example of a point-group symmetry that is often imposed on calculations that describe collective phenomena emerging in triaxially deformed nuclei, we provide the group-theoretical derivation of relations between the intermediate matrix elements that are integrated, which permits a further significant reduction of the computational cost of the method. These simplifications are valid regardless of the number parity of the quasiparticle states and therefore can be used in the description of even-even, odd-mass, and odd-odd nuclei.

Conclusions: The quantum-number projection technique is a versatile and efficient method that permits to restore the symmetry of any arbitrary many-body wave function. Its numerical implementation is relatively simple and accurate. In addition, it is possible to use the conserved symmetries of the reference states to reduce the computational burden of the method. More generally, the ever-growing computational resources and the development of nuclear ab-initio methods opens new possibilities of applications of the method.

Figure 1

Evolution of the weight $|c^{N_0}\left(M_{\phi }\right){|}^2$ of the numerically projected states as a function of the number of neutrons $N_0$ on which one projects for different choices for the number of discretization points $M_{\phi }$ for the quasiparticle states $|{\mathrm{\Phi }}_a\rangle$ [panel (a)], $|{\mathrm{\Phi }}_b\rangle$ [panel (b)], and $|{\mathrm{\Phi }}_c\rangle$ [panel (c)] as specified in the text. On each panel, the results obtained with the largest value of $M_{\phi }$ cannot be distinguished from exact results for the values of $N_0$ shown. Points calculated with the same $M_{\phi }$ are connected by straight lines to guide the eye.

Figure 2

Same as panel (a) of Fig. 1, but in logarithmic scale and some additional values of $M_{\phi }$.

Figure 3

Deviation (78), in logarithmic scale, of the sum rule for the components $|c^{N_0}\left(M_{\phi }\right){|}^2$ of the state $|{\mathrm{\Phi }}_a\rangle$ from the analytical value 1, as a function of $M_{\phi }$. See text for details.

Figure 4

Comparison between the weight $|c^{N_0}\left(M_{\phi }\right){|}^2$ of the numerically projected states (markers) and its Gaussian approximation (79) (solid line) as a function of the number of neutrons $N_0$ on which one projects for the quasiparticle states $|{\mathrm{\Phi }}_a\rangle$ [panel (a)], $|{\mathrm{\Phi }}_b\rangle$ [panel (b)], and $|{\mathrm{\Phi }}_c\rangle$ [panel (c)] as specified in the text. On each panel, the number of points $M_{\phi }$ used is the same as the largest one displayed in Fig. 1.

Figure 5

Evolution of the deviation of the expectation value of the neutron number from the value it is projected on (a) and dispersion of the neutron number (b) for states numerically projected on particle number $N_0$ from the same state with $\langle {\mathrm{\Phi }}_a\left|\widehat{N}\right|{\mathrm{\Phi }}_a\rangle =24$, as specified in Table 1, for the number of discretization points $M_{\phi }$ as indicated. Points calculated with the same $M_{\phi }$ are connected by straight lines to guide the eye.

Figure 6

Evolution of the projected energy, for states with $N_0=24$, 70, and 138, projected from states with the same mean particle number as specified in Table 1, as a function of the total number of points $M_{\phi }$ used for the discretized projection operator.

Figure 7

Evolution of ${\mathrm{\Xi }}_K(M_{\alpha },M_{\beta },M_{\gamma })$ as a function of the value of $K$ on which one projects for different choices for the number of discretization points $M_{\alpha }=M_{\gamma }$ for the Slater determinants $|{\mathrm{\Phi }}_a\rangle$ [panel (a)] and $|{\mathrm{\Phi }}_b\rangle$ [panel (b)], as specified in Table 2. Points calculated with the same $M_{\alpha }=M_{\gamma }$ are connected by straight lines to guide the eye. On each panel, the results obtained with the largest value of $M_{\alpha }=M_{\gamma }$ cannot be distinguished from exact results for the values of $J_0$ shown.

Figure 8

Evolution of the weight $|c^{J_0{K}_0}\left(M_{\beta }\right){|}^2$ of the numerically projected states as a function of the angular momentum $J_0$ on which one projects for different choices for the number of discretization points $M_{\beta }$ for the Slater determinants $|{\mathrm{\Phi }}_c\rangle$ [panel (c)] and $|{\mathrm{\Phi }}_d\rangle$ [panel (d], as specified in Table 2. Points calculated with the same $M_{\beta }$ are connected by straight lines to guide the eye. On each panel, the results obtained with the largest value of $M_{\beta }$ is the first one for which the projection is exact for all values of $J_0$ plotted.

Figure 9

Evolution of the difference between the numerical values for the weight $|c^{J_0{K}_0}\left(M_{\beta }\right){|}^2$ at $M_{\beta }$ as indicated and the converged values $|c^{J_0{K}_0}\left(M_{\beta }^c\right){|}^2$ obtained with $M_{\beta }^c=11$ in logarithmic scale as a function of angular momentum $J_0$ on which one projects for the Slater determinants $|{\mathrm{\Phi }}_c\rangle$ [panel (c)] and $|{\mathrm{\Phi }}_d\rangle$ [panel (d)], as specified in Table 2. Points calculated with the same $M_{\beta }$ are connected by straight lines to guide the eye.

Figure 10

Evolution in logarithmic scale of the difference $J_0^e-J_0$, where $J_0^e$ is extracted from Eq. (123), for the state $|{\mathrm{\Phi }}_d\rangle$ as a function of the angular momentum $J_0$ onto which one projects for different choices of the number of discretization points $M_{\beta }$. Points calculated with the same $M_{\beta }$ are connected by straight lines to guide the eye.

Figure 11

Norm kernel as a function of the Euler angle $\beta$ on a logarithmic scale for axial reflection-symmetric states of ${}^{240}\mathrm{Pu}$ with dimensionless quadrupole moment ${\beta }_{20}$ and dispersion of angular momentum $\langle {\mathrm{\Phi }}_a\left|{\widehat{J}}_{\perp }^2\right|{\mathrm{\Phi }}_a\rangle$ perpendicular to the symmetry axis as indicated. Dots indicate calculated values for the kernel using $M_{\beta }=96$ points, whereas the dotted lines represent the estimate defined in Eq. (124). For axial reflection-symmetric states, the norm kernel is a symmetric function around $\beta =\pi /2$ $\iff$ $cos\left(\beta \right)=0$, $\langle {\mathrm{\Phi }}_a|\widehat{R}(0,\beta ,0)|{\mathrm{\Phi }}_a\rangle =\langle {\mathrm{\Phi }}_a|\widehat{R}(0,\pi -\beta ,0)|{\mathrm{\Phi }}_a\rangle$, cf. Eqs. (132b) and (132c), such that only half the integration interval for $\beta$ is shown.

Figure 12

Decomposition of the weights $|c^{J_{0}0}{|}^2$ of the projected components as defined in Eq. (121) on a logarithmic scale for axial reflection-symmetric states of ${}^{240}\mathrm{Pu}$ with dimensionless quadrupole moment ${\beta }_{20}$ and number of discretization points $M_{\beta }$ as indicated. The number in parentheses indicates the number of Euler angles whose contribution is at most ten orders of magnitude smaller than the largest one. Because of the reflection symmetry imposed on the quasiparticle vacua, the $|c^{J_{0}0}{|}^2$ are exactly zero for odd angular momenta and therefore not plotted.

Figure 13

Top: quantity ${\mathrm{\Xi }}^J$ for the selected quasiparticle states discussed in the text. Bottom: same for the quantity ${\mathrm{\Xi }}_J$.

Figure 14

Energy spectrum built from the expectation values $E_{iK}^{JNZ}$ for different values of $K$ obtained in the decomposition of the “axial state” as defined in Table 3.

Figure 15

Same as Fig. 14 for the decomposition states of the states $|{\mathrm{\Phi }}_a\rangle$ [panel (a)] and $|{\mathrm{\Phi }}_b\rangle$ [panel (b)] as defined in Tab. 4. The colors of the levels indicate the value of $K$, whereas the labels next to the levels indicate the value of $J$.

Figure 16

Spectrum of $J=9/2$ states before and after $K$ mixing for the states $|{\mathrm{\Phi }}_{\text{a}}\rangle$ and $|{\mathrm{\Phi }}_{\text{b}}\rangle$ as defined in Table 4. The energies $E_{iK}^{JNZ}$ (155) of $K$ components are drawn in the same color code as used in Fig. 15, whereas energy levels after $K$ mixing are drawn in black. Note that the three lowest $K$ components of state $|{\mathrm{\Phi }}_{\text{a}}\rangle$ are quasidegenerate, such that their levels fall on top of each other on the scale of the plot.

Figure 17

Summed weight ${\mathrm{\Xi }}^J$ as defined in Eq. (157) of components with given $J$ projected out from one-quasiparticle states for ${}^{25}\mathrm{Mg}$ obtained by blocking single particles in originally time-reversed partner states of opposite $x$ signatures $\eta$, and cranked to different values of $J_c$.

Figure 18

Projected energies, as defined in Eq. (158), of yrast states projected from two one-quasiparticle states for ${}^{25}\mathrm{Mg}$ obtained from blocking two originally time-reversed partners of opposite $x$ signatures ${\eta }_a$ cranked to different values of $J_c$ as indicated.