Unitary correlation operator method and similarity renormalization group: Connections and differences

We discuss relations and differences between two methods for the construction of unitarily transformed effective interactions, the Similarity Renormalization Group (SRG) and Unitary Correlation Operator Method (UCOM). The aim of both methods is to construct a soft phase-shift equivalent effective interaction which is well suited for many-body calculations in limited model spaces. After contrasting the two conceptual frameworks, we establish a formal connection between the initial SRG-generator and the static generators of the UCOM transformation. Furthermore we propose a mapping procedure to extract UCOM correlation functions from the SRG evolution. We compare the effective interactions resulting from the UCOM-transformation and the SRG-evolution on the level of matrix elements, in no-core shell model calculations of light nuclei, and in Hartree-Fock calculations up to ${}^{208}\mathrm{Pb}$. Both interactions exhibit very similar convergence properties in light nuclei but show a different systematic behavior as function of particle number.

Figure 1

Illustration of the SRG evolution of the momentum-space matrix elements and the deuteron wave function starting from the Argonne V18 potential. The four columns correspond to different values of the flow parameter: (a) $\overline{\alpha }=0\hspace{0.5em}{\mathrm{fm}}^4$, (b) $0.001\hspace{0.5em}{\mathrm{fm}}^4$, (c) $0.01\hspace{0.5em}{\mathrm{fm}}^4$, and (d) $0.04\hspace{0.5em}{\mathrm{fm}}^4$. The upper two rows depict the matrix elements $V_{\alpha }(q,q^{\text{'}})$ for the ${}^3{S}_1$ and the ${}^3{S}_1{-}^3{D}_1$ partial waves, respectively, in units of $\mathrm{MeV}{\mathrm{fm}}^3$. The bottom row shows the radial coordinate-space wave functions ${\varphi }_L(r)$ of the deuteron ground state obtained with the respective SRG-evolved matrix elements: (blue line) $L=0$, (red dashed line) $L=2$.

Figure 2

Radial wave function ${\varphi }_0(r)$ and resulting correlation function $R_{+}(r)-r$ for the ${}^1{S}_0$ partial wave for different flow parameters: $\overline{\alpha }=0.02\hspace{0.5em}{\mathrm{fm}}^4$ (blue line), $0.04\hspace{0.5em}{\mathrm{fm}}^4$ (red dashed line), $0.06\hspace{0.5em}{\mathrm{fm}}^4$ (green dot-dashed line), $0.08\hspace{0.5em}{\mathrm{fm}}^4$ (violet dotted line). The thin solid curve in panel (a) shows the corresponding wave function for the initial potential used for the mapping.

Figure 3

Radial wave functions of the $S$ and $D$-wave component, ${\varphi }_0(r)$ and ${\varphi }_2(r)$ and resulting central correlation function $R_{+}(r)-r$ as well as the tensor correlation function $\vartheta (r)$ for the ${}^3{S}_1-{}^3{D}_1$ partial waves for different flow parameters: $\overline{\alpha }=0.02\hspace{0.5em}{\mathrm{fm}}^4$ (blue line), $0.04\hspace{0.5em}{\mathrm{fm}}^4$ (red dashed line), $0.06\hspace{0.5em}{\mathrm{fm}}^4$ (green dot-dashed line), $0.08\hspace{0.5em}{\mathrm{fm}}^4$ (violet dotted line). The thin solid curve in panel (a) shows the corresponding wave functions for the initial potential used for the mapping.

Figure 4

Central correlation functions $R_{+}(r)-r$ derived from the ground state (blue line), the second (red dashed line), the fourth (green dot-dashed line), and the sixth excited states (violet dotted line) in the ${}^1{S}_0$ partial wave for the flow parameter $\overline{\alpha }=0.04\hspace{0.5em}{\mathrm{fm}}^4$. The correlation functions are plotted only up to the position of the first zero of the radial wave functions.

Figure 5

Central correlation functions $R_{+}(r)-r$ for different spin and isospin channels obtained from an energy minimization (blue line) and from the SRG-mapping (red dashed line).

Figure 6

Tensor correlation functions $\vartheta (r)$ obtained from an energy minimization (blue line) and from the SRG-mapping (red dashed line).

Figure 7

Momentum-space matrix elements $\langle q(\mathit{LS})\mathit{JT}|\hspace{0.5em}ˆ\hspace{0.5em}|q^{\text{'}}(L^{\text{'}}S)\mathit{JT}\rangle$ (in units of $\mathrm{MeV}\hspace{0.5em}{\mathrm{fm}}^3$) for the ${}^1{S}_0$ partial wave obtained from (a) the initial AV18 potential, (b) the SRG-evolved interaction ($\overline{\alpha }=0.03\hspace{0.5em}{\mathrm{fm}}^4$), (c) the UCOM-transformed interaction using the SRG-generated correlators ($\overline{\alpha }=0.04\hspace{0.5em}{\mathrm{fm}}^4$), and (d) the UCOM-transformed interaction using the variationally optimized correlators.

Figure 8

Relative harmonic-oscillator matrix elements $\langle n(\mathit{LS})\mathit{JT}|\hspace{0.5em}ˆ\hspace{0.5em}|n^{\text{'}}(L^{\text{'}}S)\mathit{JT}\rangle$ (in units of $\mathrm{MeV}$) for an oscillator frequency $\hslash \Omega =20\hspace{1em}\mathrm{MeV}$ in the ${}^1{S}_0$ partial wave obtained with the same interactions as in Fig. 7.

Figure 9

Convergence behavior of the ground-state energy of ${}^3\mathrm{H}$ obtained in no-core shell model calculations as function of $\hslash \Omega$ for different potentials. The various curves correspond to different $N_{max}\hslash \Omega$ model spaces in the range $N_{max}=6,12,18,\dots ,48$ as indicated by the labels. The different potentials are (a) untransformed AV18 potential (note the different energy scale). (b) SRG-evolved AV18 potential for $\overline{\alpha }=0.03\hspace{0.5em}{\mathrm{fm}}^4$. (c) UCOM-transformed AV18 potential using the SRG-generated correlators for $\overline{\alpha }=0.04\hspace{0.5em}{\mathrm{fm}}^4$. (d) UCOM-transformed AV18 potential using the standard correlators constructed by energy minimization. The dashed horizontal line indicates the experimental ground-state energy.

Figure 10

Convergence behavior of the ground-state energy of ${}^4\mathrm{He}$ obtained in no-core shell model calculations as function of $\hslash \Omega$ for different potentials. The various curves correspond to different $N_{max}\hslash \Omega$ model spaces in the range $N_{max}=0,2,4$,..., 16 as indicated by the labels. The different potentials are as described in Fig. 9.

Figure 11

Hartree-Fock ground-state energies per particle and charge radii obtained with the SRG-evolved AV18 potential (blue circle), the UCOM-transformed AV18 using SRG-generated correlators (red square), and the UCOM-transformed interaction using the variationally optimized correlators (green diamond). The parameters of all transformed interactions are the same as in the NCSM calculations. Experimental data are represented by black bars [22, 23].