Matrix elements and few-body calculations within the unitary correlation operator method

We employ the unitary correlation operator method (UCOM) to construct correlated, low-momentum matrix elements of realistic nucleon-nucleon interactions. The dominant short-range central and tensor correlations induced by the interaction are included explicitly by an unitary transformation. Using correlated momentum-space matrix elements of the Argonne V18 potential, we show that the unitary transformation eliminates the strong off-diagonal contributions caused by the short-range repulsion and the tensor interaction and leaves a correlated interaction dominated by low-momentum contributions. We use correlated harmonic oscillator matrix elements as input for no-core shell model calculations for few-nucleon systems. Compared to the bare interaction, the convergence properties are dramatically improved. The bulk of the binding energy can already be obtained in very small model spaces or even with a single Slater determinant. Residual long-range correlations, not treated explicitly by the unitary transformation, can easily be described in model spaces of moderate size allowing for fast convergence. By varying the range of the tensor correlator we are able to map out the Tjon line and can in turn constrain the optimal correlator ranges.

Figure 1

(Color online) Application of the central and tensor correlators to a deuteron-like two-body wave function. Panels (a), (c), and (e) depict the uncorrelated, the central correlated, and the fully correlated radial wave functions, respectively. The panels (b) and (d) show the corresponding central and tensor correlation functions (see text).

Figure 2

(Color online) Optimal central correlation functions $R_{+}(r)-r$ for the AV18 potential according to the parameters given in Table . The curves correspond to the different spin-isospin channels: $(S,T)=(0,1)$ (), (1, 0) (), (0, 0) (), (1, 1) ().

Figure 3

(Color online) Optimal tensor correlation functions $\vartheta (r)$ for different values of the range constraint $I_{\vartheta }$. (a) Correlation functions for $T=0$ with $I_{\vartheta }=0.06,0.09$, and $0.12{\text{fm}}^3$. (b) Correlation functions for $T=1$ with $I_{\vartheta }=0.01,0.03$, and $0.06{\text{fm}}^3$. The arrows indicate the direction of increasing $I_{\vartheta }$.

Figure 4

(Color online) Relative momentum-space matrix elements of the bare AV18 potential (left-hand column), the central correlated AV18 potential (center column), and the fully correlated AV18 potential. The different rows correspond to different partial waves: ${}^1{S}_0$ (top row), ${}^3{S}_1$ (middle row), and ${}^3{S}_1-{}^3{D}_1$ (bottom row). The optimal tensor correlator for $I_{\vartheta }=0.09{\text{fm}}^3$ is used. The red dots mark the plane of vanishing matrix elements. The momenta are given in fm${}^{-1}$ and the matrix elements in MeV.

Figure 5

(Color online) Momentum-space matrix elements of the correlated AV18 potential for different ranges of the tensor correlator (solid lines): $I_{\vartheta }=0.06{\text{fm}}^3,0.09{\text{fm}}^3,0.12{\text{fm}}^3$. The arrow indicates the direction of increasing correlator range. The dashed line represents the matrix elements of the bare potential. The upper panel shows diagonal matrix elements in the ${}^3{S}_1$ channel as function of $q=q^{\text{'}}$. The lower panel depicts the off-diagonal ${}^3{S}_1-{}^3{D}_1$ matrix elements for fixed $q=0$ as function of $q^{\text{'}}$.

Figure 6

(Color online) Comparison of the diagonal momentum space matrix elements of the correlated interaction $v_{\mathrm{UCOM}}$ (red lines) with the $V_{\text{low}k}$ matrix elements (blue dots) for the AV18 potential. The dashed line corresponds to the matrix elements of the bare potential. The optimal tensor correlators for $I_{\vartheta }=0.08{\text{fm}}^3$ are used and the $V_{\text{low}k}$ momentum cutoff is $\Lambda =2.1{\text{fm}}^{-1}$.

Figure 7

(Color online) Ground-state energy of ${}^4\mathrm{He}$ as function of the oscillator parameter $\hslash \Omega$ for different model model-space sizes ${\mathcal{N}}_{max}=0,2,\dots ,16$ as indicated by the labels. The upper panel shows results for the bare AV18 potential, the lower panel corresponds to the correlated potential $v_{\mathrm{UCOM}}$ for $I_{\vartheta }=0.09{\text{fm}}^3$. The horizontal lines represent the exact binding energy for the bare potential taken from Ref. 29.

Figure 8

(Color online) Ground-state energy of ${}^3\mathrm{H}$ and ${}^4\mathrm{He}$ as function of the oscillator parameter $\hslash \Omega$ for the correlated AV18 potential ($I_{\vartheta }=0.09{\text{fm}}^3$) obtained in a no-core shell-model diagonalization. The different curves correspond to different model-space sizes ${\mathcal{N}}_{max}$ as indicated by the labels. The horizontal lines represent the exact binding energies for the bare potential taken from Ref. 29.

Figure 9

(Color online) Binding energies of ${}^4\mathrm{He}$ versus ${}^3\mathrm{H}$. The blue circles show the results of exact Faddeev calculations obtained by A. Nogga et al. [29] using different modern NN-potentials. The green diamonds show results obtained by including simple three-body forces in addition to the realistic two-nucleon potentials [29]. The red triangles are the converged no-core shell model results for the correlated AV18 potential for different values $I_{\vartheta }=0.03,\dots ,0.12{\text{fm}}^3$ of the range-constraint for the tensor correlator.