Operator evolution for ab initio theory of light nuclei

The past two decades have seen a revolution in ab initio calculations of nuclear properties. One key element has been the development of a rigorous effective interaction theory, applying unitary transformations to soften the nuclear Hamiltonian and hence accelerate the convergence as a function of the model space size. For consistency, however, one ought to apply the same transformation to other operators when calculating transitions and mean values from the eigenstates of the renormalized Hamiltonian. Working in a translationally invariant harmonic oscillator basis for the two- and three-nucleon systems, we evolve the Hamiltonian, square radius, and total dipole strength operators by the similarity renormalization group (SRG). The inclusion of up to three-body matrix elements in the ${}^4$He nucleus all but completely restores the invariance of the expectation values under the transformation. We also consider a Gaussian operator with adjustable range; short ranges have the largest absolute renormalization when including two- and three-body induced terms, while at long ranges the induced three-body contribution takes on increased relative importance.

Figure 1

${}^3\mathrm{H}$ rms radius as a function of the SRG evolution parameter, $\lambda$. Shown are results obtained with wave functions from two Hamiltonians: $\mathit{NN}+3N$-induced (blue [gray] dashed line) and $\mathit{NN}+3N$ (red [gray] solid line), and three levels of operator evolution: bare operator (circles), operator evolved in the two-body space (squares), and operator evolved in the three-body space (triangles). The dotted line is the rms radius calculated using the bare Hamiltonian and bare operator.

Figure 2

Calculations of ${}^4\mathrm{He}$ ground-state energy (a), rms radius (b), and total strength of the dipole transition (c) for $N_{\max }=18$, with a range of $\lambda$ from $1.5\mathrm{fm}{}^{-1}$ to $3.0\mathrm{fm}{}^{-1}$. The purple (gray) dot-dashed line indicates results obtained with the $\mathit{NN}$-only Hamiltonian. The dotted line in panel (b) indicates the rms radius computed using the bare Hamiltonian and bare operator. See also caption of Fig. 1.

Figure 3

Renormalization percent as a function of range of a Gaussian operator [see Eq. (6)] for three values of the SRG parameter, $\lambda$: $1.5$ $\mathrm{fm}{}^{-1}$ (solid line), $2.0$ $\mathrm{fm}{}^{-1}$ (dashed line), and $2.5$ $\mathrm{fm}{}^{-1}$ (dot-dashed line). Symbols as in the caption of Fig. 1. The wave function is from the $\mathit{NN}+3N$ SRG evolved Hamiltonian.