Similarity renormalization group for nucleon-nucleon interactions

The similarity renormalization group (SRG) is based on unitary transformations that suppress off-diagonal matrix elements, forcing the Hamiltonian toward a band-diagonal form. A simple SRG transformation applied to nucleon-nucleon interactions leads to greatly improved convergence properties while preserving observables and provides a method to consistently evolve many-body potentials and other operators.

Figure 1

Contour plots of $V_s(k,k^{\prime })$ illustrating the evolution with $\lambda \equiv s^{-1/4}$ for ${}^1\mathrm{S}$${}_0$ (left) and ${}^3\mathrm{S}$${}_1$ (right). The initial potential on the left is a chiral N${}^3\mathrm{LO}$ potential with a 600-MeV cutoff [12] and on the right is an N${}^3\mathrm{LO}$ potential with a 550-MeV cutoff on the Lippmann-Schwinger equation and a 600-MeV cutoff on a regularized spectral representation of two-pion exchange [13].

Figure 2

$S$-wave phase shifts from the two chiral EFT N${}^3\mathrm{LO}$ potentials from Fig. 1. For each initial potential, the phase shifts for different λ agree to within the widths of the lines at all energies shown.

Figure 3

Matrix elements of the evolved SRG potentials at $\lambda =2{\mathrm{fm}}^{-1}$ for ${}^1\mathrm{S}$${}_0$ (top, diagonal elements) and ${}^3\mathrm{S}$${}_1$ (bottom, off-diagonal elements) for the same initial potentials as in Fig. 1. Also shown is the $V_{\mathrm{low}k}$ potential with a smooth (exponential) regulator for momentum cutoff $\Lambda =2{\mathrm{fm}}^{-1}$, evolved from the two potentials (600 MeV above and 550/600 MeV below).

Figure 4

The largest repulsive Weinberg eigenvalues as a function of λ in the ${}^1\mathrm{S}$${}_0$ channel and the ${}^3\mathrm{S}$${}_1{‐}^3$D${}_1$ coupled channel for the same initial potentials as in Fig. 1.

Figure 5

The absolute error vs. λ of the predicted deuteron binding energy from a variational calculation in a fixed-size basis of harmonic oscillators ($N_{\max }\hslash \omega$ excitations). The initial potential is from Ref. [13].

Figure 6

The variational binding energy for selected λ of the triton with two-nucleon interactions only, as a function of the size of the harmonic oscillator space ($N_{\max }\hslash \omega$ excitations), for the same initial potentials as in Fig. 1.