Similarity renormalization group for nuclear forces

Abstract

Renormalization group methods generate low-resolution Hamiltonians that are more diagonal and easier to solve. This chapter reviews the similarity renormalization group for nuclear Hamiltonians, which is a popular method for generating low-resolution nuclear forces. It presents the similarity renormalization group flow equations, analyzes how the similarity renormalization group drives the Hamiltonian towards the diagonal, and studies the effect of induced many-body interactions. It concludes by highlighting the progress in first-principles calculations of nuclei driven by low-resolution nuclear Hamiltonians.

Figures (6)

• Figure 1: Jacobi momentum-space nucleon-nucleon potential matrix elements $V(p',p)$ in the neutron-proton $^1\mathrm{S}_0$ channel for various Hamiltonians. We consider two initial Hamiltonians: the "EM500" nucleon-nucleon potential (top) from chiral effective field theory at next-to-next-to-next-to-leading order Entem2003PRC_EM500; and the phenomenological "AV18" nucleon-nucleon potential (bottom) Wiringa1995PRC_AV18. Due to its repulsive short-range core, AV18 clearly couples low and high momenta strongly with nonzero matrix elements $V(0, p)$ beyond $p=5\:\mathrm{fm}^{-1}$. On the other hand, the $\Lambda=500\:\mathrm{MeV}$ cutoff of the EM500 potential regularizes high-momentum behavior such that matrix elements with $p, p' \gtrsim 3.2\:\mathrm{fm}^{-1}$ are approximately 0. For both potentials, we show the SRG transformation to lower resolution scales $\lambda$, down to $\lambda=1.8\:\mathrm{fm}^{-1}$. We see that the potentials are driven towards a band-diagonal form, and matrix elements $V(0, p)$ with $p>\lambda$ are suppressed. Once SRG transformed to low resolution, AV18 and EM500 have very similar low-momentum matrix elements.
• Figure 2: Matrix element suppression for the matrix elements $V_\lambda(0, p)$ visualized through the ratio $V_\lambda(0, p)/ V_\infty(0, p)$. The initial Hamiltonian is the EM500 potential in the neutron-proton $^1\mathrm{S}_0$ channel. As the EM500 potential is SRG transformed to lower resolution scales $\lambda$, the off-diagonal matrix elements with $p>\lambda$ are exponentially suppressed as predicted in Eq. \ref{['eq:SRGSuppression']}.
• Figure 3: The ground-state energies of $^3$H (left) and $^4$He (right) computed with nuclear Hamiltonians from chiral effective field theory that have been transformed using the similarity renormalization group to lower resolution scales $\lambda$. We compare the behavior of the ground-state energy for various $\lambda$ when only SRG-evolved nucleon-nucleon potentials are used (NN-only, black line), when the SRG-induced three-nucleon interactions are included (NN + NNN-induced, red line), and when a three-nucleon potential is included in the initial Hamiltonian (NN + NNN, blue line). For $^3$H, accounting for induced three-nucleon interactions is required for SRG invariance of the ground-state energy, and the inclusion of explicit three-nucleon forces is required to reproduce the experimental value. For $\lambda \geq 1.8\:\mathrm{fm}^{-1}$, the predicted ground-state energy of $^4$He is approximately independent of $\lambda$, indicating that induced four-body forces are small. Figure adapted from Jurgenson2009PRL_SRG_ManyBody.
• Figure 4: Ground-state energies of $^4$He (top) and $^6$Li (bottom) computed with the no-core shell model using Hamiltonians from chiral effective field theory that have been transformed to lower resolution scales $\lambda$. The SRG scales are $\lambda = 2.2\:\mathrm{fm}^{-1}$ (dark blue circles), $2.1\:\mathrm{fm}^{-1}$ (red diamonds), $2.0\:\mathrm{fm}^{-1}$ (green triangles), $1.9\:\mathrm{fm}^{-1}$ (purple squares), and $1.6\:\mathrm{fm}^{-1}$ (teal stars). In no-core shell model calculations, the Hamiltonian is expanded in a basis truncated based on the parameter $N_\mathrm{max}$. The ground-state energies are shown as a function of $N_\mathrm{max}$, where the extrapolation to $N_\mathrm{max}\to\infty$ is indicated on the very right. As in Fig. \ref{['fig:TritonFlow']}, we consider cases where only the nucleon-nucleon force has been SRG transformed (NN only), where the SRG-induced three-nucleon interactions are accounted for (NN+3N-induced), and where the Hamiltonian also includes an initial three-nucleon force before the SRG transformation (NN+3N-full). As in Fig. \ref{['fig:TritonFlow']}, consistently transforming both two- and three-nucleon potentials is required to achieve SRG invariance and the inclusion of the the initial three-nucleon force is important for agreement with experiment. Comparing $\lambda = 2.2\:\mathrm{fm}^{-1}$ and $1.6\:\mathrm{fm}^{-1}$ (dark blue circles and teal stars, respectively), the ground-state energy converges much more quickly in $N_\mathrm{max}$ for $\lambda=1.6\:\mathrm{fm}^{-1}$. This improved convergence substantially reduces the computational cost required to deliver converged calculations and makes calculations in heavier nuclei up to around $^{16}$O possible. Figure from Roth2011PRL_SRG3N.
• Figure 5: Ground-state energies of $^{40}$Ca computed with the in-medium similarity renormalization group using Hamiltonians from chiral effective field theory that have been transformed to lower resolution scales $\lambda$. The SRG scales are unevolved, i.e., $\lambda = \infty$, (orange), $2.2\:\mathrm{fm}^{-1}$ (green), $2.0\:\mathrm{fm}^{-1}$ (blue), and $1.8\:\mathrm{fm}^{-1}$ (red). In in-medium similarity renormalization group calculations, the Hamiltonian is expanded in a single-particle basis truncated based on the parameter $e_\mathrm{max}$. As in Fig. \ref{['fig:TritonFlow']}, we consider cases where only the nucleon-nucleon force has been SRG transformed (NN-only), where the SRG-induced three-nucleon interactions are accounted for (NN + 3N-induced), and where the Hamiltonian also includes an initial three-nucleon force before the SRG transformation (NN + 3N-full). Unevolved Hamiltonians converge slowly in $e_\mathrm{max}$ in all cases, while SRG-transformed Hamiltonians converge quickly. This improved convergence substantially reduces the computational cost required to deliver converged calculations. Figure adapted from Hoppe2019PRC_ChiralMedMass.