Advanced density matrix renormalization group method for nuclear structure calculations

We present an efficient implementation of the Density Matrix Renormalization Group (DMRG) algorithm that includes an optimal ordering of the proton and neutron orbitals and an efficient expansion of the active space utilizing various concepts of quantum information theory. We first show how this new DMRG methodology could solve a previous 400 keV discrepancy in the ground state energy of ${}^{56}\mathrm{Ni}$. We then report the first DMRG results in the $pf+g9/2$ shell model space for the ground $0^{+}$ and first $2^{+}$ states of ${}^{64}\mathrm{Ge}$ which are benchmarked with reference data obtained from a Monte Carlo shell model. The corresponding correlation structure among the proton and neutron orbitals is determined in terms of two-orbital mutual information. Based on such correlation graphs we propose several further algorithmic improvement possibilities that can be utilized in a new generation of tensor network based algorithms.

Figure 1

DMRG ground state energy for ${}^{56}\mathrm{Ni}$ and ${}^{64}\mathrm{Ge}$ in the $pf$ shell and ${}^{64}\mathrm{Ge}$ in the $pf+g9/2$ valence space as functions of $1/M$; diamonds (DMRG), crosses (exact diagonalization), circle (MCSM). The solid lines are the fits that produce our extrapolated value, while the dotted lines joining the points not used in the fit are just to guide the eye.

Figure 2

Single-site entropy (a) and mutual information (b) obtained with DMRG for ${}^{64}\mathrm{Ge}$ in the $pf+g9/2$ shell. The mutual information matrix elements are shown on a ladder topology with time-reversed pairs in the rungs. Circles and diamonds label proton and neutron orbitals, respectively, and sites are denoted by $l,j,m$ with $+m$ outside the ladder and $-m$ inside.

Figure 3

Energy gaps between the 0 and $2^{+}$ states for ${}^{64}\mathrm{Ge}$ with $pf$ (a) and $pf+g9/2$ (b) valence space as a function of $1/M$; circles (DMRG), cross (exact diagonalization), diamond (MCSM). The solid lines are our fits.