Tensor-decomposition techniques for ab initio nuclear structure calculations. From chiral nuclear potentials to ground-state energies

TL;DR

This paper explores tensor decomposition methods, especially tensor hypercontraction, to efficiently approximate nuclear Hamiltonians from chiral effective field theory, enabling accurate and scalable ground-state energy calculations in nuclear physics.

Contribution

It demonstrates that tensor hypercontraction can compress two-body matrix elements effectively, allowing controlled approximations of nuclear energies with minimal error.

Findings

Tensor hypercontraction yields low tensor ranks for nuclear Hamiltonians.

Decomposition error remains small for nuclei like helium, oxygen, and calcium.

Normal-ordered three-body contributions do not hinder data compression efficiency.

Abstract

The impact of applying state-of-the-art tensor factorization techniques to modern nuclear Hamiltonians derived from chiral effective field theory is investigated. Subsequently, the error induced by the tensor decomposition of the input Hamiltonian on ground-state energies of closed-shell nuclei calculated via second-order many-body perturbation theory is benchmarked. With the aid of the factorized Hamiltonian, the second-order perturbative correction to ground-state energies is decomposed and the scaling properties of the underlying tensor network are discussed. The employed tensor formats are found to lead to an efficient data compression of two-body matrix elements of the nuclear Hamiltonian. In particular, the sophisticated \emph{tensor hypercontraction} (THC) scheme yields low tensor ranks with respect to both harmonic-oscillator and Hartree-Fock single-particle bases. It is found that the tensor rank depends on the two-body total angular momentum $J$ for which one performs the decomposition, which is itself directly related to the sparsity the corresponding tensor. Furthermore, including normal-ordered two-body contributions originating from three-body interactions does not compromise the efficient data compression. Ultimately, the use of factorized matrix elements authorizes controlled approximations of the exact second-order ground-state energy corrections. In particular, a small enough error is obtained from low-rank factorizations in $^{4}$He, $^{16}$O and $^{40}$Ca.