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

Abstract

Background:The computational resources needed to generate the ab initio solution of the nuclear many-body problem for increasing mass number and/or accuracy necessitates innovative developments to improve upon (1) the storage of many-body operators and (2) the scaling of many-body methods used to evaluate nuclear observables. The storing and efficient handling of many-body operators with high particle ranks is currently one of the major bottlenecks limiting the applicability range of ab initio studies with respect to mass number and accuracy. Recently, the application of tensor decomposition techniques to many-body tensors has proven highly beneficial to reduce the computational cost of ab initio calculations in quantum chemistry and solid-state physics.Purpose: 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. Methods:The first proof-of-principle application of tensor-decomposition techniques to the nuclear Hamiltonian is performed. Two different tensor formats are investigated by systematically benchmarking the approximation error on matrix elements stored in various bases of interest. The analysis is achieved while including normalordered three-nucleon interactions that are nowadays used as input to the most advanced ab initio calculations in medium-mass nuclei. 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. Results: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 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. Conclusions:It is presently demonstrated that tensor-decomposition techniques can be efficiently applied to systematically approximate the nuclear many-body Hamiltonian in terms of lower-rank tensors. Beyond the input Hamiltonian, tensor-decomposition techniques can be envisioned to scale down the cost of state-of-the-art non-perturbative man...

Many-Body Perturbation Theories for Finite Nuclei

“…However, due to the increasing number of basis states these efforts will require significant computational resources as well as extensive formal developments. To control the increase of computational requirements (memory and runtime), socalled tensor factorization techniques have been proposed recently where high-mode tensors are decomposed into sums of product of lower-rank ones [107,119]. While initial proof-of-principle applications have demonstrated the high potential, extensive additional research is required in this direction.…”

Novel chiral Hamiltonian and observables in light and medium-mass nuclei

“…Here one-and two-body operators are always included in full. However, this is not feasible for three-body interactions due to exponential increase in the number of their matrix elements and therefore 12 these are restricted to e 3max = 16 < 3 e max . The dependence on the basis parameters was tested by computing ground-state observables for different harmonic oscillator frequencies, Ω, and model space sizes, e max .…”

“…Next let us examine rms charge radii r 2 ch 1/2 . In the present approach rms charge radii are computed from rms 12 For what concerns absolute radii, a large variation between the different interactions is observed in all cases. For oxygen isotopes, studies with NN+3N (400) and NNLO sat exist in the literature [16], where was shown that already in these light systems NN+3N (400) leads to a strong underestimation of the size of nuclei.…”

A Guided Tour of ab initio Nuclear Many-Body Theory

“…The aforementioned steps make use of prior theoretical knowledge, e.g., to identify desired decoupling patterns in interactions, or define analytical measures for the importance of basis states. If such knowledge is not available, or we want to avoid bias, we can leverage a myriad of Principal Component Analysis (PCA) methods to factorize interactions or intermediate quantities in many-body calculations [271,272]. This can potentially even give us control over the computational scaling of nuclear many-body methods (see, e.g., [273][274][275][276][277]).…”