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Few-body systems. Supplementum

Effective Linear Two-Body Method for Many-Body Problems in Atomic and Nuclear Physics

Y.E. Kim, Alexander L. Zubarev

Published Jan 1, 2000• 0 citations
Publisher
DOI
10.1007/978-3-7091-6287-3_2

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Effective field theories have established themselves as key pillars of modern nuclear physics. They enable a quantitative understanding of the strong nuclear force, provided low-energy constants that parametrize short-distance physics can be determined from experimental data. This, however, often becomes prohibitively expensive due to a significant computational cost of solving the A-body problem. The computational challenge is particularly severe for three-body forces, which are at the frontier of nuclear and atomic physics and play an important role in the equation of state of neutron stars. Here we prove that for a parametric low-rank update of a Hamiltonian, the A-body problem at a fixed energy exactly reduces to a low-dimensional matrix equation regardless of the size of the Hilbert space. As a proof-of-principle, we present exact and computationally cheap snapshot-based emulators for few-body scattering and bound states. Unlike alternatives, our emulators can be used far away from the snapshot region without loss of precision and yield accurate results for parameter values not accessible using conventional solution techniques. Our approach is not restricted by the interaction type, number of particles, and methods for generating snapshots and can be applied to mitigate the computational burden of the A-body problem to a broad class of problems in nuclear, atomic, and molecular physics.
The quantum three body problem generally has its own complexities and is difficult to solve. It requires rigorous steps and iterations to find the solution. We often encounter three body system in various branches of physics including Nuclear Physics, Particle, Atomic, Molecular physics, and condensed matter. The underlying physics involved is quite complicated. Faddeev in 1961 made a successful attempt and formulated theory of three particle scattering in a rather rigorous mathematical way. Efimov developed a method that enabled one to obtain the universal properties of three particle system and is quite independent of the two body potentials. This article attempts to introduce Efimov effect. The formalism and the salient features are also discussed.
In most nuclear many-body methods, observables are calculated using many-body wave functions explicitly. The variational two-particle reduced density matrix method is one of the few exceptions to the rule. Ground-state energies of both closed-shell and open-shell nuclear systems can indeed be evaluated by minimizing a constrained linear functional of the two-particle reduced density matrix. However, it has virtually never been used in nuclear theory, because nuclear ground states were found to be well overbound, contrary to those of atoms and molecules. Consequently, we introduced new constraints in the nuclear variational two-particle reduced density matrix method, developed recently for atomic and molecular systems. Our calculations then show that this approach can provide a proper description of nuclear systems where only valence neutrons are included. For the nuclear systems where both neutrons and protons are active, however, the energies obtained with the variational two-particle reduced density matrix method are still overbound. The possible reasons for the noticed discrepancies and solutions to this problem will be discussed.