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Computational Science & Discovery

Computational aspects of nuclear coupled-cluster theory

David J. Dean, G. Hagen, Morten Hjorth-Jensen, T. Papenbrock

Published Nov 20, 2008• 6 citations
Publisher

Abstract

Coupled-cluster (CC) theory represents an important theoretical tool that we use to solve the quantum many-body problem. CC theory also lends itself to computation in a parallel computing environment. In this paper, we present selected results from ab initio studies of stable and weakly bound nuclei utilizing computational techniques that we employ to solve CC theory. We also outline several perspectives for future research directions in this area.

DOI
10.1088/1749-4699/1/1/015008

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Coupled-cluster (CC) theory represents an important theoretical tool that we use to solve the quantum many-body problem. CC theory also lends itself to computation in a parallel computing environment. In this paper, we present selected results from ab initio studies of stable and weakly bound nuclei utilizing computational techniques that we employ to solve CC theory. We also outline several perspectives for future research directions in this area.
Solving the nuclear many-body problem engenders significant theoretical and computational challenges. I will briefly discuss the science that goes into and comes out of the nuclear problem, and then turn to the numerical implementations of coupled-cluster theory for the problem. Specific to the nuclear problem is the presence of real three-body forces which are computationally represented by six-dimensional tensors. Both memory and communications become issues as one begins to scale up the underlying coupled-cluster algorithms to thousands of processors. Another interesting feature of the nuclear problem is weak binding that occurs in very unstable nuclei. When the single-particle continuum is present, the nuclear Hamiltonian becomes complex. I also will describe our attempts to solve the complex coupled-cluster problem in the presence of complex interactions. Efficient parallelism requires spreading the interaction matrices globally and the coupled-cluster amplitudes in a more local fashion.
We discuss computational aspects of the spherical coupled-cluster method specific to the nuclear many-body problem. Using chiral nucleon-nucleon interaction at next-to-next-to-next-to leading order (N3LO) with cutoff Λ = 500MeV, we present coupled-cluster results for the ground state of 40Ca. Scaling and performance studies are presented together with challenges we meet with when extending the coupled-cluster effort to nuclei mass hundred and beyond. The low-energy nuclear many-body problem is a challenging undertaking, how-ever, in the last decade there has been significant progress in first principle cal-culations of nuclei.1), 2), 3), 4), 5), 6), 7), 8), 9) With the advance of chiral effective field theory10), 11), 12), 13), 14), 15) which allows for a systematic and consistent derivation of
The phenomenon of pairing in atomic and nuclear many-body systems gives rise to a great number of different physical properties of matter, from areas as seemingly diverse as the shape of stable nuclei to superconductivity in metals and superfluidity in neutron stars. With the experimental realisation of the long sought BCS-BEC crossover observed in trapped atomic gases - where it is possible to fine tune the s-wave scattering length a of a many-fermion system between a dilute, correlated BCS-like superfluid of Cooper pairs and a densely packed BEC of composite bosons - pairing problems in atomic physics have found renewed interest in recent years. Given the high precision techniques involved in producing these trapped gas condensates, we would like to employ a suitably accurate many-body method to study such systems, preferably one which goes beyond the simple mean-field picture.The Coupled Cluster Method (CCM) is a widely applied and highly successful ab initio method in the realm of quantum many-body physics and quantum chemistry, known to be capable of producing extremely accurate results for a wide variety of different many-body systems. It has not found many applications in pairing problems however, at least not in a general sense. Our aim, therefore, is to study various models of pairing using a variety of CCM techniques - we are interested in studying the generic features of pairing problems and in particular, we are especially interested in probing the collective modes of a system which exhibits the BCS-BEC crossover, in either the BCS or BEC limit. The CCM seems a rather good candidate for the job, given the high precision results it can produce.
The coupled cluster method (CCM) is one of the most powerful and most successful fully microscopic, ab initio formulations available for quantum N-body theory, with N finite or infinite. It has probably been applied to more systems in quantum chemistry, nuclear, condensed matter and other areas of physics, and quantum field theory than any other competing method. In nearly all such cases the numerical results are either the best or among the best available. The CCM can deal with ground- and excited-state energies of closed- and open-shell systems, density matrices and hence other properties, sum rules, and the sub-sum-rules that follow from imbedding linear response theory within it. Extensions exist to deal with systems at nonzero temperature and out of equilibrium. At the formal level it provides an exact mapping of the quantum-mechanical problem onto a classical Hamiltonian phase space where the multiconfigurational canonical classical coordinates have specific cluster and locality properties. In this way it can provide exact hierarchical generalizations of mean-field theory and the random phase approximation. We discuss here both the formalism itself and a selection of its applications.
What has since become known as the coupled cluster method (CCM) was invented some 35 years ago to calculate ground-state energies of closed-shell atomic nuclei.The method has since been extended in many different ways to the point where it is nowadays recognized as providing one of the most universally applicable, most powerful, and numerically most accurate of all ab initio methods in the microscopic quantum many-body theory of finite or infinite systems.We review here the applications within nuclear physics, and thereby show how the CCM provides a unifying approach of almost unique stature.The early calculations on both closed-and open-shell nuclei and on infinite nuclear matter are thus related to the recent extensions which provide both extremely accurate calculations for light nuclei and a framework for handling various problems in nuclear field theory.The latter include pion-nucleon field theory and such Hamiltonian lattice gauge field theories as the Abelian U(l) model and the non-Abelian SU(2) model.In all cases we stress the underlying concepts and the high quality of the results obtained.