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arXiv:1203.3765 · Progress of Theoretical Physics Supplement

Computational Aspects of Nuclear Coupled-Cluster Theory

Gaute Hagen, Hai Ah Nam

Published Jan 1, 2012• 6 citations
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Abstract

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= 500 MeV, 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.

DOI
10.1143/ptps.196.102

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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= 500 MeV, 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.
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
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.
Background: Ab initio many-body methods have been developed over the past 10 yr to address closed-shell nuclei up to mass $A\ensuremath{\approx}130$ on the basis of realistic two- and three-nucleon interactions. A current frontier relates to the extension of those many-body methods to the description of open-shell nuclei. Several routes to address open-shell nuclei are currently under investigation, including ideas that exploit spontaneous symmetry breaking.Purpose: Singly open-shell nuclei can be efficiently described via the sole breaking of U(1) gauge symmetry associated with particle-number conservation as a way to account for their superfluid character. While this route was recently followed within the framework of self-consistent Green's function theory, the goal of the present work is to formulate a similar extension within the framework of coupled cluster theory.Methods: We formulate and apply Bogoliubov coupled cluster (BCC) theory, which consists of representing the exact ground-state wave function of the system as the exponential of a quasiparticle excitation cluster operator acting on a Bogoliubov reference state. Equations for the ground-state energy and the cluster amplitudes are derived at the singles and doubles level (BCCSD) both algebraically and diagrammatically. The formalism includes three-nucleon forces at the normal-ordered two-body level. The first BCC code is implemented in mmm scheme, which will permit the treatment of doubly open-shell nuclei via the further breaking of SU(2) symmetry associated with angular momentum conservation.Results: Proof-of-principle calculations in an Nmax=6{N}_{\text{max}}=6Nmax​=6 spherical harmonic oscillator basis for 16,18O^{16,18}\mathrm{O}16,18O and 18Ne^{18}\mathrm{Ne}18Ne in the BCCD approximation are in good agreement with standard coupled cluster results with the same chiral two-nucleon interaction, while 20O^{20}\mathrm{O}20O and 20Mg^{20}\mathrm{Mg}20Mg display underbinding relative to experiment. The breaking of U(1) symmetry, monitored by computing the variance associated with the particle-number operator, is relatively constant for all five nuclei, in both the Hartree-Fock-Bogoliubov and BCCD approximations.Conclusions: The newly developed many-body formalism increases the potential span of ab initio calculations based on single-reference coupled cluster techniques tremendously, i.e., potentially to reach several hundred additional midmass nuclei. The new formalism offers a wealth of potential applications and further extensions dedicated to the description of ground and excited states of open-shell nuclei. Short-term goals include the implementation of three-nucleon forces at the normal-ordered two-body level. Midterm extensions include the approximate treatment of triples corrections and the development of the equation-of-motion methodology to treat both excited states and odd nuclei. Long-term extensions include exact restoration of U(1) and SU(2) symmetries.
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.
This topic deals with the study of correlation of ground and excited states of even nuclei like 16O and 4He. The main objective of present work is to develop more theoretical techniques applicable in nuclear physics. The work is also extended to discrete excited states as well as odd even nuclei. The work is useful for the calculation of nuclear many body problems for spherically symmetric nuclear quantization representation. The ground state calculation of 16O and 4He are done using G matrix, which also help in calculation of ground state binding energy and one body two body densities.