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arXiv:1204.2207 · WORLD SCIENTIFIC eBooks

Superfluid Local Density Approximation: A Density Functional Theory Approach to the Nuclear Pairing Problem

Aurel Bulgac

Published Mar 1, 2013• 1 citation
arXivHTMLPDF

Abstract

I describe the foundation of a Density Functional Theory approach to include pairing correlations, which was applied to a variety of systems ranging from dilute fermions, to neutron stars and finite nuclei. Ground state properties as well as properties of excited states and time-dependent phenomena can be achieved in this manner within a formalism based on microscopic input.

DOI
10.1142/9789814412490_0008

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I describe the foundation of a Density Functional Theory approach to include pairing correlations, which was applied to a variety of systems ranging from dilute fermions, to neutron stars and finite nuclei. Ground state properties as well as properties of excited states and time-dependent phenomena can be achieved in this manner within a formalism based on microscopic input.
Density functional theory provides the basis for uncounted studies of ground-state properties of many-body systems. However, the connection between the energy density functional and the underlying microscopic interactions of a given theory is not fully understood. In this thesis we use renormalization-group techniques in combination with density functional theory to study many-body systems from microscopic interactions. We apply our formalism to different one-dimensional systems. We start with systems of identical fermions interacting via a short-range repulsive and long-range attractive interaction which serves as a simple one-dimensional toy model for nuclei. After that we study systems of spin-1/2 fermions, where we assume interactions only between fermions with different spins and consider both a non-local interaction as in the previous case and a contact interaction. In particular, the contact interaction plays a prominent role for ultracold Fermi gases. We calculate ground-state properties such as the energy, density, intrinsic density and density correlation functions and compare our results to values obtained from other approaches. Moreover, we show how energies of excited states and the absolute square of the ground-state wavefunction can be extracted from the density-density correlation function. The relation between our formalism and conventional density functional theory as well as many-body perturbation theory is discussed which may help to guide the development of ab initio functionals for quantitative studies of nuclei in the future.
I describe the time-dependent superfluid local density approximation, which is an adiabatic extension of the density functional theory to superfluid Fermi systems and their real-time dynamics. This new theoretical framework has been used to describe several phenomena in cold atomic gases and nuclear collective motion: excitation of the Higgs modes in strongly interacting Fermi superfluids, generation of quantized vortices, crossing and reconnection of vortices, excitation of the superflow at velocities above the critical velocity, excitation of quantum shock waves and domain walls in the collisions of superfluid atomic clouds, and excitation of collective states in nuclei.
In cold atoms and in the crust of neutron stars, the pairing gap can reach values comparable with the Fermi energy. While in nuclei the neutron gap is smaller, it is still of the order of a few percent of the Fermi energy. The pairing mechanism in these systems is due to short range attractive interactions between fermions and the size of the Cooper pair is either comparable to the inter‐particle separation or it can be as big as a nucleus, which is still relatively small in size. Such a strong pairing gap is the result of the superposition of a very large number of particle–particle configurations, which contribute to the formation of the Cooper pairs. These systems have been shown to be the host of a large number of remarkable phenomena, in which the large magnitude of the pairing gap plays an essential role: quantum shock waves, quantum turbulence, Anderson–Bogoliubov–Higgs mode, vortex rings, domain walls, soliton vortices, vortex pinning in neutron star crust, unexpected dynamics of fragmented condensates and role of pairing correlations in collisions on heavy‐ions, Larkin–Ovchinnikov phase as an example of a Fermi supersolid, role of pairing correlations in controlling the dynamics of fissioning nuclei.
Over the past two decades, many studies in the density functional theory context revealed new aspects and properties of strongly correlated superfluid quantum systems in numerous configurations that can be simulated in experiments. This was made possible by the generalization of the local density approximation to superfluid systems by Bulgac [Phys. Rev. C 65, 051305(R) (2002); Phys. Rev. A 76, 040502(R) (2007)]. In the present work, we propose an extension of the superfluid local density approximation, systematically improvable and applicable to a large range of many-body quantum problems getting rid of the fitting procedures of the functional parameters. It turns out that only the knowledge of the density dependence of the quasiparticle properties, namely, the chemical potential, the effective mass, and the pairing gap function, are enough to obtain an explicit and accurate local functional of the densities without any adjustment a posteriori. This opens the way toward an effective field theory formulation of the density functional theory in the sense that we obtain a universal expansion of the functional parameters entering in the theory as a series in pairing gap function. Finally, we discuss possible applications of the developed approach allowing precise analysis of experimental observations. In that context, we focus our applications on the static structure properties of superfluid vortices.
Abstract A density-functional theory for superfluid systems is developed in the framework of the functional renormalization group based on the effective action formalism. We introduce the effective action for the particle-number and non-local pairing densities and demonstrate that the Hohenberg–Kohn theorem for superfluid systems is established in terms of the effective action. The flow equation for the effective action is then derived, where the flow parameter runs from 000 to 111, corresponding to the non-interacting and interacting systems. From the flow equation and the variational equation that the equilibrium density satisfies, we obtain the exact expression for the Kohn–Sham potential generalized to include the pairing potentials. The resultant Kohn–Sham potential has a nice feature in that it expresses the microscopic formulae of the external, Hartree, pairing and exchange–correlation terms separately. It is shown that our Kohn–Sham potential gives the ground-state energy of the Hartree–Fock–Bogoliubov theory by neglecting the correlations. An advantage of our exact formalism lies in the fact that it provides ways to improve the correlation part systematically.