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arXiv:1304.1161 · Journal of Physics G Nuclear and Particle Physics

Towards a renormalization group approach to density functional theory—general formalism and case studies

Sandra Kemler, Jens Braun

Published Jul 10, 2013• 33 citations
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Abstract

We discuss a two-particle point-irreducible (2PPI) approach to many-body physics which relies on a renormalization group (RG) flow equation for the associated effective action. In particular, the general structure and properties of this RG flow equation are analyzed in detail. Moreover, we discuss how our 2PPI-RG approach relates to density functional theory and argue that it can in principle be used to study ground-state properties of non-relativistic many-body systems from microscopic interactions, such as (heavy) nuclei. For illustration purposes, we use our formalism to compute the ground-state properties of two toy models. Communicated by Achim Schwenk

DOI
10.1088/0954-3899/40/8/085105

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We discuss a two-particle point-irreducible (2PPI) approach to many-body physics which relies on a renormalization group (RG) flow equation for the associated effective action. In particular, the general structure and properties of this RG flow equation are analyzed in detail. Moreover, we discuss how our 2PPI-RG approach relates to density functional theory and argue that it can in principle be used to study ground-state properties of non-relativistic many-body systems from microscopic interactions, such as (heavy) nuclei. For illustration purposes, we use our formalism to compute the ground-state properties of two toy models. Communicated by Achim Schwenk
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.
Abstract In nuclear physics, density functional theory (DFT) provides the basis for state-of-the art studies of ground-state properties of heavy nuclei. However, the direct relation of the density functional underlying these calculations and the microscopic nuclear forces is not yet fully understood. We present a combination of DFT and renormalization group (RG) techniques which allows to study selfbound many-body systems from microscopic interactions. We discuss its application with the aid of systems of identical fermions interacting via a long-range attractive and short-range repulsive two-body force in one dimension. We compute ground-state energies, intrinsic densities, and density correlation functions of these systems and compare our results to those obtained from other methods. In particular, we show how energies of excited states as well as the absolute square of the ground-state wave function can be extracted from the correlation functions within our approach. The relation between many-body perturbation theory and our DFT-RG approach is discussed and illustrated with the aid of the calculation of the second-order energy correction for a system of N identical fermions interacting via a general two-body interaction. Moreover, we discuss the control of spuriously emerging fermion self-interactions in DFT studies within our framework. In general, our approach may help to guide the development of energy functionals for future quantitative DFT studies of heavy nuclei from microscopic interactions.
We propose a nonperturbative renormalization-group (NPRG) approach to fermion systems in the two-particle-irreducible (2PI) effective action formalism, based on an exact RG equation for the Luttinger-Ward functional. This approach enables us to describe phases with spontaneously broken symmetries while satisfying the Mermin-Wagner theorem. We show that it is possible to choose the Hartree-Fock random phase approximation theory as initial condition of the RG flow and argue that the 2PI-NPRG is not restricted to the weak-coupling limit. An expansion of the Luttinger-Ward functional about the minimum of the 2PI effective action including only the two-particle 2PI vertex leads to nontrivial RG equations where interactions between fermions and collective excitations naturally emerge.
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.
We derive functional renormalization group (fRG) schemes for Fermi systems which are based on the two-particle irreducible approach to the quantum many-body problem. In a first step, the cutoff is introduced in the non-interacting propagator as it is commonly done in fRG based on one-particle irreducible vertex functions. The most natural truncation of the resulting infinite hierarchy of flow equations is shown to be fully equivalent to self-consistent perturbation theory. An earlier suggested alternative truncation strategy is considered as well. In a second step, the cutoff is introduced in the two-particle interaction. Again two truncation procedures are investigated, one of which was derived before. In the latter, the mean-field solution of the many-body problem is considered as the starting point of the renormalization group flow. We compare the performance and the required numerical resources for solving the coupled flow equations for all the approximate schemes by applying them to the problem of the quantum anharmonic oscillator. In a functional integral representation, this model has a formal similarity to the quantum many-body problem. The perspectives for applying the derived two-particle irreducible fRG approaches to zero- and one-dimensional systems of correlated fermions are discussed.
In the present thesis superfluid many-fermion systems are investigated in the framework of the Renormalization Group (RG). Starting from an experimentally determined two-body interaction this scheme provides a microscopic approach to strongly correlated many-body systems at low temperatures. The fundamental objects under investigation are the two-point and the four-point vertex functions. We show that explicit results for simple separable interactions on BCS-level can be reproduced in the RG framework to high accuracy. Furthermore the RG approach can immediately be applied to general realistic interaction models. In particular, we show how the complexity of the many-body problem can be reduced systematically by combining different RG schemes. Apart from technical convenience the RG framework has conceptual advantage that correlations beyond the BCS level can be incorporated in the flow equations in a systematic way. In this case however the flow equations are no more explicit equations like at BCS level but instead a coupled set of implicit equations. We show on the basis of explicit calculations for the single-channel case the efficacy of an iterative approach to this system. The generalization of this strategy provides a promising strategy for a non-perturbative treatment of the coupled channel problem. By the coupling of the flow equations of the two-point and four-point vertex self-consistency on the one-body level is guaranteed at every cutoff scale.