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arXiv:nucl-th/0302081

Renormalization Group Methods for the Nuclear Many-Body Problem

A. Schwenk, Bengt L. Friman, G. E. Brown

Published Feb 27, 2003• 0 citations
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Abstract

The application of renormalization group methods to microscopic nuclear many-body calculations is discussed. We present the solution of the renormalization group equations in the particle-hole channels for neutron matter and the application to S-wave pairing. Furthermore, we point out that the inclusion of tensor and spin-orbit forces leads to spin non-conserving effective interactions in nuclear matter.

DOI
10.48550/arxiv.nucl-th/0302081

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The application of renormalization group methods to microscopic nuclear many-body calculations is discussed. We present the solution of the renormalization group equations in the particle-hole channels for neutron matter and the application to S-wave pairing. Furthermore, we point out that the inclusion of tensor and spin-orbit forces leads to spin non-conserving effective interactions in nuclear matter.
Renormalization group methods can be applied to the nuclear many-body problem using the approach proposed by Shankar. We start with the two-body low momentum interaction V_{low k} and use the RG flow from the particle-hole channels to calculate the full scattering amplitude in the vicinity of the Fermi surface. This is a new straightforward approach to the many-body problem which is applicable also to condensed matter systems without long-ranged interactions, such as liquid 3He. We derive the one-loop renormalization group equations for the quasiparticle interaction and scattering amplitude at zero temperature. The RG presents an elegant method to maintain all momentum scales and preserve the antisymmetry of the scattering amplitude. As a first application we solve the RG equations for neutron matter. The resulting quasiparticle interaction includes effects due to the polarization of the medium, the so called induced interaction of Babu and Brown. We present results for the Fermi liquid parameters, the equation of state of neutron matter and the 1S0 superfluid pairing gap.
We show that the renormalization group decimation of modern nucleon potential models to low momenta results in a unique nucleon interaction V_{low k}. This interaction is free of short-ranged singularities and can be used directly in many-body calculations. The RG scaling properties follow directly from the invariance of the scattering phase shifts. We discuss the RG treatment of Fermi liquids. The RG equation for the scattering amplitude in the two particle-hole channels is given at zero temperature. The flow equations are simplified by retaining only the leading term in an expansion in small momentum transfers. The RG flow is illustrated by first studying a system of spin-polarized fermions in a simple model. Finally, results for neutron matter are presented by employing the unique low momentum interaction V_{low k} as initial condition of the flow. The RG approach yields the amplitude for non-forward scattering, which is of great interest for calculations of transport properties and superfluid gaps in neutron star interiors. The methods used can also be applied to condensed matter systems in the absence of long-ranged interactions.
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.
Conventional many-body quantum theory considers, as a rule, a system of particles in the mean field interacting through two-body forces. Recently it was suggested that in nuclear physics many-body forces, first of all three-body ones, are important for saturation of nuclear matter and for many details of nuclear structure. We consider possible influence of three-body forces, regardless of their origin (bare nucleon interactions or effective medium phenomena), on many-body dynamics. The new effects include, but are not limited to, renormalization of pairing and other two-body forces, enhancement of anharmonicity for collective modes, and special features of shell model calculations.
Many-body Green functions are a very efficient formulation of the many-body problem. We review the application of this method to nuclear physics problems. The formulas which can be derived are of general applicability, e.g., in self-consistent as well as in nonself-consistent calculations. With the help of the Landau renormalization, one obtains relations without any approximations. This allows to apply conservation laws which lead to important general relations. We investigate the one-body and two-body Green functions as well as the three-body Green function and discuss their connection to nuclear observables. The generalization to systems with pair correlations are also presented. Numerical examples are compared with experimental data.
Renormalization group methods generate low-resolution Hamiltonians that are more diagonal, with reduced coupling between low- and high-energy states, and thus easier to solve. This chapter reviews the similarity renormalization group for nuclear Hamiltonians, which is a popular method for generating low-resolution nuclear forces. It presents the similarity renormalization group flow equations, analyzes how the similarity renormalization group drives the Hamiltonian towards the diagonal, and studies the effect of induced many-body interactions. It concludes by highlighting the progress in first-principles calculations of nuclei driven by low-resolution nuclear Hamiltonians.