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arXiv:1207.5735 · Physical Review C

Functional renormalization group for few-nucleon systems: SU(4) symmetry and its breaking

Michael C. Birse, Boris Krippa, Niels R. Walet

Published May 8, 2013• 7 citations
arXivHTMLPDF

Abstract

We apply the functional renormalization group to few-nucleon systems. Our starting point is a local effective action that includes three- and four-nucleon interactions, expressed in terms of nucleon and two-nucleon boson fields. The evolution of the coupling constants in this action is described by a renormalization group flow. We derive these flow equations both in the limit of exact Wigner SU(4) symmetry and in the realistic case of broken symmetry. In the symmetric limit we find that the renormalization flow equations decouple and can be combined into two sets, one of which matches the known results for bosons, and the other result matches the one for a single flavor of spin $1/2$ fermions. The equations show universal features in the unitary limit, which is obtained when the two-body scattering length tends to infinity. We calculate the spin-quartet neutron-deuteron scattering length and the deuteron-deuteron scattering lengths in the spin-singlet and spin-quintet channels.

DOI
10.1103/physrevc.87.054001

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We apply the functional renormalization group to few-nucleon systems. Our starting point is a local effective action that includes three- and four-nucleon interactions, expressed in terms of nucleon and two-nucleon boson fields. The evolution of the coupling constants in this action is described by a renormalization group flow. We derive these flow equations both in the limit of exact Wigner SU(4) symmetry and in the realistic case of broken symmetry. In the symmetric limit we find that the renormalization flow equations decouple and can be combined into two sets, one of which matches the known results for bosons, and the other result matches the one for a single flavor of spin 1/21/21/2 fermions. The equations show universal features in the unitary limit, which is obtained when the two-body scattering length tends to infinity. We calculate the spin-quartet neutron-deuteron scattering length and the deuteron-deuteron scattering lengths in the spin-singlet and spin-quintet channels.
We apply a functional renormalization group to systems of four bosonic atoms close to the unitary limit. We work with a local effective action that includes a dynamical trimer field, and we use this field to eliminate structures that do not correspond to the Faddeev-Yakubovsky equations. In the physical limit, we find three four-body bound states below the shallowest three-body state. The values of the scattering lengths at which two of these states become bound are in good agreement with exact solutions of the four-body equations and experimental observations. The third state is extremely shallow. During the evolution we find an infinite number of four-body states based on each three-body state which follow a double-exponential pattern in the running scale. None of the four-body states shows any evidence of dependence on a four-body parameter.
We use the functional renormalization group to study the spectrum of three- and four-body states in bosonic systems around the unitary limit. Our effective action includes all energy-independent contact interactions in the four-atom sector, and we introduce a running trimer field to eliminate couplings that involve the atom-atom-dimer channel. The results show behavior qualitatively similar to those from exact approaches. The truncated action we use leads to overbinding of the two four-body states seen in those treatments. It also generates a third state, although only for a very narrow range of two-body scattering lengths.
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.
We apply a field-theoretic functional renormalization-group technique to the few-body (vacuum) physics of nonrelativistic atoms near a Feshbach resonance. Three systems are considered: one-component bosons with a U(1) symmetry, two-component fermions with a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{SU}(2)$ symmetry, and three-component fermions with a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{SU}(3)$ symmetry. We focus on the scale-invariant unitarity limit of an infinite scattering length. The exact solution for the two-body sector is consistent with the unitary fixed-point behavior of the considered systems. Nevertheless, the numerical three-body solution in the sss-wave sector develops a limit cycle scaling in case of U(1) bosons and SU(3) fermions. The Efimov parameter for the one-component bosons and the three-component fermions is found to be $s\ensuremath{\approx}1.006$, consistent with the result of Efimov.
The appropriate power counting for the effective field theory of NN interactions is reviewed. It is more subtle than in most effective field theories since in the limit that the S-wave NN scattering lengths go to infinity it is governed by a nontrivial fixed point. The leading two body terms in the effective field theory for nucleon self interactions are scale invariant and invariant under Wigner SU(4) spin-isospin symmetry in this limit. Higher body terms with no derivatives (i.e., three and four body terms) are automatically invariant under Wigner symmetry.
We consider pairing phenomena in the framework of the exact renormalization group. We introduce a boson field to describe pairing effects, and take a simple ansatz for the effective action. We derive a set of approx. flow equations for the effective coupling including boson and fermionic fluctuations. The initial conditions are obtained by renormalizing the interaction to fit the scattering length in vacuum. At some crit. value of the running scale, the numerical solns. show a phase transition to a gapped phase. Std. results are recovered if we omit the boson loops. [on SciFinder (R)]