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arXiv:1011.5852 · Physical Review A

Convergence of a renormalization-group approach to dimer-dimer scattering

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

Published Feb 28, 2011• 15 citations
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Abstract

We study the convergence of a functional renormalization group technique by looking at the ratio between the fermion-fermion scattering length and the dimer-dimer scattering length for a system of nonrelativistic fermions. We find that in a systematic expansion in powers of the fields there is a rapid convergence of the result that agrees with known exact results.

DOI
10.1103/physreva.83.023621

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We study the convergence of a functional renormalization group technique by looking at the ratio between the fermion-fermion scattering length and the dimer-dimer scattering length for a system of nonrelativistic fermions. We find that in a systematic expansion in powers of the fields there is a rapid convergence of the result that agrees with known exact results.
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Abstract. We consider the applications of functional renormalisation group to few and many-body systems. As an application to the few-body dynamics we study the ratio between the fermion-fermion scattering length and the dimer-dimer scattering length for systems of few nonrelativistic fermions. We find a strong dependence on the cutoff function used in the renormalisation flow for a two-body truncation of the action. Adding a simple three-body term substantially reduces this dependence. In the context of many-body physics we study the dynamics of both symmetric and asymmetric many-fermion systems using the same functional renormalisation technique. It is demonstrated that functional renormalisation group gives sensible and reliable results and provides a solid theoretical ground for the future studies. Open questions as well as lines of further developments are discussed. Nonperturbative or Exact Renormalisation Group (ERG) has become a popular tool to study strongly interacting dy-namics [1]. It has been used to study a variety of physical systems, from nonrelativistic two-body [2] and many-body [3] systems to effective quark models [4] and gauge theo-ries [5]. The main object of the ERG is the scale-dependent average effective action Γk, where k is an auxiliary running
We study the dimer-dimer scattering length a4{a}_{4}a4​ for a two-component Fermi mixture in which the different fermions have different masses ${m}_{\ensuremath{\uparrow}}$ and ${m}_{\ensuremath{\downarrow}}$. This is made in the framework of the exact field-theoretic method. In the large mass ratio domain the equations are simplified enough to lead to an analytical solution. In particular we link a4{a}_{4}a4​ to the fermion-dimer scattering length a3{a}_{3}a3​ for the same fermions and obtain the very simple relation a4=a3/2{a}_{4}={a}_{3}/2a4​=a3​/2. The result ${a}_{4}\ensuremath{\simeq}{a}_{3}/2$ is actually valid whatever the mass ratio with quite good precision. As a result we find an analytical expression providing a4{a}_{4}a4​ with fairly good precision for any mass. To dominant orders for large mass ratio it agrees with the literature. We show that in this large mass ratio domain, the dominant processes are the repeated dimer-dimer Born scatterings, considered earlier by Pieri and Strinati [Phys Rev. B 61, 15370 (2000)]. We conclude that their approximation of retaining only these processes is a fairly good one whatever the mass ratio.
We consider two-component fermions with short-range interactions and large scattering length. This system has universal properties that are realized in several different fields of physics. In the limit of large fermion–fermion scattering length aff and zero-range interaction, all properties of the system scale proportionally with aff. For the case with shallow bound dimers, we calculate the dimer–dimer scattering phase shifts using lattice effective field theory. We extract the universal dimer–dimer scattering length add/aff=0.618(30) and effective range rdd/aff=−0.431(48). This result for the effective range is the first calculation with quantified and controlled systematic errors. We also benchmark our methods by computing the fermion–dimer scattering parameters and testing some predictions of conformal scaling of irrelevant operators near the unitarity limit.
The functional renormalization group is applied to the effective action for scattering of two nonrelativistic fermions. The resulting physical effective action is shown to contain the correct threshold singularity. The corresponding ``bare'' action respects Galilean invariance up to second order in momenta. Beyond that order it contains terms that violate this symmetry and, for the particular regulator considered, nonanalytic third-order terms. The corresponding potential can be expanded around a nontrivial fixed point using the power counting appropriate to a system with large scattering length.