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Few-body physics with functional renormalization

Sergej Moroz

Published Jan 1, 2011• 4 citations
Publisher

Abstract

In this thesis nonrelativistic few-body quantum physics is investigated with the method of functional renormalization. In particular, we discuss different theoretical aspects of few-body physics related to the Efimov effect. Our central finding is that this effect manifests itself as a renormalization group limit cycle. First, we treat the problem of a singular, inverse square potential in quantum mechanics. Then we examine the original Efimov problem of three particles interacting via a short-range two-body potential. Subsequently, we present a first step towards the renormalization group solution of the four-body quantum problem for bosons. Finally, a general theoretical feature of the Efimov physics, composite operators with complex scaling dimensions, are discussed.

DOI
10.11588/heidok.00011602

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In this thesis nonrelativistic few-body quantum physics is investigated with the method of functional renormalization. In particular, we discuss different theoretical aspects of few-body physics related to the Efimov effect. Our central finding is that this effect manifests itself as a renormalization group limit cycle. First, we treat the problem of a singular, inverse square potential in quantum mechanics. Then we examine the original Efimov problem of three particles interacting via a short-range two-body potential. Subsequently, we present a first step towards the renormalization group solution of the four-body quantum problem for bosons. Finally, a general theoretical feature of the Efimov physics, composite operators with complex scaling dimensions, are discussed.
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.
Physics is often said to be universal when microscopically distinct systems show the same lowenergy behavior.In critical phenomena, the universality of the second-order phase transition is elucidated by the renormalization-group (RG) method, which provides a powerful computational tool for the critical exponents of the power-law behavior of observables.In particular, the attraction of different RG-flow trajectories to a small-dimensional sub-theory space accounts for the universal nature.The attraction structure motivates us to extract universal physical observables by investigating the sub-theory space called the renormalized trajectory, onto which RG flows are attracted.To put the motivation forward, we consider the universality in quantum few-body physics, in which the RG exhibits another characteristic flow than the fixed points.The Efimov physics, which is renowned for self-similar trimer states, shows an RG limit-cycle behavior that refers to a periodic RG-flow trajectory.In this thesis, we investigate the relation between the RG limit cycle and the universality in Efimov physics.Firstly, to verify that the RG limit cycle is the renormalized trajectory in Efimov physics, we demonstrate that the RG-flow trajectories of microscopically distinct systems universally exhibit the RG limit-cycle behavior at sufficiently low energy, thereby giving a numerical evidence of the attraction structure of the RG-flow trajectories to the RG limit cycle; in other words, we show that the RG limit cycle behaves as an attractor of various RG flows.To address this point unbiasedly, we perform an exact RG calculation for various effective models of identical bosons that features Efimov physics at low energy.We apply an exact functional renormalization-group (FRG) method to effective separable models for various systems of identical bosons with short-range resonant interactions.For a given microscopic three-body model, we numerically demonstrate for the first time that the RG-flow trajectory converges at the RG limit cycle in the low-energy regime.Physical observables such as Efimov's scaling factor and the three-body parameter can be extracted from the RG limit cycle.Secondly, we investigate the role of RG limit cycle in a four-body extension of Efimov physics, which has attracted considerable attention in recent years.Although the universality in four-body physics has been established both theoretically and experimentally, the relationship between the RG limit cycle and the four-body physics remains an open question.By performing an FRG calculation for a simple effective field theory, we obtain for the first time the RG limit cycle in the four-body sector that reproduces the universality of the four-body physics.As the first attempt, we develop a simple separable pole approximation of the four-body FRG equation and successfully reproduce qualitative features of the four-body physics.A systematic improvement of the approximation quantitatively reproduces the numbers and the binding energies of tetramer states that emerge universally at low energy.2 The restriction r 0 < ∼ R < ∼ a on the length scale limits the binding energy E t of the Efimov trimers as ℏ ma 2 < ∼ E t < ∼ ℏ mr 2 0 .3 A similar discrete scale-invariant phenomenon is known for the doubly excited H -ion, in which two valence electrons are excited from their ground states [16].In the doubly excited H -ion, an interaction between an electron-
This thesis deals with systems of interacting particles with very low energy in the limit where the particle-particle scattering is much larger than the range of the interactions. We use a quantum-field-theory approach which allows us to study both few-body and dense-matter systems in a unified framework. This allows to introduce composite fields of two and three particles (when appropriate). The quantum corrections are calculated nonperturbatively with the Functional RenormalisationGroup.We deal with three types of systems. First we study systems with three and four scalar particles. For three-particle systems our framework describes the Efimov effect. During the FRG flow in the scaling limit, the four-particle system has an infinite sequence of (unphysical) four-particle states on top of each Efimov trimer. This is a case of super Efimov behaviour. Three of these four-particle states survive to the physical limit. Two of these three states have been found in exact quantum-mechanical calculations, and have also been observed in gases of ultracold atoms.Next, this thesis studies systems of three and four spin-1/2 particles. In the scaling limit, we find attractive fixed points for the three- and four-particle systems. Out of the scaling limit, we study atom-molecule scattering and molecule-molecule scattering, in particular their scattering length. Finally, we study dense-matter systems of spin-1/2 particles. This calculation includes all the two-, three-, and four-particle interactions. These systems show spontaneous symmetry breaking: the two-particle field has a finite classical value. We find the value of the atom gap in units of the chemical potential.
We revisited how Weinberg's ideas in Nuclear Physics influenced our own work and lead to a renormalization group invariant framework within the quantum mechanical few-body problem, and we also update the discussion on the relevant scales in the limit of short-range interactions. In this context, it is revised the formulation of the subtracted scattering equations and fixed-point Hamiltonians applied to few-body systems, in which the original interaction contains point-like singularities, such as Dirac-delta and/or its derivatives. The approach is being illustrated by considering two-nucleons described by singular interactions. This revision also includes an extension of the renormalization formalism to three-body systems, which is followed by an updated discussion on the applications to four particles.
The Efimov effect is the only experimentally realized universal phenomenon that exhibits the renormalization-group limit cycle with the three-body parameter parametrizing a family of universality classes. Recent experiments in ultracold atoms have unexpectedly revealed that the three-body parameter itself is universal when measured in units of an effective range. By performing an exact functional renormalization-group analysis with various finite-range interaction potentials, we demonstrate that the onset of the renormalization-group flow into the limit cycle is universal, regardless of short-range details, which connects the missing link between the two universalities of the Efimov physics. A close connection between the topological property of the limit cycle and few-body physics is also suggested.
Effective Field Theory (EFT) provides a powerful framework that exploits a separation of scales in physical systems to perform systematically improvable, model-independent calculations. Particularly interesting are few-body systems with short-range interactions and large two-body scattering length. Such systems display remarkable universal features. In systems with more than two particles, a three-body force with limit cycle behavior is required for consistent renormalization already at leading order. We will review this EFT and some of its applications in the physics of cold atoms and nuclear physics. In particular, we will discuss the possibility of an infrared limit cycle in QCD. Recent extensions of the EFT approach to the four-body system and N-boson droplets in two spatial dimensions will also be addressed.