Few-body physics with functional renormalization [Elektronische Ressource] / presented by Sergej Moroz

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Dissertationsubmitted to theCombined Faculties for the Natural Sciences andfor Mathematicsof the Ruperto-Carola University of Heidelberg,Germanyfor the degree ofDoctor of Natural Sciencespresented bySergej Morozborn in Konchezero, RussiaOral examination: 25.01.2011Few-body PhysicswithFunctional RenormalizationReferees: Prof. Dr. Christof WetterichProf. Dr. Jan Martin PawlowskiWenig-Teilchen Physik mit funktionalerRenormierungZusammenfassungIndieserDissertationwirdmitHilfederMethodederfunktionalenRenor-mierungdienicht-relativistischeQuantenphysikwenigerTeilchenuntersucht.Im speziellen diskutieren wir verschiedene theoretische Aspekte der PhysikwenigerTeilchen,diemitdemEfimov-Effektverbundensind. UnserzentralesErgebnisist,dasssichdieserEffektimAuftretenvonRenormierungsgruppen-Limitzyklen offenbart. Zuerst wird das Problem eines singulären Poten-1tial V(r) ∼ behandelt. Hierauf untersuchen wir das Efimov Problem2rdreierTeilchen,diedurcheinkurzreichweitigesZwei-Teilchen-Potentialwech-selwirken. Anschließend präsentieren wir die ersten Schritte in Richtungder Lösung des quantenmechanischen Vier-Teilchen Problems mit Hilfe derRenormierungsgruppe für Bosonen. Abschließend werden zusammengeset-zte Operatoren mit komplexer Skalendimension diskutiert, welche ein allge-meiner Bestandteil der Efimov Physik sind.

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Dissertation
submitted to the
Combined Faculties for the Natural Sciences and
for Mathematics
of the Ruperto-Carola University of Heidelberg,
Germany
for the degree of
Doctor of Natural Sciences
presented by
Sergej Moroz
born in Konchezero, Russia
Oral examination: 25.01.2011Few-body Physics
with
Functional Renormalization
Referees: Prof. Dr. Christof Wetterich
Prof. Dr. Jan Martin PawlowskiWenig-Teilchen Physik mit funktionaler
Renormierung
Zusammenfassung
IndieserDissertationwirdmitHilfederMethodederfunktionalenRenor-
mierungdienicht-relativistischeQuantenphysikwenigerTeilchenuntersucht.
Im speziellen diskutieren wir verschiedene theoretische Aspekte der Physik
wenigerTeilchen,diemitdemEfimov-Effektverbundensind. Unserzentrales
Ergebnisist,dasssichdieserEffektimAuftretenvonRenormierungsgruppen-
Limitzyklen offenbart. Zuerst wird das Problem eines singulären Poten-
1tial V(r) ∼ behandelt. Hierauf untersuchen wir das Efimov Problem2r
dreierTeilchen,diedurcheinkurzreichweitigesZwei-Teilchen-Potentialwech-
selwirken. Anschließend präsentieren wir die ersten Schritte in Richtung
der Lösung des quantenmechanischen Vier-Teilchen Problems mit Hilfe der
Renormierungsgruppe für Bosonen. Abschließend werden zusammengeset-
zte Operatoren mit komplexer Skalendimension diskutiert, welche ein allge-
meiner Bestandteil der Efimov Physik sind.
Few-body physics with functional
renormalization
Abstract
In this thesis nonrelativistic few-body quantum physics is investigated
with the method of functional renormalization. In particular, we discuss dif-
ferent 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
potentialinquantummechanics. ThenweexaminetheoriginalEfimovprob-
lem of three particles interacting via a short-range two-body potential. Sub-
sequently, we present a first step towards the renormalization group solution
of the four-body quantum problem for bosons. Finally, a general theoreti-
cal feature of the Efimov physics, composite operators with complex scaling
dimensions, are discussed.Contents
1 Introduction 1
2 Functional Renormalization Group 5
2.1 Functional renormalization: what is it about and what is it for? 5
2.2 The Wetterich flow equation . . . . . . . . . . . . . . . . . . . 6
2.3 Aspects of functional renormalization . . . . . . . . . . . . . . 8
2.4 Zero-dimensional field theory. . . . . . . . . . . . . . . . . . . 11
3 Quantum Mechanics with Inverse Square Potential 28
3.1 The model and the flow equation . . . . . . . . . . . . . . . . 31
3.2 General analysis of the flow equation . . . . . . . . . . . . . . 35
3.3 Complex extension . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Complex RG flows on the Riemann sphere . . . . . . . . . . . 54
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Universal Three-body Problem: Efimov Effect 59
4.1 Vertex expansion and definition of the models . . . . . . . . . 61
4.2 Vacuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3 Two-body sector: exact solution . . . . . . . . . . . . . . . . . 67
4.4 Three-body Sector: flow equations . . . . . . . . . . . . . . . . 72
4.5ody sector: pointlike approximation . . . . . . . . . . 77
4.6 Three-body systems I and II . . . . . . . . . . . . . . . 79
4.7ody sector: system III . . . . . . . . . . . . . . . . . . 81
4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Four-body Problem: Universal Tetramers 85
5.1 Derivative expansion and definition of the model . . . . . . . . 88
5.2 Two-body sector . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 Three-body sector . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4 Four-body sector . . . . . . . . . . . . . . . . . . . . . . . . . 94
iCONTENTS
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6 Operators with Complex Scaling Dimensions 100
6.1 Composite operatorO =ψψ . . . . . . . . . . . . . . . . . . . 102
′′ ′6.2 Energy Green function G (⃗r ,⃗r ;ω) . . . . . . . . . . . . . . 105D
6.3 Two-particle propagator iG . . . . . . . . . . . . . . . . . . . 107
O
(l)6.4 Composite operatorsO with higher angular momentum . . . 110
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7 Conclusions 113
A Z(j) in zero-dimensional field theory 116
B Galilean and nonrelativistic conformal symmetry 117
C Bound state approximation and separable potential 119
D Effective range expansion 122
E Scaling dimension of composite operator O =ψψ 126
iiChapter 1
Introduction
Problems related to the motion of few interacting bodies are ubiquitous in
physics. After the advent of the Newton’s classical mechanics astronomers
were able to predict the motion of planets in the Solar System. To the
first approximation the trajectory of the planet is determined only by the
gravitational force between the Sun and the planet – a simple two-body
problem. This trajectory is, however, slightly perturbed by the neighboring
planets and for a more precise prediction one must solve three-, four- and
higher-body problems. The triumph of few-body classical physics was the
discoveryofNeptune. Theexistenceofthisplanetwastheoreticallypredicted
in 1845 from unexpected changes of the orbit of its neighbor Uranus. In
the early days of quantum mechanics few-body physics was of a key interest.
After solving problem of the hydrogen atom in 1925−1926, people started to
tackle more difficult quantum three-body problems such as the helium atom,
wheretwoelectronsorbitaroundthenucleus. Althoughtheatomcan
not be solved analytically, it served as an excellent stimulus for developing
perturbationandvariationalmethodsinquantummechanics. Alsonowadays
few-bodyproblemsareofcentralimportanceindifferentareasofsciencesuch
as physical chemistry, atomic physics and nuclear physics.
Themaintopicofthisthesisisnonrelativisticfew-bodyquantumphysics.
More specifically, we will be mainly interested in quantum problems of few
electrically neutral atoms. Two neutral atoms interact with each other via
the van der Waals potential V(r) which falls off at large distances r as
C
lim V(r) =− . (1.1)
6r→∞ r
¡ ¢1/4mCOne can now associate a length scale l = with the van der WaalsvdW 2
~
potential. HereC isaconstant,mdenotesthemassofinteractingatomsand
~ stands for the reduced Planck constant. Importantly, at collision energy
1Introduction
2
~E ≪ the interaction can be described accurately by a simple short-2ml
vdW
range potential. In other words, at low energies neutral atoms behave like
pointlike particles. Their two-body scattering is determined by the s-wave
1scattering length which will be denoted by a.
Generically, physical observables of a few-particle system will depend on
the scattering length a, but also on other (microscopic) parameters of the
interaction potential such as the effective range, the range of interaction, etc.
However, in the regime of a≫ all other length scales, physical properties of
the system will depend only on the scattering length. This is what is known
under the name of few-body universality (for a review see [1]). Few-body
universality is a very useful concept, since in the universal regime physical
observables are insensitive to microscopic details of the interaction potential
which are often difficult to measure precisely. Hence, prediction of a few-
body universal theory are very simple and can be straightforwardly tested
experimentally. As an example of such prediction, we consider a two-body
(dimer) bound state. For a> 0 the universal theory predicts the presence of
2
~a dimer with the binding energy E =− , which is in a good agreementD 2ma
with numerous experiments with ultracold atomic gases.
In general, we can enter the universal regime by a fine-tuning of some
interaction parameter. The fine-tuning can be either accidental or experi-
mental. Accidentally fine-tuned systems are gifts of Nature, for which “by
accident“ thescatteringlengthhappenedtobemuchlargerthanotherlength
4scales. In atomic physics, for example, a pair of He atoms can be described
fairly well with the universal theory due to the accidental fine-tuning. In
nuclear physics, pn and αα systems are good examples of the accidental
fine-tuning. Nowadays it is possible to change the scattering length over a
wide range in experiments with ultracold atomic gases. A so-called Feshbach
resonance (for a review see [2]) gives experimentalists a unique opportunity
to enter the universal regime by simply tuning an external magnetic field.
TheadventofFeshbachresonancesrevolutionizedthefieldoffew-bodyquan-
tum physics and made it a very active area of theoretical and experimental
research.
In this thesis we focus our attention mainly on the specific part of few-
body quantum physics known as the Efimov effect. In 1970 Vitaly Efimov
solved a quantum problem of three bosons interacting via a two-body short-
2range attractive potential [3]. In the universal regime Efimov found an
1A notable exception is a collision of two identical fermions which can not scatter in
the s-wave channel. We do not discuss this case in the thesis.
2NotethatinordertomaketheEfimovproblemwell-defined, inadditiontothescatter-
ing length a, one must necessarily introduce a dimensionless three-body parameter which
encodes details of the microscopic interaction [1]. In the context of the Efimov effect we
2