Functional Renormalization and Ultracold Quantum Gases

199 Pages·2010·4.626 MB·English

by  Stefan Flörchinger (auth.)

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Stefan Flörchinger
Functional Renormalization and
Ultracold Quantum Gases
Doctoral Thesis accepted by
Heidelberg University, Germany
123
Author Supervisor
Dr. Stefan Flörchinger Prof.Christof Wetterich
Institutfür Theoretische Physik Institutfür Theoretische Physik
Universität Heidelberg Universität Heidelberg
Philosophenweg 16 Philosophenweg 16
69120Heidelberg 69120Heidelberg
Germany Germany
[email protected] [email protected]
ISSN 2190-5053 e-ISSN2190-5061
ISBN 978-3-642-14112-6 e-ISBN 978-3-642-14113-3
DOI 10.1007/978-3-642-14113-3
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2010931607
(cid:2)Springer-VerlagBerlinHeidelberg2010
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Supervisor’s Foreword
The field of ultracold quantum gases has seen an enormous progress in
experimental possibilities and methods in the last years. After the first realization
ofBose–Einsteincondensationindilute gasesofrubidiumandsodium, it became
possible to investigate many other elements. Particularly interesting is the
possibility to tune the interaction strength in a wide range by making use of
Feshbach resonances. By using tightly confining traps one can realize situations
withthree,twooronespacedimensions.Itisalsopossibletouseopticallatticesto
simulate elements of solid state physics.
Due to this pandora of experimental possibilities, atomic gases became an
interestingtestingfieldformanytheoreticalideasandmethodsthatwereoriginally
developed in high energy quantum field theory or in solid state physics. Quanti-
tative precise comparison between theory and experiments for a number of
interesting many-body observables could become possible in the next years.
The present thesis develops an approach to many-body physics of cold atomic
gases that is based on the functional renormalization group method. The ideas of
renormalizationareverycentralinourunderstandingofquantumfieldtheoryand
the modern functional formulation provides a valuable tool for theoretical physi-
cists.Themethodisbasedonanexactrenormalizationgroupflowequationwhich
constitutes an convenient starting point for approximations. In the last years,
functionalrenormalizationhasbeenappliedtostudyvarioustopicsinquantumand
statistical field theory. A few examples are universal critical exponents and
amplitude ratios of classical phase transitions, the properties of the chiral phase
transition in QCD or condensed matter problems as the Hubbard model or the
Kondo effect.
Coldatomicquantumgasesareanotherfieldofapplicationbutcanalsobeseen
as a particularly interesting testing field. In contrast to other applications one has
the big advantage that the microscopic physics is well known and can be con-
trolled experimentally to a large extend. In some cases one expects a far going
universality of the macroscopic properties. An example for this is the BCS–BEC
crossover close to broad Feshbach resonances. Apart from the temperature, the
physics is described by a single parameter, the scattering length a in units of the
v
vi Supervisor’sForeword
average inter-particle distance. Observables such as the single particle gap or the
critical temperature of the transition to the superfluid state in units of the inter-
particledistancearepredictedtobethesamefordifferentmaterialssuchaslithium
orpotassium.Itshouldbepossibletousethisforaprecisioncomparisonbetween
theory and experiment in the quantum many body regime similar to the universal
critical exponents that are dominated by thermal fluctuations close to classical
second order phase transitions.
Heidelberg, May 2010 Christof Wetterich
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Flow Equations to Solve an Integral . . . . . . . . . . . . . . . . . . . . 4
1.2 Functional Integral Representation of Quantum Field Theory . . . 8
1.2.1 From the Lattice to Field Theory . . . . . . . . . . . . . . . . . 9
1.2.2 Expectation Values, Correlation Functions. . . . . . . . . . . 10
1.2.3 Functional Derivatives, Generating Functionals . . . . . . . 11
1.2.4 Microscopic Actions in Real Time and Analytic
Continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.5 Matsubara Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 14
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 The Wetterich Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Scale Dependent Schwinger Functional . . . . . . . . . . . . . . . . . . 17
2.2 The Average Action and its Flow Equation . . . . . . . . . . . . . . . 19
2.3 Functional Integral Representation and Initial Condition . . . . . . 20
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Generalized Flow Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1 Scale-dependent Bosonization. . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Flowing Action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 General Coordinate Transformations . . . . . . . . . . . . . . . . . . . . 29
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Truncations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 Symmetries as a Guiding Principles. . . . . . . . . . . . . . . . . . . . . 34
4.2 Separation of Scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Derivative Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Cutoff Choices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
vii
viii Contents
6 Investigated Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.1 Bose Gas in Three Dimensions. . . . . . . . . . . . . . . . . . . . . . . . 43
6.1.1 Lagrangian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.2 Bose Gas in Two Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . 46
6.3 BCS–BEC Crossover. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.3.1 Lagrangian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.4 BCS–Trion–BEC Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.4.1 Lagrangian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.1 Derivative Expansion and Ward Identities . . . . . . . . . . . . . . . . 59
7.1.1 Propagator and Dispersion. . . . . . . . . . . . . . . . . . . . . . 63
7.2 Noethers Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8 Truncated Flow Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
8.1 Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
8.1.1 Flow Equations for the Effective Potential. . . . . . . . . . . 73
8.1.2 Kinetic Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.2 BCS–BEC Crossover. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8.2.1 Flow of the Effective Potential. . . . . . . . . . . . . . . . . . . 79
8.3 BCS–Trion–BEC Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 82
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9 Few-Body Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
9.1 Repulsive Interacting Bosons . . . . . . . . . . . . . . . . . . . . . . . . . 86
9.1.1 Vacuum Flow Equations and their Solution for d = 3 . . . 86
9.1.2 Logarithmic Running in Two Dimensions . . . . . . . . . . . 89
9.2 Two Fermion Species: Dimer Formation . . . . . . . . . . . . . . . . . 91
9.2.1 Two-body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
9.2.2 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9.2.3 Binding Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
9.2.4 Dimer–Dimer Scattering . . . . . . . . . . . . . . . . . . . . . . . 96
9.3 Three Fermion Species: Efimov Effect. . . . . . . . . . . . . . . . . . . 97
9.3.1 SU(3) Symmetric Model . . . . . . . . . . . . . . . . . . . . . . . 98
9.3.2 Experiments with Lithium . . . . . . . . . . . . . . . . . . . . . . 107
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
10 Many-Body Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
10.1 Bose–Einstein Condensation in Three Dimensions. . . . . . . . . . . 117
10.1.1 Different Methods to Determine the Density . . . . . . . . . 117
10.1.2 Quantum Depletion of Condensate . . . . . . . . . . . . . . . . 118
10.1.3 Quantum Phase Transition. . . . . . . . . . . . . . . . . . . . . . 120
Contents ix
10.1.4 Thermal Depletion of Condensate. . . . . . . . . . . . . . . . . 123
10.1.5 Critical Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 126
10.1.6 Zero Temperature Sound Velocity. . . . . . . . . . . . . . . . . 127
10.1.7 Thermodynamic Observables . . . . . . . . . . . . . . . . . . . . 129
10.2 Superfluid Bose Gas in Two Dimensions. . . . . . . . . . . . . . . . . 144
10.2.1 Flow Equations at Zero Temperature. . . . . . . . . . . . . . . 144
10.2.2 Quantum Depletion of Condensate . . . . . . . . . . . . . . . . 146
10.2.3 Dispersion Relation and Sound Velocity . . . . . . . . . . . . 147
10.2.4 Kosterlitz–Thouless Physics. . . . . . . . . . . . . . . . . . . . . 149
10.3 Particle–Hole Fluctuations and the BCS–BEC Crossover . . . . . . 154
10.3.1 Particle–Hole Fluctuations . . . . . . . . . . . . . . . . . . . . . . 154
10.3.2 Bosonization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10.3.3 Critical Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.3.4 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.3.5 Crossover to Narrow Resonances . . . . . . . . . . . . . . . . . 164
10.4 BCS–Trion–BEC Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 166
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
11 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
12 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
12.1 Appendix A: Some Ideas on Functional Integration and
Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
12.1.1 Functional Integral with Probability Measure. . . . . . . . . 178
12.1.2 Correlation Functions from Functional Probabilities . . . . 184
12.1.3 Conservation Laws for On-Shell Excitations . . . . . . . . . 187
12.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
12.2 Appendix B: Technical Additions . . . . . . . . . . . . . . . . . . . . . . 189
12.2.1 Flow of the Effective Potential for Bose Gas. . . . . . . . . 189
12.2.2 Flow of the Effective Potential for
BCS–BEC Crossover. . . . . . . . . . . . . . . . . . . . . . . . . . 191
12.2.3 Hierarchy of Flow Equations in Vacuum. . . . . . . . . . . . 196
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Chapter 1
Introduction
Functionalrenormalizationinitsmodernformulationcontributesacentralpartand
a valuable tool to our understanding of theoretical physics. It describes how dif-
ferent theories, each of them valid on some momentum scale, are connected to
each other. In our modern understanding most theories of physics are ‘‘effective’’
theories.Theydescribephenomenaconnectedwithsometypicalmomentumscale
ktoagoodapproximation—oftenwithveryhighprecision.Ontheotherside,they
neglect phenomena that are not relevant at this momentum scale. Even the Stan-
dard Model of elementary particle physics is of this kind, since it neglects e.g.
gravity. Often, the relevant degrees of freedom change with the scale. For
example,inquantumchromodynamics(QCD)thehighenergytheoryisdescribed
in terms of quarks and gluons, while the low energy limit is governed by mesons
andbaryons.Functionalrenormalizationdescribesthistransitionbetweendifferent
descriptions—from one (effective) theory to another.
Thefunctionalrenormalizationgroupmethodisalsousefulforadifferenttask—
the statistical description of complex systems with many particles. In atomic sys-
tems,thephysicsatthemomentumscalegivenbytheinverseBohrradiusk¼ (cid:2)h is
a0
wellknown.ThesolutionofSchrödinger’sequationdeterminesthestationarywave
functionsforelectrons,theorbitals.Fromthestructureoftheorbitals,theelectro-
static,magneticandspin-relatedproperties,onecancalculatepredictionsforscat-
teringexperiments,bindingenergiesandsoon.However,ifweincreasethenumber
ofparticles,thecomplexityoftheproblemrapidlyincreasesaswell.Itisimpossible
to find exact solutions to the Schrödinger equation for several—say ten—atoms
includingalltheirelectrons.Tomakeprogress,sensibleapproximationsareneeded.
Finding a way to describe complex systems in terms of simple, but nevertheless
accurate effective theories is not easy. A lot of physical intuition and insight is
neededtomaketherightapproximations.Functionalrenormalizationhelpsusinthis
importanttask.Anexactrenormalizationgroupflowequationconnectsfieldtheories
ondifferentscales.Itscloseinvestigationoftenshowswhichtermsbecomerelevant
orirrelevantifthecharacteristicmomentumscaleischanged.
S.Flörchinger,FunctionalRenormalizationandUltracoldQuantumGases, 1
SpringerTheses,DOI:10.1007/978-3-642-14113-3_1,
(cid:2)Springer-VerlagBerlinHeidelberg2010