Functional renormalization and ultracold quantum gases [Elektronische Ressource] / presented by Stefan Flörchinger

Dissertationsubmitted to theCombined Faculties for the Natural Sciences and for Mathematicsof the Ruperto-Carola University of Heidelberg, Germanyfor the degree ofDoctor of Natural Sciencespresented byStefan Fl¨orchingerborn in Friedrichshafen, GermanyOral examination: 24. June 2009Functional renormalizationandultracold quantum gasesReferees: Prof. Dr. Christof WetterichProf. Dr. Holger GiesFunktionale Renormierung und ultrakalte QuantengaseZusammenfassungDie Funktionale Renormierungsgruppen-Methode wird zur theoretischen Untersuchung ultra-kalter Quantengase angewandt. Flussgleichungen werden abgeleitet fur¨ Bosonen mit n¨aherungs-weise punktf¨ormiger Wechselwirkung, fur¨ Fermionen mit zwei (Hyperfein-) Komponenten imCrossover vom Bardeen-Cooper-Schrieffer (BCS) Zustand zu einem Bose-Einstein Kondensat(BEC) sowie fur¨ ein Gas von Fermionen mit drei Komponenten. Die L¨osungen der Flussgle-ichungen bestimmen die Eigenschaften dieser Systeme sowohl fur¨ wenige Teilchen als auch imthermischen Gleichgewicht.Im Fall der Bosonen werden die Eigenschaften sowohl fur¨ drei als auch fur¨ zwei ra¨umlicheDimensionen diskutiert, insbesondere das Quantenphasendiagramm, der kondensierte und su-perfluide Anteil, die kritische Temperatur, die Korrelationsla¨nge, die spezifische W¨arme unddie Schallausbreitung.
Publié le : jeudi 1 janvier 2009
Lecture(s) : 40
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Source : ARCHIV.UB.UNI-HEIDELBERG.DE/VOLLTEXTSERVER/VOLLTEXTE/2009/9615/PDF/DISSERTATION_STEFAN_FLOERCHINGER.PDF
Nombre de pages : 187
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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
Stefan Fl¨orchinger
born in Friedrichshafen, Germany
Oral examination: 24. June 2009Functional renormalization
and
ultracold quantum gases
Referees: Prof. Dr. Christof Wetterich
Prof. Dr. Holger GiesFunktionale Renormierung und ultrakalte Quantengase
Zusammenfassung
Die Funktionale Renormierungsgruppen-Methode wird zur theoretischen Untersuchung ultra-
kalter Quantengase angewandt. Flussgleichungen werden abgeleitet fur¨ Bosonen mit n¨aherungs-
weise punktf¨ormiger Wechselwirkung, fur¨ Fermionen mit zwei (Hyperfein-) Komponenten im
Crossover vom Bardeen-Cooper-Schrieffer (BCS) Zustand zu einem Bose-Einstein Kondensat
(BEC) sowie fur¨ ein Gas von Fermionen mit drei Komponenten. Die L¨osungen der Flussgle-
ichungen bestimmen die Eigenschaften dieser Systeme sowohl fur¨ wenige Teilchen als auch im
thermischen Gleichgewicht.
Im Fall der Bosonen werden die Eigenschaften sowohl fur¨ drei als auch fur¨ zwei ra¨umliche
Dimensionen diskutiert, insbesondere das Quantenphasendiagramm, der kondensierte und su-
perfluide Anteil, die kritische Temperatur, die Korrelationsla¨nge, die spezifische W¨arme und
die Schallausbreitung. Die Diskussion des Fermionen-Gases im BCS-BEC-Crossover konzentri-
ert sich auf den Effekt von Teilchen-Loch Fluktuationen, betrifft aber das gesamte Phasendia-
gramm. Fur¨ drei verschiedene Fermionen zeigen die Flussgleichungen einen Grenzzyklus sowie
das Efimov-Spektrum von Dreiteilchen-Bindungszusta¨nden. Angewandt auf Lithium kann ein
kur¨ zlich beobachteter Dreiteilchen-Verlust erkl¨art werden. Mit Hilfe eines Kontinuita¨ts-Argu-
mentes findet sich fur¨ drei Fermionen-Komponenten in der Na¨he einer gemeinsamen Resonanz
eine neue Trionen-Phase, welche die BCS- und die BEC-Phase trennt.
Etwas formaler ist die Herleitung einer neuen exakten Flussgleichung fur¨ skalenabh¨angige
zusammengesetzte Operatoren. Diese erm¨oglicht etwa eine verbesserte Behandlung gebundener
Zusta¨nde.
Functional renormalization and ultracold quantum gases
Abstract
Themethodoffunctionalrenormalizationisappliedtothetheoreticalinvestigationofultracold
quantum gases. Flow equations are derived for a Bose gas with approximately pointlike inter-
action, for a Fermi gas with two (hyperfine) spin components in the Bardeen-Cooper-Schrieffer
(BCS) to Bose-Einstein condensation (BEC) crossover and for a Fermi gas with three compo-
nents. The solution of the flow equations determine the properties of these systems both in the
few-body regime and in thermal equilibrium.
For the Bose gas this covers the quantum phase diagram, the condensate and superfluid
fraction, the critical temperature, the correlation length, the specific heat or sound propagation.
The properties are discussed both for three and two spatial dimensions. The discussion of the
Fermi gas in the BCS-BEC crossover concentrates on the effect of particle-hole fluctuations but
addresses the complete phase diagram. For the three component fermions, the flow equations
in the few-body regime show a limit-cycle scaling and the Efimov tower of three-body bound
states. Applied to the case of Lithium they explain recently observed three-body loss features.
Extending the calculations by continuity to nonzero density, it is found that a new trion phase
separates a BCS and a BEC phase for three component fermions close to a common resonance.
More formal is the derivation of a new exact flow equation for scale dependent composite
operators. This equation allows for example a better treatment of bound states.
iiiContents
1. Introduction 1
1.1. Flow equations to solve an integral . . . . . . . . . . . . . . . . . . . . . . 3
1.2. Functional integral representation of quantum field theory . . . . . . . . . 7
2. The Wetterich equation 14
2.1. Scale dependent Schwinger functional . . . . . . . . . . . . . . . . . . . . 14
2.2. The average action and its flow equation . . . . . . . . . . . . . . . . . . . 15
2.3. Functional integral representation and initial condition . . . . . . . . . . . 17
3. Generalized flow equation 18
3.1. Scale-dependent Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2. Flowing action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3. General coordinate transformations . . . . . . . . . . . . . . . . . . . . . . 23
4. Truncations 26
5. Cutoff choices 30
6. Investigated models 34
6.1. Bose gas in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.2. Bose gas in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.3. BCS-BEC Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.4. BCS-Trion-BEC Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7. Symmetries 45
7.1. Derivative expansion and ward identities . . . . . . . . . . . . . . . . . . . 45
7.2. Noethers theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8. Truncated flow equations 55
8.1. Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.2. BCS-BEC Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8.3. BCS-Trion-BEC Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 65
9. Few-body physics 67
9.1. Repulsive interacting bosons . . . . . . . . . . . . . . . . . . . . . . . . . . 68
9.2. Two fermion species: Dimer formation . . . . . . . . . . . . . . . . . . . . 73
9.3. Three fermion species: Efimov effect . . . . . . . . . . . . . . . . . . . . . 79
9.3.1. SU(3) symmetric model . . . . . . . . . . . . . . . . . . . . . . . . 79
ivContents
9.3.2. Experiments with Lithium . . . . . . . . . . . . . . . . . . . . . . . 88
10.Many-body physics 96
10.1.Bose-Einstein Condensation in three dimensions. . . . . . . . . . . . . . . 96
10.1.1. Different methods to determine the density . . . . . . . . . . . . . 96
10.1.2. Quantum depletion of condensate . . . . . . . . . . . . . . . . . . . 97
10.1.3. Quantum phase transition . . . . . . . . . . . . . . . . . . . . . . . 99
10.1.4. Thermal depletion of condensate . . . . . . . . . . . . . . . . . . . 102
10.1.5. Critical temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 104
10.1.6. Zero temperature sound velocity . . . . . . . . . . . . . . . . . . . 105
10.1.7. Thermodynamic observables. . . . . . . . . . . . . . . . . . . . . . 108
10.2.Superfluid Bose gas in two dimensions . . . . . . . . . . . . . . . . . . . . 122
10.2.1. Flow equations at zero temperature . . . . . . . . . . . . . . . . . 123
10.2.2. Quantum depletion of condensate . . . . . . . . . . . . . . . . . . . 124
10.2.3. Dispersion relation and sound velocity . . . . . . . . . . . . . . . . 126
10.2.4. Kosterlitz-Thouless physics . . . . . . . . . . . . . . . . . . . . . . 128
10.3.Particle-hole fluctuations and the BCS-BEC Crossover . . . . . . . . . . . 132
10.3.1. Particle-hole fluctuations . . . . . . . . . . . . . . . . . . . . . . . 132
10.3.2. Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10.3.3. Critical temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10.3.4. Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
10.3.5. Crossover to narrow resonances . . . . . . . . . . . . . . . . . . . . 141
10.4.BCS-Trion-BEC Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 143
11.Conclusions 146
A. Some ideas on functional integration and probability 151
B. Technical additions 162
B.1. Flow of the effective potential for Bose gas . . . . . . . . . . . . . . . . . . 162
B.2. Flow of the effective potential for BCS-BEC Crossover . . . . . . . . . . . 164
B.3. Hierarchy of flow equations in vacuum . . . . . . . . . . . . . . . . . . . . 168
Bibliography 172
v1. Introduction
Functional renormalization in its modern formulation contributes a central part and
a valuable tool to our understanding of theoretical physics. It describes how different
theories,eachofthemvalidonsomemomentumscale,areconnectedtoeachother. Inour
modern understanding most theories of physics are “effective” theories. They describe
phenomena connected with some typical momentum scale k to a good approximation
– often with very high precision. On the other side, they neglect phenomena that are
not relevant at this momentum scale. Even the Standard 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, in Quantum Chromodynamics (QCD) the
highenergytheoryisdescribedintermsofquarksandgluons, whilethelowenergylimit
is governed by mesons and baryons. Functional renormalization describes this transition
between different descriptions – from one (effective) theory to another.
The functional renormalization group method is also useful for a different task – the
statistical description of complex systems with many particles. In atomic systems, the
~physics at the momentum scale given by the inverse Bohr radius k = is well known.
a0
ThesolutionofSchr¨odinger’sequationdeterminesthestationarywavefunctionsforelec-
trons, the orbitals. From the structure of the orbitals, the electrostatic, magnetic and
spin-relatedproperties, onecancalculatepredictionsforscatteringexperiments, binding
energies and so on. However, if we increase the number of particles, the complexity of
the problem rapidly increases as well. It is impossible to find exact solutions to the
Schr¨odinger equation for several – say ten – atoms including all their electrons. To
make progress, sensible approximations are needed. 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 needed to make the right approximations. Func-
tional renormalization helps us in this important task. An exact renormalization group
flow equation connects field theories on different scales. Its close investigation often
shows which terms become relevant or irrelevant if the characteristic momentum scale
is changed.
The renormalization group was first developed for the description of critical phenom-
ena close to phase transitions in statistical systems [1–3]. Subsequently, it was realized
that the idea of a renormalization group flow as a continuous version of Kadanoff’s
block-spin transformation [4] is of great value also for quantum field theory, see e. g.
[5]. The development of the renormalization group allowed for a deeper understanding
of the formalism including the “mysterious divergences” in perturbation theory and the
“renormalization of coupling constants” by introducing counterterms.
The modern formulation of functional renormalization uses the concept of the aver-
age action (or flowing action) as a modification of the quantum effective action [6–8].
11. Introduction
The effective action is the generating functional of the one-particle irreducible correla-
tion functions, see e. g. [9]. A simple, intuitive, but nevertheless exact renormalization
group flow equation describes the evolution from microscopic to macroscopic scales [10].
The approach has proven to be successful in many applications ranging from Quantum
Chromodynamics (QCD) to critical phenomena, for reviews see [11–17]. However, the
methodisnotyetdevelopedcompletely. Openissuesconcernforexamplethedescription
on non-equilibrium dynamics or the improvement of various approximation schemes. A
conceptualpointaddressedinthisthesisisaflowequationforscale-dependentcomposite
operators which allows for a flow from ultraviolet to infrared degrees of freedom [18, 19].
Ultracoldquantumgasesprovideanidealtestinggroundfortheflowequationmethod.
The ultraviolet physics in the form of atomic physics is well known. In the few-particle
sector, exact results are available from quantum mechanical treatments. Many param-
eters, such as interaction strengths or more obvious temperature and density, can be
tuned experimentally over a wide range. Using tightly confining trap potentials, it is
even possible to realize different dimensionalities of space. The experimental methods
were developed rapidly and improved steadily in the last years and allow for the prepa-
ration of very clean samples, nearly without any impurities. It can be expected that
future developments lead to more and more accurate measurement techniques which
would allow for precision tests of theoretical predictions.
At the current stage, many experiments are rather well described by perturbative
theories for small coupling constants. Examples are the free theory where interaction
effects are neglected completely, or Gaussian approximations such as Bogoliubov theory,
Hartree-Fock, or various variants of Mean-Field theories. In principle, the flow equa-
tion method can reproduce the results of all these approaches in the regime where the
corresponding approximations are valid. Moreover, it can give corrections and also de-
scribe (non-perturbative) features that are not captured in a Gaussian treatment, for
example critical phenomena. Experimentally, these corrections should become relevant
for strongly interacting systems such as fermions in the BCS-BEC crossover or for lower
dimensional systems where fluctuation effects are more important. Also the regions
around phase transitions – either quantum or classical – are interesting in this respect.
Flow equations have the potential to constitute a systematic extension of perturbative
treatments. In contrast to Monte-Carlo methods, the numerical effort is very small.
Physical insight is easier to gain from inspecting flow equations then from complex nu-
merical simulations. Even exact statements can sometimes be made from considering
the flow equations in an interesting limit where they can be solved exactly.
A large part of the original work presented in this thesis has been published in differ-
ent articles. For the Bose gas these are “Functional renormalization for Bose-Einstein
condensation” [20], “Superfluid Bose gas in two dimensions” [21] and “Nonperturba-
tive thermodynamics of an interacting Bose gas” [22] and for the BSC-BEC crossover
“Particle-hole fluctuations in BCS-BEC crossover”[23]aswellas“Functional renormal-
ization group approach for the BCS-BEC crossover” [24]. The work on three-component
fermions is published in “Functional renormalization for trion formation in ultracold
fermion gases” [25], “Efimov effect from functional renormalization” [26] and “Three-
body loss in lithium from functional renormalization” [27]. Finally, the new exact flow
21. Introduction
equation for composite operators is published in “Exact flow equation for composite
operators” [28].
In this thesis, the first chapters are devoted to more conceptual issues while concrete
applicationstoultracoldquantumgasesarediscussedinlaterchapters. Intheremainder
ofthepresentchapter, weexplainsomemathematicalideasunderlyingtheflowequation
method at the example of a one-dimensional integral. The functional integral formu-
lation of quantum field theory including the Matsubara formalism is briefly reviewed
thereafter. Chapter 2 discusses the flow equation first obtained by C. Wetterich. We
re-derive it starting from the functional integral representation of the partition function.
A somewhat generalized form is derived in chapter 3. In chapter 4 we discuss the idea
and use of truncations as an approximate method to solve the flow equation. Chapter
5 is devoted to the choice of the appropriate cutoff function with an emphasis on the
particular problems occurring in nonrelativistic field theories.
Contact with concrete physics is first made in chapter 6 where we introduce the mi-
croscopic models investigated in this thesis. Besides the repulsive Bose gas in three and
two dimensions, this includes two component fermions in the BCS-BEC crossover and
fermions with three hyperfine species (“BCS-Trion-BEC transition”). Chapters 7 and
8 are again a bit more formal and discuss the different symmetries of our models and
theusedapproximationschemes. Resultsconcerningthefew-bodyphysicsarepresented
in chapter 9. However, the results of this thesis concern mainly the many-body regime
and are presented in chapter 10. We discuss the phase diagram and thermodynamic
observables for bosons in three spatial dimensions, the superfluidity of an interacting
Bose gas in two dimensions, the phase diagram of a two-component Fermi gas in the
BCS-BEC crossover with emphasis on the particle-hole fluctuations and the BCS-Trion-
BEC transition expected for a gas of three fermion species close to a common Feshbach
resonance. Finally, we draw some conclusions in chapter 11.
InappendixAwepresentsomeideasconcerningtheconnectionbetweenthefunctional
integral and probability in the foundations of quantum theory. More specific, we discuss
areformulationofthefunctionalintegralrepresentationintermsof(quasi-)probabilities.
More technical additions such as concrete flow equations for the effective potential and
the proof of a theorem concerning the flow equations in vacuum, are given in appendix
B.
1.1. Flow equations to solve an integral
Inthisintroductorysectionwedevelopamethodtocalculatesimpleone-dimensionalin-
tegrals. Thismaynotseemveryusefulsincemanyintegrationtechniquesareknownand
the method we devise here is not particularly simple. However, it has some advantages,
the most important of which is that it can be generalized easily to higher-dimensional or
even infinite-dimensional (functional) integrals. No attempt is made to present the fol-
lowingdiscussioninmathematicalrigor. Itshouldbeseenasanintroductorywarm-upto
come into contact with some tools and ideas used in later chapters. In particular we will
assume that all involved functions have nice enough properties concerning smoothness
31. Introduction
and convergence.
Our goal is to calculate an integral of the formZ ∞
Z = dxf(x). (1.1)
−∞
For simplicity we take the function f to be positive semi-definite, f(x)≥ 0. One may
think of f as describing a probability distribution of the variable x. (For Z = 1 this
probabilitydistributionwouldbenormalized.) Forconvenienceweintroducethefunction
S(x) defined by
−S(x)f(x)=e . (1.2)
The special case where S is quadratic in x
1 2 2S(x)= m x (1.3)
2
2withm >0 can be treated exactly. In that case the integral in Eq. (1.1) is of Gaussian
form and we obtain Z ∞ √1 2 2− m x
2Z = dxe = 2π/m. (1.4)
−∞
It is useful to generalize Z somewhat by introducing the “source” j. We defineZ ∞
−S(x)+jxZ(j)= dxe . (1.5)
−∞
From Z(j) we can easily derive expectation values, for exampleZ ∞1 1 ∂−S(x)+jxhxi= dxxe = Z(j). (1.6)
Z(j) Z(j)∂j−∞
Higher momenta of the probability distribution are obtained as
n1 ∂nhx i= Z(j). (1.7)
nZ(j)∂j
From the “Schwinger function”W(j)=lnZ(j) we can directly obtain the cumulants of
the probability distribution. For example, the variance is given by
2∂2 2σ =hx i−hxi = W(j). (1.8)
2∂j
More general, the n-th cumulant is given by
n∂
κ = W(j). (1.9)n n∂j
For this reason the functions Z(j) and W(j) are also known as the moment-generating
function and cumulant generating function, respectively.
4

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