Dynamical mean-field theory approach for ultracold atomic gases [Elektronische Ressource] / Irakli Titvinidze

Sujets

• Physik

Informations

• Publié par: goethe_universitat_frankfurt_am_main
• Publié le: 01 janvier 2009
• Nombre de lectures: 25
• Langue: English
• Poids de l'ouvrage: 13 Mo

Extrait

Dynamical Mean-Field Theory Approach for
Ultracold Atomic Gases
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften
vorgelegt am Fachbereich Physik
der Goethe-Universität in Frankfurt am Main
Irakli Titvinidze
aus Tiflis
Frankfurt am Main
2009vom Fachbereich Physik
der Goethe-Universität als Dissertation angenommen
Dekan Prof. Dr. Dirk-Hermann Rischke
Gutachter Prof. Dr. Walter Hofstetter
Prof. Dr. Maria-Roser Valentí
Datum der
Disputation 12.10.2009Dedicated to the memory of my father
12Contents
1 Introduction 5
2 Ultracold atomic Physics 7
2.1 Short Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Cooling Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 BCS-BEC Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 The Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6.1 The Fermionic Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . 21
2.6.2 The Bose-Fermi Model . . . . . . . . . . . . . . . . . . . . . . 22
3 Method 25
3.1 Dynamical Mean-Field Theory (DMFT) . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Generalized Dynamical Mean-Field Theory (GDMFT) . . . . . . . . . . . . . . 30
3.2.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.2 Energy Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Real-Space Dynamical Mean-Field Theory (R-DMFT) . . . . . . . . . . . . . . 35
3.4 Impurity Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.1 Exact Diagonalization (ED) . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.2 Numerical Renormalization Group (NRG) . . . . . . . . . . . . . . . . . 39
4 Mixtures of Fermions and Bosons in Optical Lattices 45
4.1 Mixtures of Spinless Fermions and Bosons in Optical Lattices . . . . . . . . . . 46
4.1.1 Half-Filled Mixture of Spinless Fermions and Bosons . . . . . . . . . . . 48
4.1.2 3/2-Filled Bosons and Half-Filled Spinless Fermions . . . . . . . . . . . 51
4.2 Mixtures of Hard-Core Bosons and Two-Component Fermions in an Optical
Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5 Ultra-Cold Atoms in a Harmonic Trap 59
5.1 Fermions in a Trap . . . . . . . . . . . . . . . . . . . . . . 59
5.1.1 Balanced Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1.2 Imbalanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Mixtures of Spinless Fermions and Bosons in a Harmonic Trap . . . . . . . . . 67
34 CONTENTS
6 Resonance Superfluidity in an Optical Lattice 71
6.1 Microscopic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7 Summary 81
8 Zusammenfassung 83
A Relation between Experimental and Hubbard Parameters 87
B Derivation of the DMFT Effective Action 91
C The Equation of Motion and Green’s Functions 93
D Derivation of the Self-Energy for a Bose-Fermi Mixture 95
E Derivation of the Kinetic Energy 99
F Iterative Diagonalization within NRG 103
Bibliography 107
Acknowledgments 117
Curriculum Vitae 119Chapter 1
Introduction
The impressive experimental progress in the field of ultracold atoms in the last decade has
brought it to the forefront of research on strongly correlated quantum many-body systems.
The possibility to confine and manipulate atoms in optical lattices created by standing waves
of laser light gives the opportunity to realize some of the model Hamiltonians of condensed-
matter physics, and this way shed light on notoriously difficult problems [1–3]. Going beyond
that, also systems without clear analog in “conventional” condensed matter systems can be
realized. In particular, cold atomic gases offer the possibility to realize mixtures of fermions
and bosons [4–19]. This yields a very rich system, which at this moment is far from fully
explored.
Thissystembearssomeanalogywiththewell-knowntwo-componentFermi-Fermimixture,
but is in fact much richer. By replacing one of the fermionic components by bosons, one
keeps the instability of half-filled fermions towards charge-density wave (CDW) ordering. For
historical reasons we keep this terminology throughout this thesis, although the fermionic
atoms under consideration do not carry a charge. At the same time the bosonic species can
be superfluid, allowing for supersolid behavior, where diagonal CDW order coexists with off-
diagonal superfluid long-range order. Several previous theoretical works have studied mixtures
of fermions and bosons in an optical lattice [20–45].
Investigating a strongly correlated Bose-Fermi mixture in an optical lattice is a difficult
problem, to which powerful numerical and analytical techniques have been applied. In one
dimension this involved Bethe-Ansatz technique [25], bosonization [26, 28], Density Matrix
Renormalization Group [32, 35], and quantum Monte Carlo (QMC) [23, 36–40]. In higher di-
mensions, however, non-perturbative calculations are sparse. In two dimensions Renormaliza-
tion Group studies [24, 31] have been carried out. Although able to describe non-perturbative
effects, this technique is limited to weak couplings. Another powerful technique that has
been applied in two [29], and recently also three dimensions [33, 34] is to integrate out the
fermions. In this way one generates a long-ranged, retarded interaction between the bosons,
which means that the resulting bosonic problem is still hard to solve. Important progress has
recently been made in mapping out the Mott-insulating lobes. A composite fermion approach
[30] was used to qualitatively describe possible quantum phases of the Bose-Fermi mixture.
InthisthesisweintroduceandapplyGeneralizedDynamicalMean-FieldTheory(GDMFT)
to study this problem. This is a non-perturbative method which becomes exact in infinite di-
mensions and is a good approximation for three spatial dimensions (see section 3.2). The only
small parameter is 1=z, where z is the coordination number. For this reason, the method
56 1. Introduction
reliably describes the full range from weak to strong coupling. The advantage of this method
is that in contrast to QMC calculations, this method works in high dimension and not only
allows to map out the phase borders but also gives reliable results away from it (in contrast
to the Refs. [29, 33, 34]). As we will show in section 3.2 GDMFT has a systematic derivation
in contrast to the composite fermion approach.
We apply the GDMFT to a variety of cases. In particular, we study commensurate mixture
of the spinless (spin-polarized) fermions and bosons, as well as a mixture of hard-core bosons
and two-component fermions in an optical lattice. The reason why we chose commensurate
filling is that in this case interesting phases, like the supersolid can occur, which break the
translation symmetry. We also take into account the effect of the harmonic trap. For this
purpose we develop Real Space Dynamical Mean-Field Theory (R-DMFT) (see section 3.3).
This thesis is structured as follows: In chapter 2 we present a short overview of the
physics of ultracold atoms. We start with a short historical outlook (section 2.1). Then we
briefly review the cooling methods (section 2.1), Feshbach resonance 2.3), BCS-BEC
crossover (section 2.4), optical lattices (section 2.5) and in the end of the chapter we derive
the model Hamiltonians that will be dealt with the rest of the thesis (section 2.6).
In chapter 3 we describe the methods used in this thesis. In particular, dynamical mean
field theory (section 3.1), GDMFT (section 3.2), R-DMFT (section 3.3). In this chapter we
also consider impurity solvers which we use during our calculation: exact diagonalization
(section 3.4.1) and numerical renormalization theory (section 3.4.2).
In chapter 4 we study the mixture of the spinless fermions and bosons for commensurate
fillings. In particular, in section 4.1.1 we study this mixture when both of them are half-filled,
while in the section 4.1.2 we study the case when the fermions are again half-filled while the
filling of the bosons is 3=2. In this chapter we also study a mixture of hard-core bosons and
two-component fermions (section 4.2).
In chapter 5 we study the effect of the harmonic trap. First we study a repulsively in-
teracting two component Fermi gas in a trap. We investigate the stability of the
antiferromagnetic order against the presence of the harmonic potential (section 5.1). Later
on we consider a mixture of spinless fermions and bosons and investigate the stability of the
supersolid in the presence of the trap (section 5.2).
In chapter 6 we study an ultracold atomic gas of fermionic atoms and bosonic molecules
close to a Feshbach resonance. We consider the process when due to the Feshbach resonance
two fermionic atoms with opposite spin can form a boson