Hubbard Model for Asymmetric Ultracold Fermionic ... - KOMET 337

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coupling regime. The trapping potential is treated within the local density approximation.In chapter 4 we study the influence of an imbalance in the Fermi-mixture on superfluidityin a Hubbard model with spin-dependent hopping at weak attractive coupling. In viewof experimental applications, we focus on superfluid states, disregarding possible competingphases (like, e.g., charge density waves). In our discussions we concentrate on ground stateproperties.In chapter 5 the strong coupling limit of the asymmetric Hubbard model is presented.We derive effective Hamiltonians for translationally invariant systems for both attractive andrepulsive interactions.iv
Contents1 Introduction 11.1 Ultracold quantum gases and the Hubbard model . . . . . . . . . . . . . . . . 11.2 Unbalanced Fermi-mixtures: a brief experimental overview . . . . . . . . . . 42 Formalism at weak coupling 72.1 Weak (attractive) coupling limit at Hartree-Fock level . . . . . . . . . . . . . 72.2 Diagonalization of the grand canonical Hamiltonian . . . . . . . . . . . . . . 72.2.1 Fourier-transformation in the k-space . . . . . . . . . . . . . . . . . . 82.2.2 Bogoliubov-transformation . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Self-consistency equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Interaction-free densities of states . . . . . . . . . . . . . . . . . . . . . 112.3.2 Self consistency equations in the thermodynamic limit . . . . . . . . . 122.4 Properties of the self-consistency equations . . . . . . . . . . . . . . . . . . . 142.4.1 Range of the parameters n, m and ∆ . . . . . . . . . . . . . . . . . . . 142.4.2 Situation at half filling . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.3 Non-polarized solutions of first type . . . . . . . . . . . . . . . . . . . 152.4.4 The grand potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Unbalanced Fermi-mixtures 193.1 Broken translational invariance and the LDA . . . . . . . . . . . . . . . . . . 193.2 Properties of the self-consistency equations at T = 0 . . . . . . . . . . . . . . 203.2.1 Uniqueness of n and m at fixed ∆ . . . . . . . . . . . . . . . . . . . . 213.2.2 Properties of the third self-consistency equation . . . . . . . . . . . . . 253.2.3 Graphical illustration of the phases . . . . . . . . . . . . . . . . . . . . 263.3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Systems with finite temperatures . . . . . . . . . . . . . . . . . . . . . . . . . 343.5.1 Infinite spatial extension . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5.2 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Model with spin-dependent hopping 354.1 Charge density wave states and the repulsive model . . . . . . . . . . . . . . 354.1.1 Special particle-hole transformation . . . . . . . . . . . . . . . . . . . 354.1.2 Grand potential at half filling . . . . . . . . . . . . . . . . . . . . . . . 364.2 Superfluidity away from half filling . . . . . . . . . . . . . . . . . . . . . . . . 374.2.1 The quasiparticle energies . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.2 Hartree-Fock density of states . . . . . . . . . . . . . . . . . . . . . . . 384.3 Numerical method for the problem at fixed parameter n . . . . . . . . . . . . 394.4 Numerical results for the ground state . . . . . . . . . . . . . . . . . . . . . . 41v
Page 1: Hubbard Model for AsymmetricUltraco
Page 6 and 7: 4.5 Critical temperatures . . . . .
Page 8 and 9: 2 CHAPTER 1. INTRODUCTIONlattice si
Page 10 and 11: 4 CHAPTER 1. INTRODUCTION1.2 Unbala
Page 12 and 13: 6 CHAPTER 1. INTRODUCTION
Page 14: 8 CHAPTER 2. FORMALISM AT WEAK COUP
Page 17 and 18: 2.3. SELF-CONSISTENCY EQUATIONS 11w
Page 19 and 20: 2.3. SELF-CONSISTENCY EQUATIONS 13w
Page 21 and 22: 2.4. PROPERTIES OF THE SELF-CONSIST
Page 25 and 26: Chapter 3Unbalanced Fermi-mixturesI
Page 33 and 34: 3.3. NUMERICAL METHOD 27- In Figure
Page 35 and 36: 3.3. NUMERICAL METHOD 29Input of
Page 37 and 38: 3.4. NUMERICAL RESULTS 31x-position
Page 39 and 40: 3.4. NUMERICAL RESULTS 33x-position
Page 41 and 42: Chapter 4Model with spin-dependent
Page 43 and 44: 4.2. SUPERFLUIDITY AWAY FROM HALF F
Page 45 and 46: 4.3. NUMERICAL METHOD FOR THE PROBL
Page 47 and 48: 4.4. NUMERICAL RESULTS FOR THE GROU
Page 49 and 50: 4.5. CRITICAL TEMPERATURES 430.4∆
Page 51 and 52: 4.5. CRITICAL TEMPERATURES 454.5.2
Page 53 and 54: Chapter 5Strong coupling limitIn th
Page 55 and 56:
5.2. GENERAL PROCEDURE 49with an an
Page 57 and 58:
= ∑ ijlσ= − ∑ ijlσ= ∑ ijl
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5.4. LOW TEMPERATURE LIMIT 53At fir
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5.4. LOW TEMPERATURE LIMIT 555.4.2
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Summary and OutlookIn this thesis w
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Bibliography[1] C. Wieman and E. Co
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DanksagungAn erster Stelle möchte
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