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

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4.5 Critical temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.5.1 Superfluidity in non-magnetized solutions . . . . . . . . . . . . . . . . 434.5.2 Phase transition at T = T C . . . . . . . . . . . . . . . . . . . . . . . . 455 Strong coupling limit 475.1 Introduction: Exactly solvable 2-site model . . . . . . . . . . . . . . . . . . . 475.1.1 Diagonalization of the Hamiltonian . . . . . . . . . . . . . . . . . . . . 475.1.2 “Band structure” at strong coupling . . . . . . . . . . . . . . . . . . . 485.2 General procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2.1 Second order perturbation theory . . . . . . . . . . . . . . . . . . . . . 495.2.2 Choice of the transformation operators . . . . . . . . . . . . . . . . . . 505.2.3 Structure of the second order term . . . . . . . . . . . . . . . . . . . . 515.3 Treatment of pure nearest neighbor hopping . . . . . . . . . . . . . . . . . . . 525.4 Low temperature limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.4.1 Repulsive U model at half filling . . . . . . . . . . . . . . . . . . . . . 525.4.2 Attractive U model with even number of fermions . . . . . . . . . . . 55vi
Chapter 1IntroductionAs an introduction, we show how the Hubbard model may be used to describe the physics ofultracold atoms loaded into an optical lattice. We also discuss the main motivation of thisthesis, namely the recent interest in unbalanced Fermi-mixtures.1.1 Ultracold quantum gases and the Hubbard modelAtoms loaded into optical lattices feel an effective potential, induced by the interaction oftheir dipole moment with the electric field of the coherent laser beam. In order to create anoptical lattice for an atomic species, the laser frequency must be far away from any transitionfrequency, so that the laser does not induce any transitions of the atoms’ internal state. Inthis case the effective potential arises from the AC-Stark-shift, and for hypercubic (d = 1,2,3)lattices it takes the form [2] [5] [7]:V (x) = − V 0dd∑i=1( cos 2 πx) ia, (1.1)where a = λ 2 is the lattice constant, λ is the wavelength of the laser beam and V 0 is proportionalto the laser’s intensity. As a consequence, for non-interacting particles, the correspondingSchrödinger equation can always be solved with the help of Floquet’s theorem inone dimension [21] [22]. Since the d-dimensional potential V (x) is a sum of one-dimensionalterms, the d-dimensional wave function is just a product of one-dimensional wave functions(see standard literature about quantum mechanics, [23] for example). Since we are onlyinterested in treating the tight-binding caseV 0 ≫22ma 2 , (1.2)where m is the atom’s mass, and want to derive a one-band Hubbard model, we present analternative and much shorter derivation here. We assume the validity of(1.2) and call eachpoint x i which satisfiesV (x i ) = −V 0 (1.3)a lattice site with lattice vector i. Of course we have the free choice to decide which pointon the d-dimensional lattice is identified with the origin i = 0, since we assume the systemto be translationally invariant here. The components of i are discrete (i ∈ N d ) and can beidentified with the difference vector from the lattice site to the origin. If we consider a fixed1
Page 1: Hubbard Model for AsymmetricUltraco
Page 4 and 5: coupling regime. The trapping poten
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
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= ∑ 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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