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

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34 CHAPTER 3. UNBALANCED FERMI-MIXTURES7. The occupation number n depends very weakly on the phase occurring.We also see that superfluidity favors half filling, which is not surprising, since the superfluidorder parameter can only be maximized at half filling (see section 2.4.1). In further investigationswe have never been able to find the magnetized superfluid phase, so that we mayassume that this phase does not occur at least at T = 0.These results could be directly observed in experiments with the help of in-situ imaging.It is not possible to image the superfluid order parameter directly, but the difference betweenthe occupation numbers of the different spin-species can be directly observed [16] [17].3.5 Systems with finite temperaturesWe will now briefly discuss some topics about systems at low temperatures and mention somedifferences and similarities to systems in the ground state.3.5.1 Infinite spatial extensionMathematically the difference arises from the fact that at finite temperatures the systemsare described with the help of a Fermi-function, which at T = 0 is replaced by a unit stepfunction. In our LDA calculation we have used that for some critical value of µ < µ C theoccupation numbers vanish exactly:n = 0 ∧ m = 0 ∧ ∆ = 0 (∀ µ < µ C ) . (3.22)This fact arises from the sharp edge at the Fermi-level, which causes all states to be empty.At T > 0 the Fermi-function has no sharp edge, so that there is no µ C in that sense. As aconsequence in our LDA-calculation we get:n ∝ e −βµ ∝ e −β(i·Ω·i) , ( |i | → ∞) , (3.23)so that we have to define an abortion criterion, n < n C = 10 −4 for example to avoid aninfinite loop.3.5.2 Phase transitionsThe phase transition in systems with B = 0 is of second order, since∂Ω∂ 2 ∣Ω ∣∣∣∆=0∂∆∣ = 0 and∆=0∂∆ 2 < 0 (∀ T < T C ) , (3.24)so that ∆ = 0 is not a minimum of the grand potential. That means for T = 0, that wefind a superfluid order parameter ∆ ≠ 0 as thermodynamically stable solution of the selfconsistencyequations. Increasing T causes ∆ to decrease until a critical temperature T C isreached, where ∆ → 0. At T > T C there is no other solution than ∆ = 0 for (3.5). In systemswith B ≠ 0, phase transitions are of first order, so that there is no similar argumentationas in non-magnetized systems. With B ≠ 0 we have to take into account that transitions offirst order between a non-magnetized superfluid phase, a magnetized superfluid phase and anormal phase may occur. More detailed information is found in section 4.5.
Chapter 4Model with spin-dependent hoppingIn this chapter we will treat effects arising from spin-dependent hopping in translationallyinvariant systems. The magnetic field is switched off (B = 0) and hopping is spin-dependent(t ↑ ≠ t ↓ ) during this chapter. First we explain our focus on superfluid solutions. We thendiscuss superfluidity in the attractive U model away from half filling with the resulting magnetizationat T = 0 and critical temperatures of the phase transition at half filling. Allnumerical calculations have been done for three-dimensional systems.4.1 Charge density wave states and the repulsive modelIn this section we compare the attractive U model (2.1) at B = 0 with its repulsive version athalf filling. We show their equivalence and compose a“dictionary”of corresponding propertiesin order to motivate our focus on superfluid phases in this thesis neglecting possible competingphases, like, e.g., charge density wave phases (CDW).4.1.1 Special particle-hole transformationHere we present the equivalence of the model (2.1) at attractive interaction and B = 0 withthe repulsive model at half filling in a magnetic field. The proof is done by performing aspecial particle-hole transformation:c † i↑→ c † i↑c † i↓→ exp(iQ · i)c i↓(4.1)Q = (π,π,... ,π) T .As was the case for the general particle-hole transformation, studies in (2.56), the specialparticle-hole transformation (4.1) leaves the hopping-term H t unchanged, but the interactionterm H U changes its sign. The Zeeman-term in the repulsive model is transformed into anadditional chemical potential term in the attractive model. We show the transformation from35
Page 1: Hubbard Model for AsymmetricUltraco
Page 4 and 5: coupling regime. The trapping poten
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: 3.4. NUMERICAL RESULTS 33x-position
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
Page 59 and 60: 5.4. LOW TEMPERATURE LIMIT 53At fir
Page 61 and 62: 5.4. LOW TEMPERATURE LIMIT 555.4.2
Page 63 and 64: Summary and OutlookIn this thesis w
Page 65 and 66: Bibliography[1] C. Wieman and E. Co
Page 67 and 68: DanksagungAn erster Stelle möchte

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