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

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42 CHAPTER 4. MODEL WITH SPIN-DEPENDENT HOPPING-0.5µ-2.5µ-1-2.6-2.7-1.5-2.8-5 -4 -3 -2 -1-2.5U-5 -4 -3 -2 -1-2.9-3.1-3.2U0.08m0.8m0.060.60.040.40.020.2-5 -4 -3 -2 -1U-5 -4 -3 -2 -1U0.4∆0.14∆0.30.120.10.20.080.060.10.040.02-5 -4 -3 -2 -1U-5 -4 -3 -2 -1UFigure 4.6: µ, m and ∆ as a function of U near half filling on the left, away from half fillingon the right.As parameters for these calculations we have used µ(U = 0) = −0.5, approximatively correspondingto n ≈ 0.79, for the solutions near half filling and µ(U = 0) = −2.5, approximativelycorresponding to n ≈ 0.18, for the solutions far away from half filling. We have alsodone a calculation exactly at half filling, where we have m = 0 and µ = U 2(see section 2.4.3).The order parameter ∆ as function of U behaves as follows:
4.5. CRITICAL TEMPERATURES 430.4∆0.30.20.1-5 -4 -3 -2 -1UFigure 4.7: Superfluid order parameter ∆ as a function of U. For small values of U, the orderparameter becomes exponentially small.The numerical results can now be summarized as follows:- At half filling we are able to show analytically that ∆ > 0 is the correct solution.- Numerically we are not able to detect ∆ for small values of U (|U| ≪ t ↑ + t ↓ ), since ∆becomes exponentially small.- Near half filling (n ≈ 0.79) we need U to be greater than a critical value in order to getsuperfluidity.- Superfluid solutions are non-magnetized solutions of the second type near half filling.- Far away from half filling (n ≈ 0.18) we need U to be even greater than in the situationnear half filling.- Superfluid solutions are magnetized far away from half filling.With these calculations we have been able to see the basic effects arising from spin-dependenthopping. It is not surprising that, the further we are away from half filling the more oursystem tends to be magnetized and non-superfluid, since the symmetric pairing is suppressedby magnetization and the non-interacting system is rigorously magnetized.4.5 Critical temperaturesIn this section we present the behavior of the critical temperature of the superfluid orderparameter ∆ in the superfluid phase and the critical temperature of the order parameter s ofthe CDW phase at half filling. We first show that at 0 ≤ T < T C,∆/s the superfluid and theCDW solution always have a lower free energy then the non-symmetry-broken one. Then wepresent the self-consistency equation at T = T C,∆/s in order to calculate T C,∆/s numerically.4.5.1 Superfluidity in non-magnetized solutionsWe now show that non-magnetized solutions of first type (see section 2.4.3) are always superfluidbelow a certain critical temperature T C,∆/s . This will be demonstrated for the modelwith spin-dependent hopping, half filling and B = 0. The proof at spin-independent hoppingand B = 0 is analogous for the superfluid case. (The filling is unimportant here, since awayfrom half filling we do not expect magnetization to occur at B = 0.) In order to show this
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 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: 4.4. NUMERICAL RESULTS FOR THE GROU
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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