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

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56 CHAPTER 5. STRONG COUPLING LIMITwhere n tot is the total number of hard-core bosons and Z is the number of nearest neighborsper site (Z = 2d for hypercubic lattices). Putting these results into (5.35), and neglectingthe constant contributions H U and the one arising from (5.71), we obtain:H = (t2 ↑ + t2 ↓ )|U|∑(ij)ñ i ñ j− 2t ↑t ↓|U|∑b † j b i, (5.72)so that we have mapped our model onto a hard-core boson model with repulsive nearestneighbor interaction with the strength Ũ = (t2 ↑ +t2 ↓ )|U|and nearest neighbor hopping with thestrength ˜t = 2t ↑t ↓|U|. The parameters always satisfy Ũ ≥ ˜t. In the isotropic Hubbard limit(t ↑ = t ↓ ) we have Ũ = ˜t, and in the Falicov-Kimball limit (t ↑ = 0 or t ↓ = 0) hopping iscompletely suppressed (˜t = 0).(ij)
Summary and OutlookIn this thesis we have analyzed different forms of asymmetric Hubbard models whichcan be used to describe ultracold asymmetric Fermi-mixtures. We have diagonalized theHartree-Fock Hamiltonian (U < 0) for superfluid phases for both asymmetric hopping andspin-dependent chemical potentials and we have derived the self-consistency equations.At symmetric hopping we have rigorously shown, which solutions the self-consistencyequations may have, and we have implemented an algorithm, which respects these properties,in order to solve the self-consistency equations. Within the local density approximation(LDA) we have shown that, at T = 0 in unbalanced Fermi-mixtures in a parabolic potential,phase separation takes place. A superfluid non-magnetized phase and a non-superfluidmagnetized phase were found. The third phase predicted as possible by our analytical calculationsnever minimized the grand potential and was therefore not thermodynamically stableat T = 0.For asymmetric hopping we presented arguments, why CDW phases in translationallyinvariant systems can be neglected, and we restricted consideration to superfluid phases. AtT = 0 and away from half filling we found both magnetized and non-magnetized superfluidphases. The critical temperatures at half filling were calculated numerically as a function ofthe hopping asymmetry.Finally we analyzed the Hubbard Hamiltonian with spin-dependent hopping at strong couplingfor both attractive and repulsive interaction. As a result of second order perturbationtheory we obtained effective models for the strong coupling limit. We derived ab anisotropicH XXZ -Heisenberg model for repulsive interaction at half filling, which reduces to the isotropicHeisenberg model in the Hubbard model limit and to an Ising model in the Falicov-Kimballmodel limit. For the attractive model at strong coupling with an even number of fermions,we derived a hard-core boson model with nearest-neighbor hopping and nearest-neighbor repulsiveinteraction for the bosons.In this thesis we performed numerical calculations in the weak coupling limit at Hartree-Focklevel, and we derived effective models in the strong coupling limit. Since, forultracold quantum gases, experimentalists are able to tune the relative interaction strength,weak coupling limit calculations may be very useful. Yet, on the other hand, it is importantto understand the physics beyond the weak coupling limit, so that the application ofnon-perturbative methods, for example DMFT or QMC, to asymmetrical Hubbard models isindispensible for reaching the intermediate interaction regime.In order to treat the trapping potential we used the LDA. Instead of using the LDAformalism,it is possible to treat the parabolic potential exactly by restricting to finite-sizelattices, where the size of the lattices may be estimated by the LDA-results. This methodis especially useful for situations in which the LDA breaks down, for small particle numbersor for a parabolic potential that varies fast in space. The exact treatment of the parabolicpotential in combination with non-perturbative treatment of the interaction is also of greatinterest, of course.57
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Hubbard Model for AsymmetricUltraco
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coupling regime. The trapping poten
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4.5 Critical temperatures . . . . .
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2 CHAPTER 1. INTRODUCTIONlattice si
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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
Page 59 and 60: 5.4. LOW TEMPERATURE LIMIT 53At fir
Page 61: 5.4. LOW TEMPERATURE LIMIT 555.4.2
Page 65 and 66: Bibliography[1] C. Wieman and E. Co
Page 67 and 68: DanksagungAn erster Stelle möchte
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