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

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38 CHAPTER 4. MODEL WITH SPIN-DEPENDENT HOPPINGWhile the quasiparticle energy for symmetric hopping is decreasing with increasing parameterε, for asymmetric hopping the quasiparticle energies are not monotonic for both spinspecies. So there is the possiblility of additional zeroes in the quasiparticle-energy functionof the spin species with the lower hopping amplitude. This causes the proof of section 3.2 tobe not valid anymore, since in the proof it has been assumed that the quasiparticle energiesE σ change their sign at most once. In the spin-dependent hopping case this is only valid, ifthe asymmetry does not cause additional zero points in the quasiparticle energy. However,even if the proof of section 3.2 breaks down for t ↑ ≠ t ↓ , there still seems to be uniquenessof the solution of the self-consistency equations: In numerical studies of the model we neverobserved multiple solutions of n and m at fixed ∆.E ↓ (ε) , E ↑ (ε)7.552.5-3 -2 -1 1 2 3-2.5ε-5-7.5-10Figure 4.1: Quasiparticle energies for asymmetric hopping. The curve corresponding to thespin species with the lower hopping amplitude is not monotonic (red curve).4.2.2 Hartree-Fock density of statesThe absence of the symmetry in the quasiparticle energies influences the (Hartree-Fock) DOSof the up and down particles, which is defined as:ν d,σ (E) :=∫ ∞−∞dεν d (ε)δ ( E − E σ (ε) ) . (4.12)The asymmetric hopping now causes the spin-dependent DOS to have additional singularitiesof 0 th order (i.e., jumps). If we assume t ↓ < t ↑ , the following changes can be established:- The square-root singularity at the superfluid gap in ν d,↑ is replaced by a 0 th ordersingularity.- The DOS ν d,↓ gets an additional singularity of 0 th order.- The superfluid gap in ν d,↓ becomes smaller than 2|U∆|.The normalization of ν d,σ is not affected, of course. Note that the following Figures are madefor the parameters: U = −5, t ↑ = 3, t ↓ = 0.5, µ = −2, n = 1, m = 0.4, ∆ = 0.3. Theseparameters are not meant to be useful in the sense of solving the self-consistency equations,they are just for illustration of the DOS.
4.3. NUMERICAL METHOD FOR THE PROBLEM AT FIXED PARAMETER N 39ν 3,↑ (E)0.120.10.080.060.040.02-15 -10 -5 5 10 15 20EFigure 4.2: Hartree-Fock DOS of the spin species with the greater hopping amplitude. Insteadof a square-root singularity there is a 0 th order singularity at the border of the superfluidgap.ν 3,↓ (E)10.80.60.40.2-6 -5 -4 -3 -2 -1 1EFigure 4.3: Hartree-Fock DOS of the spin-species with the lower hopping-amplitude. Anadditional 0 th order singularity appears and the superfluid gap is smaller than 2|U∆|.So there are more effects arising from spin-dependent hopping away from half filling thanjust the possibility of magnetization. The different widths of the superfluid gaps could beobserved experimentally, for example.4.3 Numerical method for the problem at fixed parameter nIn this section we will present a numerical method for solving the self-consistency equationsat fixed n. Since we fix the occupation number, µ becomes a function of U, µ → µ(U). Weanalyze the magnetization m and the superfluid order parameter ∆ in the ground state.
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: 4.2. SUPERFLUIDITY AWAY FROM HALF F
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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