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

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6 CHAPTER 1. INTRODUCTION
Chapter 2Formalism at weak couplingIn this chapter we will derive the self-consistency equations for the problem of an attractiveon-site interaction. The grand-canonical Hamiltonian is given by:H gk = − ∑ (ij)σt σc † iσ c jσ + U ∑ in i↑ n i↓ − µ ∑ iσn iσ + B ∑ iσσn iσ . (2.1)In this context we identify the spin indices with ↑≡ + and ↓≡ −. We have to discuss the termsappearing in H gk . The first term describes as usual the (in general spin-dependent) hoppingof the fermions. The second one describes the on-site interaction, where U is negative. Thethird term is as usual the chemical potential term which mainly controls the average totalnumber of particles in the system. The fourth one is the Zeeman term which is mainly usedto control the unbalance between particles with spin ↑ and spin ↓. Of course controling thetotal number of particles and their unbalance cannot be done independently, as we will seelater. We use the grand canonical ensemble especially at attractive interaction, since we wantto treat superfluidity, which is described by pair creation/annihilation terms at Hartree-Focklevel, and therefore it is more comfortable to use a formulation in Fock space with a variablenumber of particles.2.1 Weak (attractive) coupling limit at Hartree-Fock levelIn order to treat the problem at Hartree-Fock level, taking into account superfluidity, we haveto decouple the interaction term as follows:n i↑ n i↓ → 〈n i↑ 〉n i↓ + 〈n i↓ 〉n i↑ − 〈n i↑ 〉〈n i↓ 〉 + ∆ ∗ i c i↓ c i↑ + ∆ i c† i↑ c† i↓ − |∆ i |2 . (2.2)The superfluid order parameter ∆ iis defined as the average of the pair annihilation operator:∆ i = 〈c i↓c i↑〉 ; ∆ ∗ i = 〈(c i↓ c i↑ )† 〉 = 〈c † i↑ c† i↓ 〉 . (2.3)Analogously, its complex conjugate is defined as the average of the pair creation operator.With these tools we are able to derive self-consistency equations. A derivation of the Hartree-Fock approximation by using first order perturbation theory with the help of Lagrangianfactors can be found in the appendix of [26].2.2 Diagonalization of the grand canonical HamiltonianIn the following we will assume that translational invariance is not broken, so that all averagescan be taken as i-independent:〈n iσ 〉 → n σ ; ∆ i → ∆ . (2.4)7
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Page 25 and 26: Chapter 3Unbalanced Fermi-mixturesI
Page 33 and 34: 3.3. NUMERICAL METHOD 27- In Figure
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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
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Page 49 and 50: 4.5. CRITICAL TEMPERATURES 430.4∆
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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
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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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