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

Contents1 Introduction 11.1 Ultracold quantum gases and the Hubbard model . . . . . . . . . . . . . . . . 11.2 Unbalanced Fermi-mixtures: a brief experimental overview . . . . . . . . . . 42 Formalism at weak coupling 72.1 Weak (attractive) coupling limit at Hartree-Fock level . . . . . . . . . . . . . 72.2 Diagonalization of the grand canonical Hamiltonian . . . . . . . . . . . . . . 72.2.1 Fourier-transformation in the k-space . . . . . . . . . . . . . . . . . . 82.2.2 Bogoliubov-transformation . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Self-consistency equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Interaction-free densities of states . . . . . . . . . . . . . . . . . . . . . 112.3.2 Self consistency equations in the thermodynamic limit . . . . . . . . . 122.4 Properties of the self-consistency equations . . . . . . . . . . . . . . . . . . . 142.4.1 Range of the parameters n, m and ∆ . . . . . . . . . . . . . . . . . . . 142.4.2 Situation at half filling . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.3 Non-polarized solutions of first type . . . . . . . . . . . . . . . . . . . 152.4.4 The grand potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Unbalanced Fermi-mixtures 193.1 Broken translational invariance and the LDA . . . . . . . . . . . . . . . . . . 193.2 Properties of the self-consistency equations at T = 0 . . . . . . . . . . . . . . 203.2.1 Uniqueness of n and m at fixed ∆ . . . . . . . . . . . . . . . . . . . . 213.2.2 Properties of the third self-consistency equation . . . . . . . . . . . . . 253.2.3 Graphical illustration of the phases . . . . . . . . . . . . . . . . . . . . 263.3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Systems with finite temperatures . . . . . . . . . . . . . . . . . . . . . . . . . 343.5.1 Infinite spatial extension . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5.2 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Model with spin-dependent hopping 354.1 Charge density wave states and the repulsive model . . . . . . . . . . . . . . 354.1.1 Special particle-hole transformation . . . . . . . . . . . . . . . . . . . 354.1.2 Grand potential at half filling . . . . . . . . . . . . . . . . . . . . . . . 364.2 Superfluidity away from half filling . . . . . . . . . . . . . . . . . . . . . . . . 374.2.1 The quasiparticle energies . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.2 Hartree-Fock density of states . . . . . . . . . . . . . . . . . . . . . . . 384.3 Numerical method for the problem at fixed parameter n . . . . . . . . . . . . 394.4 Numerical results for the ground state . . . . . . . . . . . . . . . . . . . . . . 41v