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

28 CHAPTER 3. UNBALANCED FERMI-MIXTURESStart with random values n 1 and m 1Calculation of n i+1 and m i+1by integration using n i and m i ; i → i + 1Check if equations approximatively solved|n i − n i−1 | < ǫ n ∧ |m i − m i−1 | < ǫ mnoyesEnd of iteration scheme, output of n i and m iFigure 3.11: Iteration scheme for solving (3.3) and (3.4) at fixed ∆. The abort-criteriaare chosen as: ǫ n = 10 −4 , ǫ m = 10 −3 .In Figure 3.11 we see the numerical solution scheme for (3.3) and (3.4) at fixed ∆. It hasbeen used that (3.3) and (3.4) are a contraction in the sense of (3.7), so that this approximationscheme is always able to solve the equations numerically with the method of successiveapproximations. We have observed that the scheme 3.11 finishes within less than 20 iterationsteps for interaction values up to |U|t= 4, so that (3.18) is a very restrictive convergencecriterion (many rigorous estimations were done). Since with increasing order parameter ∆,the parameter n approaches half filling and the magnetization vanishes, we can conclude thatconvergence improves with increasing values of ∆.