By Stefan Flörchinger

Modern options from quantum box conception are utilized during this paintings to the outline of ultracold quantum gases. This results in a unified description of many phenomena together with superfluidity for bosons and fermions, classical and quantum section transitions, diversified dimensions, thermodynamic homes and few-body phenomena as certain nation formation or the Efimov impression. The non-perturbative remedy with renormalization staff movement equations can account for all identified proscribing instances through fixing one unmarried equation. It improves past effects quantitatively and brings qualitatively new insights. for instance, new quantum part transitions are came across for fermions with 3 spin states. Ultracold atomic gases could be noticeable as a fascinating version for good points of excessive power physics and for condensed subject idea. The study suggested during this thesis is helping to resolve the tricky complexity challenge in sleek theoretical physics.

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**Sample text**

4)], plug this solution into the second equation for g2, solve it, and so on. In praxis one has to stop at some i = l, of course. The only error in this solution comes from the fact that it may be necessary to solve the (ordinary) differential equations for the gi numerically. What we obtain by this process is not an exact solution for the complete functional Ck ½U but for the coefficients gi with i B l. However, all observables that depend on these couplings only, can be determined exactly. 4) is indeed found for the nonrelativistic few-body problem.

3 General Coordinate Transformations It is sometimes useful to perform a change of coordinates in the space of fields during the renormalization flow. 20) under such a change of the basis for the fields. We follow here the calculation in [2, 3]. For simplicity we restrict the discussion to bosonic fields. It is straightforward to transfer this to fermions as well [2, 3]. 28). Let us consider a transformation of the form U ! W½U: ð3:32Þ Here we denote by U the original fields. The functional W½U is a k-dependent map of the old coordinates to the new ones.

4 BCS–Trion–BEC Transition 53 The many-body phase diagram is far less understood. Not too close to the resonance one expects a superfluid ground state which is similar to the BCS ground state for a \ 0 or a BEC-like ground state for a [ 0. However, there are also some important differences. While in the two-component case the order parameter is a singlet under the corresponding SU(2) spin symmetry, the order parameter for the three component case with SU(3) spin symmetry is a (conjugate) triplet.