By Stefan Flörchinger
Modern ideas 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 part transitions, varied dimensions, thermodynamic homes and few-body phenomena as sure kingdom formation or the Efimov impression. The non-perturbative therapy with renormalization staff move equations can account for all identified proscribing situations by means of fixing one unmarried equation. It improves earlier effects quantitatively and brings qualitatively new insights. as an instance, new quantum section transitions are discovered for fermions with 3 spin states. Ultracold atomic gases should be obvious as an attractive version for beneficial properties of excessive strength physics and for condensed topic idea. The study pronounced during this thesis is helping to resolve the tricky complexity challenge in smooth theoretical physics.
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This shows that our model is equivalent to a purely fermionic theory with an interaction term & ' Z h2 Sint ¼ À wÃ ðp0 Þw ðp1 Þ Pu ðp1 þ p2 Þ 1 1 1 ð6:15Þ p1 ;p2 ;p01 ;p02 Â wÃ2 ðp02 Þw2 ðp2 Þdðp1 þ p2 À p01 À p02 Þ; 52 6 Investigated Models where p ¼ ðp0 ; ~Þ p and the classical inverse boson propagator is given by Pu ðqÞ ¼ iq0 þ ~ q2 þ mK À 2l: 2 ð6:16Þ On the microscopic level the interaction between the fermions is described by the tree level expression kw;eff ¼ À h2 : Àx þ 12~ q2 þ mK À 2l ð6:17Þ Here, x is the real-time frequency of the exchanged boson u: It is connected to the Matsubara frequency q0 via analytic continuation x = -iq0.
7). 9) with respect to j yields À Á ~ s i Q þ Ruk sr Àu jr À dr h~ u vr i ¼ h~ u u Â À ÁÃ ¼ u ðQuÞr þ ðdj dj Wk Þ Q þ Ruk r Àu lr À dr : ð3:14Þ We now turn to the scale-dependence of Wk[g, j]. In addition to Rwk and Ruk also Q and H are k-dependent. For H we assume ok Hab ¼ ðok Fq ÞHqab ð3:15Þ where we take the dimensionless matrix F to be symmetric for simplicity. 15) Á 1 1 À ok Wk ¼ À w ok Rwk w À u ok Ruk u 2 2 n o 1 À STr ðdg dg Wk Þ ok Rwk 2 À ÁÉ 1 È À Tr ðdj dj Wk Þ ok Ruk 2 À Á 1 ÈÂ À Tr Qðok QÀ1 ÞRuk þ Ruk ok QÀ1 Q 2 þ Ruk ðok QÀ1 ÞRuk þ Ruk QÀ1 ðok FÞðQ þ Rk Þ À Á Ã É þ Q þ Ruk ðok FÞQÀ1 Ruk ðdj dj Wk Þ Ã 1Â þ l ðok QÀ1 ÞQ þ QÀ1 ðok FÞQ u 2 Â Ã þ 12u Qðok QÀ1 Þ þ Qðok FÞQÀ1 l Ã 1Â À l ok QÀ1 þ QÀ1 ðok FÞ þ ðok FÞQÀ1 l 2 Ã É 1 ÈÂ þ Tr ok QÀ1 þ QÀ1 ðok FÞ þ ðok FÞQÀ1 Ruk 2 É 1 È þ Tr Qok QÀ1 : 2 ð3:18Þ The supertrace STr contains the appropriate minus sign in the case that wa are fermionic Grassmann variables.
The remainder of this section is devoted to the discussion of the microscopic model in the SU(3) symmetric case. 14) Z & 1 S¼ wy ðos À D À lÞw þ uy os À D À 2l þ mu u 2 x 1 þ vÃ os À D À 3l þ mv v 3 1 þ hijk uÃi wj wk À ui wÃj wÃk 2 ' À Ã Ã Á Ã þ g ui wi v À ui wi v : The (Grassmann valued) fermion field has now three components w = (w1, w2, w3) and similar the boson field u ¼ ðu1 ; u2 ; u3 Þ¼ðw ^ 1 w2 ; w2 w3 ; w3 w1 Þ: In addition we also include a single component fermion field v.