The method of continuous unitary transformations (CUTs) is applied to the Anderson impurity and the Kondo model aiming at the systematic derivation of convergent effective models. If CUTs are applied in a conventional way, diverging differential equations occur. Similar to poor man's scaling the energy scale, below which the couplings diverge, corresponds to the Kondo temperature TK . We present a way to apply CUTs to the Kondo and to the Anderson impurity model so that no divergences occur but a converged effective low-energy model is derived with small finite parameters at arbitrarily small energies. The ground state corresponds to a bound singlet with a binding energy given by the Kondo temperature TK .
The long-time and long-distance asymptotic behavior of the x spin correlations at finite temperature in an anisotropic spin-1/2 XY chain is determined numerically. The decay of the correlations is exponential in both space and time. Similar exponential decay of correlations was already found earlier in the special case of the isotropic model, where analytical expressions for the decay rates could be derived via a mapping to a different model. While no such mapping is known for the anisotropic model, the asymptotic correlations can be very well approximated by a natural generalization of the known analytic results for the isotropic case.
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