Renormalization group methods become natural for quantum many-body systems where there is an emergence of new energy scales like as Kondo scale in case of Kondo model. It was the Numerical Renormalization Group which gave the full quantitative understanding of the strong coupling fixed point of Kondo model and also of Anderson Impurity model. However, it was Andersons poor man scaling method which gave the analytical insights into the scaling behaviour of quantum impurity models. That makes poor man scaling method very important, and hence it has been used extensively in carrying out renormalization studies of quantum impurity models. This method has been generalized and also applied to other quantum many-body Hamiltonians. In this paper, we give a detailed presentation of poor man scaling method and apply it to study the scaling behaviour of Kondo model and Anderson impurity model as well. Though there are many reviews of perturbative renormalization of Kondo model, in this paper we have also presented a detailed discussion about scaling analysis of Anderson impurity model. We also present another method called flow equation method which is a generalization of poor man scaling method and has recently been applied to study the quantum many-body dynamics both in equilibrium and out-of equilibrium. We also present a numerical procedure to extract Kondo scale from flow equations of Kondo model. The importance of this procedure is that it can be applied to calculate the Kondo scale for situations where poor mans scaling method can not be applied.
We use the poor man's scaling approach to study the phase boundaries of a pair of quantum impurity models featuring a power-law density of states $\ensuremath{\rho}(\ensuremath{\varepsilon})\ensuremath{\propto}{|\ensuremath{\varepsilon}|}^{r}$, either vanishing (for
r>0) or diverging (for
r<0) at the Fermi energy $\ensuremath{\varepsilon}=0$, that gives rise to quantum phase transitions between local-moment and Kondo-screened phases. For the Anderson model with a pseudogap (i.e.,
r>0), we find the phase boundary for (a)
0<r<1/2, a range over which the model exhibits interacting quantum critical points both at and away from particle-hole (p-h) symmetry, and (b)
r>1, where the phases are separated by first-order quantum phase transitions that are accessible only for broken p-h symmetry. For the p-h-symmetric Kondo model with easy-axis or easy-plane anisotropy of the impurity-band spin exchange, the phase boundary and scaling trajectories are obtained for both
r>0 and
r<0. Throughout the regime of weak-to-moderate impurity-band coupling in which poor man's scaling is expected to be valid, the approach predicts phase boundaries in excellent qualitative and good quantitative agreement with the nonperturbative numerical renormalization group, while also establishing the functional relations between model parameters along these boundaries.
Recently, the issue of whether the Kondo problem in quantum dots at large bias is a weak-coupling problem or not has been raised. In this paper, we revisit this problem by carefully analyzing a corresponding model in the solvable limit --- the Emery-Kivelson line, where various crossover energy scales can be easily identified. We then try to extract the scaling behavior of this problem from various physical correlation functions within the spirit of ``poor man's scaling.'' Our conclusions support some recent suggestions made by Coleman et al. [Phys. Rev. Lett. 86, 4088 (2001)], which are obtained by perturbative analysis: The voltage acts as a cutoff of the renormalization group flow for only half of the impurity so that the low-temperature physics is controlled by a strong-coupling fixed point. But the low-temperature response functions in general show one-channel Kondo behaviors with two-channel Kondo behaviors occurring only through proximity to a quantum critical point.
The Kondo effect in a repulsively interacting electron system (Tomonaga-Luttinger liquid) is studied. By using the poor man's scaling method it is shown that the Kondo coupling in this model flows to the strong-coupling regime not only for the antiferromagnetic but also for the ferromagnetic case. The ground state is governed by stable strong-coupling fixed points where the impurity spin is completely screened; the fixed-point Hamiltonian consists of two semi-infinite Tomonaga-Luttinger liquids and a spin singlet. Specific heat, susceptibility, and conductance are calculated for low temperatures.
Using a previously derived effective theory we explore the conductance in the Luttinger model with one impurity. A new approach to the renormalization group (RG) analysis of this model is developed. It is based on the original Gell-Mann-Low formulation of RG. We sum up infrared logarithmic contributions to conductance in the leading and a few subsequent approximations. We analyze the validity of the widely used 'poor man's' scaling approach and find that it is applicable only in the leading approximation. Our results for corrections to this approximation are different from results obtained in other papers. This should be expected beforehand, as the Gell-Mann-Low function of the model is not regularization scheme invariant. For this reason the observed quantity (e.g. conductance) can not satisfy the Gell-Mann-Low equation beyond the 'leading-log' approximation as it is supposed in the 'poor man's' approach. We formulate the method to calculate the conductance from the renormalized hamiltonian in post-leading approximations and match our results to the case of weak impurity where the answer is known in any order of electron-electron interaction.
We calculate the dynamical spin-spin correlation functions of a Kondo dot coupled to two noninteracting leads held at different chemical potentials. To this end we generalize a recently developed real-time renormalization group (RG) method in frequency space [H. Schoeller, Eur. Phys. J. Spec. Top. 168, 179 (2009)] to allow the calculation of dynamical correlation functions of arbitrary dot operators in systems describing spin and/or orbital fluctuations. The resulting two-loop RG equations are analytically solved in the weak-coupling regime. This implies that the method can be applied provided either the voltage
V through the dot or the external magnetic field
h0 are sufficiently large, $\mathrm{max}{V,{h}_{0}}\ensuremath{\gg}{T}_{K}$, where the Kondo temperature
TK is the scale where the system enters the strong-coupling regime. Explicitly, we calculate the longitudinal and transverse spin-spin correlation and response functions as well as the resulting fluctuation-dissipation ratios. The correlation functions in real-frequency space can be calculated in Matsubara space without the need of any analytical continuation. We obtain analytic results for the line shape, the small- and large-frequency limits, and several other features like the height and width of the peak in the transverse susceptibility at $\ensuremath{\Omega}\ensuremath{\approx}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{h}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{h}$ denotes the renormalized magnetic field. Furthermore, we discuss how the developed method can be generalized to calculate dynamical correlation functions of other operators involving reservoir degrees of freedom as well.