Poor Man's Scaling for the Nonequilibrium Kondo Problem and Antiferromagnetic Correlations in Optical Lattices

Joseph Maciejko

We revisit the problem of a single quantum impurity on the edge of a two-dimensional time-reversal invariant topological insulator and show that the zero-temperature phase diagram contains a large local moment region for antiferromagnetic Kondo coupling, which was missed by previous poor man's scaling treatments. The combination of an exact solution at the so-called decoupling point and a renormalization group analysis \`a la Anderson-Yuval-Hamann allows us to access the regime of strong electron-electron interactions on the edge and strong Kondo coupling. We apply similar methods to the problem of a regular one-dimensional array of quantum impurities interacting with the edge liquid. When the edge electrons are at half-filling with respect to the impurity lattice, the system remains gapless unless the Luttinger parameter of the edge is less than $1/2$ in which case two-particle backscattering effects drive the system to a gapped phase with long-range Ising antiferromagnetic order. This is in marked contrast with the gapped disordered ground state of the ordinary half-filled one-dimensional Kondo lattice.

Anton S. Buyskikh, Andrew J. Daley, Stephan Langer

Rukhsan Ul Haq

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.

Tokuro Hata

Dmytro Tarasevych, Jan Krieg, Peter Kopietz

We show how the one-loop ``poor man's scaling'' equations for the Kondo model with arbitrary impurity spin can be obtained within the framework of the functional renormalization group approach for quantum spin systems recently developed by Krieg and Kopietz (arXiv:1807.02524). We argue that our method supersedes the poor man's scaling approach and can also be used to study the strong coupling regime of the Kondo model.

Leena Chandran, H. R. Krishnamurthy, Ciriyam Jayaprakash