# P5: RG approaches to the non-equilibrium Kondo problem

## Project Description

The Kondo effect in quantum dots is among the most well-known correlation effects in mesoscopic physics. In equilibrium it can be described by exact or numerically exact methods such as the Bethe Ansatz or the numerical renormalization group (RG) applied to the Kondo model. In non-equilibrium, the regime of weak exchange coupling between the localized spin and the conduction electrons is accessible by perturbative RG [1] or the flow equations method [2]. In contrast, it is difficult to describe the non-equilibrium strong coupling regime in the part distant from the Fermi-liquid fixed point [3]. It was possible to reach this regime by the real time RG starting from weak coupling and imposing appropriate strong coupling boundary conditions [4]. The results of this approach for the voltage dependence of the conductance seem reliable. Those for the temperature dependence are, however, not as fully convincing. Further investigations of this regime are required.

The functional RG (FRG) has proved to provide a powerful approach to correlation physics in low dimensional Fermi systems [5]. However, we know of only one study in which this method was applied to the Kondo model [6], as opposed to the single-impurity Anderson model (see, e.g., [7]). The authors report on problems in correctly describing the spin relaxation rate in the strong coupling regime. Concerning the choice of the spin representation and the cut-off, other approaches than those chosen in Ref. [6] are conceivable and ought to be tested [8,9].

During his time as doctoral researcher in the RTG, Jan Rentrop formulated an FRG approach to the Kondo model using the drone variant of a Majorana fermion representation for the local spin [10,11] and a band cut-off for the conduction electrons [12]. This groundwork was in Matsubara formalism and used a Polchinski-type formulation of the FRG [5]. Jan Diekmann extended it to the Keldysh formalism and the one-particle irreducible formulation of the FRG in his Master’s thesis. After successfully reproducing Anderson's poor man's scaling equations, he formulated a frequency dependent generalization which includes the relax-ation of the local spin. Currently, he is analyzing the numerical solution of the flow equations.

The prospects of these investigations are threefold: Firstly, they may contribute to a better understanding of the strong coupling regime in non-equilibrium. In this case, the study may be extended to include a magnetic field. Secondly, a profound methodological comparison between real time RG and FRG may be possible. Usually, these two methods are seen as complementary as they use different expansion parameters. For the Kondo model in contrast, both are based on the same expansion parameter, namely, the exchange coupling. In this new situation a comparison of the approximation procedures can lead to a deeper understanding of the relationship of the methods. Thirdly, insights gained in the FRG treatment of the Kondo model may be transferred to the single impurity Anderson model, when the hybridization is taken as expansion parameter of the latter. This could allow to pursue the approach taken in Ref. [13] in order to access the strong coupling regime.

[1] A. Rosch, J. Paaske, J. Kroha, and P. Wölfle, Phys. Rev. Lett. **90**, 076804 (2003)

[2] S. Kehrein, Phys. Rev. Lett. **95**, 056602 (2005)

[3] J. Eckel, F. Heidrich-Meisner, S. G. Jakobs, M. Thorwart, M. Pletyukhov, and R. Egger, New J. Phys. **12**, 043042 (2010)

[4] F. Reininghaus, M. Pletyukhov, and H. Schoeller, Phys. Rev. B **90**, 085121 (2014)

[5] W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, and K. Schönhammer, Rev. Mod. Phys. **84**, 299 (2012)

[6] H. Schmidt and P. Wölfle, Ann. Phys. (Berlin) **19**, 60 (2010)

[7] S. G. Jakobs, M. Pletyukhov, and H. Schoeller, Phys. Rev. B **81**, 195109 (2010)

[8] S. G. Jakobs, V. Meden, and H. Schoeller, Phys. Rev. Lett. **99**, 150603 (2007)

[9] S. G. Jakobs, M. Pletyukhov, and H. Schoeller, J. Phys. A: Math. Theor. **43**, 103001 (2010)

[10] W. Mao, P. Coleman, C. Hooley, and D. Langreth, Phys. Rev. Lett. **91**, 207203 (2003)

[11] A. Shnirman and Y. Makhlin, Phys. Rev. Lett. **91**, 207204 (2003)

[12] J. F. Rentrop, PhD-thesis, RWTH Aachen University (2016)

[13] M. Kinza, J. Ortloff, J. Bauer, and C. Honerkamp, Phys. Rev. B **87**, 035111 (2013)