# P11: Spectral densities in non-equilibrium

## Project Description

The investigation of dynamical properties of strongly correlated quantum systems in non-equilibrium plays a central role in a variety of modern fields in condensed matter physics, like e.g. dissipative quantum mechanics, nanoelectronics, quantum information theory, and cold atom systems. A theoretical understanding of the dynamics in such systems is of fundamental interest. Whereas the properties in linear response are well understood, the calculation of dynamical quantities in non-equilibrium still poses a major theoretical challenge. The central issue of this project is the development of methods to calculate dynamical correlation functions in non-equilibrium, which will be applied to the Kondo model and the Anderson impurity model. These models are of fundamental interest because they describe the exchange of charge and spin with the environment and interesting many-body effects arise from an enhancement of charge and spin fluctuations, which among other effects lead to the appearance of a Kondo peak in the spectral density. One of the aims of this project is the clarification whether this Kondo peak splits by an applied bias voltage, an issue which has also been investigated experimentally^{(1),(2)} but not yet been satisfactorily answered from the theoretical side. The theoretical methods used so far are either only applicable for large bias voltages ^{(3),(4)} or approximations are used for which the range of validity has still to be systematically analyzed^{(5),(6)}. Here, we will use the combination of a generalization of the numerical renormalization group (NRG) to non-equilibrium systems together with the analytical method of real-time renormalization group (RTRG). For equilibrium systems, the NRG method is numerically exact and can be used to calculate spectral densities very reliably^{ (7)}. In non-equilibrium, a generalization of this method is planned, which is based on an improvement of a recently developed NRG method using scattering states cite{Anders08}. This development can provide important insights for projects P8 and P10. Complementary to the numerical approach, we will also develop and apply the RTRG method ^{(8)} to calculate spectral densities in non-equilibrium. This technique has already been applied successfully to describe transport properties of the non-equilibrium Kondo model in weak ^{(9),(10)} and strong coupling ^{(11)} in agreement with recent experiments ^{(12),(13)}.

In this project the method will be generalized to the calculation of correlation functions in the strong coupling limit. The analysis can profit from the study of Ward identities in project P16. Since the RTRG method is based on an expansion in effective vertices which are not small in the strong coupling limit, the reliability of the results has to be checked by comparing different truncation orders of the renormalization group equations. Since it is not clear whether the series converges by improving the truncation order, it is very important to obtain further insight from the NRG method, first in equilibrium where exact results are already available, and secondly for the non-equilibrium regime, where no theoretical benchmarks exist at the moment. Therefore, this project will essentially profit from the combined use of two independent approaches.

(1) R. Leturcq, L. Schmid, K. Ensslin, Y. Meir, D. C. Driscoll and A. C. Gossard,

Phys. Rev. Lett.** 95**, 126603 (2005)

(2) S. De Franceschi, R. Hanson, W. G. van der Wiel, J. M. Elzerman, J. J. Wijpkema, T. Fujisawa, S. Tarucha

and L.P. Kouwenhoven, Phys. Rev. Lett. **89**, 156801 (2002)

(3) A. Rosch, T. A. Costi, J. Paaske and P. Wölfle, Phys. Rev. B **68**, 014430 (2003)

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

(5) J. König, H. Schoeller and G. Schön, Phys. Rev. Lett. **76**, 1715 (1996)

(6) Y. Meir, N. S. Wingreen and P. A. Lee, Phys. Rev. Lett. **70**, 2601 (1993)

(7) R. Bulla, T. A. Costi and Th. Pruschke, Rev. Mod. Phys. **80**, 395 (2008)

(8) H. Schoeller, Eur. Phys. J. Special Topics **168**, 179 (2009)

(9) M. Pletyukhov, D. Schuricht and H. Schoeller, Phys. Rev. Lett. **104**, 106801 (2010)

(10) D. Schuricht and H. Schoeller, Phys. Rev. B **80**, 075120 (2009)

(11) M. Pletyukhov and H. Schoeller, Phys. Rev. Lett. **108**, 260601 (2012)

(12) A. V. Kretinin, H. Shtrikman, D. Goldhaber-Gordon, M. Hanl, A. Weichselbaum, J. von Delft,

T. A. Costi and D. Mahalu, Phys. Rev. B **84**, 245316 (2011)

(13) A. V. Kretinin, H. Strikman and D. Mahalu, Phys. Rev. B **85**, 201301(R) (2012)