# P10: Time dynamics in correlated multi-level quantum dots

## Project Description

The investigation of the time dynamics of correlated quantum systems is an important subject of increasing interest in a variety of fields such as quantum information processing, nanoelectronic systems, dissipative quantum mechanics, and cold atom systems. Starting from an out-of-equilibrium state, the aim is to calculate the relaxation into the stationary state which can be either an equilibrium state or a non-equilibrium one if the system is in contact with several reservoirs at different chemical potentials or temperatures. Motivated by several achievements within the first funding period of the RTG, the aim of this project is to consider a correlated multi-level quantum dot in contact with one or several multi-channel reservoirs and to study the time dynamics of the dot populations, the current and the correlation functions. Two complementary methods will be used: the time-dependent numerical renormalization group (TD-NRG) [1], and the analytical real-time renormalization group method (RTRG) [2]. Over the course of the first funding period, the TD-NRG was further developed [3] and successfully applied to the quench dynamics of a qubit coupled to an environment modeled by the ohmic spin boson model [4], as well as to periodic driving within the single level Anderson impurity model [5]. The RTRG has been applied in many pre-vious publications to calculate the time dynamics of the Kondo model [6], the interacting resonant level model [7,8,9], the ohmic spin boson model at zero bias [10], and, most recently, within the first funding period, the ohmic spin boson model at arbitrary bias [11]. In an additional publication [12] resulting from the first funding period, it has been shown that correlated quantum dots with N levels reveal SU(N)-Kondo physics in the co-tunneling regime of a singly occupied quantum dot with new interesting fixed points in the non-equilibrium case. Furthermore, the reliability of the RTRG method has been verified in comparison to the NRG in equilibrium and the functional renormalization group method in non-equilibrium (by C. Lindner).

Whereas the latter projects considered the stationary properties of multi-level quantum dots, within this project, we aim to calculate the time dynamics with RTRG and TD-NRG. If the stationary state is an equilibrium one, then we expect that the physics of the SU(N)-Kondo fixed point model will show up in the long-time dynamics. This is expected from various previous works supporting SU(N)-Kondo physics for N=2 [13,14] and N=3 [11,15] in stationary equilibrium and linear response. The aim is to calculate the fingerprint of SU(N)-Kondo physics in the time evolution. In a second stage of the project, we will also consider several reservoirs with different chemical potentials and will try to analyze the fingerprints in the time dynamics of the new non-equilibrium non-Kondo fixed points proposed recently [12,14]. In addition, we plan to use recent developments of the TD-NRG method [16,17] to further improve the calculation of non-equilibrium spectral functions of the Anderson impurity model and to compare them with predictions from the RTRG method. This project is strongly linked to P8 which also deals with driven few-level quantum systems.

[1] F. B. Anders and A. Schiller, Phys. Rev. Lett. **95**, 196801 (2005)

[2] H. Schoeller, Eur. Phys. J. Special Topics **168**, 179 (2009)

[3] H. T. M. Nghiem and T. A. Costi, Phys. Rev. B **89**, 075118 (2014)

[4] H. T. M. Nghiem, D. M. Kennes, C. Klöckner, V. Meden, and T. A. Costi, Phys. Rev. B **93**, 165130 (2016)

[5] H. T. M. Nghiem and T. A. Costi, Phys. Rev. B **90**, 035129 (2014)

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

[7] C. Karrasch, S. Andergassen, M. Pletyukhov, D. Schuricht, L. Borda, V. Meden, and H. Schoeller, Europhys. Lett. **90**, 30003 (2010)

[8] S. Andergassen, M. Pletyukhov, D. Schuricht, H. Schoeller, and L. Borda, Phys. Rev. B **83**, 205103 (2011)

[9] D. M. Kennes, O. Kashuba, M. Pletyukhov, H. Schoeller, and V. Meden, Phys. Rev. Lett. **110**, 100405 (2013)

[10] O. Kashuba and H. Schoeller, Phys. Rev. B **87**, 201402(R) (2013)

[11] C.J. Lindner and H. Schoeller, submitted to Phys. Rev. B, arXiv:1802.09846

[12] C.J. Lindner, F.B. Kugler, H. Schoeller, and J. von Delft, submitted to Phys. Rev. B, arXiv:1802.09976

[13] V. Kashcheyevs , A. Schiller, A. Aharony, and O. Entin-Wohlman, Phys. Rev. B **75**, 115313 (2007)

[14] S. Göttel, F. Reininghaus, and H. Schoeller, Phys. Rev. B **92**, 041103(R) (2015)

[15] R. Lopez, T. Rejec, J. Martinek, and R Zitko, Phys. Rev. B **87**, 035135 (2013)

[16] H. T. M. Nghiem and T. A. Costi, Phys. Rev. Lett. **119**, 156601 (2017)

[17] H. T. M. Nghiem and T. A. Costi, submitted to Phys. Rev. B, arXiv:1803.04098