Clustermethods for correlated systems out of equilibrium

The understanding of the nonequilibrium behavior of strongly correlated quantum many-body systems is a long standing challenge, both in theory as well as in experiments. The enormous progress and level of control achieved in various fields such as quantum optics, quantum simulation, heterostructures, nanotechnology and spintronics, renders nonequilibrium properties increasingly relevant. In many of the investigated systems, strong correlations play a crucial role to fully understand their physical properties. In this project, we plan to develop a new numerical approach that allows to calculate nonequilibrium steady state properties of strongly correlated quantum many-body systems. We have recently published preliminary test calculations of this method on the preprint archive. The approach is formulated in the framework of Keldysh Green`s functions and is based on the ideas of the variational cluster approach (VCA), which has been successfully applied to a variety of strongly correlated many-body systems in equilibrium. This broad applicability also generalizes to the nonequilibrium method proposed here. In particular, the proposed approach can treat both fermions and bosons, is applicable in any spatial dimension, and can treat systems, where the strongly correlated region is spatially extended. Furthermore, the nonequilibrium method is neither perturbative in the many-body interaction nor in the field, that drives the system out of equilibrium. As in equilibrium VCA, one crucial aspect appears to be the variational procedure, consisting in a self-consistent adjustment of the equilibrium reference system to the nonequilibrium target state. A detailed analysis of the various options and of their performance will be one of the aspects dealt with in this project. The method shall be tested and applied to a number of physically interesting models. We expect our proposed method to provide valuable insight into the nonequilibrium physics of strongly correlated many-body systems, complementary to the ones obtained by approaches developed up to now. A sound understanding of fundamental nonequilibrium properties of quantum many-body systems might help to pave the way for the enhancement and invention of several high-tech applications, which might be rooted in the fields of material science or quantum information.

Our everyday life is coined by out-of-equilibrium phenomena. Blood is moving through our cardiovascular system as we drive to work in a car or ride on a bus contributing to the traffic system on the road network. We do so watching the touch-screen of our navigation screen and listening to the radio which both are ultimately operated thanks to moving charge carriers. Non-equilibrium systems typically consist of a large number of individual entities and are characterized by a current in the system.The advancement of microelectronics used in modern computer technology via miniaturization is especially viable for the design of tomorrow's applications in telecommunications, intelligent pocket devices and wearables. Pushing the miniaturization further saves energy and allows for a more elaborate device design and functionality as well as higher computational power and storage density. It becomes possible by artificially created nano-structures or molecular junctions. In such systems the transport of electrons is used to store logical information and ultimately perform complex operations. Harnessing the complicated strong interactions between the individual particles is a difficult but opens new routes for future functionalities and new applications.Microscopic particles are described by the quantum theory which relies on a dual interpretation of an electron as a wave as well as a particle. To guide recent experiments on the nano-scale, faithful theoretical predictions are required. The theoretical description of a quantum system consisting of a large number of particles in a non-equilibrium setup is however a challenging task. Due to the mutual interaction of the individual particles the governing equations become intractable and insightful approximations need to be devised in order to be able to claim truly predictive power.In this project we successfully developed such approximate computational methods. Non-equilibrium quantum many-body cluster methods rely on a mapping of the full system which needs to be solved into a number of smaller systems which can be solved exactly. The properties of the original system, like the current, magnetization or charge density can then be obtained based on these solutions. In particular we presented the steady-state cluster perturbation theory and its improvement, the steady-state variational cluster approach. The latter implements a self-consistent feedback to the non-equilibrium state which significantly improves the results. In addition we presented the master equation based cluster perturbation theory which marries the previous approach with the very successful master equation approach to out-of-equilibrium transport. Finally we presented the auxiliary master equation approach which does not rely on a perturbative treatment of the governing equations but constructs an auxiliary open quantum system which can be solve exactly. Besides these approaches to the non-equilibrium steady-state, we employed a real-time evolution in the quasi-exact framework in a so-called matrix product state Ansatz to obtain the transient behaviour of interacting quantum junctions.We applied the newly developed and improved methods to study transport across atomically sized quantum dots and small molecules. Quantum dots are artificially created nano-structures which consist of a small number of electrons in a geometrically confined region which enhances their quantum nature. These dots are then contacted to a source, drain and gate lead to apply a bias voltage, like in a transistor or diode. The same can be done with small molecules by sticking them to metal electrodes using anchor groups. We obtained the current-voltage characteristics of an interacting single-quantum dot and studied the transport through ring-shaped molecules in a magnetic field. Furthermore we used the methods to study the negative differential resistance arising from current-blocking effects in a quantum dot with polarized leads and a ring transistor. We applied our methods to study transport through two-dimensional heterostructures. An emphasize of our work was to gauge the effects of strong electronic interactions on transport. An advantage of the presented methods is that electronic interactions can be treated more accurately than in standard approaches.