Abstract
We introduce a systematic approximation for an efficient evaluation of Born–Markov master equations for steady state transport studies in open quantum systems out of equilibrium: the energy resolved master equation approach. The master equation is formulated in the eigenbasis of the open quantum system and build successively by including eigenstates with increasing grandcanonical energies. In order to quantify convergence of the approximate scheme we introduce quality factors to check preservation of trace, positivity and hermiticity. Furthermore, we discuss different types of master equations that go beyond the commonly used secular approximation in order to resolve coherences between quasi-degenerate states. For the discussion of complete positivity we introduce a canonical Redfield–Bloch master equation and compare it to a previously derived master equations in Lindblad form with and without using the secular approximation. The approximate scheme is benchmarked for a six orbital quantum system which shows destructive quantum interference under the application of a bias voltage. The energy resolved master equation approach presented here makes quantum transport calculations in many-body quantum systems numerically accessible also beyond six orbitals with a full Hilbert space of the order of ∼106.
Highlights
In the breakdown of Moore’s law when reaching the quantum limit, the study of quantum effects on electronic transport is becoming increasingly important
General form of a master equation generating a universal dynamical map Having defined the general requirements of a UDM we examine their implications on the form of its generator, the master equation
After introducing the theoretical concept of a UDM which led to the Lindblad equation and before discussing several types of Born–Markov master equations we introduce the basic idea of the energy resolved master equation approach, namely a physically motivated constructive way to represent the action of the superoperator in a small basis such that the determination of the steady state or relevant dynamics is possible without evaluating the full superoperator
Summary
In the breakdown of Moore’s law when reaching the quantum limit, the study of quantum effects on electronic transport is becoming increasingly important. The other is a microscopic derivation where one starts from a model Hamiltonian (e.g. derived from DFT ab–initio calculations in molecular electronics) and tries to find a Lindblad–like master equation by applying perturbation theory and further approximations Because the latter approach may face problems concerning the perturbative character of the derivation and in some cases it does not lead to a Lindblad equation, so that complete positivity is violated, a lot of research has been done to tackle those problems. Both the size n of the full density matrix of a system with electron–electron interactions (n = 4l) as well as the number of equations N in the master equation (N = 42l) scale exponentially with increasing number of systems sites (orbitals) l To climb this exponential wall we here introduce a scheme to suitably truncate such large Hilbert spaces, the socalled energy resolved master equation approach (ERMEA).
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