Conclusion and Outlook
This chapter gives a summary of the fundamental ideas, principles and (computational) achievements covered in the book and presents some prospects of future applications of the method. It is to be expected, that these applications will deepen the understa
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Conclusion and Outlook
This monograph was devoted to quantum systems consisting of a finite number of identical particles in equilibrium and nonequilibrium. Examples were electrons in small atoms and molecules as well as electrons in quantum dots. We were interested in the treatment of correlation effects and their dynamics during and after an external excitation—problems of high current interest in many fields. Our approach was based on second quantization and has used the nonequilibrium Green’s functions as the central quantities. The main advantages of this method are its internal consistency, the fulfillment of conservation laws and the existence of systematic approximation schemes that can be derived from Feynman diagrams. While most previous numerical works using NEGFs focused on spatially homogeneous systems, here we concentrated on the numerical analysis of spatially inhomogeneous systems. This situation is more challenging because the one-particle NEGF, in general, depends on two coordinates and two times. Below we summarize the main results and outline future developments.
7.1 Summary • It has been shown that direct numerical solutions of the two-time Kadanoff-Baym equations are feasible for various spatially inhomogeneous quantum many-body systems with Coulomb interaction, including electrons in atoms and molecules. Another example were electrons in “artificial atoms”, i.e., in quantum dots. Compared to Coulomb interaction, systems with short range interaction should be even simpler to treat. • It has been demonstrated that reasonable results that include two-particle correlations can already be obtained on the level of the second Born approximation, which is among the most simple conserving approximations for the one-particle self-energy beyond the mean-field level. When including direct and exchange terms, the results are also quantitatively reliable in the case of weak to moderate coupling. K. Balzer, M. Bonitz, Nonequilibrium Green’s Functions Approach to Inhomogeneous Systems, Lecture Notes in Physics 867, DOI 10.1007/978-3-642-35082-5_7, © Springer-Verlag Berlin Heidelberg 2013
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Conclusion and Outlook
• For inhomogeneous systems, the finite element-discrete variable representation (FE-DVR), as introduced in Chap. 3, has allowed for a drastic simplification of the self-energy computation. The main advantage is that the matrix elements of the binary interaction become highly diagonal and that, hence, the integration or summation over the vertex points in the perturbation diagrams can, at least partly, be performed analytically. • With the efficient MPI parallelization of the two-time propagation, longer propagation times and (or) calculations on larger systems become feasible. • A further possibility to reduce computer storage requirements is to reconstruct the two-time NEGFs from their values on the time diagonal. To this end, we have employed the generalized Kadanoff-Baym ansatz with Hartree-Fock propagators for the second Born approximation which retains the conservation laws. It
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