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Non-Lattice Systems

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Book cover Nonequilibrium Green's Functions Approach to Inhomogeneous Systems

Part of the book series: Lecture Notes in Physics ((LNP,volume 867))

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Abstract

Chapter 6 contains nonequilibrium Green’s function (NEGF) applications for non-lattice prototypes of inhomogeneous systems out of equilibrium: atoms, molecules, and electrons in semiconductor heterostructures exposed to weak and strong external fields.

The first part of the chapter focuses on small atoms and molecules and includes the description of their ground states in Hartree-Fock and second Born (2B) approximation. In particular, it is shown that the 2B approximation captures a good portion of the correlations. Importantly, this also holds in nonequilibrium situations. The second part deals with correlation-induced double excitations and their description by the 2B approximation and the generalized Kadanoff-Baym ansatz. To this end, optical absorption spectra are computed for a four-electron system confined in a quantum well.

For the one-dimensional examples discussed in this chapter, the finite element-discrete variable representation (introduced in Chap. 4) is used and demonstrates its benefits regarding the two-time solution of the Kadanoff-Baym equations. Moreover, the NEGF results are supplemented by thorough comparisons to exact diagonalization and solutions of the time-dependent Schrödinger equation.

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Notes

  1. 1.

    We note that there also may exist discontinuities which are induced by boundary conditions or other constraints.

  2. 2.

    SI units, SI: Système International d’unités.

  3. 3.

    A x (t) denotes the x-component of the vector potential.

  4. 4.

    Without the contour delta function.

  5. 5.

    Recall definition (2.39).

  6. 6.

    Note, that we treat different spin degrees of freedom only by a degeneracy factor as outlined in Sect. 3.3.4.

  7. 7.

    In 2B approximation, this means we, in addition to the correlation functions, have also to propagate the mixed Green’s functions, which are obsolete in a mean-field (HF) description.

  8. 8.

    At least for the helium atom, where single and double excitations are energetically well separated from one another.

  9. 9.

    Both spectral functions are obtained by the real-time propagation method.

  10. 10.

    Adapted from Ref. [132].

  11. 11.

    A non-zero slope of the interaction potential at |z i −z j |=0.

  12. 12.

    This is indicated by the primes in Eq. (6.15).

  13. 13.

    This applies also to other multiply-excited states.

  14. 14.

    See the lower panel of Fig. 6.7 and the last paragraph in Sect. 6.2.1.

  15. 15.

    More precisely, they have zero transition dipole moment for λ ∗=0, cf. Sect. 6.2.1.

  16. 16.

    Notice, that this behavior is not a specialty of the GKBA as full 2B calculations do confirm this behavior, cf. the dots.

  17. 17.

    Here, the TDM is nearly constant for λ ∗<1. Note that both behaviors can be understood by standard perturbation theory.

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Authors and Affiliations

  1. Max Planck Research Department for Structural Dynamics, University of Hamburg, CFEL, Hamburg, Germany

    Karsten Balzer

  2. Institut für Theoretische Physik und Astrophysik, University of Kiel, Kiel, Germany

    Michael Bonitz

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  1. Karsten Balzer
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Balzer, K., Bonitz, M. (2013). Non-Lattice Systems. In: Nonequilibrium Green's Functions Approach to Inhomogeneous Systems. Lecture Notes in Physics, vol 867. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35082-5_6

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