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Representations of the Nonequilibrium Green’s Function

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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

An efficient utilization of the numerical resources available is crucial for successful nonequilibrium Green’s function (NEGF) calculations for spatially inhomogeneous systems. Typically, the limiting factor is the computer memory. This fact has its origin in the non-Markovian structure of the Kadanoff-Baym equations which requires to remember (and hence to store) the dynamics history.

After a brief review of NEGF applications for homogeneous systems, the third chapter discusses the pros and cons of grid and basis representations of the nonequilibrium Green’s function in respect of reducing the computational effort and, in turn, of enabling larger propagation times and/or system sizes. A combination of both (grid and basis) strategies emerges as being particularly promising and advantageous. This is exemplified by using the finite element-discrete variable representation which remarkably simplifies the computation of the self-energies.

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Notes

  1. 1.

    This means, that the Green’s function (or self-energy) does not decay sufficiently fast when “looking” back into the past from the current time.

  2. 2.

    Aside from the generic two-time structure.

  3. 3.

    Note, that the correlation functions are in general complex. Assuming double precision, we need 16 bytes for a single complex number.

  4. 4.

    Potentially, including an adaptive time step size.

  5. 5.

    In Coulomb systems, the correlation time is typically of the order of τ cor≈2π|ω pl|. At weak coupling, the relaxation time of the Wigner distribution is typically significantly larger, τ rel≫τ cor [11, 95, 123].

  6. 6.

    The resulting equations of motion for the 1pNEGF are sometimes called “generalized” Kadanoff-Baym equations, e.g., [121, 124].

  7. 7.

    In the presence of correlations, that is beyond the HF level.

  8. 8.

    Note that the runtime itself can be a bottleneck at large memory consumption as we have to account for matrix instead of scalar multiplications.

  9. 9.

    Quadratic integrability is required to well-define operator matrix elements.

  10. 10.

    Similar to methods using B-splines.

  11. 11.

    As the roots of Legendre polynomials are symmetric about x=0, the same symmetry applies to the Gauss-Lobatto points (weights) x m (w m ).

  12. 12.

    With the short-hand notation \(\delta _{mm'}^{ii'}=\delta_{ii'}\delta_{mm'}\).

  13. 13.

    The same result holds for any other operator that is local in space.

  14. 14.

    This becomes clear after a thorough analysis of Eg. (3.21).

  15. 15.

    For a spin-polarized system, we have ξ=1.

  16. 16.

    Restore all orbital indices and insert the full matrix for w (2).

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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). Representations of the Nonequilibrium Green’s Function. 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_3

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