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Complex Quantum Systems and Random Matrix Theory

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Book cover Statistical Benchmarks for Quantum Transport in Complex Systems

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

Complex systems are ubiquitous in a wide variety of scientific disciplines. The terminology is profoundly present in fields ranging from social sciences (Sawyer, Social emergence: societies as complex systems, Cambridge University Press, Cambridge, 2005, Sawyer 2005), over economics (Arthur, Science 284:107–109 1999, Arthur 1999) and neuroscience (Hopfield, PNAS 79:2554–2558, 1982, Hopfield 1982), to engineering (Ottino, Nature 427:399–399, 2004, Ottino 2004), biology (Odum and Barrett, Fundamentals of ecology, Thomson Brooks/Cole, Belmont, 2005, Odum and Barrett 2005) and physics (Richter and Rost, Komplexe Systeme, Fischer-Taschenbuch-Verl, Frankfurt am Main, 2004, Richter and Rost 2004). Given the multitude of different themes considered in these different fields, it is legitimate to wonder whether there are general properties that unify the “complex systems” of these different disciplines.

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Notes

  1. 1.

    Since patterns can occur over long ranges of data, it is difficult to practically estimate the required measurement time. In the literature, one often assumes that the available string of data is infinitely long (Crutchfield and Young 1989).

  2. 2.

    One may also consider other types of complex networks, but these are the most well-known examples.

  3. 3.

    For an more complete introduction to the genesis of quantum chaos, see (Gutzwiller 2007).

  4. 4.

    The notation \( \overline{A}\) represents the component-wise complex conjugate, rather than the complex transpose. Notice also that \(K^2=\mathbbm {1}\) implies that \(KAK = \overline{A}\).

  5. 5.

    The last step is obtained by a very short “iterative expansion”.

  6. 6.

    This ensemble will not be considered in our subsequent work and thus we refer the reader to (Mehta 2004) for further details.

  7. 7.

    Starting from Eq. (3.22), with \(b=0\), we first note that \(\text{ tr }H^2 = \sum _{i,j} H_{ij}H_{ji}\). Because we consider the GOE \(H_{ij}=H_{ji}\), thus we find that \(\text{ tr }H^2 = \sum _i {H_{ii}}^2 + 2\sum _{i>j} {H_{ij}}^2\). Inserting this in (3.22) leads to \(P(H_{11}, \dots , H_{NN}) = c \prod _i \exp \big ( - a {H_{ii}}^2 \big )\prod _{i>j}\exp \big ( - 2 a {H_{ij}}^2 \big )\). The results (3.25) and (3.26) follow immediately from straightforward integration.

  8. 8.

    This dependence appears via factors \(\left|\left\langle \eta _i,\phi \right\rangle \right|^2\), where \(\phi \) is the state vector of the system and \(\eta _i\) is an eigenvector of the Hamiltonian.

  9. 9.

    This does imply that the notation using the same “P” is somewhat unfortunate, since we are really talking about different functions. Still, to avoid notational overhead, we stick to the P wherever a probability density is treated.

  10. 10.

    This would be similar to sampling the value 0.12345 for a stochastic quantity and trying to use that value to identify the probability distribution from which that value was sampled.

  11. 11.

    Note that this is exactly what is done in (Bohigas et al. 1984), where one considers the local properties the spectra of three different systems to acquire the data presented in this publication’s Fig. 3.1. Another example is (Walschaers 2011), where the density of states of a single, large (\(5000 \times 5000\)) random matrix is sufficient to accurately estimate the Marchenko-Pastur statistics (Marčenko and Pastur 1967).

  12. 12.

    Even though the moments contain all the information about the probability distribution, that does not imply that it is straightforward to find an explicit expression for the probability density function based on the moments.

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Authors and Affiliations

  1. Laboratoire Kastler Brossel, Sorbonne University, CNRS, École Normale Supérieure, and Collège de France, Paris, France

    Mattia Walschaers

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Correspondence to Mattia Walschaers .

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Walschaers, M. (2018). Complex Quantum Systems and Random Matrix Theory. In: Statistical Benchmarks for Quantum Transport in Complex Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-93151-7_3

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