Of symmetries, symmetry classes, and ­symmetric spaces

From disorder and quantum chaos to topological insulators

  • Martin R. Zirnbauer
  • 09 / 2012 Seite: 41

Quantum mechanical systems with some degree of complexity due to multiple scattering behave as if their Hamiltonians were random matrices. Such behavior, while originally surmised for the interacting many-body system of highly excited atomic nuclei, was later discovered in a variety of situations including single-particle systems with disorder or chaos. A fascinating theme in this context is the emergence of universal laws for the fluctuations of energy spectra and transport observables. After an introduction to the basic phenomenology, the talk highlights the role of symmetries for universality, in particular the correspondence between symmetry classes and symmetric spaces that led to a classification scheme dubbed the “Tenfold Way”. Perhaps surprisingly, the same scheme has turned out to organize also the world of topological insulators.

Let me begin by expressing that I feel greatly honored to be this year’s recipient of the Max-Planck medal, and I appreciate the opportunity to give a talk on some of the work that may have earned me this distinction. To set the stage and give you a flavor of what is to come, let me remind you of the old but still fascinating story of universal conductance fluctuations (UCF). Predicted theoretically in the middle of the 1980s by Altshuler [1] and by Lee and Stone [2], UCF was investigated in a large number of experiments. It was found that in a great variety of different mesoscopic systems − such as a small gold ring for example, or an even smaller silicon MOSFET − the electrical conductance displays characteristic fluctuations of the order of one when expressed in units of the conductance quantum e2/h (Fig. 1). What is most remarkable is that the size of the fluctuations in a broad range of parameters does not depend on the system dimension, the disorder strength, etc., but only on a few fundamental symmetries.
It was realized early on that there exists a close connection with the fluctuations that had been observed decades earlier in the scattering cross section of slow neutrons on atomic nuclei. This far reaching connection is at the very root of what I have to say. It led, among other things, to the development of a broad framework in which to model and calculate mesoscopic effects such as UCF. ...

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