Phase correlation in the eigenfunctions of a class of quantum chaotic systems

W. D. Heiss, A. A. Kotz�

Sudhir R. Jain, Benoît Grémaud, Avinash Khare

Even as we understand for long that the world is quantal and buried in it is classical dynamics which is chaotic, finding eigenfunctions analytically from the the Schroedinger equation has turned out to be a near-impossibility. Here, we discover a class of chaotic quantum systems for which we obtain some analytically exact eigenfunctions in closed form. This paves way to an exact classical and quantum mechanical treatment of chaotic systems. Furthermore, we bring out connections, underlying the discovery, between different areas of physics and mathematics related to universality in fluctuations observed in a wide variety of complex quantum systems.

Yoana R. Chorbadzhiyska, Peter A. Ivanov, Charlie Nation

Arnd Bäcker, Roman Schubert

We study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits. We obtain an expansion of the autocorrelation function in terms of the correlation distance. For localized states in billiards, like bouncing ball modes or states living on tori, a simple model using only classical input gives good agreement with the exact result. In particular, a prediction for irregular eigenfunctions in mixed systems is derived and tested. For chaotic systems, the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales.

G. P. Berman, Andrey R. Kolovsky

Leonid Andreevich Pastur, Mira Shamis

The paper deals with the asymptotic behavior of a widely used correlation characteristic in large quantum systems. The correlation is quantum entanglement, the characteristic is entanglement entropy, and the system is an ideal gas of lattice fermions. If the one-body Hamiltonian of fermions is an ergodic finite difference operator with an exponentially decaying spectral projection, then the large-block form of the entanglement entropy is the so-called area law. However, the only class of one-body Hamiltonians for which this spectral condition was verified consists of discrete Schrödinger operators with random potential. In this paper, we prove the area law for several classes of Schrödinger operators whose potentials are ergodic but not random. We begin with quasiperiodic and limit-periodic operators and then move to a highly non-trivial case of potentials generated by subshifts of finite type. These arose in the theory of dynamical systems when studying chaotic phenomena. The corresponding asymptotic study requires involved spectral analysis, which therefore constitutes the bulk of the paper. Specifically, we prove uniform localisation of the eigenfunctions for the Maryland model and exponential decay of the eigenfunction correlator for various models. We believe these properties are of significant independent interest.