Matrix models, one-dimensional fermions, and quantum chaos

Parametric correlations versus fidelity decay: the symmetry breaking case

We provide formulas for fidelity decay and parametric energy correlations for random matrix ensembles where time-reversal invariance of the original Hamiltonian is broken by the perturbation. Like in the case of a symmetry conserving perturbation a simple relation between both quantities exists. Fidelity freeze is observed for systems with even and odd spin.

Surprising relations between parametric level correlations and fidelity decay

Relations among fidelity, cross-form-factor (i.e., parametric level correlations), and level velocity correlations are found both by deriving a Ward identity in a two-matrix model and by comparing exact results, using supersymmetry techniques, in the framework of random matrix theory. A power law decay near Heisenberg time, as a function of the relevant parameter, is shown to be at the root of revivals recently discovered for fidelity decay. For cross-form-factors the revivals are illustrated by a numerical study of a multiply kicked Ising spin chain.

Origin of quantum chaos for two particles interacting by short-range potentials

We address the problem of two confined one-dimensional particles of arbitrary masses interacting by general short-range potentials. We study under what conditions quantum chaos emerges for the system by analyzing its spectrum statistics. We show that these conditions are directly connected with a specific feature of the underlying classical dynamics, namely, the ergodicity in the changes of the particles momenta. Quantum mechanically this prevents one from obtaining the exact wave function through the Bethe ansatz. Possible extensions for many-body systems are also discussed.

Alternative technique for complex spectra analysis

The choice of a suitable random matrix model of a complex system is very sensitive to the nature of its complexity. The statistical spectral analysis of various complex systems requires, therefore, a thorough probing of a wide range of random matrix ensembles which is not an easy task. It is highly desirable, if possible, to identify a common mathematical structure among all the ensembles and analyze it to gain information about the ensemble properties. Our successful search in this direction leads to the Calogero Hamiltonian, a one-dimensional quantum Hamiltonian with inverse-square interaction, as the common base. This is because both the eigenvalues of the ensembles and a general state of the Calogero Hamiltonian evolve in an analogous way for arbitrary initial conditions. The varying nature of the complexity is reflected in different forms of the evolution parameter in each case. A complete investigation of the Calogero Hamiltonian can then help us in the spectral analysis of complex systems.