Statistical Properties of Response Functions in Quantum Chaotic Systems

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Statistical properties of response functions in quantum chaotic systems have been investigated concerning the following two points :(1) Corrrelation between Berry's geometric and quantum chaosAn instantaneous eigenfunction acquires a phase of geometrical origin (Berry's phase) during the adiabatic change of parameters included in hamiltonian. This phase can be expressed in a form of flux due to fictious "magnetic fields" originated from acccidental degeneracies of adiabatic eigenenergies in a parameter space. The acccidental degeneracies in quantum mechanics corresponds to non-integrability in classical mechanics. The non-integrability, in turn, implies chaotic behavior. Hence, the possible correlation is expected between quantal Berry's phase and classical chaos. Numerical simulations with use of a simple model hamiltonian demonstrated this correlation : The "magnetic fields" show large variations in a chaotic region while they are negligible in a regular region. It is an interesting … More future problem whether the behavior of Berry's phase in chaotic regions show universalities as the case of energy level statistics.(2) Validity of random matrix models in quantum chaotic scatteringsConductance (one example of response functions) is expressed in terms of scattering matrices in mesoscopic systems. Statistical properties of scattering matrices have been theoretically studied with use of various kind of random matrices in place of more complicated original hamiltonians. However the degree of validity of random matrix models in scaterings is not clear becasue trajectories with short dwell time in a scattering region are likely to contribute to scattring matrices as well as long period ergodic trajectories. In order to clarify this point, results of numercial simulations with use of a quasi-one-dimensional tight binding Anderson model are compared with those with randam matrix models. The comparison shows the better agreement as the effective coupling strength between a scatering region and external regions become weaken. How possible universal behavior described by random matrices can be separated from a specific feature of individual systems are a compelling future problem. Less