Elizabeth J. Austin

A statistical topographic model for exciton luminescence spectra

Some features of the absorption and luminescence spectra of excitons in disordered 2D semiconductors appear to be nearly universal over a wide range of samples. In particular, the offset of the peaks of these two spectra is proportional to their linewidths, over a range of two orders of magnitude. The authors show that the relationship between these spectra can be understood in terms of the statistical properties of a Gaussian random function: the absorption spectrum is proportional to the probability distribution of the function itself, whereas the luminescence spectrum is proportional to the distribution of the heights of the minima of the function.

A random-matrix model for the non-perturbative response of a complex quantum system.

We consider the dynamics of a complex quantum system subjected to a time-dependent perturbation, using a random matrix approach. The dynamics are described by a diffusion constant characterizing the spread of the probability distribution for the energy of a particle which was initially in an eigenstate. We discuss a system of stochastic differential equations which are a model for the Schrodinger equation written in an adiabatic basis. We examine the dependence of the diffusion constant D on the rate of change of the perturbation parameter, X. Our analysis indicates that D alpha X2, in agreement with the Kubo formula, up to a critical velocity X*; for faster perturbations, the rate of diffusion is lower than that predicted from the Kubo formula. These predictions are confirmed in numerical experiments on a banded random matrix model. The implications of this result are discussed.

Phase-space lattices with threefold symmetry

This paper presents a semiclassical analysis of the spectrum of the Hamiltonian ir = cos(P - e) +cos ;ci+v/3 5, +COS f(P -A 5) which is a model for Bloch electrons in a magnetic field. The energy levels are determined, to a first approximation, by Bohr-Sommerfeld quantisation. Because of the translational symmetry in phase space, this leads to a lattice of infinitely degenerate states. This degeneracy is lifted by tunnelling effects, which can be described by an effective Hamiltonian of the same form as that above, but with different values of 0 and h. The calculation of the effective Hamiltonian therefore defines a renormalisation-group transformation, and it predicts that the spectrum has a complex recursive structure, which is confirmed by numerical experiments. The results complement earlier work on lattices with fourfold symmetry, in which a similar type of spectrum occurs: the greater complexity of the problem in lattices with threefold symmetry necessitated developing simpler, canonically invariant, methods of analysis.

Distribution of matrix elements of a classically chaotic system

The statistics of matrix elements of a perturbation of a classically chaotic quartic oscillator system are investigated. Our numerical results confirm that the local variance of the matrix elements can be obtained from a classical correlation function. The probability distribution of the matrix elements (normalised using their local variance) was investigated. This is expected to be a Gaussian distribution in the semi-classical (high-energy) limit. The expected Gaussian form emerges slowly as the energy is increased. The form of the distribution at lower energies is identified.