In quantum chaos one considers systems whose classical motion is chaotic, i.e., depends sensitively on the initial conditions. One then investigates the quantum mechanical properties of these systems, for example their energy levels. It turns out that there are deep connections between the classical and quantum mechanical behaviour. Many aspectsI of this quantum behaviour are universal and independent of peculiarities of the system. For example, the statistics of energy levels becomes universal, and the different energy levels of a chaotic system always have a tendency to repel each other. These universal properties agree with phenomenological predictions from random matrix theory. A central topic of my research is to understand the reasons (and, importantly, the conditions and limitations) of this universality. This can be studied using semiclassical approximations, i.e., by relating quantum properties of chaotic systems to their classical trajectories (in particular the periodic ones). It turns out that pairs of very similar trajectories are of crucial importance. This includes pairs identified by my Bristol colleague Martin Sieber as well as Klaus Richter; here one trajectory includes a self-crossing with a small angle and the second trajectory narrowly avoids then crossing but otherwise closely follows the first trajectory. Ultimately the importance of such pairs of trajectories is due to interference effects. More generally interference effects arise between trajectories that differ in arbitrarily many so-called “encounters” where arbitrarily many parts of the trajectories come close. I was involved in extending these ideas to evaluate the two-point correlation function of the level density for quantum chaotic systems and show agreement with random matrix theory.  A recent research focus connected to this direction, with Martin Sieber and our joint students, has been to study the role of symmetries in quantum chaos and how they affect the interference mechanisms above.

Another focus of my research has been to study problems from condensed matter physics using methods established in mathematical physics, such as semiclassics, random matrix theory and techniques for disordered systems. One line of work in this direction involved using interference between trajectories to study the properties of mesoscopic chaotic conductors. More recently I have been interested in studying spectral properties of many body systems, using an adaptation of semiclassical methods to second quantised systems, as well as the embedded ensembles of random matrix theory.

PhD projects

Possible PhD topics evolve continuously as the state of the art improves. They can range from more mathematical problems (using random matrix theory, representation theory, or combinatorics) to condensed matter physics. Two possible directions are listed below but I am also open to supervising projects in other parts of the fields listed above.

Many particle systems

The aim of this project is to further pursue the semiclassical approach to many particle systems, in first and/or second quantisation. Challenges involve for example to take into account the indistinguishability of particles, interactions, and symmetries specific to many particle systems. Alternatively a student interested in this line of research could concentrate on the random matrix approach to many particle systems, or on the connection between semiclassics and random matrix theory. The emphasis of this project could be either on spectral statistics or transport properties. I am linking some references for the semiclassical approach, its general context, and the random matrix approach.

Anderson localisation in chaotic wires

The conductance of a wire is strongly reduced if the wire contains irregularities. This effect is well understood if the irregularity is due to disorder inside the wire; it is an example for a phenomenon called Anderson localisation. However the conductance is also suppressed if the wire is completely clean and just the shape of the boundary is irregular, leading to chaotic classical motion. In this project the behaviour of clean chaotic wires will be studied using semiclassical techniques. The conductance is a quantum mechanical property but in the semiclassical limit it can be expressed through interference effects between pairs of classical trajectories. One has to pay attention to fully capture the properties of the classical dynamics. In contrast to small conducting cavities one has to take into account effects due to the long length of the wire. It is expected that this dynamics over large distances can by modelled by a diffusion equation. This project will study in detail the interplay between possible types of classical dynamics and their signatures in quantum mechanics. The results could also be extended to spectral and wave function statistics. Useful references are here and here, with a general overview of the semiclassical approach here.