Research Projects
Fractional Quantum Hall Effect
The quantum Hall effect is a prototypical example of quantum topological phases, the discovery and study of which led to three Nobel prizes (1985, 1998, 2016). The fractional quantum Hall effect (FQHE) is a many-body phenomenon, a first and so far the only experimentally realized class of topological phases resulting from strong electron-electron interaction. The zoology of particle-flux composites, and the emergence of anyonic/non-Abelian statistics, are not only profoundly interesting theoretically, but also of great practical potentials for storing and processing quantum information.
Our group specializes in the microscopic theories of the FQHE, including pseudopotential formalism, Jack-like polynomials, and with state-of-the-art numerical simulations. We are interested in understanding the underlying mechanisms for the incompressibility/stability of various FQH phases, especially with the interplay of topology and emergent geometry in these systems. Here are some of the questions we are looking into: a). how does geometry stabilize topological phases or drive phase transitions; b). how do we design novel experimental knobs for tuning experimental systems with geometric degrees of freedom; c). what is the nature of competing non-Abelian topological phases in systems with or without boundaries, especially for particle-hole symmetric Hilbert space (e.g. the Moore-Read state).
Geometric Engineering in Low-dimensional Systems
Breaking of rotational invariance has crucial consequences in two-dimensional electron gas systems (2DEGS). Though in many cases rotational invariance is assumed theoretically for its technical convenience, anisotropy is actually ubiquitous in experimental systems (e.g. band mass anisotropy, in-plane magnetic field, mechanical strain and folding). Our group is interested in a systematic characterization of anisotropy in real systems, so instead of treating it as a technical complication, we can judiciously utilize anisotropy for geometric engineering of novel quantum devices. For systems with non-trivial pseudospins or emergent orbital angular momenta, coupling of local "angular momentum" density to the spatially varying two-dimensional manifold can lead to additional Berry phase that is not only useful, but also illuminating to the fundamental local gauge degrees of freedom in 2+1 dimensional space-time.
Emergent Behaviors in Classical Complex Systems and Management of Urban Transportation
Even without quantum mechanics, classical systems consisting of a large number of interacting components can exhibit intriguing emergent behaviours; most interestingly they are in general independent of microscopic details. The spatiotemporal evolution of traffic jams is a simple example.
Our group tries to understand many aspects of such complex systems, including emergent dynamical phases, self-organised behaviours, critical phenomena, and universal statistical distributions. A particular focus is on urban systems, in which the interacting components are commuters and different modes of transportation (buses, taxis and private cars, etc.). We would like to understand the dynamics of the transportations, and thus devise means and algorithms to actively manage and optimise such systems.
We are also interested in analysing large scale empirical data, and using it to tune mathematical models of the transport systems, as well as developing modern machine learning techniques for data driven optimisation.