The Centre for Mathematical Physics is an interdisciplinary research centre designed to foster interaction between Mathematics and Physics at The University of Queensland, and with outside institutions. Many breakthroughs in the development of physical theories, particularly in the realm of quantum physics, have been
underpinned by the application of novel and sophisticated mathematical techniques. The research conducted by members of the Centre covers a broad spectrum from areas of pure mathematics such as Lie algebras, quantum algebras, supersymmetry, low dimensional topology and category theory through to applications in areas of Bose-Einstein condensates, superconductivity, condensed matter systems, quantum nanoscience and
quantum information science. The Centre is also active in promoting Mathematical Physics through regular workshops and seminar series.

The members of the Centre are:

  • Prof. Mark Gould (Director, Mathematics)
    Research interests: quantum groups and representation theory, knot theory, quantum chemistry, integrable quantum systems.
  • Prof. Peter Drummond (Deputy Director, Physics)
    Research interests: quantum many-body theory, quantum optics, computational physics, photonics and information, quantum solitons, Bose-Einstein condensates.
  • Dr. Stephen Bartlett (Physics)
    Research interests: quantum information theory, quantum computing,
    foundations of quantum physics.
    Publications
  • Prof. Tony Bracken (Mathematics)
    Research interests: quantum randon walks, phase space description of quantum mechanics relativistic localisation, coherent states, Wigner distribution functions.
  • Dr. Andrew Doherty (Physics)
    Research interests: quantum information theory, open quantum systems, quantum measurement.
    Publications
    Research
  • Dr. Cathy Holmes (Mathematics)
    Research interests: quantum dynamics, quantum chaos, Bose-Einstein condensates.
  • Dr. Jon Links (Mathematics)
    Research interests: algebraic structures in mathematical physics, category theory, integrable and Bethe ansatz solvable systems, quantum information theory.
    Publications
  • Prof. Ross McKenzie (Physics)
    Research interests: systems of strongly correlated electrons, quantum phase transitions, disordered systems, mesoscopic electronic devices, nano-materials.
    Publications
  • Prof. Gerard Milburn (Physics)
    Research interests: quantum control and measurement, quantum electromechanical systems, quantum chaos, quantum information theory, atomic Bose-Einstein
    condensates.
    Publications
  • Prof. Michael Nielsen (Physics)
    Research interests: quantum information science, quantum computation, entanglement, quantum phase transitions.
    Publications
  • Dr. Yao-Zhong Zhang (Mathematics)
    Research interests: quantum algebras and representation theory,.correlation functions, integrable and exactly solvable systems, disordered systems, quantum and conformal field theories.
    Publications
  • Dr. Huan-Qiang Zhou (Mathematics)
    Research interests: atomic and molecular Bose-Einstein condensates,.strongly correlated electronic systems, Bethe ansatz and integrable models, random matrix theory.

Below are some examples of the projects that may be taken in the Centre
for Mathematical Physics.

Projects:
Title: Current superalgebra and conformal field theory approach to disordered systems
Supervisor: Yao-Zhong Zhang

Superalgebras have emerged in a wide variety of areas ranging from high energy and condensed matter physics such as topological field theory, logarithmic conformal field theories (CFTs), the integer quantum Hall transition, sigma models on supergroup manifolds and superstring theory (the only available candidate for unifying all four
forces in nature), just to mention a few. In particular, the replica or supersymmetric treatment of disordered systems has revealed that the random critical points in two dimensions are described by current superalgebras with zero superdimension and their corresponding non-unitary CFTs.

This project aims to investigate such current superalgebras and CFTs as well as their applications in disordered systems and the integer quantum Hall transition.

Title: Quantum correlations in quantum field theory and integrable systems
Supervisor: Yao-Zhong Zhang

The study of correlation functions is the major open problem in the theory of exactly soluble models in condensed matter physics and integrable quantum field theories. There is much interest in this field
internationally, leading to high-profile research activity at present. A key part of this project is to evaluate correlation functions and form factors of physical significance. The recent development of sophisticated
algebraic and analytic methods makes the computation of such correlators feasible.

A great success in the understanding of physical phenomena such as phase transitions in quantum systems has come through the study of exactly soluble models. As is well known, most problems of physical interest are of a non-perturbative and non-linear nature and therefore are very difficult to solve. Under proper approximations, these complex physical problems reduce to less complex but physically non-trivial models that
can be solved exactly. Such exactly soluble models inherit many physical features of, and therefore provide important insight into, the original systems. One of the consequences of such approximations is that form
factors and correlation functions can be computed in closed form, so providing essential information for the full description of the original systems.

This project is to capitalize on our recent success in the evaluation of correlators of some soluble models by undertaking a through and systematic investigation into the algebraic formulation of correlation functions and form factors. This is to be achieved by further developing the vertex operator method as well as the factorizing F-matrix method. The latter approach will enable correlation functions and form factors to be expressed in determinant representations.


Title: Elliptic quantum groups and integrable many-body systems with long-range interactions
Supervisors: Yao-Zhong Zhang, Mark D. Gould and Wen-Li Yang

Elliptic quantum groups are algebraic structures underlying elliptic or dynamical Yang-Baxter equation. They can be obtained by quasi-Hopf twistings from the ordinary quantum groups, under the framework of
quasi-Hopf algebras invented by the Fields medalist, Drinfeld. Elliptic quantum groups play an important role in the study of quantum many-body integrable models with long-range interactions such as elliptic Gaudin
models, a very exciting subject in mathematical physics due to their role in establishing the integrability of the Seiberg-Witten theory and the BCS theory of small metallic grains.

The overall objectives of this project are to investigate these mathematical structures and physical properties of dynamical elliptic Gaudin models. We have already made significant progress on various aspects of the field, and we are looking for students to join us to work on the project.

Title: Quantum frames of reference in information processing
Supervisors: Stephen Bartlett and Jon Links

Theoretical physics research in the field of quantum information processing (which includes quantum communication, cryptography and computation) continues to motivate experiments, develop new techniques in
informatics, and introduce novel information processing tasks that is expected to lead to the emergence of powerful information technologies. In the operational approach to quantum mechanics, the elements of the mathematical formalism are associated with experimental operations, defined with respect to an external reference frame (RF). To date, such RFs have mostly been treated classically. For distributed information
processing involving two or more parties (for example, when two parties are communicating over long distances), these RFs must be shared in order to correlate preparations, transformations and measurements by one party
with those of another. These shared RFs, which have been traditionally presumed, are a resource that can be quantified, traded and consumed. First, there is a communication cost to establishing a shared RF through
the exchange of physical systems. The resulting physical correlate of the shared RF will require a generic quantum mechanical description. Second, repeated use of this shared RF by both parties to perform information
processing tasks (e.g., quantum teleportation, communication, or cryptography) will likely degrade this shared RF, ultimately resulting in errors in the protocols. This project aims to define, characterise and quantify shared quantum RFs in terms of their applications to quantum information processing. Specific goals of this research are to:

(1) Develop a formalism for the description of quantum RFs, including a quantisation procedure for traditional (i.e. classical) RFs;

(2) Quantify the communication cost, in terms of the exchange of physical systems, needed to establish shared RFs, and develop a quantitative measure for this result;

(3) Investigate the description of measurements with respect to a quantum RF, including the possibility of degradation of the frame;

(4) Assess the RF requirements for current quantum information protocols.

Title: Quasi Lie (super)algebras
Supervisor: Mark Gould

There have been many recent attempts to generalise the concept of Lie (super)algebras to provide a wider range of algebraic structures for application to physics. One such approach is based on adjoint orbits in Hopf (super)algebras, by analogy with the fact that a normal Lie algebra occurs as an adjoint orbit in it's own enveloping algebra. However these structures suffer from the draw back that their enveloping algebra is not
generally a Hopf (super)algebra (important for coupling of physical systems). Recently we have developed a new approach based on co-algebras (the dual objects of associative algebras), which overcome this latter problem. Moreover we have extended the theory of affinization to these structures to determine quasi Kac-Moody (super)algebras. However so far we have found very few examples of such algebras that were not already known.
We are interested in the systematic construction of new quasi Lie (super)algebras and their application to physics, particularly the determination of new R-matrices and corresponding lattice models with quasi Lie (super)algebra symmetry. A related problem is to understand the nature of the affinization procedure at a fundamental level.

Title: Correlations, entanglement and teleportation in many-body systems
Supervisors: Gerard Milburn, Ross McKenzie, Jon Links

The phenomenon of quantum teleportation is a consequence of entanglement which can exist in quantum systems. Entanglement, coupled with the ability to perform two-qubit measurements, is a key component in implementing quantum computation. An important issue to understand is the precise relation between the entanglement of a quantum channel, and the fidelity of teleporting across that channel, particularly in the cases where the quantum channel is some state of a complex many-body system. While it can be shown that there exists states of the system which provide perfect fidelity of teleportation, relationships between the entanglement of generic states and the fidelity of teleportation form an open problem. Since we cannot expect to know the explicit wavefunction of a general state, we are forced to seek some entanglement measure which can be related to observable quantities. This project will explore this problem in the context of correlations in states of many-body systems.