Computational Methods for Strongly Correlated Models in Two Dimensions
(Supervisor: A/Prof. Chris Hamer)

The treatment of strongly correlated quantum lattice models in two dimensions is one of the ‘grand challenges’ for theorists. Such models may represent high-temperature superconductors, organic conductors, exotic magnetic materials, or even gauge models in particle physics. Quantum fluctuations are especially large in two dimensions, and strong correlations are predicted to lead to exotic phenomena such as ‘spin liquid’ states, ‘deconfined’ quantum phase transitions with fractional excitations, as well as high-temperature superconductivity and other new effects. Unfortunately, our traditional techniques of numerical calculation have proved inadequate to confirm these predictions. Recently, new techniques involving matrix product representations of wave functions by Cirac, Verstraete and Vidal have attracted great interest for these two-dimensional models. The project is to develop, test and apply an alternative technique involving localized cluster correlation coefficients on the lattice, which may be more efficient than the matrix product approach. The work will suit those with a computational bent.

High-temperature superconductivity.
Supervisor Prof. O. P. Sushkov

The quest for understanding the mechanism of high-temperature superconductivity in the copper oxide materials has been at the forefront of condensed matter physics research for two decades. The experimental situation in the field has advanced dramatically during the past several years. These advancements represent a challenge for theory, and, more importantly,  have created a great opportunity for theory to provide deeper insights into the physical phenomena. The superconductivity project of our group can accommodate 2-3 PhD students.
PhD research projects are aimed at:
1)Spin dynamics in high-temperature superconductors.
2)Charge dynamics in high-temperature superconductors.

Lyapunov Modes and Correlation time scales
Supervisor A/Professor Gary Morriss

The study of the Lyapunov stability of classical systems of particles has revealed that as well as a spectrum of eigenvalues, called the Lyapunov spectrum, there are the associated eigenvectors of the time evolution which have some unexpected properties. The eigenvectors associated with the smallest positive and negative exponents are delocalised and represent global motion of the system. The time dependence of these modes is likewise unexpected and, in at least one case, related to the decay of correlations. There are a number of possible research directions based on the initial studies of Taniguchi & Morriss.

Atomic clocks and fundamental physics.
Supervisors: Prof. V. V. Flambaum, Dr. V. A. Dzuba

The study of atomic clocks is a rapidly developing area of research due to important role atomic clocks play in metrology, navigation and fundamental research. For example, the metric second is defined as duration of 9,192,631,770 periods of oscillations between the two hyperfine levels of the ground state of the cesium atom. The error of this clock is less than a second in 60 million years! In spite of this incredible accuracy atomic clocks are still not sufficiently accurate for some navigation tasks, e.g. automatic aircraft landing. There are many proposals to build even more accurate clocks by using atomic optical transitions or appropriate nuclear transitions.

Fundamental research would also benefit a lot from improving the accuracy of atomic clocks. For example, some theories unifying gravity with other interactions allow the fundamental constants to very. Variation of fundamental constants may be an indication of extra dimensions in our Universe or even parallel universes. The variation of fundamental constants in space may explain famous fine tuning of fundamental constants which is need for life to appear. We appeared in the area of the Universe where the fundamental constants are consistent with existence of life.

The temporal variation of the fundamental constants can be studied by comparing different atomic clocks over long period of time. The best current result of this kind comes from comparison of optical transitions in aluminium and mercury ions. It reads that if the fine structure constant changes in time than it changes no faster than tiny fraction of 10-17 a year!

We search for the ways to improve the accuracy of atomic clocks, perform atomic and nuclear calculations to help in the clocks design and for the interpretation of the measurements. The student may choseto be involved in any aspects of this work on his/her and the supervisor's discretion.

A scholarship top-up is possible for exceptionally good applicants.

Calculation of isotope shift in many-electron atoms and study of fundamental interactions.
Supervisors: Prof. V. V. Flambaum, Dr. V. A. Dzuba

Isotope shift is the difference in optical spectra of different isotopes of the same atom. This difference is due to difference in mass and nuclear structure. Examining the effect of isotope shift on the atomic spectra of distant objects in the universe provides an opportunity to study isotope abundance evolution in early universe and test theories of nuclear reactions in stars and supernova. On the other hand, study of isotope shift in heavy atoms can be used to obtain valuable information about nuclear structure. This information is to be used to reduce uncertainty of experimental study of fundamental interactions in heavy many-electron atoms. In both cases accurate atomic calculations are needed for interpretation of experimental results.

The project involves using existing and developing new computer codes in Fortran, running these codes and combining theoretical and experimental results to extract information about fundamental interactions.

A scholarship top-up is possible for exceptionally good applicants.

Study of relativistic and quantum electrodynamics effects in many-electron atoms.
Supervisors: Prof. V. V. Flambaum, Dr. V. A. Dzuba

Electrons in heavy atoms move with speeds close to the speed of light. Therefore they should be treated relativistically for accurate results. Dominant relativistic corrections are usually included by replacing Schrödinger equations for single-electron states by Dirac equations. Breit and quantum-electrodynamic (QED) corrections are smaller relativistic effects which are not included in Dirac equation but still play important role in heavy atoms. Breit interaction is the difference between exact relativistic expression for the inter-electron interaction and its non-relativistic Coulomb approximation (e^2/r). Leading terms in this difference are magnetic interaction and retardation. QED corrections are due to interaction of atomic electrons with vacuum fluctuations.

Project involves developing and running Fortran computer codes for accurate relativistic calculations for heavy atoms. The results would contribute to the relativistic theory of atoms and help in interpretation of experimental investigation of fundamental interactions in heavy atoms.

A scholarship top-up is possible for exceptionally good applicants.

Effects of variation of fundamental constants of Nature from Big Bang to atomic clocks.
Supervisors:Prof. V. V. Flambaum, Dr. V. A. Dzuba

Variation of fundamental constants (speed of light, electron electric charge, etc.) in space and time is suggested by theories unifying gravity with other interactions. Another argument for the variation comes from the anthropic principle. There must be very fine tuning of the fundamental constants which allows humans (and any life) to appear. This fine tuning can be naturally explained by the spatial variation of the fundamental constants. We appeared in the area of the Universe where the values of the fundamental constants are consistent with our existence.

The aim of this project is to search for the manifestation of the variation and perform necessary calculations of observable effects. For example, a change of the fundamental constants influences outcome of the Big Bang nucleosynthesis. The primordial amounts of deuterium, helium and lithium have been measured by astronomers. Comparing the calculations and measurements one can determine values of the fundamental constants after Big Bang. Variation of the fundamental constants also influences quasar spectra and ticking of different atomic clocks. A number of such measurements are now in progress. To interpret these measurements we should perform calculations of the variation effects for atomic transition frequencies. These calculations can be done using computer codes developed in our group. It is also very important to find new, enhanced effects of the variation and suggest new measurements.

This project may involve both analytical and numerical calculations in different areas of physics and cosmology, and may accommodate several PhD students.

A scholarship top-up is possible for exceptionally good applicants.

Violation of fundamental symmetries in atoms and test of grand unification theories.
Supervisors:Prof. V. V. Flambaum, Dr. V. A. Dzuba

Recently great progress has been made in experiments on violation of parity and time reversal invariance in atoms. The aim of this project is to perform necessary atomic and nuclear calculations of the observed effects which are needed to test theories unifying all interactions of Nature. Different theories predict different strengths of weak interactions which violate parity and time reversal invariance, and comparison between the calculations and measurements will help to select correct theory.

It is also very important to find new, enhanced effects of the violation of the fundamental symmetries and suggest new measurements.

This project may involve both analytical and numerical calculations in atomic, nuclear and particle physics, and may accommodate several PhD students. Atomic calculations can be done using computer codes developed in our group.

A scholarship top-up is possible for exceptionally good applicants.

Reaction-Diffusion Models
(Supervisor: A/Prof. Chris Hamer)

Reaction-diffusion models describe particles which can travel over a grid, or evaporate from or condense onto the grid. They can be used to model chemical reactions, or adsorption or desorption of particles from a surface, or even the flow of traffic. They can be formulated in terms of a master equation like the Schroedinger equation, but with a non-Hermitian Hamiltonian, in the general case.

The project, formulated in collaboration with Dr. R. Stinchcombe from the University of Oxford, is to calculate the phase structure and properties of these models using series expansion techniques. Our group has great expertise in perturbation series expansion techniques for the usual Hermitian systems; but the treatment of non-Hermitian systems requires the development of new techniques and algorithms. The work would involve advanced computational techniques. This is a new venture for our group, and there is considerable scope for future work in this area.

Application of the PEPS method to Strongly Correlated Systems in Two Dimensions
(Supervisor: A/Prof. Chris Hamer)

The treatment of strongly correlated lattice models in two dimensions presents a special challenge to theorists. Such models may represent high-temperature superconductors, organic conductors, exotic magnetic materials, or even gauge models in particle physics. Quantum fluctuations are especially large in two dimensions, and strong correlations are predicted to lead to exotic phenomena such as ‘spin liquid’ states, ‘deconfined’ quantum phase transitions with fractional excitations, and other new effects. Unfortunately, our traditional techniques of numerical calculation have proved inadequate to confirm these predictions.
Recently a new technique called ‘PEPS’ involving matrix product representations of wave functions has been formulated by Cirac, Verstraete and Vidal, which may provide much improved results for these two-dimensional models. The project is to develop, test and apply these techniques to some of these interesting models. The work will suit those with a computational bent.

Quantum properties of black holes.
Supervisor: Dr. Michael Kuchiev

Black holes are surrounded by event horizons, which represent a boundary between the outside world and the inside region. Well known classical description of the problem reveals that a particle, which approaches a black hole, crosses the horizon quite smoothly; the probability to penetrate inside is 100%. However, it was demonstrated recently in Refs.[1,2] that quantum corrections change this conclusion qualitatively. Quantum effects make it possible reflection from the event horizon, which is a surprising result. The reflection is possible for any particle, being stronger for long-wave particles. For sufficiently large wavelengths the particle cannot cross the horizon at all, being predominantly reflected. In other words, the black hole in this situation behaves as a good mirror.

The project aims to study relations of this new property of black holes with their other properties, in particular with the Hawking radiation, the information paradox and related phenomena.

References:
1. M.Yu.Kuchiev. Reflection from black holes and space-time topology, Europhys. Lett. 65, 445 (2004)
2. M.Yu.Kuchiev, Reflection, radiation and interference for black holes, Phys. Rev. D 69, 124031 (2004); gr-qc/0310051
3. M.Yu.Kuchiev and V.V.Flambaum, Scattering of scalar particles by a black hole, Phys. Rev. D 70, 044022 (2004); gr-qc/0312065