- Self-consistent perturbation theory-based modelling of superconducting and magnetic materials
Suitable for students who have have completed or are currently enrolled in Modern Physics, Linear Algebra and an introductory computer programming class
Superconductivity and magnetism are examples of phenomena that emerge as result of interactions between the constituent particles in materials, namely a small set of mobile electrons and quasistatic ionic cores. However, the properties of interacting systems of particles can never be determined exactly using formal mathematics. Typically, some type of approximation scheme must be employed and, even though a calculation is only approximate, a realistic approximation will almost always require the use of one or many computers to obtain numerical estimates of physical properties and processes. This is especially true for magnetic and superconducting materials as accurate results for these materials require the use of quantum mechanical laws for modeling the motion of the particles; models based on conceptually simpler Newtonian laws of motion.
Our approach is to use self-consistent many-body perturbation theory. In this method, one starts by assuming that interaction effects are so weak that the properties of a non-interacting set of particles (which can be determined exactly) can be used as a starting point. Interaction effects are introduced gradually, but systematically with the assistance of a diagramatic trick developed by Richard Feynman. The key idea is that properly drawn diagrams (created using a well-defined set of rules) can be used to determine exact mathematical representations of correction terms due to interaction effects. An example of a diagram that we use in our work is shown here:
This diagram roughly represents the scattering of electrons that meet at some position, x1, and their subsequent motions before and after the scattering event. This diagram is translated into a mathematical equation that becomes part of a complex computer program that analyzes the effect of this simplex process in concert with other more complex ones.
Even though self-consistent many-body perturbation theory is not exact, completing a calculation based on this method can prove challening even with the assistance of the most powerful computers. When we develop our computer programs we have always made sure that the programs take advantage of parallelism - the ability of many computers to work on one problem and achieve a solution faster by dividing the work. While at one time executing parts of a computer program in parallel was an infrequently used step used for only the most difficult problems, adopting parallell schemes are becoming more commonplace on account of multicore architectures in PC cpus and graphics cards that effectively makes a single computer today behave like a small set of computers did just a few years ago.
- Simulations of magnetic and superconducting systems
Suitable for undergraduates at all levels and beginning graduate students
In this work you use existing computer codes to simulate the
electronic properties of superconducting and magnetic systems.
Of special interest is the evolution of these properties
as a function of the atomic positions in the crystal.
You would be responsible for running these simulation on
laboratory PC workstations and subsequently
for analyzing the
data from thses simulations.
- Performance analysis of numerical and computational methods
Suitable for undergraduates at all levels and beginning graduate students
Analyze the computational effeciency of numerical algorithms
for common mathematical operations used in
these calculations
(such as linear algebra and Fourier transforms)
on PC workstations, parallel computing systems and graphics cards.
|
|
