- I study phase transitions, with an emphasis on using numerical techniques such as Monte Carlo simulations (see the next item) to give a physical understanding of the problem. The ultimate goal of these simulations is not to calculate some well understood property to higher and higher accuracy but to rather to gain a qualitative and semi-quantitative understanding of the physics of problems where little is currently understood.
- A Simple Introduction to Monte Carlo Simulations,
- including a direct proof that the Monte Carlo algorithm converges to the equilibrium distribution. See also an elementary account of the jackknife and boostrap data resampling methods frequently used in statistical analysis of Monte Carlo data.
- Here are some links to Monte Carlo simulations that you can see on the web:
- Java applet showing a Monte Carlo Simulation for the Ising model.
- From this URL the source code for some Monte Carlo programs (with graphic display) can be downloaded.
- Random Systems, especially Spin Glasses
- One of my major areas of interest is phase transitions in random systems, particularly systems where the disorder is so severe that the system is "frustrated", i.e. no configuration of the system simultaneously minimizes each term in the energy. Such highly disordered magnetic systems are called "spin glasses". One reason why there is so much interest in spin glasses is that ideas first developed for that problem turn out to be relevant in many other areas of science, such as brain function (neural networks) optimization problems in computer science (simulated annealing) and in high temperature superconductors (the proposed vortex glass transition). The combination of frustration and disorder makes the problem very difficult, so much of what we know has come from numerical work.
One of my important earlier pieces of work was to show that the expected vortex glass transition in superconductors in a magnetic field is rounded out because of screening of the interaction, between the flux lines, (see also this ref).
My recent contributions include:
- I showed that there is no "Almeida-Thouless" transition in a magnetic field in spin glasses in three dimensions. The question of whether or not there is a transition in a field has been hotly debated for many years, and is one of the main differences between the "droplet picture" and the "RSB picture" of the spin glass state. According to RSB there is such a transition, whereas according the droplet picture there is no transition in a field. Subsequently I showed that there is likely to be an AT line in higher dimensions (d > 6).
- In several papers, cond-mat/0302371 , cond-mat/0703770 , arXiv:0804.3988 , I have looked at the critical behavior of vector (rather than Ising) spin glasses, finding a single transition at which "chiralities" and spins both order, rather than a transition only in the chiralities which had been proposed by Kawamura.
- Quantum Phase Transitions
- Another broad area where much remains to be done concerns phase transitions which are driven by quantum fluctuations rather than thermal fluctuations. These quantum phase transitions occur when the system is changed at (essentially) zero temperature, as a function of varying some other parameter, which might, for example, be the degree of disorder. There are many examples of such transitions; the most commonly studied being the metal-insulator transition, in which a metal is driven to an insulating state by adding more and more disorder. It has also been proposed that high temperature superconductors may be close to a quantum phase transition. Other theoretical techniques have not yielded very much information about quantum phase transitions in the presence of disorder, so one of the most useful approaches is numerical simulations.
In recent work in this area, I have used quantum Monte Carlo simulations to to try to understand the "complexity" of the quantum adiabatic algorithm, which has been proposed as a general purpose algorithm for a quantum computer. By looking at the size dependence of the minimum gap as the algorithm is evolved, I estimated how the running time varies with system size for much larger sizes than was possible before. Up to N=128 the running times increases only with a power of the system size rather than exponentially. Work is ongoing to determine if this "polynomial" complexity continues to larger sizes or whether there is a "crossover" to exponential complexity.
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