My prime focus is in condensed matter physics. This covers both "hard" (electronic) and "soft" (essentially classical) systems. This area is very diverse and is consistently shifting. As a result, my research spans many problems. Students are encouraged to join in on any one of the topics outlined below.
The application of simple statistical mechanics and classical mechanics ideas to graph theory and satistifiability problems has led to a remarkably simple algorithms for old problems. Our ideas rest on dynamics in high dimensions where slow covergence to the solution can be avoided.
The fascinating field of topological quantum orders [partially motivated by prospects of fault tolerant quantum computing] forces us to rethink anew our basic notions of order and topology. It is one of the most rapidly growning fields in condensed matter physics. Many of the tools used borrow heavily from field theory: condensed matter systems offer a direct realization of many ideas. In addition, they offer examples of emergent phenomena which appear at low energy scales.
Recently, the BCS-BEC transitions have been observed in cold dilute atomic gases. A long standing issue is what exactly transpires at the transition. Although the problem is easy to pose exactly, rigorous results are scarce. Remarkable numerical work has given us much insight into the problem. We hoped to obtain these results by direct calculation. Our work on the BCS-BEC transition in arbitrary dimensions and suggestions therein later paved the way for the remarkable tour de force epsilon expansions by Y. Nishida and D. Thanh Son about the d=2,4 dimensional problems which we exactly solved. To date, this dimensional extension is the only method enabling controlled expansions to this problem.
One of my main passions has been the study of the glass transition which is a problem of considerable practical importance and is also very deep. I collaborate with Ken Kelton and Tyrone Daulton in our department on this problem.
I spent much time thinking about "quantum critical points" which have by now been seen in many compounds. We have come up with simple model Hamiltonians which are exactly solvable and have many of the properties long predicted for such systems. Along with Stuart Solin in our department we are examining anew related systems. Much research is also done on the order of the atomic orbitals in transition metal oxides and on elastic defect dynamics. Additional work investigates single spin dynamics in superconducting junctions (where new nutations were predicted), "stripes" in high temperature superconductors, noise spectroscopy for examining fluctuations in small quantum systems, supersolids, avoided critical points, incommensurate phases, the role of competing orders in electronic systems, exact dimensional reductions resulting from symmetries, and the finding of new percolation crossovers in lattice gauge theories.