Most of my research falls within condensed matter physics, particularly the physics of electrons in solids. In such systems, it is possible for the electrons to organize themselves into highly-correlated collective states with properties which depart radically from those of an individual electron. Some well-known examples include superconductivity and magnetism. Another, perhaps less well-known, example is the quantum Hall effect. Unlike the other two examples, which are spontaneously broken-symmetry states, quantum Hall states are topological phases of matter. In such a phase, all measurable quantities are topological invariants at low energies and temperatures. In more technical terms, topological phases are those states of matter in which the low-energy effective field theory is a topological field theory. Perhaps the most striking feature of such phases is that the excitations above the ground state are localized quasiparticles which have exotic braiding statistics -- when one quasiparticle goes around another, a non-trivial phase can result, unlike in the case of bosons or fermions. Such particles are called anyons. At present, quantum Hall states are not merely the best but are, in fact, the only examples of topological phases in nature. Other topological phases have been hypothesized and have been found in theoretical models but have not yet been observed experimentally. Two of the major problems around which my work revolves are: (1) under what conditions will a given topological phase occur? and (2) by what experimental measurements can we identify a topological phase?
Interest in topological phases has recently been driven by its potential as a platform for fault-tolerant quantum computation: since low-energy physics is topologically-invariant, it is impervious to local imperfections and local interactions with the environment, which would cause errors in other quantum computing architectures. Another question which animates my work is: how can we build a topological quantum computer? Thus, although I describe myself as a condensed matter physicist, my work is now influenced strongly by the imperatives of quantum computation. A semi-popular account can be found in ``Topological Quantum Computation,'' S. Das Sarma, M. Freedman, C. Nayak, Physics Today 59, 32-38 (2006).
At present, most of my attention is focussed on the observed 5/2 and 12/5 quantum Hall states. These are the two quantum Hall states which are most likely to be non-Abelian topological phases. In such a phase, there is a degenerate set of multi-quasiparticle states. Hence, when one quasiparticle goes around another, the state of the system may transform non-trivially within this state space, rather than merely acquire a phase. Since the transformations corresponding to different quasiparticle braids need not commute, such quasiparticles are called non-Abelian anyons. Since braiding results in non-trivial unitary transformations of the state space, we can hope to thereby implement the type of unitary transformations needed for Shor's algorithm, for instance.
Unusual broken-symmetry states are another, unrelated
subject of my research. My interest grew out of
an exploration of ordered states which are formed by the
condensation of particle-hole pairs in a non-zero angular-momentum
channel. Together with a number of collaborators as well
as students and post-docs, I have been trying to understand
many of the unusual properties of the cuprate superconductors
starting from the hypothesis that the condensation
of such an order parameter occurs in the underdoped part
of the phase diagram. The order parameter was conjectured
to be of $d_{{x^2}-{y^2}}$ form, dubbed `d-density-wave'
(DDW) order, and its competition with superconductivity
was argued to explain the suppression of superconductivity
in this part of the phase diagram. A number of experiments appear
to be consistent with this hypothesis, but none of them
can rule unambiguously one way or another.