Condensed Matter Physics is arguably the broadest, most loosely defined discipline in Physics. My work is, in that sense, quintessential condensed matter theory, ranging from studies of transport in disordered superconductors to models for the motion of flocks of birds. The unifying theme of this work, to the extent that there is one, is the study of long distance and long time properties of strongly fluctuating systems with many degrees of freedom. The three topics I am currently most actively studying are:
1) Birds; or, the theory of flocking: I've recently collaborated in the development of a theory of the collective motions of large swarms of organisms that attempt to follow their neighbors. The continuum equations of motion we propose should be quite generally applicable to the motions of herds of wildebeest, schools of fish, flocks of birds, singular of rhinoceri, and swarms of bacteria. Indeed, it should describe the collective motions of any organisms - or self-propelled automata - that follow their neighbors. Our model combines features of the Navier-Stokes equation for a simple compressible fluid and a simple relaxational model for spins in a ferromagnet. In addition, it exhibits the unusual phenomenon known as the breakdown of linearized hydrodynamics: the failure of the linearized equations of motion to correctly predict even the scaling of response and correlations. This breakdown is induced by strong fluctuations; despite it, we are able to predict exactly the scaling exponents characterizing the long wavelength behavior of the flock. Our most surprising result is that although a million physicists, all standing in a plain and able to see only a few of their neighbors, could not manage to all point in the same direction, a million wildebeest can quite easily all move in the same direction.
2) Tubules: a new phase of fluctuating tethered membranes. A tethered membrane is a thin sheet of elastic material (e.g., a piece of paper, a biological membrane, etc.) with free boundaries. It was conjectured nearly a decade ago that such membranes could undergo a "crumpling" transition with increasing temperature, from a low temperature flat phase to a high temperature crumpled phase. I have recently been involved in demonstrating that when the membrane is anisotropic (e.g., harder to bend along one axis than another), an intermediate "tubule" phase always appears between the crumpled and flat phases. In this tubule phase, as the name suggests, the membrane is crumpled in one direction but extended in the other; i.e., it is shaped like a long, messy tube. The detailed properties of this phase have so far only been worked out using two uncontrolled approximations, so much work remains to be done. Future projects in this area include a controlled e-expansion study, consideration of the effects of disorder and long-ranged interactions, and formulation of a theory of the crumpled to tubule and tubule to flat phase transitions.
3) Liquid crystals: Smectic A liquid crystals consist of a periodic stack of liquid (translationally disordered) layers; thus, they are liquid-like in two directions (within the layers) but solid-like in one (normal to the layers). This combination leads to huge fluctuations which have been the subject of a great deal of theoretical and experimental study over the past two decades. I am currently studying two problems involving these systems.
(A) Dynamically X-ray scattering. It has recently become experimentally feasible to perform time (or frequency) resolved X-ray scattering from smectics A. This makes possible the study of the dynamical evolution of the aforementioned large fluctuations. I am using the hydrodynamic equations for smectics A to predict the results of these experiments, which are being performed by a colleague, Professor Stephen Kevan .
(B) Smectics in disordered media: I am attempting to predict the behavior of smectics A in aerogel, an experimental system that has been much studied recently. P.E. Lammert, D.S. Rokhsar, and J. Toner. "Topology and nematic ordering. I. A gauge theory". Phys. Rev. E25:1778 (1995).
J. Toner, P.E. Lammert, and D.S. Rokhsar"Topology and nematic ordering. II. Observable critical behavior".
J. Toner. "Anomalous elasticity of unidirectionally tethered and rippled (PB') membranes". To appear in Phys. Rev. B.
Y. Tu and J. Toner. "How birds fly together: Long-ranged order in a two-dimensional dynamical XY model". To appear in Phys. Rev. Lett.
L. Radzihovsky and J. Toner. "A new phase of tethered membranes: Tubules". To appear in Phys. Rev. Lett.