My research is primarily in the area of exactly solvable models. I study these in the context of the broader area of condensed matter theory. Condensed matter theory, broadly speaking, is interested with the study of materials (or matter) and their properties. This includes subjects like magnetism, superconductivity, behavior of gases and liquids. In the recent past as more exotic material properties are being discovered, both theoretically and experimentally, condensed matter physics as an area is getting larger and more inclusive. The study of both macroscopic (the colors of birds feathers, for example), and microscopic (the behaviour of a quantum dot), now fall under this broad umbrella, leading to a variety of very complex and interesting open problems.

My work primarily deals with understanding the dynamics of integrable models. Integrable models, for practical purposes, are models which can be solved exactly, i.e., one can obtain the complete spectrum of allowed states. While not always obvious, one can show that integrable models have many conserved quantities (like energy, momentum, etc.), many more than a model that is not integrable. These models have special properties that are not believed to exist in other models. Further, because these are cases where we have full access to the behavior and properties of the model, it leads us to a deeper understanding of the physics. All of this would of course be of not much use to the real world, but luckily, a lot of systems can be modeled quite well using integrable systems. Deviations from integrability can then be studied as perturbations around integrability. Also, one can study so-called "universal limits" of these models to obtain universal properties such as critical exponents for phase transitions and so on.

Several integrable models of interest have been studied in great detail within the last eighty or so years. Sophisticated mathematical techniques, starting with the Bethe Ansatz have been developed over the course of these years in order to solve these models. However, understanding the dynamics has still been elusive due to several technical complications. In particular, understanding the time evolution of systems after a "quench" has been of much interest in the recent past. A quench (also quantum quench, interaction quench) is, like it's metallurgical origin suggests, a sudden change in some parameter of the system, which in cases of interest to us is usually an interaction strength or the external potentials on the system. Experiments that study such questions have become increasingly accessible with technological developments in optical lattices and cold atoms, and nanotechnology with quantum dots.

I'm part of a group that is currently interested in studying these questions in the context of one dimensional exactly solvable models. Most exactly solvable models in Physics tend to be one dimensional (i.e., one spatial and one time dimension). Also one dimensional systems behave very differently from their higher dimensional counterparts. Consider for example a gas. In two or more spatial dimensions, the particle can avoid each other quite easily, however, in one dimension, particles don't have an option but to encounter each other as they move about. Beyond this, one can show using simple arguments based on energy and entropy that one-dimensional systems cannot have spontaneous long-range order (barring some examples of quasi-long range order) develop at temperatures above zero. In other words, these systems exhibit phase transitions only at zero temperature.

One might think that this is all fine but not relevant since real-life systems are seldom one-dimensional. This is certainly true, but for one, several systems behave as though they are one-dimensional. Thin wires for example, where the transverse directions are much smaller than the lontudinal one, and so they aren't dynamically relevant degrees of freedom and can be ignored. But most fascinating is that in the last decade or two, experimentalist have succesfully constructed nearly exactly one dimensional systems by trapping gases using lasers. One stands to gain tremendous insight from studying the nature of these quantum phase transitions and other dynamical behavior occuring in these systems.

With this motivation, we employ integrability as a powerful tool to study some models exactly. One of the main difficulties with studying the dynamics of these systems exactly have had to do with representing experimentally viable initial states in a form that is amenable to study using exact solutions. In these models, the exact solution (eigenstates and eigenenergies) appears in a rather unusual and non-intuitive basis, that is usually not invertible in terms of the physical space of states. The approach we use essentially alleviates this issue for some models (more are under study right now). Some results are published here. We just put out a longer version of this with technical details, and spelling out a bunch of subtleties regarding quench dynamics more carefully here.

I am also currently involved in a project with Matthew Foster who recently moved to Rice. We are interested in studying the effect of interactions on the multifractal spectra of disordered two dimensional electrons. More about this, including a guide to multifractals coming soon!