This award supports theoretical research and education on quantum phases and phase transitions as well as the PI's constant efforts to communicate developments in his field to a wide audience using a variety of vehicles like summer school courses, dialogs, public lectures, books, and review articles.
Quantum phase transitions occur when the ground state of a system undergoes a change of phase, say from superconductor to insulator, in response to a changing parameter in the Hamiltonian. Equally interesting as the transitions, are the phases themselves: the excitations they support, their degeneracies and their responses to external probes. The PI has developed two tools to this end: (i) the renormalization group for fermions in which low-energy states near the Fermi surface are systematically eliminated to display the ultimate properties of the ground state and its instabilities (ii) and the Hamiltonian Theory of the Fractional Quantum Hall Effect which is an operator treatment of the problem that allows the computation of gaps, relaxation rates etc. He plans to extend and apply the renormalization group in many ways: to describe the singular Coulomb interaction, to describe ferromagnetism, to describe Fermions with more than one momentum direction normal to the Fermi surface, to describe the vexing but important problem of how to formulate a renormalization group for bosons and fermions at the same time and bilayer graphene. He plans to apply the Hamiltonian theory to describe the response of the Hall system near half-filling to microwave radiation. Finally he proposes to pursue a rather tantalizing connection between Lie groups and fermion motion on several lattices- where Lie groups generate the very lattice and determine its unusual spectrum.
The PI plans to continue disseminating ideas here and abroad, to a broad audience, ranging from school children to advanced graduate students in summer schools, from professionals to the lay public via review articles, public lectures, web-based lectures, and his third text book, this time on the methods of quantum field theory in condensed matter physics. He will continue to actively disseminate his work on renormalization group to sister disciplines like Nuclear Physics and Particle Physics which have adapted it to describe matter at finite density.
NON-TECHNICAL SUMMARY This award supports theoretical research and education on quantum phase transitions and quantum states of matter. Unlike the more familiar phase transformations, like water to steam, quantum phase transitions occur at the absolute zero of temperature as an external variable, like pressure, is tuned through the transformation from one phase to another. It is believed that these phase transitions can influence the electronic properties of materials for a wide range of temperatures and hold promise to explain anomalous electronic properties of some classes of materials. The PI will further develop and apply a powerful technique first developed in particle physics but later applied to theory of phase transitions. The technique, called the renormalization group, enables one to examine the physics contained in a theory across expanding length scales. While this approach has yielded great insights into a number of important problems to condensed matter and materials physics, its application to others has proven more difficult. The PI aims to clear conceptual roadblocks and harvest the valuable insights it offers into quantum phase transitions and quantum mechanical systems of interacting particles more generally.
The PI plans to continue disseminating ideas here and abroad, to a broad audience, ranging from school children to advanced graduate students in summer schools, from professionals to the lay public via review articles, public lectures, web-based lectures, and his third text book, this time on the methods of quantum field theory in condensed matter physics. He will continue to actively disseminate his work on renormalization group to sister disciplines like Nuclear Physics and Particle Physics which have adapted it to describe matter at finite density.
The research carried out using this grant pertains to toplogical insulators. Now, insulators are known for not carrying any current when a voltage is applied: this is why we stand on them to change the fuse or light bulb. Recently it was realized that insulators actually come in two types: the standard ones that simply does not carry any current and the toplogical insulators that carry no current in their bulk but do carry current on their boundaries. For example a square chunk of toplogical insulator will carry current along its perimeter and an orange shaped chunk will carry current on its skin. The word toplogical in the name refers to the fact that there is something about the motion of electrons in the bulk that demands that current be allowed to flow at the boundaries. The boundaries themselves are unusual in that they can do certain things by virtue of being the boundary of a higher dimensional object that other materials of the same dimensionality cannot. For example, an isolated wire carries current in both directions while a one dimensional boundary of a two-dimensional toplogical insulator carries current in only one sense (clockwise or counter-clockwise). Toplogical insulators exhibit the Quantum Hall Effect (QHE). The familar Hall effect, predicted in classical electromagnetic theory, refers to the fact that when a voltage is applied to a two dimensional system of charges in one direction in the plane it will cause a current to flow in a perpendicular direction, if the system is immersed in a magnetic field perpendicular to the plane. What is remarkable about the toplogicsl insulators is that they do the same thing but in the absense of any external field. In addition, the ratio of current to violtage, the conductance, is quantized to an integer multiple of a basic unit of conductance, e2/h where e is the electronic charge and h is Planck's constant. This behavior, the Integer Quantum Hall Effect has already been observed in two -dimensional electron gases, but in an external magnetic field. The latter also exhibit the Fractional Quantum Hall Effect: as the number of electrons is reduced, the conductance still ssumes discrete quantized values, but now in certain rational fractional multiples of e2/h. There is evidence, based on computer calculation on small systems, that that toplogical insulators will also exhibit the Fractional Quantum Hall Effect. The present project was devoted to developing anlytical means for describing such an effect in considerable detail. The results are approximate but still far more detailed than is possible in any analytical scheme. It was done by adapting the ideas the PI and Professor Ganpathy Murthy of the University of Kentucky had developed for describing the FQHE in the two-dimensional electron gas. A toplogical insulator can display the QHE not only in the absence of an external magnetic field, but also at much higher temperatures than the two -dimensional electron gas. This makes them very valuable for metrology and quantum computing based on quantum Hall states.
