This award supports theoretical research and education to advance understanding of topological order in various forms. The PI seeks to advance understanding of their underlying mathematical structure.
The PI's specific goals are: i) to obtain a direct proof of the existence of gapless excitations in the Haldane-Rezayi quantum Hall state, ii) to identify a new and general connection between fractional quantum Hall states and integrable models, and in doing so open a new avenue to the problem of classifying quantum Hall states, iii) to develop a deeper understanding of the "occupation number basis" structure of FQH wave functions in the torus geometry, which is generally indispensable to the study of different kinds of topological order, iv) to develop a new method to systematically study the correlations of short-range resonating valence bond spin-1/2 wave functions, the proposed prototype wave functions for a long sought spin liquid, v) to prove the uniqueness of the nearest neighbor valence bond ground states in generalized Klein models, and to establish the feasibility of an SU(2)-invariant spin liquid on the kagome lattice, vi) to develop an argument that determines theoretically whether or not a supersolid state exists.
The PI will also generate new advanced course material. The PI will continue to give public outreach lectures where the broader ideas of quantum mechanics and of quantum condensed matter systems in particular are presented in perspective with some of the main questions driving research supported through this award.
NON-TECHNICAL SUMMARY
This award supports theoretical research and education on how electrons and atoms organize themselves into ordered states of matter. The self-organization of atoms into a regular array in crystalline materials and the alignment of atomic scale constituents of magnetism to become a magnet are examples of ordered states of matter. Research on quantum Hall systems, electrons confined to two dimensions in artificial semiconductors and placed in a strong perpendicular magnetic field, has led to the notion of a new kind of order known as topological order which is reflected in topological insulators. Like ordinary insulators, for example rubber, topological insulators do not conduct electricity though the interior of the material. Unlike ordinary insulators, topological insulators are able to conduct electricity on their edges or boundaries through the formation of a new state of matter. Among the known topological insulators are compounds made of the elements bismuth and selenium, and bismuth and tellurium.
The PI will develop new methods to study model systems exhibiting topological order both in fractional quantum Hall systems as well as in quantum magnets. These methods will borrow from the idea that the fundamental physics is faithfully contained in the properties of its edge, for example the surface metallic state of the topological insulator. The PI will also investigate the organizing principles for other new states of matter.
The PI will also generate new advanced course material. The PI will continue to give public outreach lectures where the broader ideas of quantum mechanics and of quantum condensed matter systems in particular are presented in perspective with some of the main questions driving research supported through this award.
