Physical Properties of Strongly Correlated Quantum Liquids

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TECHNICAL SUMMARYThis award supports theoretical research and education on the notion of order, a fundamental concept in condensed matter physics. Research in last 20 years suggests that Landau's symmetry breaking theory only describes a subset of possible ordered states that matter can realize. The possible ordered states of matter may be much richer than imagined before. The PI introduced the concepts of topological order and quantum order to describe the new types of ordered states that are not encompassed by the concept of broken symmetry. In this project, the PI plans to continue his research on topological/quantum order and to work towards building a comprehensive theory for these kinds of order. In particular, the PI will work in the following areas:(a) Based on the string-net picture, the PI has developed a comprehensive theory for non-chiral topological order based on the tensor category theory. The PI plans to combine the pattern-of-zeros approach, the vertex algebra approach, and the effective theory approach from projective construction to develop a comprehensive theory for chiral topological order in quantum Hall states. This will enable the study of phases and phase transitions for non-Abelian quantum Hall states and will enable the prediction of new non-Abelian states, for example in double-layer systems.(b) The PI plans to develop a new type of approach based on tensor network. The previous work has demonstrated the effectiveness of tensor network approach in obtaining topological phases and topological phase transitions. The previous work also reveals the directions that one needs to improve the tensor network approach. The PI plans to use the new approach to study frustrated quantum systems to discover more topological phases in real materials.(c) The PI plans to study a new class of topological phases, symmetry protected topological phases, which exist only for Hamiltonians with certain symmetries. A preliminary theory for these phases has been developed based on projective symmetry group. The Haldane phase for spin-1 chain and topological insulators/superconductors are special examples of symmetry protected topological phases. The PI plans to concentrate on phase transitions and gapless states on the interfaces. Such studies may lead to device applications for topological phases.The PI's emerging theory of topological/quantum order has the potential for high impact on many areas of physics and mathematics. The proposed research will result in new approaches for calculating phase diagrams of strongly correlated systems. Predicting topological/quantum phases lies outside the reach of traditional methods. This project will train students in advanced methods and concepts of theoretical condensed matter physics. NONTECHNICAL SUMMARYThis award supports theoretical research and education that extends a fundamental concept of materials. The notion of order is an important cornerstone in the foundation of our understanding of the world around us. For example when a liquid becomes a solid, the atoms may organize themselves in a periodic array to form a crystal lattice. This is an example of an ordered state of matter; there are many other diverse examples, some more exotic and subtle. They can be organized and the transitions among them described by the standard theory of phase transitions. The discoveries of new materials and phases, such as the high temperature superconductors or the quantum Hall phases, which arise when electrons are confined to two dimensions in a high magnetic field, have led to questions about the fundamental nature of order and whether the concept of order is more general. The PI has proposed new kinds of order that are not contained in the standard theory of phase transitions, but yet would have significant consequences on how we understand materials. This award supports research that aims to develop further a theory of transformations involving these new ordered states and to discover new ordered states of matter. The theoretical prediction of new materials-related phenomena may also result from this work. This project influences how we understand the world around us and could have potential impact on future technologies and other scientific disciplines. The possibility of utilizing some of these states of matter to form the basis of computation provides a possible way to make a quantum computer which would have impact on information technology. This project also involves students and will help train the next generation of condensed matter theorists in advanced concepts and techniques.