This 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 SUMMARY
This 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.
Report for the period: 09/01/2013 to 08/31/2014 In past year we have posted 15 research papers: 1. arXiv:1408.6514 Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy Tian Lan, Juven Wang, Xiao-Gang Wen 2. arXiv:1408.1676 $U(1) imes U(1)$ Symmetry Protected Topological Order in Gutzwiller Wave Functions Zheng-Xin Liu, Jia-Wei Mei, Peng Ye, Xiao-Gang Wen 3. arXiv:1407.0869 Design local spin models for Gutzwiller-projected parton wave functions Jia-Wei Mei, Xiao-Gang Wen 4. arXiv:1406.5090 Stochastic local transformations, emergence of unitarity, long-range entanglement, gapped quantum liquids, and topological order Bei Zeng, Xiao-Gang Wen 5. arXiv:1405.7689 Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond Juven Wang, Zheng-Cheng Gu, Xiao-Gang Wen 6. arXiv:1405.5858 Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions Liang Kong, Xiao-Gang Wen 7. arXiv:1404.7854 Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory Juven Wang, Xiao-Gang Wen 8. arXiv:1404.4618 Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations Heidar Moradi, Xiao-Gang Wen 9. arXiv:1404.2818 Realization of 2-Dimensional Bosonic Topological Insulators Zheng-Xin Liu, Zheng-Cheng Gu, Xiao-Gang Wen 10. arXiv:1403.5256 Bosonic Anomalies, Induced Fractional Quantum Numbers and Degenerate Zero Modes: the anomalous edge physics of Symmetry-Protected Topological States Juven Wang, Luiz H. Santos, Xiao-Gang Wen 11. arXiv:1401.5557 Modular Matrices as Topological Order Parameter by Gauge Symmetry Preserved Tensor Renormalization Approach Huan He, Heidar Moradi, Xiao-Gang Wen Phys. Rev. B 90, 205114 (2014) 12. arXiv:1401.0518 Universal Wave Function Overlap and Universal Topological Data from Generic Gapped Ground States Heidar Moradi, Xiao-Gang Wen 13. arXiv:1311.5539 Universal symmetry-protected topological invariants for symmetry-protected topological states Ling-Yan Hung, Xiao-Gang Wen Phys. Rev. B 89, 075121 (2014) 14. arXiv:1311.1784 Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models Tian Lan, Xiao-Gang Wen Phys. Rev. B 90, 115119 (2014) 15. arXiv:1309.7032 Lattice Model for Fermionic Toric Code Zheng-Cheng Gu, Zhenghan Wang, Xiao-Gang Wen Phys. Rev. B 90, 085140 (2014) The goal of the current project is to study the physical properties of highly entangled many-body states. Highly entangled many-body states can produce new phases of quantum matter, such as topologically ordered states and symmetry protected topological states, beyond Landau symmetry breaking theory. Those new states of matter have some amazing properties, such as perfect conducting boundary even in presence of impurities and quasiparticles with fractional/non-Abelian statistics and fractional charges. One of the important issues for new states of matter is to find a systematical way to label them. We know that all the symmetry breaking orders are label by a pair of groups (Gh,Gpsi), the symmetry group of system (Gh) and the symmetry group of the ground state (Gpsi). Our previous research suggests that all the 2+1D topological orders are labelled (R_SL(2,Z), c), where R_SL(2,Z) is an irreducible representation of SL(2,Z) and c is real number called central charge. In the paper "Universal Wave Function Overlap and Universal Topological Data from Generic Gapped Ground States", we proposed to use wave function overlap to measure R_SL(2,Z) -- an irreducible representation of SL(2,Z). Similar approach can also to used to label symmetry protected topological orders (see "Universal symmetry-protected topological invariants for symmetry-protected topological states"). In papers "Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory" and "Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations", we proposed to use the representations and SL(3,Z) to (partially) label the 3+1D topological orders. We also use the representations and SL(3,Z) to study a total new phenomena -- the braiding statistics of string-like excitations (such as vortex lines in superconductors). We find that, just like particles which can have different braiding statistics such as Bose statistics and Fermi statistics, strings can also have different braiding statistics and even non-Abelian braiding statistics.
