Interplay of Topological Order and Symmetry In and Out of Equilibrium

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NONTECHNICAL SUMMARYThis award supports theoretical research and education on exotic low temperature states of condensed matter systems. Although quantum mechanics describes the behavior of matter at the microscopic level, quantum mechanical effects are typically not visible at the macroscopic scale. In the 1980s a new class of so-called `topologically ordered' states was discovered in studying the quantum Hall effect in which electrons are confined in a plane between two semiconductors and placed in a high magnetic field perpendicular to the plane. The quantum Hall states show quantum effects that are felt at the macroscopic scale. These include the emergence of particle like behavior that carries only a fraction of the electron charge and channels which only allow charge to flow in one direction.To gain a better understanding of what kinds of new states of electrons can be realized in materials and how they can be detected experimentally, the PI and collaborators will combine methods from theoretical condensed matter physics and the field of quantum information theory which is comprised of ideas from information theory, quantum mechanics, mathematics, and computer science. Quantum information theory provides a useful way to think about aspect of computing and information transmission involving the manipulation of quantum mechanical states. This award also supports the education of a graduate student at the frontiers of modern theoretical condensed matter physics. Furthermore, the PI will continue to develop a graduate level course that incorporates new material from quantum information theory in addition to standard condensed matter field theory. In addition, the work may have an additional positive impact in that condensed matter ideas may potentially lead to advances in quantum information and quantum computation.TECHNICAL SUMMARYThis award supports theoretical research and education on the combined effects of symmetry and topology in strongly correlated many-body quantum systems. Such strongly correlated systems can realize zero-temperature phases of matter beyond the standard Ginzburg-Landau-Wilson symmetry breaking paradigm. These can have intrinsic topological order and support fractionalized `anyon' excitations, or they can be symmetry protected.A large portion of this project will be to classify phases of matter which exhibit intrinsic topological order, symmetry-protected features, or both. This will include both equilibrium zero temperature gapped quantum phases and many-body localized (MBL) out of equilibrium systems. Specific areas of focus include: 1) classifying fermionic symmetry protected phases and understanding the strongly correlated topological surface states they can exhibit, 2) extracting universal properties of topological and symmetry protected phases from commuting projector lattice Hamiltonians and understanding the conditions under which such commuting projector Hamiltonians exist, and 3) classifying topological phases in periodically driven (Floquet) MBL quantum systems. The PI will use analytical tools, such as quantum field theory and exactly solvable models, to prove the existence of gapped phases, and will also use mathematical methods, such as topological quantum field theory and algebraic topology, to study and potentially rule out putative patterns of topological order. The PI will develop new quantum-information theoretic tools to address these problems.This project has the potential to develop fundamental understanding of topological order in and out of equilibrium. In particular, the problem of classifying topological order in non-equilibrium MBL systems is not necessarily amenable to the same field theory techniques as is the equilibrium case, but can be usefully addressed with quantum information theory methods. The PI will build on ongoing work with collaborators within the quantum information community to develop such methods, which recent results indicate will be useful for both the physics and quantum information communities.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.