Interplay of symmetry and topology in gapped phases of condensed matter systems

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NON-TECHNICAL SUMMARYThis award supports theoretical research and education to identify and classify new phases of matter termed topological phases. The concept of a phase of matter allows one to account for drastically different physical properties in systems made out of the same underlying constituents, e.g. the liquid and solid ice forms of water. Solid state materials can also typically come in one of several phases - e.g. metal versus insulator, or magnetic versus non-magnetic - and much of their usefulness derives from exploiting the different physical properties characteristic of such phases. A theoretical framework for understanding the organization of constituents of materials into phases, "ordering" that materials undergo when they transform into a phase, was established over 50 years ago and had proven extremely successful until exotic new phases - called fractional quantum Hall effect states - were discovered. These states, found when electrons are confined to two dimensions and are placed in large perpendicular magnetic fields, did not fit into the old theoretical framework and possess striking new physical properties that currently go under the name of "topological order". This research is aimed to advance fundamental understanding of topological phases and how the classification of topologically ordered phases is enriched when additional symmetries are present. For example: topological insulators, which differ from ordinary insulators by the necessary existence of a metallic state at the surface, can exist because reversing the direction of time leads to a state that looks the same as the original one. This time reversal symmetry is not obeyed by a simple ferromagnet for which time reversal flips the direction of magnetism. The research will address important questions: What interesting properties can arise with other symmetries present, or with combinations of such symmetries? How robust are they? Are there any such phases that have no analogue in conventional solid-state materials? Can the quantum mechanical connectedness of spatially separated parts of such phases be exploited to store or manipulate quantum information for future information technology? The PI will investigate these questions using analytical techniques and novel mathematical methods.This award also supports education and outreach activities including: the development of a new one-semester course focused on topological order at the advanced graduate students level; the PI will present lectures related to the research area to high school students through a program organized at Stony Brook university; the PI will organize a once-a-month physics lecture series aimed at the broader community, which will bring in outside speakers to introduce the audience to cutting-edge research in this and related fields.TECHNICAL SUMMARYThis award supports theoretical research and education to identify and classify new phases of matter, termed topological phases, with a specific focus on how symmetries enrich possible topological phases. Although topological order is robust and independent of any global symmetries, the inclusion of symmetries can lead to additional phases, for example the recently discovered topological insulators, which are robust only when time reversal symmetry is present. The PI will use analytical techniques, including those from algebraic topology, to pursue several research directions:1) Classifying symmetry-enriched phases for physically realistic symmetry groups, e.g. continuous, anti-unitary, and spatial symmetries, generalizing results for finite discrete symmetry groups.2) Classifying both symmetry-enriched and symmetry-protected phases involving fundamental fermion degrees of freedom. This research will focus on the stability of such phases accessible through the band framework, as well as on the search for inherently strongly interacting phases.3) Constructing many-body invariants, which distinguish among the states in such a classification. In particular, the PI will explore the connection between non-trivial symmetry-protected phases and anomalies in one lower dimension.4) Constructing a unified mathematical framework, based on topological quantum field theory, to tackle the above classification problems.This award also supports education and outreach activities including: the development of a new one-semester course focused on topological order at the advanced graduate students level; the PI will present lectures related to the research area to high school students through a program organized at Stony Brook university; the PI will organize a once-a-month physics lecture series aimed at the broader community, which will bring in outside speakers to introduce the audience to cutting-edge research in this and related fields.