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Composite-particle approach to Symmetry Protected Topological Phases

Periodic Reporting for period 1 - CompositeSPTphases (Composite-particle approach to Symmetry Protected Topological Phases)

Reporting period: 2015-04-01 to 2017-03-31

The sum being greater than its parts is a common theme in condensed matter physics. Materials made of large numbers of simple constituents often exhibit intriguing and markedly distinct phases of matter with properties very different from any of the individual constituents. Understanding the possible phases of matter and identifying them in real materials is the central focus of this branch of physics. Roughly speaking, two categories of phases of matter exist— conventional phases which show a geometrical pattern of order, and topological phases, where the order is more elusive and related to topological concepts. In the past three decades, topological phases have attracted a large amount of interest due to their tendency to exhibit highly robust quantum phenomena which have various applications in quantum engineering and metrology.

The current frontier in the field aims at understanding the variety of novel topological phases which arise when some extra symmetries, such as time reversal, are not allowed to be broken. In this project we explore these new type of phases using the concept of composite particles — an idea which has been extremely useful in previous studies of topological matter, but has not been applied in the symmetry protected context previously. The fundamental idea of composite particles is to view symmetry protected topological (SPT) phases as conventional or classical phases of composite objects. Besides its conceptual importance, such an approach will allow us to utilize our knowledge of conventional phases in the context of SPT phases and also derive microscopic models which realize these states of matter. It will thus increase the chance of discovering new SPT phases in nature. Such materials with novel properties may, in due time, lead to various technological advancements in the field of quantum computation and spintronics.
During this project we have provided several advancements in the field.

(I) We have developed a comprehensive picture of two dimensional SPTs using the notions of composite particles.

(II) We showed how to employ a powerful analytical technique (conformal field theory) to analyze various properties of these phases.

(III) We provided an exact mathematical framework to discuss composite particles on lattices rather than on a continuum as is standard.

(IV) We proposed that a new type classical phases of matter exists in classical magnetic systems whose phenomenology mimics that of SPTs. We have provided several classical models which exhibit such phenomena. This paves the way to realizing novel types of topological phases and provides an efficient numerical way of simulating SPTs numerically.

(V) We addressed an important question which arose from our work: Are there topological phenomena which cannot have a classical analog and, as a result, are always difficult to simulate with a classical computer? We found an intriguing results that phases that have particular responses to a change in their geometry forbid a classical description.
All aforementioned works have advanced the state of the art in the field. Two theoretical results we have obtained are likely to have a strong impact on the field in due time.

The first result, concerns our discovery classical mechanical systems which realize topological phenomena which was previously thought to be inherently quantum. Although these are currently only theoretical models, there is ongoing work by us and collaborators in ETH aimed at realizing such model in the lab. Since these are classical models, this should be far easier than realizing their quantum counterparts. Nonetheless their exotic behavior, which include stable critical behavior on surface, has never been witnessed before. New an exotic physical phenomena often find their way to technological applications.

The second result, concerns the relation we exposed between geometric responses--- motion of charge/energy following bends and twists in the geometry, and computation complexity. Roughly speaking, we showed that the way some phases of matter react when coupled to gravity implies that their equilibrium properties cannot be efficiently calculated using a classical computer. Connecting these two very different aspects of a physical system - its computation complexity and its response to gravity, may very well be of fundamental importance. As this is fundamental research it is premature to talk about specific technological implications.

From a much broader point of view this projects deals with an ubiquitous phenomena--- the fact that many simple interacting components, in our case spins or electrons, often lead to strange and diverse behaviors. Our world is becoming more advance and complex exactly from those reasons: Many brokers interacting in the Stock market, many artificial neurons interacting in artificial intelligence architectures, and many different processes interacting to create our ecological systems. Condensed matter physics, while not dealing directly with such real world scenario, has provided important conceptual tools to tackle such complex problems and will most likely do so also in the future.
A conceptual illustration of a three dimensional classical topological paramagnet.
Adding boundaries, torsion, and curvature to a classical lattice model.
Space-time twist in a material exhibiting a global gravitational anomaly