Nonlinear waves in deformed optical honeycomb lattices are investigated. Discrete couple mode equations are used to find higher order continuous nonlinear Dirac systems which are employed to describe key underlying phenomena. For weak deformation and nonlinearity the wave propagation is circular–ellliptical. At strong nonlinearity the diffraction pattern becomes triangular in structure which is traced to appropriate nonequal energy propagation in momentum space. At suitably large deformation the dispersion structure can have nearby Dirac points or small gaps. The effective dynamics of the wave packets is described by two maximally balanced nonlocal nonlinear Schrödinger type equations.

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We give a transparent algebraic formulation of our pictorial approach to the reflection positivity (RP), that we introduced in a previous paper. We apply this quantization to the 2+1 Levin–Wen model to obtain 1+1 anyonic/quantum spin chain theory on the boundary, possibly entangled in the bulk. The reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin–Wen models with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems.

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No abstract uploaded!