Reflected entropy and Markov gap in Lifshitz theories

We study the reflected entropy in $(1+1)$-dimensional Lifshitz field theory whose groundstate is described by a quantum mechanical model. Starting from tripartite Lifshitz groundstates, both critical and gapped, we derive explicit formulas for the R\'enyi reflected entropies reduced to two adjacent or disjoint intervals, directly in the continuum. We show that the reflected entropy in Lifshitz theory does not satisfy monotonicity, in contrast to what is observed for free relativistic fields. We analytically compute the full reflected entanglement spectrum for two disjoint intervals, finding a discrete set of eigenvalues which is that of a thermal density matrix. Furthermore, we investigate the Markov gap, defined as the difference between reflected entropy and mutual information, and find it to be nonvanishing, signaling irreducible tripartite entanglement in Lifshitz groundstates. As a byproduct of our results on reflected entropy, we provide exact formulas for two other entanglement-related quantities, namely the computable cross-norm negativity and the operator entanglement entropy.Comment: 34 pages, 12 figure

TiO2 Nanocrystals Grown on Graphene as Advanced Photocatalytic Hybrid Materials

Graphene/TiO2 nanocrystals hybrid is successfully prepared by directly growing TiO2 nanocrystals on graphene oxide (GO) sheets. The direct growth of nanocrystals on GO sheets was achieved by a two-step method, in which TiO2 was coated on GO sheets by hydrolysis first and crystallized into anatase nanocrystals by hydrothermal treatment in second step. Slow hydrolysis reaction through the use of EtOH/H2O mixed solvents and addition of H2SO4 allows the selectively growing TiO2 on GO and suppressing free growth in solution. The method offers easy access to the GO/TiO2 nanocrystals hybrid with well controlled coating and strong interactions between TiO2 and the underlying GO sheets. The strong coupling could lead to advanced hybrid materials for various applications including photocatalysis. The prepared graphene/TiO2 nanocrystals hybrid has demonstrated superior photocatalytic activity in degradation of rhodamine B over other TiO2 materials, showing an impressive 3-fold photocatalytic enhancement over P25. It is expected that the hybrid material could also be promising for various other applications including lithium ion battery where strong electrical coupling to TiO2 nanoparticles is essential.Comment: Nano Research, in pres

Freestanding dielectric nanohole array metasurface for mid-infrared wavelength applications

We designed and simulated freestanding dielectric optical metasurfaces based on arrays of etched nanoholes in a silicon membrane. We showed $2\pi$ phase control and high forward transmission at mid-infrared wavelengths by tuning the dimensions of the holes. We also identified the mechanisms responsible for high forward scattering efficiency and showed that these conditions are connected with the well-known Kerker conditions already proposed for isolated scatterers. A beam deflector was designed and optimized through sequential particle swarm and gradient descent optimization to maximize transmission efficiency and reduce unwanted grating orders. Such freestanding silicon nanohole array metasurfaces are promising for the realization of silicon based mid-infrared optical elements