Pseudospin induced chirality with Staggered Optical Graphene

Pseudospin plays a very important role in understanding various interesting physical phenomena associated with 2D materials such as graphene. It has been proposed that pseudospin is directly related to angular momentum, and it was recently experimentally demonstrated that orbit angular momentum is an intrinsic property of pseudospin in a photonic honeycomb lattice. However, in photonics, the interaction between spin and pseudospin for light has never been investigated. In this Letter, we propose that, in an optical analogue of staggered graphene, i.e. a photonic honeycomb lattice waveguide with in-plane inversion symmetry breaking, the pseudospin mode can strongly couple to the spin of an optical beam incident along certain directions. The spin-pseudospin coupling, caused by the spin-orbit conversion in the scattering process, induces a strong optical chiral effect for the transmitted optical beam. Spin-pseudospin coupling of light opens door to the design of pseudospin-mediated spin or valley selective photonic devices

Stability of Gorenstein flat categories with respect to a semidualizing module

In this paper, we first introduce $\mathcal {W}_F$-Gorenstein modules to establish the following Foxby equivalence: \xymatrix@C=80pt{\mathcal {G}(\mathcal {F})\cap \mathcal {A}_C(R) \ar@[r]^{C\otimes_R-} & \mathcal {G}(\mathcal {W}_F) \ar@[l]^{\textrm{Hom}_R(C,-)}} where $\mathcal {G}(\mathcal {F})$, $\mathcal {A}_C(R)$ and $\mathcal {G}(\mathcal {W}_F)$ denote the class of Gorenstein flat modules, the Auslander class and the class of $\mathcal {W}_F$-Gorenstein modules respectively. Then, we investigate two-degree $\mathcal {W}_F$-Gorenstein modules. An $R$-module $M$ is said to be two-degree $\mathcal {W}_F$-Gorenstein if there exists an exact sequence \mathbb{G}_\bullet=\indent ...\longrightarrow G_1\longrightarrow G_0\longrightarrow G^0\longrightarrow G^1\longrightarrow... in $\mathcal {G}(\mathcal {W}_F)$ such that $M \cong$ \im(G_0\rightarrow G^0) and that $\mathbb{G}_\bullet$ is Hom$_R(\mathcal {G}(\mathcal {W}_F),-)$ and $\mathcal {G}(\mathcal {W}_F)^+\otimes_R-$ exact. We show that two notions of the two-degree $\mathcal {W}_F$-Gorenstein and the $\mathcal {W}_F$-Gorenstein modules coincide when R is a commutative GF-closed ring.Comment: 18 page

KK-theory of two-dimensional substitution tiling spaces from AFAF-algebras

Given a two-dimensional substitution tiling space, we show that, under some reasonable assumptions, the $K$-theory of the groupoid $C^\ast$-algebra of its unstable groupoid can be explicitly reconstructed from the $K$-theory of the $AF$-algebras of the substitution rule and its analogue on the $1$-skeleton. We prove this by generalizing the calculations done for the chair tiling in [JS16] using relative $K$-theory and excision, and packaging the result into an exact sequence purely in topology. From this exact sequence, it appears that one cannot use only ordinary $K$-theory to compute using the dimension-filtration on the unstable groupoid. Several examples are computed using Sage and the results are compiled in a table