## Photonic topological insulators: Building topological states of matter

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The discovery of topological insulators -- materials which are conventional insulators in the bulk but support dissipationless, "topologically protected" edge states -- has revolutionized condensed matter physics in recent years. Indeed, topological insulators have been of interest to physicists as much for their unique physics as for their plethora of potential applications, which run the gamut from nano-scale electronic circuits to the realization of Majorana fermions and large-scale quantum computers. However, the main drawback of topological insulators is that they are currently difficult to produce experimentally, and only a handful of materials supporting the topological insulator state are known. The Shvets group recently proposed an analogue of the topological insulator state in photonic crystals. In contrast to the topological insulator state in conventional materials, in which we must simply take what nature gives us, in photonics we can literally build a topological insulator. In order for this photonic topological insulator state to occur, non-zero bianisotropy is introduced. Bianisotropy simply adds another coupling

This thesis seeks to extend the photonic topological insulator model originally proposed by Shvets et. al. by adding another important term from photonics, a so-called "magneto-optic" (MO) term. This term is produced by applying an external magnetic field to the crystal, which is interesting theoretically because magnetic-fields are not time-reversal symmetric. Thus this research has a twofold purpose: (1) to determine the effects of another important photonic property on the photonic topological insulator structure and potentially exploit those effects in novel applications, thereby \emph{building} topologically insulating structures, and (2) to investigate the role of time-reversal symmetry breaking in topological insulators via photonic crystals. For the photonic topological insulator structure, I derive an effective Dirac Hamiltonian that describes the two bands of the Dirac crossing. I show that this Hamiltonian, when no MO term is present, is identical to the famous Kane-Mele Hamiltonian that introduced the topological insulator state in conventional materials. Here, bianisotropy plays the role of spin-orbit coupling, with the states