## Photonic topological insulators: Building topological states of matter

##### Abstract

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 $\chi$ to the constitutive relations for the crystal: $\v{D} = \hat\epsilon \cdot \v{E} + \hat\chi \cdot \v{H}$, $\v{B} = \hat{\chi}^\dagger \cdot \v{E}+\hat\mu \cdot \v{H}$. A photonic crystal composed of a hexagonal lattice of rods has a Dirac point crossing in its band structure at the K-point, and in the presence of a non-zero bianisotropy, this Dirac point becomes gapped, supporting topologically protected edge states.
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 $\Psi^+=E_z + H_z$ and $\Psi^-= E_z - H_z$ playing the roles of spin-up and spin-down, respectively. I demonstrate that these states are immune to disorder -- a consequence of their topological protection -- by calculating the Chern numbers of the edge states, calculating their band diagrams, and by launching these states in simulations and showing that they propagate one-way through various obstacles with no backscattering. I also calculate the new terms added by the time-reversal-symmetry-breaking MO term and calculate analytically what becomes of the system's eigenstates. As well, a weak form of Maxwell's equations is derived for use in finite-element-method simulations in COMSOL, then implemented for band-structure calculations. I find that MO adds a second mass term to the Dirac Hamiltonian which has no natural analogue in conventional topological insulators, as this mass term is not time-reversal invariant. The system's eigenstates are shown to be an admixture of spin-up and spin-down, which I refer to as $\Phi^+$ and $\Phi^-$, which become $\Psi^+$ and $\Psi^-$ in the limit of bianisotropy being much stronger than MO and become just $E_z$ and $H_z$ (TE and TM modes) in the limit of MO being much stronger than bianisotropy. MO's effect on the band structure is shown to be that it splits the gap caused by bianisotropy alone, dividing it into two gaps, one larger than the other. As the magnitude of the MO term increases, one of the gaps becomes smaller indefinitely while the other grows, a consistent result with previous literature in photonics which predicts that the MO term alone only opens \emph{one} gap (only TM waves are affected by the MO term). Several types of interfaces are explored in order to investigate the edge states in the presence of MO and bianisotropy. Remarkably, it is found that if the ratio of the MO term and the bianisotropy is kept constant across two interfacing photonic crystals, then the edge states survive. More remarkable still, they retain their topological protection, demonstrated via propagating $\Phi^\pm$ through disordered interfaces. This result is truly unexpected, as the topological protection of the edge states of topological insulators is fundamentally dependent on the system being time-reversal invariant.