# Topological Insulators and Topological Semimetals

Recently, several compounds have vaulted into the forefront of the condensed matter field after being named prime candidates for the physical realization of a new class of materials, known as topological insulators. A topological insulator has an energy gap in the bulk, however, unlike traditional insulators or semiconductors, topological insulators are topologically inequivalent to the vacuum. A fundamental consequence and principle experimental indicator of the non-trivial topology is one or more robust metallic surface states that are protected both from localization and backscattering by time-reversal symmetry. Beyond the inherent importance of exploring a complex phase of quantum matter, these systems are of great interest for device applications such as quantum computing and photonics.

With the discovery of Dirac and Weyl semimetals, the idea of topology has extended from insulators to semimetals. In **topological semimetals**, Dirac-like band crossings appear in momentum space and is protected by discrete symmetries. Topological semimetals has attracted great interest since they host quasiparticles that cannot occur in vacuum but do occur in solid-state systems. The nodal-line semimetals extend the point-degeneracies in Dirac semimetals into lines/rings of degeneracies in momentum space. These unusual band structure and topology can be captured by powerful power-law analysis of response functions, e.g. optical spectroscopy.

*Optical signatures of Dirac nodal lines in NbAs2*

**Shao et al. PNAS 116, 1168 (2019) Ref. [XXX]. **

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The 3D nodal points in Dirac and/or Weyl semimetals are in the vanguard of quantum materials research. A hallmark of these systems is the linear band dispersion (top left). This latter electronic structure gives rise to unconventional transport and optical phenomena. We demonstrate that solids with dispersive nodal lines in the electronic structure (bottom left) share many common aspects with the response of 3D nodal-points systems. We investigated NbAs2 using a combination of optical and magneto-optical techniques and have identified electromagnetic signature of dispersive nodal lines, see Shao et al, PNAS 116, 1168-1173 (2019). This particular compound has allowed us to inquire the impact of spin-orbit coupling on the universal characteristic of nodal metals. This work has been highlighted in PNAS cover, see also commentary article from Prof. Eugene J. Mele E. J. Mele, PNAS 116, 1084-1086 (2019)

*Faraday Rotation Due to Surface States in the Topological Insulator (Bi1–xSbx)2Te3s*

**Shao et al. Nano Letters 17, 980 (2017) Ref. [237]. **

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Using magneto-infrared spectroscopy, we have explored the charge dynamics of (Bi,Sb)2Te3 thin films on InP substrates. From the magneto-transmission data we extracted three distinct cyclotron resonance (CR) energies that are all apparent in the broad band Faraday rotation (FR) spectra. This comprehensive FR-CR data set has allowed us to isolate the response of the bulk states from the intrinsic surface states associated with both the top and bottom surfaces of the film. The FR data uncovered that electron- and hole-type Dirac Fermions reside on opposite surfaces of our films, which paves the way for observing many exotic quantum phenomena in topological insulators.

*Topological insulators are tunable waveguides for hyperbolic polaritons*

**Wu et al. Physical Review B 92 205430 (2015). Article [217]**

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*Sum-Rule Constraints on the Surface State Conductance of Topological Insulators*

**Post et al. Physical Review Letters 115, 116804 (2015). Article [212]**