Voigt points represent propagation directions in anisotropic crystals along which optical modes degenerate, leading to a single circularly polarized eigenmode. They are a particular class of exceptional points. Here, we report the fabrication and characterization of a dielectric, anisotropic optical microcavity based on nonpolar ZnO that implements a non-Hermitian system and mimicks the behavior of Voigt points in natural crystals. We prove the exceptional-point nature by monitoring the complex-square-root topology of the mode eigenenergies (real and imaginary parts) around the Voigt points. Polarization state analysis shows, that these artificially engineered Voigt points behave as vortex cores for the linear polarization and sustain chiral modes. Our findings apply to any planar microcavity with broken cylindrical symmetry and, thus, pave the way to exploiting exceptional points in widespread optoelectronic devices as VCSELs and RCLEDs.

The strong time-reversal symmetric (TRS) topological insulator (TI) in three space dimensions features gapless surface states in the form of massless Dirac fermions. We study these surface states with the method of bosonization, and find that the resulting bosonic theory has a topological contribution due to the parity anomaly of the surface Dirac fermions. We argue that the presence of a quantum anomaly is, in fact, the main reason for the existence of a surface state, by the principle that anomalies of a surface and bulk must cancel. Inspecting other classes of topological insulators, we argue that this principle holds in general. Moving beyond purely topological considerations, we incorporate the dynamics of the surface electron states into the bosonic theory. Additionally, we discuss the thermodynamics of the bosonic theory and propose a representation of the surface Dirac fermions in terms of the bosonic fields.

One of the hallmarks of time-reversal-symmetric topological insulators in three dimensions is the topological magnetoelectric effect (TME). So far, a time-reversal breaking variant of this effect has attracted much attention, in the sense that the induced electric charge changes sign when the direction of an externally applied magnetic field is reversed. Theoretically, this effect is described by the so-called axion term. Here, we discuss a time-reversal-symmetric TME, where the electric charge depends only on the magnitude of the magnetic field but is independent of its sign. We obtain this nonperturbative result both analytically and numerically, and suggest a mesoscopic setup to demonstrate it experimentally.Introduction. Time-reversal-symmetric (TRS) topological insulators (TIs) [1,2] are a fascinating class of electronic materials with insulating bulk and topologically protected surface states, which are either gapless, break a symmetry, or feature topological order [3]. Evidence for the existence of such surface states comes from spin textures observed in photoemission experiments [4,5] and from the observation of a half-integer quantum Hall effect [6][7][8][9].From a theoretical point of view, the hallmark response of TRS TIs in three dimensions (3D) is the topological magnetoelectric effect [10,11]. So far, a time-reversal (TR) breaking variant of this effect has attracted much attention. When TR is broken by, say, a magnetic coating with Zeeman coupling to the TI surface, a Hall conductivity σ xy =θ 2π e 2 2π arises, whereθ is quantized tõ θ = ±π. Then, the insertion of a magnetic flux tube gives rise to the accumulation of a charge |Q| = e/2 per flux quantum Φ 0 = h/e. Importantly, sgn(Q) depends on the direction of the magnetic field inside the flux tube, i.e., the response is not TRS. A consequence of the surface Hall conductivity is quantized Kerr and Faraday rotations [12,13], which have recently been confirmed experimentally [14][15][16].In the presence of TRS, the linear magnetoelectric response vanishes [17,18]. Thus, strictly speaking, all variants of the topological magnetoelectric effect are nonlinear effects, as they require an additional perturbation, say, a Zeeman coupling on the surface as above. The absence of a linear magnetoelectric response may seem to be at odds with the fact that the bulk of a 3D TRS TI has been characterized by the so-called axion action [10,11] Under a TR transformation, E → E, B → −B, and S θ → −S θ . Classically, this action breaks TRS, but quantum mechanically, only the Feynman amplitude e iS θ needs to be symmetric. If the electronic wave functions and electromagnetic fields satisfy periodic boundary conditions, one can show that this integral is quantized to integer multiples of 4π 2 /e 2 [19,20]. This implies that S θ = θ modulo 2π, hence θ = ±π would respect TRS also. Now, for a TI with boundaries, by using partial integration, S ±π can be converted into the surface quantum Hall term discussed above, and it seems naively that the axion action indeed describ...

We present a study of the transmission of electrons through a semiconductor quantum wire with strong spin-orbit coupling in proximity to an s-wave superconductor, which is Coulomb-blockaded. Such a system supports Majorana zero modes in the presence of an external magnetic field. Without superconductivity, phase lapses are expected to occur in the transmission phase, and we find that they disappear when a topological superconducting phase is stabilized. We express tunneling through the nanowire with the help of effective matrix elements, which depend on both the fermion parity of the wire and the overlap with Bogoliubov-de-Gennes wave functions. Using a modified scattering matrix formalism, that allows for including electron-electron interactions, we study the transmission phase in different regimes.

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