Adding a small subdominant quadratic in momentum term to a dominant linear Dirac dispersion curve affects conduction and valence band differently and leads to an hourglass-like structure for energy as a function of momentum. This applies to the protected surface states in topological insulators. The energies of the conduction and valence band Landau levels are also different and this leads to the splitting of optical absorption lines produced by the magnetic field, which acquire a two peak structure. It also changes the peaks in the imaginary part of the Hall conductivity into two distinct contributions of opposite signs. The real part of the circularly polarized optical conductivity however retains its single peak structure but the peaks in right and left handedness case are shifted in energy with respect to each other in contrast to the pure Dirac case. The magnitude of the semiclassical cyclotron frequency is significantly modified by the presence of a mass term as is its variation with value of the chemical potential $\mu$. Its optical spectral weight is found to decrease with increasing $\mu$ rather than increase as it does in the pure Dirac limit.Comment: 11 pages, 8 figures, to appear in PR

A monolayer of M oS2 has a non-centrosymmetric crystal structure, with spin polarized bands. It is a two valley semiconductor with direct gap falling in the visible range of the electromagnetic spectrum. Its optical properties are of particular interest in relation to valleytronics and possible device applications. We study the longitudinal and the transverse Hall dynamical conductivity which is decomposed into charge, spin and valley contributions. Circular polarized light associated with each of the two valleys separately is considered and results are filtered according to spin polarization. Temperature can greatly change the spin admixture seen in the frequency window where they are not closely in balance.

The helical Dirac states on the surface of a topological insulator are protected by topology and display significant particle-hole asymmetry. This asymmetry arises from a subdominant Schr\"{o}dinger type contribution to the Hamiltonian which provides a small perturbation to a dominant Dirac contribution. This changes the Landau levels energies in an external magnetic field ($B$) and provides modifications to the usual relativistic optical matrix elements. Nevertheless we find that the relativistic quantization of the Hall plateaux remains even when the ratio of the Schr\"{o}dinger ($E_0$) to Dirac ($E_1$) magnetic energy scale increases either through an increase in $B$, a decrease in the Schr\"{o}dinger mass or of the Dirac fermi velocity. First corrections to the optical matrix elements(OME) in the relativistic case drop out at least to order $(E_0/E_1)^3$. In the opposite limit $E_1$ small, the quantization remains classical but there is a split into two series. The first corrections to the OME in this case, cancel out at least to order $(E_1/E_0)^4$.Comment: 10 pages, 7 figures, to appear in Phys. Rev.

ARPES studies of the protected surface states in the Topological Insulator Bi2T e3 have revealed the existence of an important hexagonal warping term in its electronic band structure. This term distorts the shape of the Dirac cone from a circle at low energies to a snowflake shape at higher energies. We show that this implies important modifications of the interband optical transitions which no longer provide a constant universal background as seen in graphene. Rather the conductivity shows a quasilinear increase with a slightly concave upward bending as energy is increased. Its slope increases with increasing magnitude of the hexagonal distortion as does the magnitude of the jump at the interband onset. The energy dependence of the density of states is also modified and deviates downward from linear with increasing energy.

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