We classify all possible singularities in the electronic dispersion of two-dimensional systems that occur when the Fermi surface changes topology, using catastrophe theory. For systems with up to seven control parameters (i.e., pressure, strain, bias voltage, etc), the theory guarantees that the singularity belongs to to one of seventeen standard types. We show that at each of these singularities the density of states diverges as a power law, with a universal exponent characteristic of the particular catastrophe, and we provide its universal ratio of amplitudes of the prefactors of energies above and below the singularity. We further show that crystal symmetry restricts which types of catastrophes can occur at the points of high symmetry in the Brillouin zone. For each of the seventeen wallpaper groups in two-dimensions, we list which catastrophes are possible at each high symmetry point.

We study systems with energy bands in two dimensions, hosting higher order Van Hove singularities (HOVHS) in the presence of disorder, using standard diagrammatic techniques for impurity averaging. In the clean limit, such singularities cause power-law divergence in the density of states (DOS), and this is expected to strongly affect electronic correlation. In order to analyse the signatures of these singularities in disordered systems, we employ various Born approximations, culminating in the self-consistent (non) Born approximation. Although the divergence of the DOS is smeared, we find that the shape of the DOS, as characterized by the power law tail and the universal ratio of prefactors, is retained slightly away from the singularity. This could help us to understand current and future experiments on materials that can be tuned to host HOVHS. The impurity induced smearing is calculated and analysed for several test cases of singularities. We also study the effects of impurities on electrical conductivity and determine the regimes where the quantitative features of the power law DOS manifest in the conductivity.

A practical method is presented to detect, diagnose, and engineer higher order Van Hove singularities in multiband systems, with no restrictions on the number of bands or hopping terms. The method allows us to directly compute the Taylor expansion of the dispersion of any band at arbitrary points in momentum space, using a generalized extension of the Feynman-Hellmann theorem, which is stated and proved. Being fairly, in general scope, it also allows to incorporate and analyze the effect of tuning parameters on the low energy dispersions, which can greatly aid the engineering of higher order Van Hove singularities. A certain class of degenerate bands can be handled within this framework. The use of this method is demonstrated, by applying it to the Haldane model.

We present a practical method to detect and diagnose higher order Van Hove singularities in multiband systems, with no restrictions on number of bands and hopping terms. The method allows us to directly compute the Taylor expansion of the dispersion of any band at arbitrary k points, using an extension of the Feynman Hellmann theorem. Being fairly general in scope, it also allows us to incorporate and analyse the effect of tuning parameters on the low energy dispersions, which can greatly aid the search for higher order Van Hove singularities. A restricted class of degenerate bands can be handled within this framework, which can be readily computationally implemented. Subsequently, we demonstrate the use of the method, taking the Haldane model as an example. Contents

The energy spectrum of topological semimetals contains protected degeneracies in reciprocal space that correspond to Weyl, Dirac, or multifold fermionic states. To exploit the unconventional properties of these states, one has to access the electronic structure of the three-dimensional bulk. In this work, we resolve the bulk electronic states of candidate topological semimetal FeSi using momentum-dependent resonant inelastic X-ray scattering (RIXS) at the Fe L 3 edge. We observe a broad excitation continuum devoid of sharp features, consistent with particle-hole scattering in an underlying electronic band structure. Using density functional theory, we calculate the electronic structure of FeSi and derive a band theory formulation of RIXS in the fast collision approximation to model the scattering process. We find that band theory qualitatively captures the number and position of the main spectral features, as well as the overall momentum dependence of the RIXS intensity. Our work paves the way for targeted studies of band touchings in topological semimetals with RIXS.