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We review recent discussions regarding the parity-odd contribution to the trace anomaly of a chiral fermion. We pay special attention to the perturbative approach in terms of Feynman diagrams, comparing in detail the results obtained using dimensional regularization and the Breitenlohner–Maison prescription with other approaches.

The hallmark of Weyl semimetals is the existence of open constant-energy contours on their surface - the so-called Fermi arcs - connecting Weyl points. Here, we show that for time-reversal symmetric realizations of Weyl semimetals these Fermi arcs in many cases coexist with closed Fermi pockets originating from surface Dirac cones pinned to time-reversal invariant momenta. The existence of Fermi pockets is required for certain Fermi-arc connectivities due to additional restrictions imposed by the six $\mathbb{Z}_2$ topological invariants characterizing a generic time-reversal invariant Weyl semimetal. We show that a change of the Fermi-arc connectivity generally leads to a different topology of the surface Fermi surface, and identify the half-Heusler compound LaPtBi under in-plane compressive strain as a material that realizes this surface Lifshitz transition. We also discuss universal features of this coexistence in quasi-particle interference spectra.

We propose a different route to time-reversal invariant Weyl semimetals employing multilayer heterostructures comprising ordinary "trivial" insulators and nontrivial insulators with \textit{pairs} of protected Dirac cones on the surface. We consider both the case of weak topological insualtors, where surface Dirac cones are pinned to time-reversal invariant momenta, and of topological crystalline insulators with unpinned surface Dirac cones. For both realizations we explain phenomenologically how the proposed construction leads to the emergence of a Weyl semimetal phase. We further formulate effective low-energy models for which we prove the existence of semimetallic phases with four isolated Weyl points. Finally, we discuss how the proposed design can be realized experimentally with state-of-the-art technologies.

Three-dimensional (3D) Dirac semimetals are 3D analogues of graphene, which display Dirac points with linear dispersion in k-space, stabilized by crystal symmetry. Cd3As2 has been predicted to be 3D Dirac semimetals and was subsequently demonstrated by angle-resolved photoemission spectroscopy. As unveiled by transport measurements, several exotic phases, such as Weyl semimetals, topological insulators, and topological superconductors, can be deduced by breaking time reversal or inversion symmetry. Here, we reported a facile and scalable chemical vapor deposition method to fabricate high-quality Dirac semimetal Cd3As2 microbelts; they have shown ultrahigh mobility up to 1.15 × 105 cm2 V−1 s−1 and pronounced Shubnikov–de Haas oscillations. Such extraordinary features are attributed to the suppression of electron backscattering. This research opens a new avenue for the scalable fabrication of Cd3As2 materials toward exciting electronic applications of 3D Dirac semimetals.