Anisotropic magnetoresistance is the change tendency of resistance of a material on the mutual orientation of the electric current and the external magnetic field. Here, we report experimental observations in the Dirac semimetal ${\mathrm{Cd}}_3{\mathrm{As}}_2$ of giant anisotropic magnetoresistance and its transverse version, called the planar Hall effect. The relative anisotropic magnetoresistance is negative and up to −68% at 2 K and 10 T. The high anisotropy and the minus sign in this isotropic and nonmagnetic material are attributed to a field-dependent current along the magnetic field, which may be induced by the Berry curvature of the band structure. This observation not only reveals unusual physical phenomena in Weyl and Dirac semimetals, but also finds additional transport signatures of Weyl and Dirac fermions other than negative magnetoresistance.

Half-quantized Hall conductance is characteristic of quantum systems with the parity anomaly. Here, we investigate topological and transport properties of a class of parity anomalous semimetals, in which massive Dirac fermions coexist with massless Dirac fermions in momentum space or real space, and uncover a distinct bulk-edge correspondence in which the half-quantized Hall effect is realized via the bulk massless Dirac fermions while the nontrivial Berry curvature is provided by the massive Dirac fermions. The spatial distribution of the edge current decays away from the boundary according to a power law instead of an exponential law in the integer quantum Hall effect. We further address the physical relevance of parity anomalous semimetals to three-dimensional semimagnetic topological insulators and two-dimensional photonic crystals.

The topological states of matter and topological materials have been attracting extensive interests as one of the frontier topics in condensed matter physics and materials science since the discovery of quantum Hall effect in 1980s. So far all the topological phases such as integer quantum Hall effect and topological insulators are characterized by integer topological invariants. None is a half integer or fractional. Here we propose a type of semimetals which hosts a single cone of Wilson fermions. The Wilson fermions possess linear dispersion near the Dirac point, but break the chiral or parity symmetry such that an unpaired Dirac cone can be realized on a lattice. In order to avoid the fermion doubling problem, the chiral symmetry or parity symmetry must be broken explicitly if the hermiticity, locality and translational invariance all hold. We find that the system can be classified by the relative homotopy group, and a half-integer topological invariant. We term the nontrivial quantum phase as quantum anomalous semimetal. The work opens the door towards exploring novel states of matter with fractional topological charge.

Condensed matter realization of a single Dirac cone of fermions in two dimensions is a long-standing issue. Here we report the discovery of a single gapless Dirac cone of half-quantized Hall conductance in a magnetically doped topological insulator heterostructure. It demonstrates that the Hall conductance is half-quantized in the unit $e^2/h$ when the parity symmetry is preserved near the Fermi surface. The gapless Dirac point is stable and protected by the local parity symmetry and the topologically nontrivial band structure of the topological insulator. The one-half Hall conductance observed in a recent experiment [M. Mogi et al., Nat. Phys. 18, 390 (2022)] is attributed to the existence of the gapless Dirac cone and the parity invariance of the Fermi surface. The results suggest a condensed matter realization of a topological phase with a one-half topological invariant.

Resistivity anomaly, a sharp peak of resistivity at finite temperatures, in the transition-metal pentatellurides ${\text{ZrTe}}_5$ and ${\text{HfTe}}_5$ was observed four decades ago, and more exotic and anomalous behaviors of electric and thermoelectric transport were revealed in recent years. Here, we present a theory of Dirac polarons, composed by massive Dirac electrons and holes in an encircling cloud of lattice displacements or phonons at finite temperatures. The chemical potential of Dirac polarons sweeps the band gap of the topological band structure by increasing the temperature, leading to the resistivity anomaly. Formation of a nearly neutral state of Dirac polarons accounts for the anomalous behaviors of the electric and thermoelectric resistivity around the peak of resistivity.