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.

Negative longitudinal and positive in-plane transverse magnetoresistance have been observed in most topological Dirac/Weyl semimetals and some other topological materials. Here, we present a quantum theory of intrinsic magnetoresistance for three-dimensional Dirac fermions at a finite and uniform magnetic field $B$. In a semiclassical regime, it is shown that the longitudinal magnetoresistance is negative and quadratic of a weak field $B$ while the in-plane transverse magnetoresistance is positive and quadratic of $B$. The relative magnetoresistance is inversely quartic of the Fermi wave vector and only determined by carrier density, irrelevant to the external scatterings in the weak scattering limit. This intrinsic anisotropic magnetoresistance is measurable in systems with low carrier density and high mobility. In the quantum oscillation regime a formula for the phase shift in Shubnikov–de Haas oscillation is present as a function of the mobility and the magnetic field, which is helpful for experimental data analysis.

Magnetoresistance in many samples of Dirac semimetals and topological insulators displays nonmonotonic behavior over a wide range of magnetic fields. Here a formula of magnetoconductivity is presented for massless and massive Dirac fermions in Dirac materials due to quantum interference of Dirac fermions in scalar impurity scattering potentials. It reveals a striking crossover from positive to negative magnetoresistivity, uncovering strong competition between weak localization and weak antilocalization in multiple Cooperon channels at different chemical potentials, effective masses, and finite temperatures. This work sheds light on the important role of strong coupling of the conduction and valence bands in the quantum interference transport in topological nontrivial and trivial Dirac materials.

The chiral hinge modes are the key feature of a second-order topological insulator in three dimensions. Here we propose a quadrupole index in combination of a slab Chern number in the bulk to characterize the flowing pattern of chiral hinge modes along the hinges at the intersection of the surfaces of a sample. We further utilize the topological field theory to present a picture of three-dimensional quantum anomalous Hall effect as a consequence of chiral hinge modes. The two bulk topological invariants can be measured in electric transport and magneto-optical experiments. In this way we establish the bulk-hinge correspondence in a three-dimensional second-order topological insulator.

Exploiting the enabling power of the Lanczos method in momentum space, we determine accurately the quasiparticle and scaling properties of disordered three-dimensional Dirac semimetals surrounding the quantum critical point separating the semimetal and diffusive metal regimes. We unveil that the imaginary part of the quasiparticle self-energy obeys a common power law before, at, and after the quantum phase transition, but the power law is nonuniversal, whose exponent is dependent on the disorder strength. More intriguingly, whereas a common power law is also found for the real part of the self-energy before and after the phase transition, a distinctly different behavior is identified at the critical point, characterized by the existence of a nonanalytic logarithmic singularity. This nonanalytical correction serves as the very basis for the unusual power-law behaviors of the quasiparticles and many other physical properties surrounding the quantum critical point. Our approach also allows the ready and reliable determination of the scaling properties of the correlation length and dynamical exponents. We further show that the central findings are valid for both uncorrelated and correlated disorder distributions and should be directly comparable with future experimental observations.