Quantum Anomalies in Matter
Initially, quantum anomalies have been theoretically prediceted in the context of elementary-particle quantum-field theory. In particular, the chiral anomaly (also called Adler-Bell-Jackiw anomaly) arises when the mass of a spin-1/2 Dirac fermion is exactly zero, in which case the Dirac fermion ``falls apart'' into two Weyl fermions of opposite chirality. Since the Weyl fermions appear completely decoupled in the action, classically one would expect conservation of particle numbers of each Weyl fermion species separately. On the quantum-mechanical level however, the two species turn out to be coupled by the electromagnetic field, which application allows to change the relative particle number. Such an anomaly causes interesting and potentially very useful phenomena, such as a strongly increasing conductivity in an applied magnetic field and Fermi-arc surface states at the boundary of the anomaly-hosting space.
So far the chiral anomaly has not been experimentally observed in the context of elementary particles, all observed fermions are massive and thus don't fall apart into Weyl fermions. In condensed matter, however, some crystals allow to realize Weyl fermions as topologically protected quasiparticles (-> review paper). These so-called Weyl semimetals have been experimentally discovered in 2015 (-> discovery paper), which triggered an enormous research activity to understand the chiral anomaly in the condensed-matter context and to explore all its consequences on a fundamental level as well as with regard to novel technological applications.