Pseudo-gauge Fields in Dirac and Weyl Materials

Electrons in low-temperature solids are governed by the non-relativistic Schr$\ddot{o}$dinger equation, since the electron velocities are much slower than the speed of light. Remarkably, the low-energy quasi-particles given by electrons in various materials can behave as relativistic Dirac/Weyl fermions that obey the relativistic Dirac/Weyl equation. These materials are called "Dirac/Weyl materials", which provide a tunable platform to test relativistic quantum phenomena in table-top experiments. More interestingly, different types of physical fields in these Weyl/Dirac materials, such as magnetic fluctuations, lattice vibration, strain, and material inhomogeneity, can couple to the "relativistic" quasi-particles in a similar way as the $U(1)$ gauge coupling. As these fields do not have gauge-invariant dynamics in general, we refer to them as "pseudo-gauge fields". In this chapter, we overview the concept and the physical consequences of pseudo-gauge fields in Weyl/Dirac materials. In particular, we will demonstrate that pseudo-gauge fields can provide a unified understanding of a variety of physical phenomena, including chiral zero modes inside a magnetic vortex core of magnetic Weyl semimetals, a giant current response at magnetic resonance in magnetic topological insulators, and piezo-electromagnetic response in time-reversal invariant systems. These phenomena are deeply related to various concepts in high-energy physics, such as chiral anomaly and axion electrodynamics.