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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. We refer to these materials as 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 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.
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Artificial gauge fields are currently realized in a wide range of physical settings. This includes solid-state devices but also engineered systems, such as photonic crystals, ultracold gases and mechanical setups. It is the aim of this review to offer, for the first time, a unified view on these various forms of artificial electromagnetic fields and spin-orbit couplings for matter and light. This topical review provides a general introduction to the universal concept of engineered gauge fields, in a form which is accessible to young researchers entering the field. Moreover, this work aims to connect different communities, by revealing explicit links between the diverse forms and realizations of artificial gauge fields.
Dirac and Weyl semimetals, materials where electrons behave as relativistic fermions, react to position- and time-dependent perturbations, such as strain, as if emergent electromagnetic fields were applied. Since they differ from external electromagnetic fields in their symmetries and phenomenology they are called pseudo-electromagnetic fields, and enable a simple and unified description of a variety of inhomogeneous systems involving topological semimetals. We review the different physical ways to create effective pseudo-fields, their observable consequences as well as their similarities and differences compared to electromagnetic fields. Among these difference is their effect on quantum anomalies, the absence of a classical symmetry in the quantum theory, which we revisit from a quantum field theory and a semiclassical viewpoint. We conclude with predicted observable signatures of the pseudo-fields and the nascent experimental status.
In planar tilted Dirac cone systems, the tilt parameter can be made space-dependent by either a perpendicular displacement field, or by chemical substitution in certain systems. We show that the symmetric partial derivative of the tilt parameter generates non-Abelian synthetic gauge fields in these systems. The small velocity limit of these gauge forces corresponds to Rashba and Dresselhaus spin-orbit couplings. At the classical level, the same symmetric spatial derivatives of tilt contribute to conservative, Lorentz-type and friction-like forces. The velocity dependent forces are odd with respect to tilt and therefore have opposite signs in the two valleys when the system is inversion symmetric. Furthermore, toggling the chemical potential between the valence and conduction bands reverses the sign of the all these classical forces, which indicates these forces couple to the electric charge of the carriers. As such, these forces are natural extensions of the electric and magnetic forces in the particular geometry of the tilted Dirac cone systems.
Within a Kubo formalism, we study dc transport and ac optical properties of 3D Dirac and Weyl semimetals. Emphasis is placed on the approach to charge neutrality and on the differences between Dirac and Weyl materials. At charge neutrality, the zero-temperature limit of the dc conductivity is not universal and also depends on the residual scattering model employed. However, the Lorenz number L retains its usual value L_0. With increasing temperature, the Wiedemann-Franz law is violated. At high temperatures, L exhibits a new plateau at a value dependent on the details of the scattering rate. Such details can also appear in the optical conductivity, both in the Drude response and interband background. In the clean limit, the interband background is linear in photon energy and always extrapolates to the origin. This background can be shifted to the right through the introduction of a massless gap. In this case, the extrapolation can cut the axis at a finite photon energy as is observed in some experiments. It is also of interest to differentiate between the two types of Weyl semimetals: those with broken time-reversal symmetry and those with broken spatial-inversion symmetry. We show that, while the former will follow the same behaviour as the 3D Dirac semimetals, for the zero magnetic field properties discussed here, the latter type will show a double step in the optical conductivity at finite doping and a single absorption edge at charge neutrality. The Drude conductivity is always finite in this case, even at charge neutrality.
In quantum field theory, we learn that fermions come in three varieties: Majorana, Weyl, and Dirac. Here we show that in solid state systems this classification is incomplete and find several additional types of crystal symmetry-protected free fermionic excitations . We exhaustively classify linear and quadratic 3-, 6- and 8- band crossings stabilized by space group symmetries in solid state systems with spin-orbit coupling and time-reversal symmetry. Several distinct types of fermions arise, differentiated by their degeneracies at and along high symmetry points, lines, and surfaces. Some notable consequences of these fermions are the presence of Fermi arcs in non-Weyl systems and the existence of Dirac lines. Ab-initio calculations identify a number of materials that realize these exotic fermions close to the Fermi level.
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