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Topological materials usually possess protected gapless states in either the boundary or bulk, exhibiting various properties such as spin-momentum locking, Klein tunneling, Fermi arcs and so on. Database searches using symmetry data at high-symmetry points have catalogued thousands of topological materials revealing a magnitude of band nodes (BNs) at high-symmetry points or lying within high-symmetry lines/planes. A complete mapping from symmetry data (namely, representation of little group) in any BN to the k ⋅ p model characterizing low-energy Hamiltonian around the BN (and from the k ⋅ p model to concrete BN, inversely), is expected to complete the characterization of all BNs and gapless states. Here we first review recent progress on classifying BNs by systematically and automatically constructing k ⋅ p models based on recently completed tabulation of all irreducible (co-)representation matrices of little groups of the 1651 magnetic space groups. As one indispensable input in constructing a symmetry-allowed and generic k ⋅ p model, the expansion order, has been carefully and systematically truncated for any BN to a reasonable nonzero integer, by comparing the emanating nodal structure (ENS, including nodal point, nodal line and nodal surface) near the BN obtained by the explicitly constructed k ⋅ p model and that by pure symmetry analysis using compatibility relations (CRs). Owing to the progress, we are able to summarize all 25 different configurations of ENS near BN required by CRs, provide a complete mapping from k ⋅ p model to its realization around BN, and the corresponding ENS by CRs in an accessible file, and also reveal the protection mechanism of additional nodal lines that escape conventional analysis by CRs and is only predictable by constructing k ⋅ p model. The symmetry-based classification results on all BNs could facilitate large-scale materials prediction and hold promise for realizing topological semimetals suitable for device applications. [ABSTRACT FROM AUTHOR]

In light of the significance of non-commutative quaternionic algebra in modern physics, this study proposes the existence of the Klein paradox in the quaternionic (3+1)-dimensional space-time structure. By introducing quaternionic wave function, we rewrite the Klein–Gordon equation in extended quaternionic form that includes scalar and the vector fields. Because quaternionic fields are non-commutative, the quaternionic Klein–Gordon equation provides three separate sets of the probability density and probability current density of relativistic particles. We explore the significance of these probability densities by determining the reflection and transmission coefficients for the quaternionic relativistic step potential. Furthermore, we also discuss the quaternionic version of the oscillatory, tunnelling, and Klein zones for the quaternionic step potential. The Klein paradox occurs only in the Klein zone when the impacted particle's kinetic energy is less than 0 − m 0 c 2 . Therefore, it is emphasized that for the quaternionic Klein paradox, the quaternionic reflection coefficient becomes exclusively higher than value one while the quaternionic transmission coefficient becomes lower than zero. [ABSTRACT FROM AUTHOR]

This paper focuses on additional inspections concerning the fermionic sector of the Standard Model Extension (SME). In this context, our main effort in this contribution is to investigate effects of Lorentz-symmetry violation (LSV) on the Klein Paradox, the Zitterbewegung and its phenomenology in connection to Condensed Matter Physics, Atomic Physics, and Astrophysics. Finally, we discuss a particular realization of LSV in the Dirac equation, considering an asymmetry between space and time due to a scale factor present in the linear momentum of the fermion, but which does not touch its time derivative. We go further and extend the implications of this asymmetry in the situation the scale factor becomes space–time dependent to compute its influence on the kinematics of the Compton effect with the extended dispersion relation for the fermion that scatters the photon. • Additional inspections concerning the fermionic sector of the SME. • Effects of LSV on the Klein Paradox, the Zitterbewegung and its phenomenology. • Dirac equation with a space and time asymmetry due to a scale factor in the momentum. • The implications of the Compton effect with a space-time dependent scale factor. [ABSTRACT FROM AUTHOR]