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Relativistic Quantum Mechanics and Quantum Field Theory
Published in Xavier Oriols, Jordi Mompart, Applied Bohmian Mechanics, 2019
known as Dirac spinor, (8.88) can also be written as () [∂μ∂μ+m2]ψ(x)=0.
Topological Insulators
Published in Evgeny Y. Tsymbal, Igor Žutić, Spintronics Handbook: Spin Transport and Magnetism, Second Edition, 2019
Matthew J. Gilbert, Ewelina M. Hankiewicz
As we have mentioned, Dirac semimetals are characterized by a low-energy effective Hamiltonian that consists of two copies of counterpropagating Weyl fermions. Weyl fermions are massless fermionic quasiparticle excitations, as in the case of a Dirac electron. However, Weyl fermions have a definite chirality in real space associated with them, essentially rendering them similar to that of the edge state encountered within the integer QHE. Another manner within which one can understand the relationship between Weyl fermions and Dirac fermions is that the Weyl fermion two component spinor is exactly half of the normal four component Dirac spinor. For many years, these excitations have been sought as another example of the intimate connection between high-energy physics and condensed matter physics. To be more specific, neutrinos had been assumed to be Weyl fermions; however, it was later realized that they do, in fact, possess a small mass that invalidates them as candidate particles that possess Weyl fermion characteristics. While finding high-energy realizations of Weyl fermions has not yet resulted in confirmed examples, the search has been fruitful within the context of condensed matter realizations of quasiparticle excitations that have characteristics consistent with Weyl fermions in semimetallic materials. Weyl semimetals are characterized by exotic Fermi open surface projections, or Fermi arcs, that connect sources and sinks of Berry curvature, known as Weyl nodes, that are characterized by an integer Chern number. To date there have been several different mechanisms through which Weyl fermions have been realized in semimetals by breaking different symmetries that allow the typical Dirac spinor to be broken into a Weyl spinor. Specifically, one can break: time-reversal symmetry, as in the case of topological heterostructures [48] or HgCr2Se4 [49], inversion symmetry as in the case of TaAs and related compounds [50, 51], or within compounds that break Lorentz invariance such as W1−xMoxTe2 and LaAlGe that are also referred to as Type-II Weyl semimetals [52].
Metal–ring interactions in actinide sandwich compounds: A combined normalized elimination of the small component and local vibrational mode study
Published in Molecular Physics, 2020
Małgorzata Z. Makoś, Wenli Zou, Marek Freindorf, Elfi Kraka
The most rigorous way to include relativity in the calculation of molecular systems is to use Dirac's full 4-component (4c) formalism leading to wave functions being vectors of four complex numbers (known as bispinors) [45,46]. A large variety of approximate methods (the lower ladders in Figure 1) have been derived over the years, motivated by the assumption that the full 4c approach is computationally too demanding and cannot be applied to larger molecules. One of the most popular approximation is the so-called 2-component (2c) approximation, derived from decoupling the large and small components of the Dirac spinor. This approach forms the basis for two important families; the Douglas-Kroll-Hess (DKH) [47,48] method and the regular approximation (RA) [49]. The most famous DKH-type Hamiltonian is the second-order DKH (DKH2) [47,48], which encouraged the development of higher finite-order Hamiltonians, the so-called DKHn Hamiltonians [50–52]. Concerning the RA family, the zeroth-order RA (ZORA) has been widely used in different forms [53,54]. It is important to note that these low-order approximate relativistic Hamiltonians do not treat the core orbitals exactly.
The Dirac equation as a model of topological insulators
Published in Philosophical Magazine, 2020
Xiao Yuan, M. Bowen, P. S. Riseborough
Since the lower components of the surface states Dirac spinor are uniquely determined by the upper components through the rotation around the z-axisthe physical properties of the surface states can be expressed entirely in terms of the upper components. For example, the effective Hamiltonian for the four-component spinor, when expressed in terms of the upper components, has the form of a Rashba Hamiltonianthe factor of 2 originates from the equal contributions of the upper and lower components. Therefore, the upper components of a four-component theory are normalised to . Likewise, the matrix elements of the electromagnetic interaction can be expressed entirely in terms of the upper components aswhich represents the minimal coupling extension of the effective Rashba Hamiltonian. Hence, the electromagnetic properties can be expressed entirely in terms of the upper components, when they are normalised to unity.
Two-dimensional, finite-difference method of solving the Dirac equation for diatomic molecules revisited
Published in Molecular Physics, 2022
Jacek Kobus, Andrzej Kędziorski
The paper is organised as follows. Section 2 presents a brief derivation of the second-order equations for the large components of Dirac spinor for one-electron diatomics and the corresponding first-order equations for the small components. The discretisation of the second-order equations with a high-order numerical stencil that transforms these equations into sets of linear equations and the salient features of the finite-difference Hartree-Fock method are also discussed. Section 3 presents the results of the test calculations for a number of one-electron atomic and diatomic systems.