Landau quantization

About: Landau quantization is a research topic. Over the lifetime, 9588 publications have been published within this topic receiving 180530 citations. The topic is also known as: Landau level.

Unconventional Integer Quantum Hall Effect in Graphene

Abstract: Monolayer graphite films, or graphene, have quasiparticle excitations that can be described by $(2+1)$-dimensional Dirac theory. We demonstrate that this produces an unconventional form of the quantized Hall conductivity ${\ensuremath{\sigma}}_{xy}=\ensuremath{-}(2{e}^{2}/h)(2n+1)$ with $n=0,1,\dots{}$, which notably distinguishes graphene from other materials where the integer quantum Hall effect was observed. This unconventional quantization is caused by the quantum anomaly of the $n=0$ Landau level and was discovered in recent experiments on ultrathin graphite films.

Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals

Abstract: INTRODUCTION Condensed-matter systems have recently become a fertile ground for the discovery of fermionic particles and phenomena predicted in high-energy physics; examples include Majorana fermions, as well as Dirac and Weyl semimetals. However, fermions in condensed-matter systems are not constrained by Poincare symmetry. Instead, they must only respect the crystal symmetry of one of the 230 space groups. Hence, there is the potential to find and classify free fermionic excitations in solid-state systems that have no high-energy counterparts. RATIONALE The guiding principle of our classification is to find irreducible representations of the little group of lattice symmetries at high-symmetry points in the Brillouin zone (BZ) for each of the 230 space groups (SGs), the dimension of which corresponds to the number of bands that meet at the high-symmetry point. Because we are interested in systems with spin-orbit coupling, we considered only the double-valued representations, where a 2π rotation gives a minus sign. Furthermore, we considered systems with time-reversal symmetry that squares to –1. For each unconventional representation, we computed the low-energy k · p Hamiltonian near the band crossings by writing down all terms allowed by the crystal symmetry. This allows us to further differentiate the band crossings by the degeneracy along lines and planes that emanate from the high-symmetry point, and also to compute topological invariants. For point degeneracies, we computed the monopole charge of the band-crossing; for line nodes, we computed the Berry phase of loops encircling the nodes. RESULTS We found that three space groups exhibit symmetry-protected three-band crossings. In two cases, this results in a threefold degenerate point node, whereas the third case results in a line node away from the high-symmetry point. These crossings are required to have a nonzero Chern number and hence display surface Fermi arcs. However, upon applying a magnetic field, they have an unusual Landau level structure, which distinguishes them from single and double Weyl points. Under the action of spatial symmetries, these fermions transform as spin-1 particles, as a consequence of the interplay between nonsymmorphic space group symmetries and spin. Additionally, we found that six space groups can host sixfold degeneracies. Two of these consist of two threefold degeneracies with opposite chirality, forced to be degenerate by the combination of time reversal and inversion symmetry, and can be described as “sixfold Dirac points.” The other four are distinct. Furthermore, seven space groups can host eightfold degeneracies. In two cases, the eightfold degeneracies are required; all bands come in groups of eight that cross at a particular point in the BZ. These two cases also exhibit fourfold degenerate line nodes, from which other semimetals can be derived: By adding strain or a magnetic field, these line nodes split into Weyl, Dirac, or line node semimetals. For all the three-, six- and eight-band crossings, nonsymmorphic symmetries play a crucial role in protecting the band crossing. Last, we found that seven space groups may host fourfold degenerate “spin-3/2” fermions at high symmetry points. Like their spin-1 counterparts, these quasiparticles host Fermi surfaces with nonzero Chern number. Unlike the other cases we considered, however, these fermions can be stabilized by both symmorphic and nonsymmorphic symmetries. Three space groups that host these excitations also host unconventional fermions at other points in the BZ. We propose nearly 40 candidate materials that realize each type of fermion near the Fermi level, as verified with ab initio calculations. Seventeen of these have been previously synthesized in single-crystal form, whereas others have been reported in powder form. CONCLUSION We have analyzed all types of fermions that can occur in spin-orbit coupled crystals with time-reversal symmetry and explored their topological properties. We found that there are several distinct types of such unconventional excitations, which are differentiated by their degeneracies at and along high-symmetry points, lines, and surfaces. We found natural generalizations of Weyl points: three- and four-band crossings described by a simple k · S Hamiltonian, where S i is the set of spin generators in either the spin-1 or spin-3/2 representations. These points carry a Chern number and, consequently, can exhibit Fermi arc surface states. We also found excitations with six- and eightfold degeneracies. These higher-band crossings create a tunable platform to realize topological semimetals by applying an external magnetic field or strain to the fourfold degenerate line nodes. Last, we propose realizations for each species of fermion in known materials, many of which are known to exist in single-crystal form.

Singularities and Discontinuities of Feynman Amplitudes

Abstract: The Landau singularities of the amplitude calculated from an arbitrary Feynman graph are considered. It is shown that the discontinuity across a branch cut starting from any Landau singularity is obtained by replacing Feynman propagators by delta functions for those lines which appear in the Landau diagram. The general formula is a simple generalization of the unitarity condition. The discontinuity is then considered as an analytic function of the momenta and masses; it is shown that its singularities are a subclass of the singularities of the original amplitude which corresponds to Landau diagrams with additional lines. The general results are illustrated by application to some single loop graphs. In particular, the general formula gives an immediate calculation of the Mandelstam spectral function for fourth‐order scattering. Singularities not of the Landau type are discussed and illustrated by the third‐order vertex part.

Magneto-optical conductivity in graphene

Abstract: Landau level quantization in graphene reflects the Dirac nature of its quasiparticles and has been found to exhibit an unusual integer quantum Hall effect. In particular, the lowest Landau level can be thought of as shared equally by electrons and holes, and this leads to characteristic behaviour of the magneto-optical conductivity as a function of frequency Ω for various values of the chemical potential μ. Particular attention is paid to the optical spectral weight under various absorption peaks and its redistribution as μ is varied. We also provide results for magnetic field B as well as chemical potential sweeps at selected fixed frequencies, which can be particularly useful for possible measurements in graphene. Both diagonal and Hall conductivities are considered.

Electronic properties of graphene in a strong magnetic field

Abstract: The basic aspects of electrons in graphene (two-dimensional graphite) exposed to a strong perpendicular magnetic field are reviewed. One of its most salient features is the relativistic quantum Hall effect, the observation of which has been the experimental breakthrough in identifying pseudorelativistic massless charge carriers as the low-energy excitations in graphene. The effect may be understood in terms of Landau quantization for massless Dirac fermions, which is also the theoretical basis for the understanding of more involved phenomena due to electronic interactions. The role of electron-electron interactions both in the weak-coupling limit, where the electron-hole excitations are determined by collective modes, and in the strong-coupling regime of partially filled relativistic Landau levels are presented. In the latter limit, exotic ferromagnetic phases and incompressible quantum liquids are expected to be at the origin of recently observed (fractional) quantum Hall states. Furthermore, the electron-phonon coupling in a strong magnetic field is discussed. Although the present review has a dominant theoretical character, a close connection with available experimental observation is intended.