Plasma oscillations of edge Dirac fermions

The dispersion law of one-dimensional plasmons in a quasi-one-dimensional system of massless Dirac fermions has been calculated. Two model two-dimensional systems where bands of edge states filled with such Dirac fermions appear at the edge have been considered. Edge states in the first system, topological insulator, are due to topological reasons. Edge states in the second system, system of massive Dirac fermions, have Tamm origin. It has been shown that the dispersion laws of plasmons in both systems in the long-wavelength limit differ only in the definition of the parameters (velocity and localization depth of Dirac fermions). The frequency of plasmons is formally quantum (ω ∝ ? ^?1/2) and, in the case of the Coulomb interaction between electrons, depends slightly on the Fermi level E _F. The dependence on E _F is stronger in the case of short-range interaction. The quantum features of oscillations of massless one-dimensional Dirac fermions are removed by introducing the mass of Dirac fermions at the Fermi level and their density. Correspondence to the dispersion law of classical one-dimensional plasma oscillations in a narrow stripe of “Schrödinger” electrons has been revealed.     

On a new type of massless Dirac fermions in crystalline topological insulators

A new type of massless Dirac fermions in crystalline three-dimensional topological insulators (three-dimensional → two-dimensional situation) has been predicted. The spectrum has fourfold degeneracy at the top of the two-dimensional Brillouin zone (M point) and twofold degeneracy near the M point. Crystal symmetry along with the time reversal invariance in three-dimensional topological insulators allows fourfold degenerate Dirac cones, which are absent in the classification of topological features in R.-J. Slager et al., Nat. Phys. 9, 98 (2013). The Hamiltonian in the cited work does not contain Dirac singularities with more than twofold degeneracy. For this reason, the corresponding topological classification is incomplete. The longitudinal magnetic field in the spinless case holds the massless dispersion law of fermions and does not lift fourfold degeneracy. In the spinor case, the magnetic field lifts fourfold degeneracy, holding only twofold degeneracy, and results in the appearance of a band gap in the spectrum of fermions.     

Weak localization of two-dimensional Dirac fermions beyond the diffusion regime

We develop a microscopic theory of the weak localization of two-dimensional massless Dirac fermions, which is valid in the whole range of classically weak magnetic fields. The theory is applied to calculate the magnetoresistance caused by the weak localization in graphene and topological insulators.     

Capacitance spectroscopy of a system of gapless Dirac fermions in a HgTe quantum well

Information on the density of states of two-dimensional Dirac fermions in a 6.6-nm-thick HgTe quantum well that corresponds to a transition from the direct to inverted spectrum is obtained for the first time by means of capacitance measurements. It is found that the density of states of Dirac electrons is a linear function of the Fermi energy at E _F > 30 meV with the corresponding velocity vDF = 8.2 × 10⁵ m/s. At lower energies, this dependence deviates from the linear law, indicating a strong effect of disorder, which is associated with fluctuations of a built-in charge, on the density of states of the studied system near the Dirac point. At negative energies, a sharp increase in the density of states is observed, which is associated with the tail of the density of states of valleys of heavy holes. The described behavior is in agreement with the proposed model, which includes both the features of the real spectrum of Dirac fermions and the effect of the fluctuation potential.     

Cyclotron resonance of Dirac ferions in HgTe quantum wells

Cyclotron resonance of single-valley two-dimensional Dirac fermions in HgTe-based quantum wells has been experimentally investigated. The thickness of the wells is close to the critical value corresponding to the transition from the direct energy spectrum to the inverted spectrum. Under terahertz laser irradiation, transitions between the ground and first Landau levels, as well as between the first and second Landau levels, have been observed. Low magnetic fields corresponding to the cyclotron resonance, as well as the strong dependence of the position of the resonance on the electron density, indicate the Dirac character of the spectrum in these quantum wells. It has been shown that disorder plays an important role in the formation of the spectrum of two-dimensional Dirac fermions.     

Electric field effect tuning of electron-phonon coupling in graphene

Gate-modulated low-temperature Raman spectra reveal that the electric field effect (EFE), pervasive in contemporary electronics, has marked impacts on long-wavelength optical phonons of graphene. The EFE in this two-dimensional honeycomb lattice of carbon atoms creates large density modulations of carriers with linear dispersion (known as Dirac fermions). Our EFE Raman spectra display the interactions of lattice vibrations with these unusual carriers. The changes of phonon frequency and linewidth demonstrate optically the particle-hole symmetry about the charge-neutral Dirac point. The linear dependence of the phonon frequency on the EFE-modulated Fermi energy is explained as the electron-phonon coupling of massless Dirac fermions.     

Dynamical mean field study of the Dirac liquid

Renormalization is one of the basic notions of condensed matter physics. Based on the concept of renormalization, the Landau’s Fermi liquid theory has been able to explain, why despite the presence of Coulomb interactions, the free electron theory works so well for simple metals with extended Fermi surface (FS). The recent synthesis of graphene has provided the condensed matter physicists with a low energy laboratory of Dirac fermions where instead of a FS, one has two Fermi points. Many exciting phenomena in graphene can be successfully interpreted in terms of free Dirac electrons. In this paper, employing dynamical mean field theory (DMFT), we show that an interacting Dirac sea is essentially an effective free Dirac theory. This observation suggests the notion of Dirac liquid as a fixed point of interacting 2 + 1 dimensional Dirac fermions. We find one more fixed point at strong interactions describing a Mott insulating state, and address the nature of semi-metal to insulator (SMIT) transition in this system.     

Weak antilocalization and interaction-induced localization of Dirac and Weyl Fermions in topological insulators and semimetals

Weak localization and antilocalization are quantum transport phenomena that arise from the quantum interference in disordered metals.At low temperatures,they can give distinct temperature and magnetic field dependences in conductivity,allowing the symmetry of the system to be explored.In the past few years,they have also been observed in newly emergent topological materials,including topological insulators and topological semimetals.In contrast from the conventional electrons,in these new materials the quasiparticles are described as Dirac or Weyl fermions.In this article,we review our recent efforts on the theories of weak antilocalization and interaction-induced localization for Dirac and Weyl fermions in topological insulators and topological semimetals.     

Electron fractionalization in two-dimensional graphenelike structures

Electron fractionalization is intimately related to topology. In one-dimensional systems, fractionally charged states exist at domain walls between degenerate vacua. In two-dimensional systems, fractionalization exists in quantum Hall fluids, where time-reversal symmetry is broken by a large external magnetic field. Recently, there has been a tremendous effort in the search for examples of fractionalization in two-dimensional systems with time-reversal symmetry. In this Letter, we show that fractionally charged topological excitations exist on graphenelike structures, where quasiparticles are described by two flavors of Dirac fermions and time-reversal symmetry is respected. The topological zero modes are mathematically similar to fractional vortices in p-wave superconductors. They correspond to a twist in the phase in the mass of the Dirac fermions, akin to cosmic strings in particle physics.     

New generation of massless Dirac fermions in graphene under external periodic potentials

We show that new massless Dirac fermions are generated when a slowly varying periodic potential is applied to graphene. These quasiparticles, generated near the supercell Brillouin zone boundaries with anisotropic group velocity, are different from the original massless Dirac fermions. The quasiparticle wave vector (measured from the new Dirac point), the generalized pseudospin vector, and the group velocity are not collinear. We further show that with an appropriate periodic potential of triangular symmetry, there exists an energy window over which the only available states are these quasiparticles, thus providing a good system to probe experimentally the new massless Dirac fermions. The required parameters of external potentials are within the realm of laboratory conditions.     

Chain of Majorana states from superconducting Dirac fermions at a magnetic domain wall

We study theoretically a strongly type-II s-wave superconducting state of two-dimensional Dirac fermions in proximity to a ferromagnet having in-plane magnetization. It is shown that a magnetic domain wall can host a chain of equally spaced vortices in the superconducting order parameter, each of which binds a Majorana-fermion state. The overlap integral of neighboring Majorana states is sensitive to the position of the chemical potential of the Dirac fermions. Thermal transport and scanning tunneling microscopy experiments to probe the Majorana fermions are discussed.     

TWO-DIMENSIONAL ANGULAR MOMENTUM AND FERMION IN THE FIELD OF MAGNETIC-FLUX-TUBE

In this paper, we showed that the eigenvalues of two-dimensional angular momentum of a particle with charge a are integer valued. The problems of a Dirac particle in a field of vortex were discussed. When the vortex tends-to string, we found that the system admits of the existence of θ vacua. For massless fermions, a change in θ is equivalent to a Γ_s rotation, while the physical result is independent of θ. For massive fermions, CP invariance is broken except for θ=0,π; and the system exhibits the Witten effect.     

Angle-resolved photoemission spectroscopy studies of electron-electron interactions in graphene

Graphene is an emergent research topic that has attracted a huge amount of research interest ever since its experimental demonstration as a two-dimensional realization of Dirac fermions in 2005. In subsequent years, the research on graphene has rapidly expanded its field not only due to the new paradigm to study relativistic high energy physics in a condensed matter, but also due to its potential in the application for next generation devices. Most of the novel phenomena observed so far in graphene are attributed to its low-energy excitations, which is described by those of relativistic Dirac fermions. This article reviews recent progress in angle-resolved photoemission spectroscopy studies of electron-electron interactions in graphene.     

Parisi-Sourlas fermions from Ramond superstring

We show that the zero energy limit of the Ramond superstring naturally describes Dirac fermions with an OSP(D + 1,1¦2) or Parisi-Sourlas supersymmetry, and establish an equivalence between ordinary Dirac fermions and a massive (0 + 1)-dimensional supergravity theory.     

Orbital quantization in a system of edge Dirac fermions in nanoperforated graphene

The dependence of the electric resistance R of nanoperforated graphene samples on the position of the Fermi level E _F, which is varied by the gate voltage V _g, has been studied. Nanoperforation has been performed by irradiating graphene samples on a Si/SiO₂ substrate by heavy (xenon) or light (helium) ions. A series of regular peaks have been revealed on the R(V _g) dependence at low temperatures in zero magnetic field. These peaks are attributed to the passage of E _F through an equidistant set of levels formed by orbitally quantized states of edge Dirac fermions rotating around each nanohole. The results are in agreement with the theory of edge states for massless Dirac fermions.     

From the Dirac operator to Wess-Zumino models on spatial lattices

We investigate two-dimensional Wess-Zumino models in the continuum and on spatial lattices in detail. We show that a non-antisymmetric lattice derivative not only excludes chiral fermions but in addition introduces supersymmetry breaking lattice artifacts. We study the non-local and antisymmetric SLAC derivative which allows for chiral fermions without doublers and minimizes those artifacts. The supercharges of the lattice Wess-Zumino models are obtained by dimensional reduction of Dirac operators in high-dimensional spaces. The normalizable zero modes of the models with N=1 and N=2 supersymmetry are counted and constructed in the weak- and strong-coupling limits. Together with known methods from operator theory this gives us complete control of the zero mode sector of these theories for arbitrary coupling.     

Dirac returns: Non-Abelian statistics of vortices with Dirac fermions

Topological superconductors classified as type D admit zero-energy Majorana fermions inside vortex cores, and consequently the exchange statistics of vortices becomes non-Abelian, giving a promising example of non-Abelian anyons. On the other hand, types C and DIII admit zero-energy Dirac fermions inside vortex cores. It has been long believed that an essential condition for the realization of non-Abelian statistics is non-locality of Dirac fermions made of two Majorana fermions trapped inside two well-separated vortices as in the case of type D. Contrary to this conventional wisdom, however, we show that vortices with local Dirac fermions also obey non-Abelian statistics.     

Topology of the momentum space,Wigner transformations,and a chiral anomaly in lattice models

Lattice models that can be used to discretize the quantum field theory with massless fermions have been discussed. These models can also be used to describe Dirac semimetals. It has been shown that the axial current for general lattice models should be redefined in order for the usual expression for the chiral anomaly to remain valid. In this case, in the presence of a time-independent potential of the external electromagnetic field, the formalism of Wigner transformations allows relating the divergence of the axial current to a topological invariant in the momentum space that is defined for a system in equilibrium and is responsible for the stability of the Fermi point. The evaluated expression is the axial anomaly for general lattice models. This expression has been illustrated for models with Wilson fermions.     

Localization, Dirac fermions and Onsager universality

Disordered systems exhibiting exponential localization are mapped to anisotropic spin chains with localization length being related to the anisotropy of the spin model. This relates localization phenomenon in fermions to the rotational symmetry breaking in the critical spin chains. One of the intriguing consequence is that the statement of Onsager universality in spin chains implies universality of the localized fermions where the fluctuations in localized wave functions are universal. We further show that the fluctuations about localized nonrelativistic fermions describe relativistic fermions. This provides a new approach to understand the absence of localization in disordered Dirac fermions. We investigate how disorder affects well known universality of the spin chains by examining the multifractal exponents. Finally, we examine the effects of correlations on the localization characteristics of relativistic fermions. Received 28 September 2001 / Received in final form 30 November 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: isatija@nickel.nist.gov