The beta function of a two-dimensional massless Dirac Hamiltonian subject to a random scalar potential, which e.g., underlies the theoretical description of graphene, is computed numerically. Although it belongs to, from a symmetry standpoint, the two-dimensional symplectic class, the beta function monotonically increases with decreasing $g$. We also provide an argument based on the spectral flows under twisting boundary conditions, which shows that none of states of the massless Dirac Hamiltonian can be localized.
The spectrum of massless Dirac electrons on the side surface of a three-dimensional weak topological insulator is significantly affected by whether the number of unit atomic layers constituting the sample is even or odd; it has a finite-size energy gap in the even case while it is gapless in the odd case. The conductivity of such a two-dimensional Dirac electron system with quenched disorder is calculated when the Fermi level is located at the Dirac point. It is shown that the conductivity increases with increasing disorder and shows no clear even-odd difference when the aspect ratio of the sy…
The application of a perpendicular electric field can drive silicene into a gapless state, characterized by two nearly fully spin-polarized Dirac cones owing to both relatively large spin-orbital interactions and inversion symmetry breaking. Here we argue that since inter-valley scattering from non-magnetic impurities is highly suppressed by time reversal symmetry, the physics should be effectively single-Dirac-cone like. Through numerical calculations, we demonstrate that there is no significant backscattering from a single impurity that is non-magnetic and unit-cell uniform, indicating a sta…
Massless Dirac particles on the helicoid are theoretically investigated. With its possible application being helical graphene, we explore how the peculiarities of Dirac particles appear on the curved, screw-symmetric surface. The zweibein is used to derive the massless Dirac equation on the helicoid, as well as general curved surfaces. Bound states of massless Dirac electrons are shown to be absent on the helicoid, and then the scattering probabilities and the phase shifts on the surface are obtained from numerically calculated wave functions. We find the local density of states and the phase …
By solving two-component spinor equation for massless Dirac Fermions, we show that graphene under a periodic external magnetic field exhibits a unique energy spectrum: At low energies, Dirac Fermions are localized inside the magnetic region with discrete Landau energy levels, while at higher energies, Dirac Fermions are mainly found in non-magnetic regions with continuous energy bands originating from wavefunctions analogous to particle-in-box states of electrons. These findings offer a new methodology for the control and tuning of massless Dirac Fermions in graphene.
The magnetotransport properties of massless Dirac fermions in a gapless HgTe quantum well are investigated. In samples with narrow channels, a large negative magnetoresistance with a Lorentzian profile is observed, which is interpreted as a manifestation of electron viscosity due to electron-electron interaction. Comparison of experiment with theory yields the shear stress relaxation time of the Dirac fermions caused by electron-electron scattering.
The Datta-Das-type spin field-effect transistor, using a two-dimensional electron gas in a semiconductor heterostructure as a channel, plays a key role in spintronics. Here, we theoretically present a type of spin field-effect transistor based on massless birefringent Dirac fermions in polar Dirac semimetals. The manipulation of spin arises from the existence of the strong spin-orbit coupling, polar space groups, and Dirac cones in a single phase. The oscillatory channel conductance can be controlled by the sign of gate voltage in addition to its magnitude due to the gapless band structures of…
MoS$_2$, a member of transition metal dichalcogenides (TMDs), recently emerged as one of the fastest growing two-dimensional materials due to its fascinating mechanical, thermal, electronic and optical properties. Unlike graphene which possesses massless Dirac fermions with ultra-high electron mobility, monolayer MoS$_2$ is a direct band gap semiconductor. An interesting question arises: Can monolayer MoS$_2$ also possess massless Dirac fermions with ultra-high electron mobility? Here, using first-principles calculations, we show that a monolayer MoS$_2$ allotrope, which consists of repeated s…
The symmetries that protect massless Dirac fermions from a gap opening may become ineffective if the Dirac equation is discretized in space and time, either because of scattering between multiple Dirac cones in the Brillouin zone (fermion doubling) or because of singularities at zone boundaries. Here we introduce an implementation of Dirac fermions on a space-time lattice that removes both obstructions. The quasi-energy band structure has a tangent dispersion with a single Dirac cone that cannot be gapped without breaking both time-reversal and chiral symmetries. We show that this topological …
Dirac particles have been notoriously difficult to confine. Implementing a curved space Dirac equation solver based on the quantum Lattice Boltzmann method, we show that curvature in a 2-D space can confine a portion of a charged, mass-less Dirac fermion wave-packet. This is equivalent to a finite probability of confining the Dirac fermion within a curved space region. We propose a general power law expression for the probability of confinement with respect to average spatial curvature for the studied geometry.
Analysis of the electronic structure of an ordinary two-dimensional electron gas (2DEG) under an appropriate external periodic potential of hexagonal symmetry reveals that massless Dirac fermions are generated near the corners of the supercell Brillouin zone. The required potential parameters are found to be achievable under or close to laboratory conditions. Moreover, the group velocity is tunable by changing either the effective mass of the 2DEG or the lattice parameter of the external potential, and it is insensitive to the potential amplitude. The finding should provide a new class of syst…
We derive the effective field theory from the microscopic Hamiltonian of interacting two-dimensional (pseudo) Dirac electrons by performing a statistic gauge transformation. The quantized Hall conductance are expected to be $σ_{xy}=\frac{e^2}{h}(2k-1)$ with $k$ is arbitrary integer. There are also topological excitations which have fractional charge and obey fractional statistics.
We study the quasiparticle properties of two-dimensional massless Dirac Fermions when the many-body states possess a finite momentum density in the clean limit. The lack of Galilean invariance endows the many-body states at finite momentum density with qualitative differences from those of the system at rest. At finite carrier densities we demonstrate the appearance of a current-induced distortion of the pseudospin texture in momentum space that can be viewed as a drag of the Dirac point and the origin of which lies entirely in electron-electron interactions. We discuss the potential observati…
Semiconductor superlattices may display dispersions that are degenerate either at the zone center or zone boundary. We show that they are linear upon the wave-vector in the vicinity of the crossing point. This establishes a realisation of massless Dirac bands within semiconductor materials. We show that the eigenstates and the corresponding Wannier functions of these superlattices have peculiar symmetry properties. We discuss the stability of the properties of such superlattices versus the electron in-plane motion. As a distinct fingerprint, the inter-subband magneto-absorption spectrum for su…
We perform an analytic calculation of thermoelectric transport coefficients for massless Dirac electrons using the approach based on the Kubo--Středa formula and generalized Mott's relation. The main focus of the letter is made on the properties of the Nernst coefficient in the vicinity of the Dirac point in quantum limit. We calculate magnetic field and temperature dependencies of the Nernst coefficient and compare our results with recent experiments in $α$-(BEDT-TTF)$_{2}$I$_{3}$ organic conductor. We argue that the Zeeman splitting is important to understand the experimental data at high ma…
The fractal spectrum of magnetic minibands (Hofstadter butterfly), induced by the moiré super- lattice of graphene on an hexagonal crystal substrate, is known to exhibit gapped Dirac cones. We show that the gap can be closed by slightly misaligning the substrate, producing a hierarchy of conical singularities (Dirac points) in the band structure at rational values Phi = (p/q)(h/e) of the magnetic flux per supercell. Each Dirac point signals a switch of the topological quantum number in the connected component of the quantum Hall phase diagram. Model calculations reveal the scale invariant cond…
We study how low-energy charge carriers scatter off periodic and linear graphene grain boundaries oriented along the zigzag direction with a periodicity three times greater than that of pristine graphene. These defects map the two Dirac points into the same position, and thus allow for intervalley scattering to occur. Starting from graphene's first-neighbor tight-binding model we show how can we compute the boundary condition seen by graphene's massless Dirac fermions at such grain boundaries. We illustrate this procedure for the 3-periodic pentagon-only grain boundary, and then work out the l…
We present the theory of the inplane magnetoresistance in two-dimensional massless Dirac fermion systems including the Zeeman splitting and the electron-electron interaction effect on the Landau level broadening within a random phase approximation. With the decrease in temperature, we find a characteristic temperature dependence of the inplane magnetoresistance showing a minimum followed by an enhancement with a plateau. The theory is in good agreement with the experiment of the layered organic conductor α-(BEDT-TTF)_2I_3 under pressure. In-plane magnetoresistsnce of graphene is also discussed…
The effect of disorder on the Landau levels of massless Dirac fermions is examined for the cases with and without the fermion doubling. To tune the doubling a tight-binding model having a complex transfer integral is adopted to shift the energies of two Dirac cones, which is theoretically proposed earlier and realizable in cold atoms in an optical lattice. In the absence of the fermion doubling, the $n=0$ Landau level is shown to exhibit an anomalous sharpness even if the disorder is uncorrelated in space (i.e., large K-K' scattering). This anomaly occurs when the disorder respects the chiral …
We study particle motion in rotating saddle-shaped potentials. It is known that such rotating potentials can generate bounded motion for particles with a parabolic dispersion law through the combination of potential, centrifugal and Coriolis forces in the rotating frame. When applied to massless Dirac particles, for example electrons in graphene, such a potential is shown to lead to eigenstates that are spatially localized near the center of the saddle at certain energies. Although other states also exist at these energies, they have non-overlapping support in the oscillator basis, which tend …
Graphene and topological insulators (TI) possess two-dimensional Dirac fermions with distinct physical properties. Integrating these two Dirac materials in a single device creates interesting opportunities for exploring new physics of interacting massless Dirac fermions. Here we report on a practical route to experimental fabrication of graphene-Sb2Te3 heterostructure. The graphene-TI heterostructures are prepared by using a dry transfer of chemical-vapor-deposition grown graphene film. ARPES measurements confirm the coexistence of topological surface states of Sb2Te3 and Dirac π bands of grap…
We calculate the tunneling density-of-states (DOS) of a disorder-free two-dimensional interacting electron system with a massless-Dirac band Hamiltonian. The DOS exhibits two main features: i) linear growth at large energies with a slope that is suppressed by quasiparticle velocity enhancement, and ii) a rich structure of plasmaron peaks which appear at negative bias voltages in an n-doped sample and at positive bias voltages in a p-doped sample. We predict that the DOS at the Dirac point is non-zero even in the absence of disorder because of electron-electron interactions, and that it is then…
An unbounded massless Dirac model with two nondegenerate Dirac cones is the simplest model for Weyl semimetals, which show the anomalous electromagnetic response of chiral magnetic effect (CME) and anomalous Hall effect (AHE). However, if this model is naively used to analyze the electromagnetic response within a linear response theory, it gives the result apparently inconsistent with the persuasive prediction based on a lattice model. We show that this serious difficulty is related to the breaking of current conservation in the Dirac model due to quantum anomaly and can be removed if current …
In a typical situation, gapless surface states of a three-dimensional (3D) weak topological insulator (WTI) appear only on sides, leaving the top and bottom surfaces gapped. To describe massless Dirac electrons emergent on such side surfaces of a WTI, a two-dimensional (2D) model consisting of a series of one-dimensional helical channels is usually employed. Yet, explicit derivation of such a model from a 3D bulk Hamiltonian has been lacking. Here, we explicitly derive an effective 2D model for the WTI surface states starting from a Wilson-Dirac Hamiltonian for the bulk WTI and establish a fir…
The virial theorem for a system of interacting electrons in a crystal, which is described within the framework of the tight-binding model, is derived. We show that, in particular case of interacting massless electrons in graphene and other Dirac materials, the conventional virial theorem is violated. Starting from the tight-binding model, we derive the generalized virial theorem for Dirac electron systems, which contains an additional term associated with a momentum cutoff at the bottom of the energy band. Additionally, we derive the generalized virial theorem within the Dirac model using the …
We present extensive numerical results for the thermodynamic density of states (i.e. quantum capacitance) of a two-dimensional massless Dirac fermion fluid in a doped graphene sheet. In particular, by employing the random phase approximation, we quantify the impact of screening exerted by a metal gate located nearby a graphene flake. Finally, we calculate the spin- and circularly-symmetric Landau parameter, which can be experimentally extracted from independent measurements on the same setup of the quantum capacitance and quasiparticle velocity.
In the absence of a magnetic field, scattered wavefunction inside a circular p-n junction in graphene exhibits an interference pattern with high intensity maximum located around the caustics. We investigate the wavefunctions in the presence of a uniform magnetic field outside the circular region to show how the loci of the high intensity region changes by forming Landau level structure outside the circular region and a central high intensity region inside the circular p-n junction due to the strong reflection of massless Dirac fermions by the outside magnetic field. We conclude by suggesting e…
Intricate interplay between the periodicity of the lattice structure and that of the cyclotron motion gives rise to a well-known self-similar fractal structure of the energy eigenvalue, known as the Hofstadter butterfly, for an electron moving in lattice under magnetic field. Evolving from the $n=0$ Landau level, the central band of the Hofstadter butterfly is especially interesting since it may hold a key to the mysteries of the fractional quantum Hall effect observed in graphene. While the entire Hofstadter butterfly can be in principle obtained by solving Harper's equations numerically, the…
Dirac fermions in condensed matter physics hold great promise for novel fundamental physics, quantum devices and data storage applications. IV-VI semiconductors, in the inverted regime, have been recently shown to exhibit massless topological surface Dirac fermions protected by crystalline symmetry, as well as massive bulk Dirac fermions. Under a strong magnetic field (B), both surface and bulk states are quantized into Landau levels that disperse as B^1/2, and are thus difficult to distinguish. In this work, magneto-optical absorption is used to probe the Landau levels of high mobility Bi-dop…
Since the discovery of topological insulators (TIs)1,2, the peculiar nature of their chiral surface states has been experimentally demonstrated both in bulk and in film materials with open boundaries3,4. Closed boundary on a TI surface may intrigue more interesting phenomena such as quantum confinement of massless Dirac fermions (DFs), which is analogous to the quantum corral (QC) for massive free electrons on a metal surface5-10. To date, it keeps a highly stringent challenge to realize a true Dirac QC due to the unusual transmitting power of a massless fermion. Through heteroepitaxially grow…
We investigate the unusual properties of quasirelativistic massless fermions under a magnetic or electric field by means of nonminimal couplings. Within this approach, the spin-orbit coupling (SOC) effects are properly generated in the energy spectrum of the quasiparticles. By including a magnetic field, $B$, we show that the spin splitting of Landau Levels (LL) obeys a $\sqrt{B}$ linear dependence with SOC, typical of relativistic particles. Moreover, our calculated spectrum of LLs resembles the behavior of the three-dimensional (3D) massless Kane fermions. Using a nonminimal coupling with an…
In the 2016 experiment by Crossno et al. [Science 351, 1058 (2016)], electronic contribution to the thermal conductivity of graphene was found to violate the well-known Wiedemann-Franz (WF) law for metals. At liquid nitrogen temperatures, the thermal to electrical conductivity ratio of charge-neutral samples was more than 10 times higher than predicted by the WF law, what was attributed to interactions between particles leading to collective behavior described by hydrodynamics. Here we show, by adapting the handbook derivation of the WF law to the case of massless Dirac fermions, that signific…
Three dimensional (3D) topological insulators (TIs) are an important class of materials with applications in electronics, spintronics and quantum computing. With the recent development of truly bulk insulating 3D TIs, it has become possible to realize surface dominated phenomena in electrical transport measurements e.g. the quantum Hall (QH) effect of massless Dirac fermions in topological surface states (TSS). However, to realize more advanced devices and phenomena, there is a need for a platform to tune the TSS or modify them e.g. gap them by proximity with magnetic insulators, in a clean ma…
The polarization function of electrons with the tilted Dirac cone found in organic conductors is studied using the tilted Weyl equation. The dynamical property is explored based on the analytical treatment of the particle-hole excitation. It is shown that the polarization function as the function of both the frequency and the momentum exhibits cusps and nonmonotonic structures. The polarization function depends not only on the magnitude but also the direction of the external momentum. These properties are characteristic of the tilted Dirac cone, and are contrast to the isotropic case of graphe…
Localization properties of random-mass Dirac fermions for a realization of mass disorder, commonly referred to as Cho-Fisher model, is studied on the D-class chiral network. We show that a simple RG description captures accurately three phases: thermal metal and two insulators with quantized Hall conductances, as well as transitions between them (including critical exponents). We find that, with no randomness in phases on the links, transmission via the RG block exhibits a sizable portion of perfect resonances. Delocalization occurs by proliferation of these resonances to larger scales. Evolut…
We identify and describe how intense short light pulses couple to massless Dirac fermions in two-dimensional systems. The ensuing excitation dynamics exhibits unusual scaling with the wavelength of the light due the linear dispersion of the band structure and the fact that light coupling is efficient only close to the Dirac points. We exploit these features to achieve valley polarization of more than 70 % with simple pulse shapes. Quantitative results are given for pristine graphene.
Using the Lewis-Riesenfeld method of invariants we construct explicit analytical solutions for the massless Dirac equation in 2+1 dimensions describing quasi-particles in graphene. The Hamiltonian of the system considered contains some explicit time-dependence in addition to one resulting from being minimally coupled to a time-dependent magnetic field. The eigenvalue equations for the two spinor components of the Lewis-Riesenfeld invariant are found to decouple into a pair of supersymmetric invariants in a similar fashion as the known decoupling for the time-independent Dirac Hamiltonians.
We investigate the spin-ordered states in multilayer massless Dirac fermion systems under magnetic fields, in which the intralayer interaction is ferromagnetic owing to the exchange interaction, while the interlayer interaction is antiferromagnetic arising from the interlayer hopping and the on-site Coulomb repulsion. The possible spin-ordered states are examined within the mean field theory, and we apply it to alpha-(BEDT-TTF)2I3, which is a multilayer massless Dirac fermion system under pressure. In the weak interlayer coupling regime the system exhibits a ferromagnetically spin-ordered stat…
Massless Dirac fermions in graphene provide unprecedented opportunities to realize the Klein paradox, which is one of the most exotic and striking properties of relativistic particles. In the seminal theoretical work [Katsnelson et al., Nat. Phys. 2 620 (2006)], it was predicted that the massless Dirac fermions can pass through one-dimensional (1D) potential barriers unimpededly at normal incidence. Such a result seems to preclude confinement of the massless Dirac fermions in graphene by using 1D potential barriers. Here, we demonstrate, both experimentally and theoretically, that massless Dir…
In the variational framework, we study the electronic energy spectrum of massless Dirac fermions of graphene subjected to one-dimensional oscillating magnetic and electrostatic fields centered around a constant uniform static magnetic field. We analyze the influence of the lateral periodic modulations in one direction, created by these oscillating electric and magnetic fields, on Dirac like Landau levels depending on amplitudes and periods of the field modulations. We compare our theoretical results with those found within the framework of non-degenerate perturbation theory. We found that the …
We report on the clear evidence of massless Dirac fermions in two-dimensional system based on III-V semiconductors. Using a gated Hall bar made on a three-layer InAs/GaSb/InAs quantum well, we restore the Landau levels fan chart by magnetotransport and unequivocally demonstrate a gapless state in our sample. Measurements of cyclotron resonance at different electron concentrations directly indicate a linear band crossing at the $Γ$ point of Brillouin zone. Analysis of experimental data within analytical Dirac-like Hamiltonian allows us not only determing velocity $v_F=1.8\cdot10^5$ m/s of massl…
There are two types of intrinsic surface states in solids. The first type is formed on the surface of topological insulators. Recently, transport of massless Dirac fermions in the band of "topological" states has been demonstrated. States of the second type were predicted by Tamm and Shockley long ago. They do not have a topological background and are therefore strongly dependent on the properties of the surface. We study the problem of the conductivity of Tamm-Shockley edge states through direct transport experiments. Aharonov-Bohm magneto-oscillations of resistance are found on graphene samp…
A topological concern is addressed in view of the extensively and intensively studied topological phases of condensed matter. In this realm, the phases with topological order cannot be characterized by symmetry alone. Moreover, the relevant phase transitions do occur without spontaneous symmetry breaking, beyond the scope of Landau's theory. The first realization of such phases is the discovery of the integer quantum Hall effect (QHE), which was followed soon by a topological interpretation. Later on, a distinct, half-integer QHE was also found from graphene, which has almost spin degeneracy d…
We investigate the effect of the interlayer spin-flip tunneling for the interlayer magnetoresistance under magnetic fields in alpha-(BEDT-TTF)2I3, which is a multilayer massless Dirac fermion system under pressure. The mean field of the spin-flip correlation associated with the interlayer Coulomb interaction enables the interlayer spin-flip tunneling. Assuming the non-vertical interlayer spin-flip tunneling, we calculate the interlayer magnetoresistance using the Kubo formula. The crossover magnetic field, at which the interlayer magnetoresistance changes from positive to negative is shifted b…
Layered compounds AMnBi2 (A=Ca, Sr, Ba, or rare earth element) have been established as Dirac materials. Dirac electrons generated by the two-dimensional (2D) Bi square net in these materials are normally massive due to the presence of a spin-orbital coupling (SOC) induced gap at Dirac nodes. Here we report that the Sb square net in an isostructural compound BaMnSb2 can host nearly massless Dirac fermions. We observed strong Shubnikov-de Haas (SdH) oscillations in this material. From the analyses of the SdH oscillations, we find key signatures of Dirac fermions, including light effective mass …
The opening of a gap in single-layer graphene is often ascribed to the breaking of the equivalence between the two carbon sublattices. We show by angle-resolved photoemission spectroscopy that Ir- and Na-modified graphene grown on the Ir(111) surface presents a very large unconventional gap that can be described in terms of a phenomenological "massless" Dirac model. We discuss the consequences and differences of this model in comparison of the standard massive gap model, and we investigate the conditions under which such anomalous gap can arise from a spontaneous symmetry breaking.
It is of keen interest to researchers understanding different approaches to confine massless Dirac fermions in graphene, which is also a central problem in making electronic devices based on graphene. Here, we studied spatial confinement, magnetic localization and their interactions on massless Dirac fermions in an angled graphene wedge formed by two linear graphene p-n boundaries with an angle 34. Using scanning tunneling microscopy, we visualized quasibound states temporarily confined in the studied graphene wedge. Large perpendicular magnetic fields condensed the massless Dirac fermions in …
We present a numerical method to compute the Landauer conductance of noninteracting two-dimensional massless Dirac fermions in disordered systems. The method allows for the introduction of boundary conditions at the ribbon edges and accounts for an external magnetic field. By construction, the proposed discretization scheme avoids the fermion doubling problem. The method does not rely on an atomistic basis and is particularly useful to deal with long-range disorder, the correlation length of which largely exceeds the underlying material crystal lattice spacing. As an application, we study the …
