We consider the optical and transport properties in a model two-dimensional Hamiltonian which describes the merging of two Dirac points. At low energy, in the presence of an energy gap parameter $Δ$, there are two distinct Dirac points with linear dispersion, these are connected by a saddle point at higher energy. As $Δ$ goes to zero, the two Dirac points merge and the resulting dispersion exhibits semi-Dirac behaviour which is quadratic in the $x$-direction ("nonrelativistic") and linear the $y$-direction ("relativistic").In the clean limit for each direction ($x,y$) the contribution of the intraband and interband optical transitions are both given by universal functions of photon energy $Ω$ and chemical potential $μ$ normalized to the energy gap. We provide analytic formulas for both small and large $Ω/2Δ$ and $μ/Δ$ limits. These define, respectively, Dirac and semi-Dirac-like regions. For $Ω/2Δ$ and $μ/Δ$ of order one, there are deviations from these asymptotic behaviors. Considering optics and also transport, such as dc conductivity, thermal conductivity and the Lorenz number, such deviations provide signatures of the evolution from the Dirac to the semi-Dirac regime as the gap $Δ$ is varied.
We theoretically study the effect of the motion and the merging of Dirac cone on the interlayer magnetoresistance in multilayer graphene like systems. This merging, which could be induced by a uniaxial strain, gives rise in monolayer Dirac electron system to a topological transition from a semi-metallic phase to an insulating phase where Dirac points disappear. Based on a universal Hamiltonian proposed to describe the motion and the merging of Dirac points in two dimensional Dirac electron crystals, we calculate the interlayer conductivity of a stack of deformed graphene like layers using Kubo…
A continuous deformation of a Hamiltonian possessing at low energy two Dirac points of opposite chiralities can lead to a gap opening by merging of the two Dirac points. In two dimensions, the critical Hamiltonian possesses a semi-Dirac spectrum: linear in one direction but quadratic in the other. We study the transport properties across such a transition, from a Dirac semi-metal through a semi-Dirac phase towards a gapped phase. Using both a Boltzmann approach and a diagrammatic Kubo approach, we describe the conductivity tensor within the diffusive regime. In particular, we show that both th…
Theory predicts that graphene under uniaxial compressive strain in an armchair direction should undergo a topological phase transition from a semimetal into an insulator. Due to the change of the hopping integrals under compression, both Dirac points shift away from the corners of the Brillouin zone towards each other. For sufficiently large strain, the Dirac points merge and an energy gap appears. However, such a topological phase transition has not yet been observed in normal graphene (due to its large stiffness) neither in any other electronic system. We show numerically and analytically th…
We have studied the dynamical polarization and collective excitations in an anisotropic two-dimensional system undergoing a quantum phase transition with merging of two Dirac points. Analytical results for the one-loop polarization function are obtained at the finite momentum, frequency, and chemical potential. The evolution of the plasmon dispersion across the phase transition is then analyzed within the random phase approximation. We derive analytically the long-wavelength dispersion of the undamped anisotropic collective mode and find that it evolves smoothly at the critical merging point. …
We reexamine the existence and stability conditions of Dirac points between valence and conduction bands of 3/4 filled $α$-(BEDT-TTF)$_2$I$_3$ conducting plane. We consider the usual nearest neigbhor tight binding model with the seven transfer energies that depend on the applied pressure. Owing to the four distinct molecules (A, A', B, C) per unit cell of the Bravais lattice, the corresponding Bloch Hamiltonian is a $4\times4$ matrix ${\cal H}({\bm k})$ for each wave vector $\mathbf{k}$ of the Brillouin zone. In most previous works the study of Dirac points was achieved through direct numerica…
We propose a simple Hamiltonian to describe the motion and the merging of Dirac points in the electronic spectrum of two-dimensional electrons. This merging is a topological transition which separates a semi-metallic phase with two Dirac cones from an insulating phase with a gap. We calculate the density of states and the specific heat. The spectrum in a magnetic field B is related to the resolution of a Schrodinger equation in a double well potential. They obey the general scaling law e_n \propto B^{2/3} f_n(Delta /B^{2/3}. They evolve continuously from a sqrt{n B} to a linear (n+1/2)B depend…
We study the Friedel oscillations induced by a localized impurity in anisotropic graphene. We focus on the limit when the two inequivalent Dirac points merge. We find that in this limit the Friedel oscillations manifest very peculiar features, such as a strong asymmetry and an atypical inverse square-root decay. Our calculations are performed using both a T-matrix approximation and a tight-binding exact diagonalization technique. They allow us to obtain numerically the local density of states as a function of energy and position, as well as an analytical form of the Friedel oscillations in the…
The electric conductivity of Dirac electrons in the organic conductor $α$-STF$_2$I$_3$ (STF = bis(ethylenedithio)diselenadithiafulvalene), which has an isostructure of ET(=bis(ethylenedithio)tetrathiafulvalene), has been theoretically studied using a two-dimensional tight-binding model in the presence of both impurity and electron-phonon (e-p) scatterings.In contrast to ET, which has a Dirac cone with almost isotropic velocity, STF provides a large anisotropy owing to a Dirac point that is close to merging. As a result, $σ_{x}$ becomes much larger than $σ_{y}$, where $σ_y$ and $σ_x$ are diagon…
Bloch oscillations are a powerful tool to investigate spectra with Dirac points. By varying band parameters, Dirac points can be manipulated and merged at a topological transition towards a gapped phase. Under a constant force, a Fermi sea initially in the lower band performs Bloch oscillations and may Zener tunnel to the upper band mostly at the location of the Dirac points. The tunneling probability is computed from the low energy universal Hamiltonian describing the vicinity of the merging. The agreement with a recent experiment on cold atoms in an optical lattice is very good.
We study under which general conditions a pair of Dirac points in the electronic spectrum of a two-dimensional crystal may merge into a single one. The merging signals a topological transition between a semi-metallic phase and a band insulator. We derive a universal Hamiltonian that describes the physical properties of the transition, which is controlled by a single parameter, and analyze the Landau-level spectrum in its vicinity. This merging may be observed in the organic salt alpha-(BEDT-TTF)_2 I_3 or in an optical lattice of cold atoms simulating deformed graphene.
Moving, merging and annihilating Dirac points are studied theoretically in the tight-binding model on honeycomb lattice with up-to third-nearest-neighbor hoppings. We obtain a rich phase diagram of the topological phase transitions in the parameter space of direction-dependent hoppings. We obtain the conditions for the three Dirac points to merge and for the tricritical points. We find that only very small third-nearest-neighbor hoppings are enough for the existence of the merging of three-Dirac-points and the tricritical points, if the system is sufficiently anisotropic. The density of states…
We study the influence of disorder on the topological transition from a two-dimensional Dirac semi-metal to an insulating state. This transition is described as a continuous merging of two Dirac points leading to a semi-Dirac spectrum at the critical point. The latter is characterized by a dispersion relation linear in one direction and quadratic in the orthogonal one. Using the self-consistent Born approximation and renormalization group we calculate the density of states above, below and in the vicinity of the transition in the presence of different types of disorder. Beyond the expected dis…
We investigate the effect of an in-plane AC electric field coupled to electrons in the honeycomb lattice and show that it can be used to manipulate the Dirac points of the electronic structure. We find that the position of the Dirac points can be controlled by the amplitude and the polarization of the field for high frequency drivings, providing a new platform to achieve their merging, a topological transition which has not been observed yet in electronic systems. Importantly, for lower frequencies we find that the multi-photon absorptions and emissions processes yield the creation of addition…
The manipulation and movement of Dirac points in the Brillouin zone by the electron-electron interaction is considered within leading order perturbation theory. At the merging point, an infinitesimal interaction is shown to cause opening of the gap or splitting of the Dirac points, depending on the inter- or intrasublattice nature of the merging and the sign of the interaction. The topology of the spectrum can therefore be efficiently changed by simply tuning the interaction between particles, as opposed to the usual careful band structure engineering. This is illustrated around the merging tr…
First, we study a square fermionic lattice that supports the existence of massless Dirac fermions, where the Dirac points are protected by a hidden symmetry. We also consider two modified models with a staggered potential and the diagonal hopping terms, respectively. In the modified model with a staggered potential, the Dirac points exist in some range of magnitude of the staggered potential, and move with the variation of the staggered potential. When the magnitude of the staggered potential reaches a critical value, the two Dirac points merge. In the modified model with the diagonal hopping …
We analyze a generalized Dirac system, where the dispersion along the $k_{x}$ and $k_{y}$ axes is $N$-th power and linear along the $k_{z}$ axis. When we apply magnetic field, there emerge $N$ monopole-antimonopole pairs beyond a certain critical field in general. As the direction of the magnetic field is rotated toward the $z$ axis, monopoles move to the north pole while antimonopoles move to the south pole. When the magnetic field becomes parallel to the $z$ axis, they merge into one monopole or one antimonopole whose monopole charge is $\pm N$. The resultant system is a multiple-Weyl semime…
The spectrum of tight binding electrons on a square lattice with half a magnetic flux quantum per unit cell exhibits two Dirac points at the band center. We show that, in the presence of an additional uniaxial staggered potential, this pair of Dirac points may merge into a single one, with a topological transition towards a gapped phase. At the transition, the spectrum is linear in one direction and quadratic in the other one (a spectrum recently named "hybrid" or "semi-Dirac"). This transition is studied in the framework of a general Hamiltonian describing the merging of Dirac points. The pos…
We consider the $Z_2$ topology of the Dirac lines, i.e., lines of band contacts, on an example of graphite. Four lines --- three with topological charge $N_1=1$ each and one with $N_1=-1$ --- merge together near the H-point and annihilate due to summation law $1+1+1-1=0$. The merging point is similar to the real-space nexus, an analog of the Dirac monopole at which the $Z_2$ strings terminate.
We review different scenarios for the motion and merging of Dirac points in two dimensional crystals. These different types of merging can be classified according to a winding number (a topological Berry phase) attached to each Dirac point. For each scenario, we calculate the Landau level spectrum and show that it can be quantitatively described by a semiclassical quantization rule for the constant energy areas. This quantization depends on how many Dirac points are enclosed by these areas. We also emphasize that different scenarios are characterized by different numbers of topologically prote…
Despite much anticipation of valleytronics as a candidate to replace the ageing CMOS-based information processing, its progress is severely hindered by the lack of practical ways to manipulate valley polarization all-electrically in an electrostatic setting. Here we propose a class of all-electric-controlled valley filter, valve and logic gate based on the valley-contrasting transport in a merging Dirac cones system. The central mechanism of these devices lies on the pseudospin-assisted quantum tunneling which effectively quenches the transport of one valley when its pseudospin configuration m…
We propose to realize Dirac states in an inclined two-dimensional Su-Schrieffer-Heeger model on a square lattice. We show that a pair of Dirac points protected by space-time inversion symmetry appear in the semimetal phase. The locations of these Dirac points are not pinned to any high-symmetry points of the Brillouin zone but are tunable through parameter modulations. Interestingly, the merging of two Dirac points undergoes a topological phase transition that leads to either a weak topological insulator or a nodal-line semimetal. We provide a systematic analysis of these topological phases fr…
We show that a gap parameter can fully describe the merging of Dirac cones in semi-Dirac materials from $K$- and $K^\prime$-points into the common $M$-point in the Brillouin zone. We predict that the gap parameter manifests itself by enhancing the thermoelectric figure of merit $zT$ as the chemical potential crosses the gap followed by a sign change in the Seebeck coefficient around the same point. Subsequently, we show that there is also a trade-off feature between the maximum power delivered and the efficiency when the chemical potential crosses the gap parameter. An optimal operating point …
Topological defects in Bloch bands, such as Dirac points in graphene, and their resulting Berry phases play an important role for the electronic dynamics in solid state crystals. Such defects can arise in systems with a two-atomic basis due to the momentum-dependent coupling of the two sublattice states, which gives rise to a pseudo-spin texture. The topological defects appear as vortices in the azimuthal phase of this pseudo-spin texture. Here, we demonstrate a complete measurement of the azimuthal phase in a hexagonal optical lattice employing a versatile method based on time-of-flight imagi…
The merging or emergence of a pair of Dirac points may be classified according to whether the winding numbers which characterize them are opposite ($+-$ scenario) or identical ($++$ scenario). From the touching point between two parabolic bands (one of them can be flat), two Dirac points with the {\it same} winding number emerge under appropriate distortion (interaction, etc), following the $++$ scenario. Under further distortion, these Dirac points merge following the $+-$ scenario, that is corresponding to {\it opposite} winding numbers. This apparent contradiction is solved by the fact that…
In the presence of an external magnetic field, the optical response of two-dimensional materials, whose charge carriers behave as massless Dirac fermions with arbitrary anisotropic Fermi velocity, is investigated. Using Kubo formalism, we obtain the magneto-optical conductivity tensor for these materials, which allows to address the magneto-optical response of anisotropic Dirac fermions from the well known magneto-optical conductivity of isotropic Dirac fermions. As an application, we analyse the combined effects of strain-induced anisotropy and magnetic field on the transmittance, as well as …
We derive the low-energy Hamiltonian for a honeycomb lattice with anisotropy in the hopping parameters. Taking the reported Dirac Hamiltonian for the anisotropic honeycomb lattice, we obtain its optical conductivity tensor and its transmittance for normal incidence of linearly polarized light. Also, we characterize its dichroic character due to the anisotropic optical absorption. As an application of our general findings, which reproduce the case of uniformly strained graphene, we study the optical properties of graphene under a nonmechanical distortion.
We show that the gap parameter in \textit{semi}-Dirac material induces a large degree of sensitivity for inter-band optical conductivity with respect to the polarization direction. The optical conductivity reveals an abruptly large value at a certain frequency for light along a particular polarization direction while it is significantly suppressed along the direction orthogonal to the former. The direction-dependent optical conductivity may, in turn, be used to uniquely predict the dispersive nature of the 2D \textit{semi}-Dirac materials, in addition to other possible applications in the fiel…
We study the photoresponse of two-dimensional semi-Dirac semimetals and three-dimensional Dirac semimetals to off-resonant circularly polarized light. For two-dimensional semi-Dirac semimetals we find that incident light does not open a gap in the spectrum, in contrast to the case of purely linear dispersion. For the three-dimensional case we find that applying a circularly polarized light, one can tune from a trivial insulator to three-dimensional Dirac semimetal with an inverted gap. We propose and show that application of light leads to an intriguing platform for generation, motion and merg…
Degeneracy (exceptional) points embedded in energy band are distinct by their topological features. We report different hybrid two-state coalescences (EP2s) formed through merging two EP2s with opposite chiralities that created from the type III Dirac points emerging from a flat band. The band touching hybrid EP2, which is isolated, is induced by the destructive interference at the proper match between non-Hermiticity and synthetic magnetic flux. The degeneracy points and different types of exceptional points are distinguishable by their topological features of global geometric phase associate…
Electronic properties of two-dimensional allotropes of carbon, such as graphene and its bilayer, multi-layer epitaxial graphene, few-layer Bernal-stacked graphene, as well as of three-dimensional bulk graphite are reviewed from the viewpoint of recent optical spectroscopy studies. Attention is focused on relativistic-like character of quasi particles in these systems, which are referred to as massless or massive Dirac fermions.
Massless Dirac fermions occur as low-energy modes in several quasi-two-dimensional condensed matter systems such as graphene, the surface of bulk topological insulators, and in layered organic semiconductors. When the rotational symmetry in such systems is reduced either by an in-plane electric field or an intrinsic tilt of the Dirac cones, the allowed dipolar optical transitions evolve from a few selected transitions into a wide fan of interband transitions. We show that the Lorentz covariance of the low-energy carriers allows for a concise analysis of the emerging magneto-optical properties.…
It has recently been realized that the first-order moment of the Berry curvature, namely the Berry curvature dipole (BCD) can give rise to non-linear current in a wide variety of time-reversal invariant and non-centrosymmetric materials. While the BCD in two-dimensional Dirac systems is known to be finite only in the presence of either substantial spin-orbit coupling where low-energy Dirac quasiparticles form tilted cones or higher order warping of the Fermi surface, we argue that the low-energy Dirac quasiparticles arising from the merging of a pair of Dirac points without any tilt or warping…
1T$^\prime$-MoS$_2$ exhibits valley-spin-polarized tilted Dirac bands in the presence of external vertical electric field and undergoes a topological phase transition between the topological insulator and band insulator around the critical value of the electric field. Within the linear response theory, we theoretically investigate the anisotropic longitudinal optical conductivities of tilted Dirac bands in both undoped and doped 1T$^\prime$-MoS$_2$, including the effects of the vertical electric field. The influence of the spin-orbit coupling gap, band tilting, and vertical electric field on t…
We show that, under the effect of an external magnetic field, a photogalvanic effect and the generation of second harmonic wave can be induced in inversion-symmetric and time reversal invariant Dirac semimetals. The mechanism responsible of these non linear optical responses is the magnetochiral effect. The origin of this magnetochiral effect is the band bending of the dispersion relation in real Dirac semimetals. Some observable consequences of this phenomenon are the appearance of a dc current on the surface of the system when it is irradiated with linearly polarized light or a rotation of t…
The interband optical absorption of linearly polarised light by two-dimensional (2D) semimetals hosting tilted and anisotropic Dirac cones in the bandstructure is analysed theoretically. Super-critically tilted (type-II) Dirac cones are characterised by an absorption that is highly dependent on the incident photon polarisation and frequency, and is tunable by changing the Fermi level with a back-gate voltage. Type-II Dirac cones exhibit open Fermi surfaces and large regions of the Brillouin zone where the valence and conduction bands sit either above or below the Fermi level. As a consequence,…
We study the electronic structure and the phase diagram of non-interacting fermions confined to hexagonal optical lattices. In the first part, we compare the properties of Dirac points arising in the eigenspectrum of either honeycomb or triangular lattices. Numerical results are complemented by analytical equations for weak and strong confinements. In the second part we discuss the phase diagram and the evolution of Dirac points in honeycomb lattices applying a tight-binding description with arbitrary nearest-neighbor hoppings. With increasing asymmetry between the hoppings the Dirac points ap…
The popularity of graphene--a pseudospin-1/2 two-dimensional Dirac-Weyl material--has prompted the search for related materials and the characterization of their properties. In this work, the magneto-optical conductivity is calculated for systems that obey the general pseudospin-s two-dimensional Dirac-Weyl Hamiltonian, with particular focus on s = {1/2, 1, 3/2, 2}. This generalizes calculations that have been made for s = 1/2 and follows previous work on the optical response of these systems in zero field. In the presence of a magnetic field, Landau levels condense out of the 2s+1 energy band…
We show that the inter-band absorption of radiation in a 2D Dirac material leads to a direct electric current flowing at sample edges. The photocurrent originates from the momentum alignment of electrons and holes and is controlled by the radiation polarization. We develop a microscopic theory of such an edge photogalvanic effect and calculate the photocurrent for gapped and gapless 2D Dirac materials, also in the presence of a static magnetic field which introduces additional imbalance between the electron and hole currents. Further, we show that the photocurrent can be considerably multiplie…
We develop an analytically solvable model able to qualitatively explain nonhydrogenic exciton spectra observed recently in two-dimensional (2d) semiconducting transition metal dichalcogenides. Our exciton Hamiltonian explicitly includes additional angular momentum associated with the pseudospin degree of freedom unavoidable in 2d semiconducting materials with honeycomb structure. We claim that this is the key ingredient for understanding the nonhydrogenic exciton spectra that was missing so far.
We study magneto-optical (MO) properties of a semi-Dirac nanoribbon in presence of a perpendicular magnetic field using Kernel Polynomial Method (KPM) based on the Keldysh formalism in the experimental (terahertz frequency) regime. For comparison, we have also included results for the Dirac systems as well, so that the interplay of the band structure deformation and MO conductivities can be studied. We have found that the MO conductivity shows features in the semi-Dirac system that are quite distinct from the Dirac case near the ultra-violet and the visible regimes. The real parts of the longi…
We present a perturbative method for calculating the optical Hall conductivity in a tight-binding framework based on the Kubo formalism. The method involves diagonalization only of the Hamiltonian in absence of the magnetic field, and thus avoids the computational problems usually arising due to the huge magnetic unit cells required to maintain translational invariance in presence of a Peierls phase. A recipe for applying the method to numerical calculations of the magneto-optical response is presented. We apply the formalism to the case of ordinary and gapped graphene in a next-nearest neighb…
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…
We present a theoretical study on the Landau levels (LLs) and magneto-optical absorption of a two-dimensional semi-Dirac electron system under a perpendicular magnetic field. Based on an effective k.p Hamiltonian, we find that the LLs are proportional to the two-thirds power law of the magnetic field and level index, which can be understood as a hybridization of the LL of Schrodinger and Dirac electrons with new features. With the help of Kubo formula, we find the selection rule for interband (intraband) magneto-optical transition is anisotropic (isotropic). Specifically, the selection rules f…
Two-dimensional (2D) massive Dirac electrons possess a finite Berry curvature, with Chern number $\pm 1/2$, that entails both a quantized dc Hall response and a subgap full-quarter Kerr rotation. The observation of these effects in 2D massive Dirac materials such as gapped graphene, hexagonal boron nitride or transition metal dichalcogenides (TMDs) is obscured by the fact that Dirac cones come in pairs with opposite sign Berry curvatures, leading to a vanishing Chern number. Here, we show that the presence of spin-orbit interactions, combined with an exchange spin splitting induced either by d…
The outstanding electronic properties of relativistic-like fermions have been extensively studied in solid state systems with isotropic linear dispersions such as graphene. Here, we show that 2D and 3D Dirac-Weyl (DW) materials exhibiting tilted energy dispersions could induce drastically different transport phenomena, compared to the non-tilted case. Indeed, the Klein tunneling of DW fermions of opposite chiralities is predicted to appear along two separated oblique directions. In addition, valley filtering and beam splitting effects are easily tailored by dopant engineering techniques while …
Dirac fermion optics exploits the refraction of chiral fermions across optics-inspired Klein-tunneling barriers defined by high-transparency p-n junctions. We consider the corner reflector (CR) geometry introduced in optics or radars. We fabricate Dirac fermion CRs using bottom-gate-defined barriers in hBN-encapsulated graphene. By suppressing transmission upon multiple internal reflections, CRs are sensitive to minute phonon scattering rates. We report on doping-independent CR transmission in quantitative agreement with a simple scattering model including thermal phonon scattering. As a new s…
We investigate parity-odd non-dissipative transport in an anisotropic Dirac semi-metal in two spatial dimensions. The analysis is relevant for interacting electronic systems with merging Dirac points at charge neutrality. For such systems the dispersion relation is relativistic in one direction and non-relativistic in the other. We give a proposal how to calculate the Berry curvature for this system and use it to derive more than one odd viscosities, in contrast to rotationally invariant systems. We observe that in such a model the odd part of stress tensor is parameterised by two independent …
Fermi arc surface states, a hallmark of topological Dirac semimetals, can host carriers that exhibit unusual dynamics distinct from that of their parent bulk. Here we find that Fermi arc carriers in intrinsic Dirac semimetals possess a strong and anisotropic light matter interaction. This is characterized by a large Fermi arc optical conductivity when light is polarized transverse to the Fermi arc; when light is polarized along the Fermi arc, Fermi arc optical conductivity is significantly muted. The large surface spectral weight is locked to the wide separation between Dirac nodes and persist…
