Collective excitations and the nature of Mott transition in undoped gapped graphene

The particle-hole continuum (PHC) for massive Dirac fermions provides an unprecedented opportunity for the formation of two collective split-off states, one in the singlet and the other in the triplet (spin-1) channel, when the short-range interactions are added to the undoped system. Both states are close in energy and are separated from the continuum of free particle-hole excitations by an energy scale of the order of the gap parameter Δ. They both disperse linearly with two different velocities, reminiscent of spin-charge separation in Luttinger liquids. When the strength of Hubbard interactions is stronger than a critical value, the velocity of singlet excitation, which we interpret as a charge composite boson, becomes zero and renders the system a Mott insulator. Beyond this critical point the low-energy sector is left with a linearly dispersing triplet mode-a characteristic of a Mott insulator. The velocity of the triplet mode at the Mott criticality is twice the velocity of the underlying Dirac fermions. The phase transition line in the space of U and Δ is in qualitative agreement with our previous dynamical mean field theory calculations.     

Graphene wormholes: A condensed matter illustration of Dirac fermions in curved space

We study the properties of graphene wormholes in which a short nanotube acts as a bridge between two graphene sheets, where the honeycomb carbon lattice is curved from the presence of 12 heptagonal defects. By taking the nanotube bridge with very small length compared to the radius, we develop an effective theory of Dirac fermions to account for the low-energy electronic properties of the wormholes in the continuum limit, where the frustration induced by the heptagonal defects is mimicked by a line of fictitious gauge flux attached to each of them. We find in particular that, when the effective gauge flux from the topological defects becomes maximal, the zero-energy modes of the Dirac equation can be arranged into two triplets, that can be thought as the counterpart of the two triplets of zero modes that arise in the dual instance of the continuum limit of large spherical fullerenes. We further investigate the graphene wormhole spectra by performing a numerical diagonalization of tight-binding Hamiltonians for very large lattices realizing the wormhole geometry. The correspondence between the number of localized electronic states observed in the numerical approach and the effective gauge flux predicted in the continuum limit shows that graphene wormholes can be consistently described by an effective theory of two Dirac fermion fields in the curved geometry of the wormhole, opening the possibility of using real samples of the carbon material as a playground to experiment with the interaction between the background curvature and the Dirac fields.     

Emergent spin

Quantum mechanics and relativity in the continuum imply the well known spin–statistics connection. However for particles hopping on a lattice, there is no such constraint. If a lattice model yields a relativistic field theory in a continuum limit, this constraint must “emerge” for physical excitations. We discuss a few models where a spin-less fermion hopping on a lattice gives excitations which satisfy the continuum Dirac equation. This includes such well known systems such as graphene and staggered fermions.     

The Heisenberg equation for phonon operators and soliton solutions in nonlinear lattices

The Heisenberg equation for phonon operators in nonlinear lattices is derived establishing the interaction Hamiltonian included higher powers of particle-hole pairs in nonlinear lattices. A phonon operator consists of a particle-hole pair in the harmonic potential approximation in the two band model; it represents an up or down transition of atoms between two levels. Applying the boson transformation method to the Heisenberg equation for phonon operators, we obtain the classical dynamical equation and a linear equation with the self-consistent potential created by the extended objects in nonlinear lattices. The boson transformation leads to soliton solutions in the long wavelength limit. The linear equation can be used to obtain scattering states, bound states and translational modes for phonons.     

Gap opening in single-layer graphene in the presence of periodic scalar and vector potentials within the continuum model

Gap opening at the Dirac point of the single-layer graphene with periodic scalar and vector potentials has been theoretically investigated under the continuum model. The symmetry analysis indicates that the two-fold degeneracy at the Dirac point can be lifted when the potentials break both the chiral symmetry and the time-reversal symmetry. With the perturbation theory, we derive an analytical expression (gap equation) for gap opening at the Dirac point. Furthermore, the bandgap from the gap equation agrees well with the exact result, when the applied potentials are weak.     

A non‐uniqueness problem of the Dirac theory in a curved spacetime

The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. We do that for the standard version of the gravitational Dirac equation, and for two alternative equations based on the tensor representation of the Dirac fields. The latter equations may be defined when the spacetime is four‐dimensional, noncompact, and admits a spinor structure. We find that, for each among the three versions of the equation, the vast majority of the possible coefficient changes do not lead to an equivalent operator H, nor to an equivalent operator E, whence a lack of uniqueness. In particular, we prove that the Dirac energy spectrum is not unique. This non‐uniqueness of the energy spectrum comes from an effect of the choice of coefficients, and applies in any given coordinates.     

Landau levels in graphene layers with topological defects

In this work, we study the low-energy electronic spectrum of a graphene layer structure with a disclination in the presence of a magnetic field. We make this study using the continuum approach, where we use the geometric theory of topological defects to introduce a disclination in a graphene layer, and the electrons are described by the massless Dirac equation in this curved background. The bound states energy spectrum and eigenfunctions are also obtained and an explicit dependence was found on the parameter that characterizes the topological defect and on the magnetic field.     

石墨烯超快动态光学性质

通过建立石墨烯的光学布洛赫方程, 研究了弱光场下的单层石墨烯超快动态光学性质. 理论研究表明在太赫兹辐射光场下由于泡利不相容和能量守恒原理使得石墨烯系统建立动态非平衡载流子并达到饱和的时间是20–200 fs, 能够在1 ps之内迅速产生光电流. 研究发现√2ev_F E₀ t<0 和ω 分别对应入射光的强度和频率, t为时间, v_F是石墨烯狄拉克点附近电子的费米速度. 研究发现光子能量?ω越大, 电极化强度以及光电流越强. 我们的理论研究结果与已有的众多实验结果一致, 表明石墨烯在超快动态光学领域尤其是太赫兹领域拥有重要的研究和应用价值.     

Eigenvalues of one-dimensional non-self-adjoint Dirac operators and applications

We analyze eigenvalues emerging from thresholds of the essential spectrum of one-dimensional Dirac operators perturbed by complex and non-symmetric potentials. In the general non-self-adjoint setting, we establish the existence and asymptotics of weakly coupled eigenvalues and Lieb–Thirring inequalities. As physical applications, we investigate the damped wave equation and armchair graphene nanoribbons.     

Gauge-independent quantization of the schrödinger and radiation fields

The quantization of several Schrödinger fields interacting with the electromagnetic field is carried out without reference to a particular gauge. The canonical formalism requires a modification introduced by Dirac and Bergmann for constraints. The Coulomb interaction is separated from the radiation and it gives rise to bound states of atoms and molecules. Particle operators are represented in the usual manner in Fock space, while the radiation field can be described by state functionals. Constraints can be included in the canonical formalism by Lagrange multipliers, leading to results equivalent to those of Dirac and Bergmann.     

Bound states in disclinated graphene with Coulomb impurities in the presence of a uniform magnetic field

In this contribution, we study the effects caused by an impurity on the quantum dynamics of massive excitations in a disclinated graphene in the presence of an external magnetic field. Within a continuum approach, the problem is mathematically modeled by the definition of a special vector potential containing all the information about the topology and the interacting fields. The presence of disclination is introduced by a term in the Dirac equation that translates the appearance of a phase associated with the transport of the spinor around the apex of the cone. We solve exactly the Dirac equation for this problem and the eigenvalues are obtained. We observe the influence of the disclination on the spectrum of energy and the allowed values of magnetic field.     

Goos-Hänchen-like shifts for Dirac fermions in monolayer graphene barrier

The quantum Goos-H?nchen effect in graphene is found to be the lateral shift of Dirac fermions on the total reflection at a single p-n interface. In this paper, we investigate the lateral shifts of Dirac fermions in transmission through a monolayer graphene barrier. Compared to the smallness of the lateral shifts in total reflection, the lateral shifts can be enhanced by the transmission resonances when the incidence angle is less than the critical angle for total reflection. It is also found that the lateral shifts, as the function of the barrier’s width and incidence angle, can be negative and positive in the cases of Klein tunneling and classical motion. The modulation of the lateral shifts can be realized by changing the electrostatic potential and induced gap, which gives rise to some applications in graphene-based devices.     

Gauge-independent quantization of the schr?dinger and radiation fields

The quantization of several Schrödinger fields interacting with the electromagnetic field is carried out without reference to a particular gauge. The canonical formalism requires a modification introduced by Dirac and Bergmann for constraints. The Coulomb interaction is separated from the radiation and it gives rise to bound states of atoms and molecules. Particle operators are represented in the usual manner in Fock space, while the radiation field can be described by state functionals. Constraints can be included in the canonical formalism by Lagrange multipliers, leading to results equivalent to those of Dirac and Bergmann.This work was supported in part by Drexel University     

Nonlinear optical response in gapped graphene

We present a formulation for the nonlinear optical response in gapped graphene, where the low-energy single-particle spectrum is modeled by massive Dirac theory. As a representative example of the formulation presented here, we obtain a closed form formula for the third harmonic generation in gapped graphene. It turns out that the covariant form of the low-energy theory gives rise to peculiar logarithmic singularities in the nonlinear optical spectra. The universal functional dependence of the response function on dimensionless quantities indicates that the optical nonlinearity can be largely enhanced by tuning the gap to smaller values.     

An alternative formulation for the analysis and interpretation of the Dirac hydrogen atom

The second-order radial differential equations for the relativistic Dirac hydrogen atom are derived from the Dirac equation treated as a system of partial differential equations. The quantum operators which arise in the development are defined and interpreted as they appear. The splitting in the energy levels is computed by applying the theory of singularities for second-order differential equations to the Klein-Gordon and Dirac relativistic equations. In the Dirac radial equation additional terms appear containing a constant, which is shown to be the radius of the electron. It is concluded that the minute perturbation of the radial eigenfunction in the vicinity of the proton brought about by the extension of the elementary particles, which appears naturally out of the Dirac equations, results in the prediction of the observed splitting of the hydrogen atom energy levels by the Dirac theory. The extension of the particles arises even though the Dirac hydrogen atom is originally formulated for point charges.     

Radiation from electrons in graphene in strong electric field

We study the interaction of electrons in graphene with the quantized electromagnetic field in the presence of an applied uniform electric field using the Dirac model of graphene. Electronic states are represented by exact solutions of the Dirac equation in the electric background, and amplitudes of first-order Feynman diagrams describing the interaction with the photon field are calculated for massive Dirac particles in both valleys. Photon emission probabilities from a single electron and from a many-electron system at the charge neutrality point are derived, including the angular and frequency dependence, and several limiting cases are analyzed. The pattern of photon emission at the Dirac point in a strong field is determined by an interplay between the nonperturbative creation of electron–hole pairs and spontaneous emission, allowing for the possibility of observing the Schwinger effect in measurements of the radiation emitted by pristine graphene under DC voltage.     

Gapless spin-1 neutral collective mode branch for graphite

Using the standard tight binding model of 2D graphite with short range electron repulsion, we predict a gapless spin-1, neutral collective mode branch below the particle-hole continuum with energy vanishing linearly with momenta at the Gamma and K points in the Brillouin zone. This spin-1 mode has a wide energy dispersion, 0 to approximately 2 eV, and is not Landau damped. The "Dirac cone spectrum" of electrons at the chemical potential of graphite generates our collective mode, so we call this "spin-1 zero sound" of the "Dirac sea." Epithermal neutron scattering experiments and spin polarized electron energy loss spectroscopy can be used to confirm and study our collective mode.     

A tale of three equations: Breit, Eddington—Gaunt, and Two-Body Dirac

G. Breit's original paper of 1929 postulates the Breit equation as a correction to an earlier defective equation due to Eddington and Gaunt, containing a form of interaction suggested by Heisenberg and Pauli. We observe that manifestly covariant electromagnetic Two-Body Dirac equations previously obtained by us in the framework of Relativistic Constraint Mechanics reproduce the spectral results of the Breit equation but through an interaction structure that contains that of Eddington and Gaunt. By repeating for our equation the analysis that Breit used to demonstrate the superiority of his equation to that of Eddington and Gaunt, we show that the historically unfamiliar interaction structures of Two-Body Dirac equations (in Breit-like form) are just what is needed to correct the covariant Eddington Gaunt equation without resorting to Breit's version of retardation.