Dirac Hamiltonian Research Articles

Tangent Fermions: Dirac or Majorana Fermions on a Lattice Without Fermion Doubling

AbstractMethods to discretize the Hamiltonian of a topological insulator or topological superconductor, without giving up on the topological protection of the massless excitations (respectively, Dirac fermions or Majorana fermions) are reviewed. The method of tangent fermions, pioneered by Richard Stacey, is singled out as being uniquely suited for this purpose. Tangent fermions propagate on a dimensional space‐time lattice with a tangent dispersion: in dimensionless units. They avoid the fermion doubling lattice artefact that will spoil the topological protection, while preserving the fundamental symmetries of the Dirac Hamiltonian. Although the discretized Hamiltonian is nonlocal, as required by the fermion‐doubling no‐go theorem, it is possible to transform the wave equation into a generalized eigenproblem that is local in space and time. Applications that are discussed include Klein tunneling of Dirac fermions through a potential barrier, the absence of localization by disorder, the anomalous quantum Hall effect in a magnetic field, and the thermal metal of Majorana fermions.

Tilt and Anisotropy of the Dirac Spectrum Caused by the Overlapping of Bloch Functions

It has been shown that the overlapping of bands belonging to equivalent representation of the symmetry group is possible in systems with Dirac points appearing at the crossing of these bands. This overlapping results in the tilt and additional anisotropy of the Dirac spectrum, as well as in the renormalization of the velocity. At the same time, overlapping does not violate the general conditions of existence of the stable band crossing point. The effective Dirac Hamiltonian in the presence of band overlapping is pseudo-Hermitian and corresponds to the effective action of a massless spinor field in the curved spacetime.

Trigonal warping effects on optical properties of anomalous Hall materials

The topological nature of solids is related to the symmetries present in the material, for example, a quantum spin Hall effect can be observed in topological insulators with time-reversal symmetry, while broken time-reversal symmetry may give rise to the presence of an anomalous quantum Hall effect (AHE). Here, we consider the effects of broken rotational symmetry on the Dirac cone of an AHE material by adding trigonal warping terms to the Dirac Hamiltonian. We calculate the linear optical conductivity semianalytically to show how by breaking the rotational symmetry we can obtain a topologically distinct phase. The addition of trigonal warping terms causes the emergence of additional Dirac cones, which when combined has a total Chern number of $\ensuremath{\mp}1$ instead of $\ifmmode\pm\else\textpm\fi{}1/2$. This results in drastic changes in the anomalous Hall and longitudinal conductivity. The trigonal warping terms also activate the higher-order Hall responses which do not exist in a $\mathcal{R}$-symmetric conventional Dirac material. We found the presence of a nonzero second-order Hall current even in the absence of a Berry curvature dipole. This shift current is also unaffected by the chirality of the Dirac cone, which should lead to a nonzero Hall current in time-reversal symmetric systems.

Proposal of multidimensional quantum walks to explore Dirac and Schrödinger systems

We propose a new multi-dimensional discrete-time quantum walk (DTQW), whose continuum limit is an extended multi-dimensional Dirac equation, which can be further mapped to the Schr\"{o}dinger equation. We show in two ways that our DTQW is an excellent measure to investigate the two-dimensional (2D) extended Dirac Hamiltonian and higher-order topological materials. First, we show that the dynamics of our DTQW resembles that of a 2D Schr\"{o}dinger harmonic oscillator. Second, we find in our DTQW topological features of the extended Dirac system. By manipulating the coin operators, we can generate not only standard edge states but also corner states.

Algebra of the spinor invariants and the relativistic hydrogen atom

It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry SU(2) of the Dirac equation. The total symmetry of the Dirac equation is the symmetry SO(3)⊗SU(2). The generator of the SO(3) symmetry group is given by the total momentum operator, and the generator of SU(2) group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson–Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers.

Thermoelectricity in bilayer graphene superlattices

Low-dimensional thermoelectricity is based on the redistribution-accumulation of the electron density of states by reducing the dimension of thermoelectric structures. Superlattices are the archetype of these structures due to the formation of energy minibands and minigaps. Here, we study for the first time the thermoelectric response of gated bilayer graphene superlattices (GBGSLs). The study is based on the four-band effective Dirac Hamiltonian, the hybrid matrix method and the Landauer-Büttiker formalism. We analyze the Seebeck coefficient, the power factor, figure of merit, output power and efficiency for different temperatures and different superlattice structural parameters. We pay special attention to the impact of not only minibands and minigaps on the thermoelectric properties, but also to intrinsic resonances in bilayer graphene structures such as Breit-Wigner, Fano and hybrid resonances. In particular, we analyze the interplay between minibands and Fano resonances as a possible mechanism to improve the thermoelectric response of GBGSLs. We also compute the density of states to know if the redistribution-accumulation of electron states is implicated in the thermoelectric response of GBGSLs.

Solving non-Hermitian Dirac equation in the presence of PDM and local Fermi velocity

We present a new approach to study a class of non-Hermitian ([Formula: see text])-dimensional Dirac Hamiltonian in the presence of local Fermi velocity. We apply the well-known Nikiforov–Uvarov method to solve such a system. We discuss applications and explore the solvability of both [Formula: see text]-symmetric and non[Formula: see text]-symmetric classes of potentials. In the former case, we obtain the solution of a harmonic oscillator in the presence of a linear vector potential while in the latter case we solve the shifted harmonic oscillator problem.

Algebraic structure of Dirac Hamiltonians in non-commutative phase space

In this article we study two-dimensional Dirac Hamiltonians with non-commutativity both in coordinates and momenta from an algebraic perspective. In order to do so, we consider the graded Lie algebra generated by Hermitian bilinear forms in the non-commutative dynamical variables and the Dirac matrices in dimensions. By further defining a total angular momentum operator, we are able to express a class of Dirac Hamiltonians completely in terms of these operators. In this way, we analyze the energy spectrum of some simple models by constructing and studying the representation spaces of the unitary irreducible representations of the graded Lie algebra . As application of our results, we consider the Landau model and a fermion in a finite cylindrical well.

Special exceptional point acting as Dirac point in one dimensional -symmetric photonic crystal

We reveal that a special exceptional point (EP) can act as a Dirac point in a one-dimensional -symmetric photonic crystal and prove it in detail using our extended first-principle theory. This theory was developed by applying biorthogonal bases of the non-Hermitian Hamiltonian to the method to study the dispersion relations of non-Hermitian systems. By using this theory, we demonstrate that two linear dispersions can cross at a critical touching point, which is a special EP, and the corresponding effective Hamiltonian can be cast into a massless Dirac Hamiltonian under the biorthogonal bases of the non-Hermitian Hamiltonian; therefore, this point can be called the Dirac point, and the linear slope can also be predicted. In contrast to the Dirac point in a Hermitian system, which is induced by the degeneracy of two different eigenstates, the Dirac point here is induced by the degeneracy of parallel eigenstates in a non-Hermitian system. In addition, after increasing the non-Hermiticity, the Dirac point evolves into a pair of EPs, and their positions in the band structure can be predicted well by our theory. Our findings and theory are important for further understanding the physics of the Dirac point in non-Hermitian wave systems.

High-order Hamiltonian obtained by Foldy–Wouthuysen transformation up to the order of mα 8

Complete relativistic corrections of an effective Hamiltonian for a single-particle system in an external electromagnetic field and their unitary equivalent form up to the order of mα 8 are obtained. The derivation is based on two approaches applying Foldy–Wouthuysen (FW) transformation to the Dirac Hamiltonian for a particle in an external electromagnetic field. The results are consistent with the previous work at the mα 6 and mα 8 order correction [Phys. Rev. A 71 012503 (2005); Phys. Rev. A 100 012513 (2019)]. We also further consider the effect of anomalous magnetic moments, namely, the Dirac–Pauli equation, and obtain FW-Hamiltonians at the same order. The results obtained can be used for the subsequent calculation of relativistic and radiation effects in simple atomic and molecular systems.

High-energy Landau levels in graphene beyond nearest-neighbor hopping processes: Corrections to the effective Dirac Hamiltonian

We study the Landau level spectrum of bulk graphene monolayers beyond the Dirac Hamiltonian with linear dispersion. We consider an effective Wannier-like tight-binding model obtained from ab initio calculations that includes long-range electronic hopping integral terms. We employ the Haydock-Heine-Kelly recursive method to numerically compute the Landau level spectrum of bulk graphene in the quantum Hall regime and demonstrate that this method is both accurate and computationally much faster than the standard numerical approaches used for this kind of study. The Landau level energies are also obtained analytically for an effective Hamiltonian that accounts for up to third-nearest-neighbor hopping processes. We find an excellent agreement between both approaches. We also study the effect of disorder on the electronic spectrum. Our analysis helps to elucidate the discrepancy between theory and experiment for the high-energy Landau level energies.

On Quantum Entropy.

Quantum physics, despite its intrinsically probabilistic nature, lacks a definition of entropy fully accounting for the randomness of a quantum state. For example, von Neumann entropy quantifies only the incomplete specification of a quantum state and does not quantify the probabilistic distribution of its observables; it trivially vanishes for pure quantum states. We propose a quantum entropy that quantifies the randomness of a pure quantum state via a conjugate pair of observables/operators forming the quantum phase space. The entropy is dimensionless, it is a relativistic scalar, it is invariant under canonical transformations and under CPT transformations, and its minimum has been established by the entropic uncertainty principle. We expand the entropy to also include mixed states. We show that the entropy is monotonically increasing during a time evolution of coherent states under a Dirac Hamiltonian. However, in a mathematical scenario, when two fermions come closer to each other, each evolving as a coherent state, the total system’s entropy oscillates due to the increasing spatial entanglement. We hypothesize an entropy law governing physical systems whereby the entropy of a closed system never decreases, implying a time arrow for particle physics. We then explore the possibility that as the oscillations of the entropy must by the law be barred in quantum physics, potential entropy oscillations trigger annihilation and creation of particles.

Excitonic instability in transition metal dichalcogenides

When transition-metal dichalcogenide monolayers lack inversion symmetry, their low-energy single particle spectrum near some high-symmetry points can, in some cases, be described by tilted massive Dirac Hamiltonians. The so-called Janus materials fall into that category. Inversion symmetry can also be broken by the application of out-of-plane electric fields, or by the mere presence of a substrate. Here we explore the properties of excitons in TMDC monolayers lacking inversion symmetry. We find that exciton binding energies can be larger than the electronic band gap, making such materials promising candidates to host the elusive exciton insulator phase. We also investigate the excitonic contribution to their optical conductivity and discuss the associated optical selection rules.

Inverse Faraday effect in massive Dirac electrons

We study the inverse Faraday effect (IFE) in a Dirac Hamiltonian with random impurities using Keldysh formalism and diagrammatic perturbation theory. The mass term in the Dirac Hamiltonian is essential for IFE, where the spin magnetic moment induced by circularly polarized light is proportional to the frequency of the incident light within the THz regime. For massive Dirac electrons, the corrections due to short-range impurities on spin magnetic moment vertex exhibits mixing of the spin magnetic moment vertex and spin angular momentum vertex. The spin magnetic density response is divergently enhanced by the vertex corrections near the band edge, indicting a long-range diffusion of spin density profile in massive Dirac electrons.

Boiling Quantum Vacuum: Thermal Subsystems from Ground-State Entanglement

In certain special circumstances, such as in the vicinity of a black hole or in a uniformly accelerating frame, vacuum fluctuations appear to give rise to a finite-temperature environment. This effect, currently without experimental confirmation, can be interpreted as a manifestation of quantum entanglement after tracing out vacuum modes in an unobserved region. In this work, we identify a class of experimentally accessible quantum systems where thermal density matrices emerge from vacuum entanglement. We show that reduced density matrices of lower-dimensional subsystems embedded in $D$-dimensional gapped Dirac fermion vacuum, either on a lattice or continuum, have a thermal form with respect to a lower-dimensional Dirac Hamiltonian. Strikingly, we show that vacuum entanglement can even conspire to make a subsystem of a gapped system at zero temperature appear as a hot gapless system. We propose concrete experiments in cold atom quantum simulators to observe the vacuum entanglement induced thermal states.

Relativistic exact two-component coupled-cluster calculations of electronic g-factors for heavy-atom-containing molecules pertinent to search of new physics

Exact two-component (X2C) coupled-cluster calculations of electronic g-factors for heavy-atom-containing small molecules pertinent to search of new physics beyond the standard model (BSM) is reported. A magnetic-field-dependent unitary transformation of the Dirac Hamiltonian has been adopted to enable a simple inclusion of the quantum electrodynamics correction to the free-electron g-factor in the four-component formulation. The X2C transformation is subsequently employed to eliminate the positronic degrees of freedom to enhance computational efficiency without significant loss of accuracy. The relationship of the present scheme to alternative four- and two-component formulations is discussed. To demonstrate the accuracy and usefulness of the present X2C scheme, we report coupled-cluster calculations for electronic g-factors of representative heavy-metal-containing small molecules including those relevant to precision spectroscopic search of BSM physics.

Floquet Engineering Topological Dirac Bands.

We experimentally realized a time-periodically modulated 1D lattice for ultracold atoms featuring a pair of linear bands, each with a Floquet winding number. These bands are spin-momentum locked and almost perfectly linear everywhere in the Brillouin zone: a near-ideal realization of the 1D Dirac Hamiltonian. We characterized the Floquet winding number using a form of quantum state tomography, covering the Brillouin zone and following the micromotion through one Floquet period. Last, we altered the modulation timing to lift the topological protection, opening a gap at the Dirac point that grew in proportion to the deviation from the topological configuration.

Foldy-Wouthuysen transformation in strong magnetic fields and relativistic corrections for quantum cyclotron energy levels

We carry out a direct, iterative Foldy--Wouthuysen transformation of a general Dirac Hamiltonian coupled to an electromagnetic field, including the anomalous magnetic moment. The transformation is carried out through an iterative disentangling of the particle and antiparticle Hamiltonians, in the expansion for higher orders of the momenta. The time-derivative term from the unitary transformation is found to be crucial in supplementing the transverse component of the electric field in higher orders. Final expressions are obtained for general combined electric and magnetic fields, including strong magnetic fields. The time-derivative of the electric field is shown to enter only in the seventh order of the fine-structure constant if the transformation is carried out in the standard fashion. We put special emphasis on the case of strong fields, which are important for a number of applications, such as electrons bound in Penning traps.

Spin Hall effect driven by the spin magnetic moment current in Dirac materials

The spin Hall effect of a Dirac Hamiltonian system is studied using semiclassical analyses and the Kubo formula. In this system, the spin Hall conductivity is dependent on the definition of spin current. All components of the spin Hall conductivity vanish when spin current is defined as the flow of spin angular momentum. In contrast, the off-diagonal components of the spin Hall conductivity are non-zero and scale with the carrier velocity (and the effective $g$-factor) when spin current consists of the flow of spin magnetic moment. We derive analytical formula of the conductivity, carrier mobility and the spin Hall conductivity to compare with experiments. In experiments, we use Bi as a model system that can be characterized by the Dirac Hamiltonian. Te and Sn are doped into Bi to vary the electron and hole concentration, respectively. We find the spin Hall conductivity ($\sigma_\mathrm{SH}$) takes a maximum near the Dirac point and decreases with increasing carrier density ($n$). The sign of $\sigma_\mathrm{SH}$ is the same regardless of the majority carrier type. The spin Hall mobility, proportional to $\sigma_\mathrm{SH}/n$, increases with increasing carrier mobility with a scaling coefficient of $\sim$1.4. These features can be accounted for quantitatively using the derived analytical formula. The results demonstrate that the giant spin magnetic moment, with an effective $g$-factor that approaches 100, is responsible for the spin Hall effect in Bi.

Intrinsic anomalous Hall conductivity of Dirac band in presence of the normal magnetic field

We have studied the intrinsic contribution to anomalous Hall conductivity in presence of the normal magnetic field. A semiclassical approach for calculating the Hall conductivity in presence of the nontrivial Berry phase is used. This method is based on the semiclassical equations of motion with anomalous velocity correction and the Boltzmann transport equation. We present the formulation of anomalous Hall conductivity in this approach. We perform explicit calculations for a two-dimensional system which is described by Dirac Hamiltonian. At zero magnetic field, our results coincide with previous results which are obtained both using the semiclassical method and Kubo formalism. For the strong magnetic field, Hall conductivity decreases as a power-law. The exponent of the power-law behavior for the infinite energy band is 1/3 and for the finite energy band, there is a crossover from 1/3 to1.