Dirac Theory Research Articles

Generalized uncertainty principle in graphene

We show that, by going beyond the low-energy approximation for which the dispersion relations of graphene are linear, the corresponding emergent field theory is a specific generalization a Dirac field theory. The generalized Dirac Hamiltonians one obtains are those compatible with specific generalizations of the uncertainty principle. We also briefly comment on the compatibility of the latter with noncommuting positions, [xi,xj] ≠ 0, and on their possible physical realization.

Terahertz spin dynamics driven by a field-derivative torque

Efficient manipulation of magnetization at ultrashort time scales is of particular interest for future technology. Here, we numerically investigate the influence of the so-called field-derivative torque, which was derived earlier based on relativistic Dirac theory [Mondal et al., Phys. Rev. B 94, 144419 (2016)], on the spin dynamics triggered by ultrashort laser pulses. We find that only considering the THz Zeeman field can underestimate the spin excitation in antiferromagnetic oxide systems as, e.g., NiO and CoO. However, accounting for both, the THz Zeeman torque and the field-derivative torque, the amplitude of the spin excitation increases significantly. Studying the damping dependence of field-derivative torque we observe larger effects for materials having larger damping constants.

Quantum Cosmologies under Geometrical Unification of Gravity and Dark Energy

A Friedmann–Robertson–Walker Universe was studied with a dark energy component represented by a quintessence field. The Lagrangian for this system, hereafter called the Friedmann–Robertson–Walker–quintessence (FRWq) system, was presented. It was shown that the classical Lagrangian reproduces the usual two (second order) dynamical equations for the radius of the Universe and for the quintessence scalar field, as well as a (first order) constraint equation. Our approach naturally unified gravity and dark energy, as it was obtained that the Lagrangian and the equations of motion are those of a relativistic particle moving on a two-dimensional, conformally flat spacetime. The conformal metric factor was related to the dark energy scalar field potential. We proceeded to quantize the system in three different schemes. First, we assumed the Universe was a spinless particle (as it is common in literature), obtaining a quantum theory for a Universe described by the Klein–Gordon equation. Second, we pushed the quantization scheme further, assuming the Universe as a Dirac particle, and therefore constructing its corresponding Dirac and Majorana theories. With the different theories, we calculated the expected values for the scale factor of the Universe. They depend on the type of quantization scheme used. The differences between the Dirac and Majorana schemes are highlighted here. The implications of the different quantization procedures are discussed. Finally, the possible consequences for a multiverse theory of the Dirac and Majorana quantized Universe are briefly considered.

High spin, high derivatives

This work is dedicated to honour the memory of Victor Manuel Villanueva Sandoval. One of the academic interests shared with Victor was the disturbing fact that despite many years of work of the most talented people, there is no consistent quantum field theory formulation for interacting high spin particles. We propose here a formalism for the description of high spin particles by directly generalizing the structure of Dirac theory for spin 1/2 to the (j, 0) ⊕ (0, j) representation. We find that this generalization leads naturally to a formalism of order 2j in the derivatives of the field. For j > 1, we conjecture that this construction evades the linear instabilities of the Ostrogradski theorem for theories with higher than two time derivatives due to the constriction introduced by the projection onto subspaces of well defined parity.

Edge-dependent reflection and inherited fine structure of higher-order plasmons in graphene nanoribbons

We investigate higher-order plasmons in graphene nanoribbons, and we present how electronic edge states and wave-function fine structure influence the graphene plasmons. Based on nearest-neighbor tight-binding calculations, we find that a standing-wave model based on nonlocal bulk plasmon dispersion is surprisingly accurate for armchair ribbons of widths even down to a few nanometers, and we determine the corresponding phase shift upon edge reflection and an effective ribbon width. Wider zigzag ribbons exhibit a similar phase shift, whereas the standing-wave model describes few-nanometer zigzag ribbons less satisfactorily, to a large extent because of their edge states. We directly confirm that also the larger broadening of plasmons for zigzag ribbons is due to their edge states. Furthermore, we report a prominent fine structure in the induced charges of the ribbon plasmons, which for armchair ribbons follows the electronic wave-function oscillations induced by intervalley coupling. Interestingly, the wave-function fine structure is also found in our analogous density-functional theory calculations, and both these and tight-binding numerical calculations are explained quite well with analytical Dirac theory for graphene ribbons.

Quantum spin dynamics in external classical fields

We study the quantum and classical dynamics of spinning particles in the framework of the general-relativistic covariant Dirac theory. The exact Foldy-Wouthuysen transformation for the most general case of a fermion in arbitrary configurations of the gravitational, inertial and electromagnetic fields is derived. We demonstrate the complete consistency of the quantum and classical dynamics. As physical applications, we discuss prospects of probing spacetime structure and using spin effects for gravitational wave detection.

Nonrelativistic limit of Dirac theory from effective field theory

In this work we analyze the low-energy nonrelativistic limit of Dirac theory in the framework of effective field theory. By integrating out the high-energy modes of the Dirac field, given in terms of a combination of the two-component Weyl spinors, we obtain a low-energy effective action for the remaining components, whose equation of motion can then be compared to the Pauli–Schrödinger equation after demanding normalization of the wave function. We then discuss the relevance of the terms in the effective action in the context of an anisotropic dimensional analysis which is suitable for nonrelativistic theories.

Emergence of non-Fermi-liquid type Weyl metals driven by doped magnetic impurities in spin-orbit coupled semiconductors

Constructing an effective field theory in terms of doped magnetic impurities (described by an O(3) vector model with a random mass term), itinerant electrons of spin-orbit coupled semiconductors (given by a Dirac theory with a relatively large mass term), and effective interactions between doped magnetic ions and itinerant electrons (assumed by an effective Zeeman coupling term), we perform the perturbative renormalization group analysis in the one-loop level based on the dimensional regularization technique. As a result, we find that the mass renormalization in dynamics of itinerant electrons acquires negative feedback effects due to quantum fluctuations involved with the Zeeman coupling term, in contrast with that of the conventional problem of quantum electrodynamics, where such interaction effects enhance the fermion mass more rapidly. Recalling that the applied magnetic field decreases the band gap in the presence of spin-orbit coupling, this renormalization group analysis shows that the external magnetic field overcomes the renormalized band gap, allowed by doped magnetic impurities even without ferromagnetic ordering. In other words, the Weyl metal physics can be controlled by doping magnetic impurities into spin-orbit coupled semiconductors, even if the external magnetic field alone cannot realize the Weyl metal phase due to relatively large band gaps of semiconductors. Furthermore, we emphasize that quasiparticles do not exist in this emergent disordered Weyl metal phase due to correlations with strong magnetic fluctuations. This non-Fermi liquid type Weyl metal state may be regarded to be a novel metallic phase in the respect that a topologically nontrivial band structure appears in the vicinity of quantum criticality.

Dirac cohomology, the projective supermodules of the symmetric group and the Vogan morphism

Abstract We derive an explicit description of the genuine projective representations of the symmetric group Sn using Dirac cohomology and the branching graph for the irreducible genuine projective representations of Sn. Ciubotaru and He [D. Ciubotaru and X. He, Green polynomials of Weyl groups, elliptic pairings, and the extended index. Adv. Math., 283:1–50, 2015], using the extended Dirac index, showed that the characters of the projective representations of Sn are related to the characters of elliptic-graded modules. We derive the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties ℬe of g and are able to use Dirac cohomology to construct an explicit model for the projective representations. We also describe Vogan’s morphism for Hecke algebras in type A using spectrum data of the Jucys–Murphy elements.

Full counting statistics in the free Dirac theory

We study charge transport and fluctuations of the (3+1)-dimensional massive free Dirac theory. In particular, we focus on the steady state that emerges following a local quench whereby two independently thermalized halves of the system are connected and let to evolve unitarily for a long time. Based on the two-time von Neumann measurement statistics and exact computations, the scaled cumulant generating function associated with the charge transport is derived. We find that it can be written as a generalization of Levitov–Lesovik formula to the case in three spatial dimensions. In the massless case, we note that only the first four scaled cumulants are nonzero. Our results provide also a direct confirmation for the validity of the extended fluctuation relation in higher dimensions. An application of our approach to Lifshitz fermions is also briefly discussed.

Emergent scale invariance of nonclassical plasmons in graphene nanoribbons

Using a nearest-neighbor tight-binding model we investigate quantum effects of plasmons on few-nanometer wide graphene nanoribbons, both for zigzag and armchair edge terminations. With insight from the Dirac description we find an emerging scale-invariant behavior that deviates from the classical model both for zigzag and armchair structures. The onset of the deviation can be related to the position of the lowest parabolic band in the band structure. Dirac theory is only valid in the parameter subspace where the scale invariance holds that relates narrow ribbons with high doping to wide ribbons with low doping. We also find that the edge states present in zigzag ribbons give rise to a blueshift of the plasmon, in contrast to earlier findings for graphene nanodisks and nanotriangles.

Quantum gyroelectric effect: Photon spin-1 quantization in continuum topological bosonic phases

Topological phases of matter arise in distinct fermionic and bosonic flavors. The fundamental differences between them are encapsulated in their rotational symmetries - the spin. Although spin quantization is routinely encountered in fermionic topological edge states, analogous quantization for bosons has proven elusive. To this end, we develop the complete electromagnetic continuum theory characterizing 2+1D topological bosons, taking into account their intrinsic spin and orbital angular momentum degrees of freedom. We demonstrate that spatiotemporal dispersion (momentum and frequency dependence of linear response) captures the matter-mediated interactions between bosons and is a necessary ingredient for topological phases. We prove that the bulk topology of these 2+1D phases is manifested in transverse spin-1 quantization of the photon. From this insight, we predict two unique bosonic phases - one with even parity $C=\pm 2$ and one with odd $C=\pm 1$. To understand the even parity phase $C=\pm 2$, we introduce an exactly solvable model utilizing non-local optical Hall conductivity and reveal a single gapless photon at the edge. This unidirectional photon is spin-1 helically quantized, immune to backscattering, defects, and exists at the boundary of the $C=\pm 2$ bosonic phase and any interface - even vacuum. The contrasting phenomena of transverse quantization in the bulk, but longitudinal (helical) quantization on the edge is addressed as the quantum gyro-electric effect (QGEE). We also validate our bosonic Maxwell theory by direct comparison with the supersymmetric Dirac theory of fermions. To accelerate the discovery of such bosonic phases, we suggest two new probes of topological matter with broken time-reversal symmetry: momentum-resolved electron energy loss spectroscopy and cold atom near-field measurement of non-local optical Hall conductivity.

Particle-hole symmetry and composite fermions in fractional quantum Hall states

We study fractional quantum Hall states at filling fractions in the Jain sequences using the framework of composite Dirac fermions. Synthesizing previous work, we write down an effective field theory consistent with all symmetry requirements, including Galilean invariance and particle-hole symmetry. Employing a Fermi liquid description, we demonstrate the appearance of the Girvin--Macdonlald--Platzman algebra and compute the dispersion relation of neutral excitations and various response functions. Our results satisfy requirements of particle-hole symmetry. We show that while the dispersion relation obtained from the HLR theory is particle-hole symmetric, correlation functions obtained from HLR are not. The results of the Dirac theory are shown to be consistent with the Haldane bound on the projected structure factor, while those of the HLR theory violate it.

A first-order Lagrangian theory of fields with arbitrary spin

The bundles suitable for a description of higher-spin fields can be built in terms of a 2-spinor bundle as the basic “building block”. This allows a clear, direct view of geometric constructions aimed at a theory of such fields on a curved spacetime. In particular, one recovers the Bargmann–Wigner equations and the [Formula: see text]-dimensional representation of the angular-momentum algebra needed for the Joos–Weinberg equations. Looking for a first-order Lagrangian field theory we argue, through considerations related to the 2-spinor description of the Dirac map, that the needed bundle must be a fibered direct sum of a symmetric “main sector” — carrying an irreducible representation of the angular-momentum algebra — and an induced sequence of “ghost sectors”. Then one indeed gets a Lagrangian field theory that, at least formally, can be expressed in a way similar to the Dirac theory. In flat spacetime, one gets plane-wave solutions that are characterized by their values in the main sector. Besides symmetric spinors, the above procedures can be adapted to anti-symmetric spinors and to Hermitian spinors (the latter describing integer-spin fields). Through natural decompositions, the case of a spin-2 field describing a possible deformation of the spacetime metric can be treated in terms of the previous results.

Galilean field theories and conformal structure

We perform a detailed analysis of Galilean field theories, starting with free theories and then interacting theories. We consider non-relativistic versions of massless scalar and Dirac field theories before we go on to review our previous construction of Galilean Electrodynamics and Galilean Yang-Mills theory. We show that in all these cases, the field theories exhibit non-relativistic conformal structure (in appropriate dimensions). The surprising aspect of the analysis is that the non-relativistic conformal structure exhibited by these theories, unlike relativistic conformal invariance, becomes infinite dimensional even in spacetime dimensions greater than two. We then couple matter with Galilean gauge theories and show that there is a myriad of different sectors that arise in the non-relativistic limit from the parent relativistic theories. In every case, if the parent relativistic theory exhibited conformal invariance, we find an infinitely enhanced Galilean conformal invariance in the non-relativistic case. This leads us to suggest that infinite enhancement of symmetries in the non-relativistic limit is a generic feature of conformal field theories in any dimension.

Axially Symmetric Solution of the Weyl–Dirac Theory of Gravitation and the Problem of Rotation Curves of Galaxies

On the basis of the Poincare´–Weyl gauge theory of gravitation, a new conformal Weyl–Dirac theory of gravitation was proposed, formulated in Cartan–Weyl space-time with the Dirac scalar field representing a model of dark matter. A static approximate axially symmetric solution of the field equations in vacuum is obtained. On the basis of this solution, in the Newtonian approximation we consider the problem of rotation velocities in the spiral components of galaxies.

Measuring Zitterbewegung predicted by the Dirac equation for a free electron

The fact that the Dirac equation predicts Zitterbewegung (ZBW) for a free electron is well known from Schrodinger [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] from the very early days of Dirac's relativistic quantum theory. Schrodingerdescribed ZBW qualitatively as a persistent interference between positive and negative energy states, and the objective of this and previous papers by this author [Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] is to describe ways to measure the presence or absence ofZBW for a free (or nearly free) electron. If ZBW measurements show that it is not present for a free electron, then the Dirac Equation and Schrodinger's analysis [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] are inaccurate. If ZBW ispresent as Schrodinger's analysis [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] of the Dirac Equation asserts, then a description of a free electron emerges that is significantly different than the present very small, static point electronwith “intrinsic” properties of spin and magnetic moment accepted by most physicists [Eides et al., Phys. Reports 342, 63 (2001).] today. This paper develops a quantitative theory of the free electron directly from the Dirac Equation and Schrodinger's analyses[E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] without any other assumptions. The fact that ZBW exists for the electron in the hydrogen atom is confirmed by the Darwin term [J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley,Reading, MA, 1967)]. It seems unnatural to the author that ZBW would vanish when the electron is removed from the hydrogen atom, and becomes free. The unanswered question is whether ZBW can be confirmed or refuted by measurement for a free electron, and possible ways to measure its presenceor absence are described in this paper. The remainder of the paper describes the free electron, assuming ZBW exists. The Dirac Equation [J. H. Wilson, Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] produces a physical description of a free electron's ZBW as rapidly oscillating,quantized Center of Charge (CoC) point rotating about a fictitious Center of Mass (CoM) at a radius of 3/ 2 times the electron's Compton wavelength. The results in the remainder of this paper, as well as Schrodinger's analyses [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.)24, 418 (1930).], are only true if ZBW can be confirmed by measurement. In case measurements show no ZBW exists for a free electron, the Dirac Equation is not a correct description of a single free electron, and more serious problems arise. The static point like electron deduced fromscattering experiments [Eides et al., Phys. Rep. 342, 63 (2001)] that give estimates of the free electron's internal structure based on far field measurements can be misleading in estimating the free electron's internal structure. In Classical Electrodynamics, when one observesthe electric field from a point electron and from a sphere with the same charge spread uniformly over its surface from the far field, Gauss' Law shows that the electric fields are identical. When one tries to “observe” or “measure” the free electron's internal structurewith the high-energy photons required, the single particle free electron is completely destroyed by the measurement process. In fact, any attempt to observe or measure a free electron's structure ensures that the electron is no longer a totally free single particle, and one must be carefulto use only elastic observation methods [J. H. Wilson, Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] for this very difficult measurement. Observing a free Quark, much less its internal structure, has proven very difficult, if not impossible.

One-W-type Modules for Rational Cherednik Algebra and Cuspidal Two-sided Cells

We classify the simple modules for the rational Cherednik algebra that are irreducible when restricted to W, in the case when W is a finite Weyl group. The classification turns out to be closely related to the cuspidal two-sided cells in the sense of Lusztig. We compute the Dirac cohomology of these modules and use the tools of Dirac theory to find nontrivial relations between the cuspidal Calogero-Moser cells and the cuspidal two-sided cells.

A model-dependent approach to the non-relativistic Lamb shift

The precise observation of the Lamb shift, between the $2s_{1/2}$ and $2p_{1/2}$ levels in hydrogen, was a genuine motivation for the development of modern quantum electrodynamics. According to Dirac theory, the $2s_{1/2}$ and $2p_{1/2}$ levels should have equal energies. However, “radiative corrections” due to the interaction between the atomic electron and the vacuum, shift the $2s_{1/2}$ level higher in energy by around $4.37493\times 10^{-6}$ eV or $ 2\pi\hbar\times 1057.85$ MHz relative to the $2p_{1/2}$ level. The measurement of Lamb and Retherford provided the stimulus for renormalization theory which has been so successful in handling troublesome divergences. The Lamb shift is still a central theme in atomic physics. W.E. Lamb was the first to see that this tiny shift, so elusive and hard to measure, would clarify in a fundamental way our thinking about particles and fields. In this article, the Lamb shift for the 2s energy level in hydrogen is assessed for three different electron models by using the variational principle. It is then verified that this shift arises mostly from the interaction of a bound electron with the zero-point fluctuations of the free electromagnetic field (Welton’s interpretation). We briefly comment on the construct validity of the proposed electron models.

Momentum-space cigar geometry in topological phases

.In this paper, we stress the importance of momentum-space geometry in the understanding of two-dimensional topological phases of matter. We focus, for simplicity, on the gapped boundary of three-dimensional topological insulators in class AII, which are described by a massive Dirac Hamiltonian and characterized by an half-integer Chern number. The gap is induced by introducing a magnetic perturbation, such as an external Zeeman field or a ferromagnet on the surface. The quantum Bures metric acquires a central role in our discussion and identifies a cigar geometry. We first derive the Chern number from the cigar geometry and we then show that the quantum metric can be seen as a solution of two-dimensional non-Abelian BF theory in momentum space. The gauge connection for this model is associated to the Maxwell algebra, which takes into account the Lorentz symmetries related to the Dirac theory and the momentum-space magnetic translations connected to the magnetic perturbation. The Witten black-hole metric is a solution of this gauge theory and coincides with the Bures metric. This allows us to calculate the corresponding momentum-space entanglement entropy that surprisingly carries information about the real-space conformal field theory describing the defect lines that can be created on the gapped boundary.