Orbital Angular Momentum Operators Research Articles

Conserved spin operator of Dirac’s theory in spatially flat FLRW space-times

New conserved spin and orbital angular momentum operators of Dirac’s theory on spatially flat FLRW space-times are proposed generalizing thus the recent results concerning the role of Pryce’s spin operator in the flat case (Cotăescu in Eur Phys J C, 82, 1073, 2022). These operators split the conserved total angular momentum generating the new spin and orbital symmetries that form the rotations of the isometry groups. The new spin operator is defined and studied in active mode with the help of a suitable spectral representation giving its Fourier transform. Moreover, in the same manner is defined the operator of the fermion polarization. The orbital angular momentum is derived in passive mode using a new method, inspired by Wigner’s theory of induced representations, but working properly only for global rotations. In this approach the quantization is performed finding that the one-particle spin and orbital angular momentum operators have the same form in any FLRW spacetime regardless their concrete geometries given by various scale factors.

Quantum commutation relationship for photonic orbital angular momentum

Orbital Angular Momentum (OAM) of photons are ubiquitously used for numerous applications. However, there is a fundamental question whether photonic OAM operators satisfy standard quantum mechanical commutation relationship or not; this also poses a serious concern on the interpretation of an optical vortex as a fundamental quantum degree of freedom. Here, we examined canonical angular momentum operators defined in cylindrical coordinates, and applied them to Laguerre-Gauss (LG) modes in a graded index (GRIN) fibre. We confirmed the validity of commutation relationship for the LG modes and found that ladder operators also work properly with the increment or the decrement in units of the Dirac constant (ℏ). With those operators, we calculated the quantum-mechanical expectation value of the magnitude of angular momentum, which includes contributions from both intrinsic and extrinsic OAM. The obtained results suggest that OAM characterised by the LG modes exhibits a well-defined quantum degree of freedom.

Spin and orbital angular momentum of coherent photons in a waveguide

Spin angular momentum of a photon corresponds to a polarisation degree of freedom of lights, and such that various polarisation properties are coming from macroscopic manifestation of quantum-mechanical properties of lights. An orbital degree of freedom of lights is also manipulated to form a vortex of lights with orbital angular momentum, which is also quantised. However, it is considered that spin and orbital angular momentum of a photon cannot be split from the total orbital angular momentum in a gauge-invariant way. Here, we revisit this issue for a coherent monochromatic ray from a laser source, propagating in a waveguide. We obtained the helical components of spin and orbital angular momentum by the correspondence with the classical Ponyting vector. By applying a standard quantum field theory using a coherent state, we obtained the gauge-independent expressions of spin and orbital angular momentum operators. During the derivations, it was essential to take a finite cross-sectional area into account, which leads the finite longitudinal component along the direction of the propagation, which allows the splitting. Therefore, the finite mode profile was responsible to justify the splitting, which was not possible as far as we were using plane-wave expansions in a standard theory of quantum-electrodynamics (QED). Our results suggest spin and orbital angular momentum are well-defined quantum-mechanical freedoms at least for coherent photons propagating in a waveguide and in a vacuum with a finite mode profile.

The role of Pryce’s spin and coordinate operators in the theory of massive Dirac fermions

It is shown that the components of Pryce’s spin operator of Dirac’s theory are SU(2) generators of a representation carried by the space of Pauli’s spinors determining the polarization of the plane wave solutions of Dirac’s equation. These operators are conserved via Noether’s theorem such that new conserved polarization operators can be defined for various polarizations. The corresponding one-particle operators of quantum theory are derived showing how these are related to the isometry generators of the massive Dirac fermions of any polarization, including momentum-dependent ones. In this manner, the problem of separating conserved spin and orbital angular momentum operators is solved naturally. Moreover, the operator proposed by Pryce as a mass-center coordinate is studied, showing that after quantization, this becomes in fact the dipole one-particle operator. As an example, the quantities determining the principal one-particle operators are derived for the first time in a momentum-helicity basis.

Analytic interpolation between the Ji and Jaffe-Manohar definitions of the orbital angular momentum distribution of gluons at small x

In this work, we first introduced a generalized Wilson line gauge link that reproduces both staple and near straight links along a light cone in different parameter limits. We then studied the gauge-invariant bilocal orbital angular momentum operator with such a general gauge link. At the appropriate combination of limits, the operator reproduces both Jaffe-Manohar's and Ji's operator structures and offers a continuous analytical interpolation between the two.

Quantum field theory for spin operator of the photon

All elementary particles in nature can be classified as fermions with half-integer spin and bosons with integer spin. Within quantum electrodynamics (QED), even though the spin of the Dirac particle is well defined, there exist open questions on the quantized description of spin of the gauge field particle -- the photon. Using quantum field theory, we discover the quantum operators for the spin angular momentum (SAM) $\boldsymbol{S}_{M}=(1/c)\int d^{3}x\boldsymbol{\pi}\times\boldsymbol{A}$ and orbital angular momentum (OAM) $\boldsymbol{L}_{M}=-(1/c)\int d^{3}x\pi^{\mu}\boldsymbol{x}\times\boldsymbol{\nabla}A_{\mu}$ of the photon, where $\pi^{\mu}$ is the conjugate canonical momentum of the gauge field $A^{\mu}$. We also reveal a perfect symmetry between the angular momentum commutation relations for Dirac fields and Maxwell fields. We derive the well-known OAM and SAM of classical electromagnetic fields from the above defined quantum operators. Our work shows that the spin and OAM operators commute which is important for simultaneously observing and separating the SAM and OAM. The correct commutation relations of orbital and spin angular momentum of the photon have applications in quantum optics, topological photonics as well as nanophotonics and can be extended in the future for the spin structure of nucleons.

Orbital Hall effect in bilayer transition metal dichalcogenides: From the intra-atomic approximation to the Bloch states orbital magnetic moment approach

Using an effective Dirac model, we study the orbital Hall effect (OHE) in bilayers of transition metal dichalcogenides with 2H stacking (2H-TMD). We use first-order perturbation theory in the interlayer coupling of the bilayer system to obtain analytical expressions for the orbital Hall conductivity in the linear response regime. We use two distinct descriptions of the orbital angular momentum (OAM) operator. The first one is the intra-atomic approximation that considers only the intrasite contribution to the OAM [Cysne et al., Phys. Rev. Lett. 126, 056601 (2021)]. The second one uses the Berry phase formula of the orbital (valley) magnetic moment to describe the OAM operator [Bhowal and Vignale, Phys. Rev. B 103, 195309 (2021)]. This approach includes both intersite and intrasite contributions to the OAM. Our results suggest that the two approaches agree qualitatively in describing the OHE in bilayers of 2H-TMDs, although they present some quantitative differences. We also show that interlayer coupling plays an essential role in understanding the OHE in the unbiased bilayer of 2H-TMD. This coupling causes the Bloch states to become bonding (antibonding) combinations of states of individual layers, demanding the consideration of the non-Abelian structure of the orbital magnetic moment to the occurrence of OHE. As we discuss throughout the work, the emerging picture of transport of OAM in the unbiased bilayer of 2H-TMDs based on OHE is very different from the usual picture based on the valley Hall effect, shedding new light on previous experimental results. We also discuss the effect of the inclusion of a gate-voltage bias in the bilayer system. Our work gives support to recent theoretical predictions on OHE in two-dimensional materials.

Orbital Dynamics in Centrosymmetric Systems.

Orbital dynamics in time-reversal-symmetric centrosymmetric systems is examined theoretically. Contrary to common belief, we demonstrate that many aspects of orbital dynamics are qualitatively different from spin dynamics because the algebraic properties of the orbital and spin angular momentum operators are different. This difference generates interesting orbital responses, which do not have spin counterparts. For instance, the orbital angular momentum expectation values may oscillate even without breaking neither the time-reversal nor the inversion symmetry. Our quantum Boltzmann approach reproduces the previous result on the orbital Hall effect and reveals additional orbital dynamics phenomena, whose detection schemes are discussed briefly. Our work will be useful for the experimental differentiation of the orbital dynamics from the spin dynamics.

Revisiting the compatibility problem between the gauge principle and the observability of the canonical orbital angular momentum in the Landau problem

As is widely-known, the eigen-functions of the Landau problem in the symmetric gauge are specified by two quantum numbers. The first is the familiar Landau quantum number n, whereas the second is the magnetic quantum number m, which is the eigen-value of the canonical orbital angular momentum (OAM) operator of the electron. The eigen-energies of the system depend only on the first quantum number n, and the second quantum number m does not correspond to any direct observables. This seems natural since the canonical OAM is generally believed to be a gauge-variant quantity, and observation of a gauge-variant quantity would contradict a fundamental principle of physics called the gauge principle. In recent researches, however, Bliokh et al. analyzed the motion of helical electron beam along the direction of a uniform magnetic field, which was mostly neglected in past analyses of the Landau states. Their analyses revealed highly non-trivial m-dependent rotational dynamics of the Landau electron, but the problem is that their papers give an impression that the quantum number m in the Landau eigen-states corresponds to a genuine observable. This compatibility problem between the gauge principle and the observability of the quantum number m in the Landau eigen-states was attacked in our previous letter paper. In the present paper, we try to give more convincing answer to this delicate problem of physics, especially by paying attention not only to the particle-like aspect but also to the wave-like aspect of the Landau electron.

On the rules for filling electron shells and the properties of the atomic Hamiltonian

The aim of this work is to determine the dependence of the total energy of the atomic system on the mutual orientation of the orbital angular moments of electrons. It is shown that the minimum value of the energy is attained for those configurations that are an eigenfunction of the orbital angular momentum operator . Thus, at least for individual atoms of the periodic table it is shown that Dirac’s postulate on the necessity of a wave function to be an eigenfunction of is a consequence that follows from the properties of the Hamiltonian, and it is not a complementary condition. Additionally, it is shown that the second Hund’s rule is incorrect with respect to the mutual orientation of the electron orbital angular momentum projections.

Twisted Magnon as a Magnetic Tweezer.

Wave fields with spiral phase dislocations carrying orbital angular momentum (OAM) have been realized in many branches of physics, such as for photons, sound waves, electron beams, and neutrons. However, the OAM states of magnons (spin waves)-the building block of modern magnetism-and particularly their implications have yet to be addressed. Here, we theoretically investigate the twisted spin-wave generation and propagation in magnetic nanocylinders. The OAM nature of magnons is uncovered by showing that the spin-wave eigenmode is also the eigenstate of the OAM operator in the confined geometry. Inspired by optical tweezers, we predict an exotic "magnetic tweezer" effect by showing skyrmion gyrations under twisted magnons in the exchange-coupled nanocylinder-nanodisk heterostructure, as a practical demonstration of magnonic OAM transfer to manipulate topological spin defects. Our study paves the way for the emerging magnetic manipulations by harnessing the OAM degree of freedom of magnons.

Synthetic spin-orbit coupling mediated by a bosonic environment

We study a mobile quantum impurity, possessing internal rotational degrees of freedom, confined to a ring in the presence of a many-particle bosonic bath. By considering the recently introduced rotating polaron problem, we define the Hamiltonian and examine the energy spectrum. The weak-coupling regime is studied by means of a variational ansatz in the truncated Fock space. The corresponding spectrum indicates that there emerges a coupling between the internal and orbital angular momenta of the impurity as a consequence of the phonon exchange. We interpret the coupling as a phonon-mediated spin-orbit coupling and quantify it by using a correlation function between the internal and orbital angular momentum operators. The strong-coupling regime is investigated within the Pekar approach and it is shown that the correlation function of the ground state shows a kink at a critical coupling, that is explained by a sharp transition from the non-interacting state to the states that exhibit strong interaction with the surroundings. The results might find applications in such fields as spintronics or topological insulators, where spin-orbit coupling is of crucial importance.

Singularity of a relativistic vortex beam and proper relativistic observables

We have studied the phase singularity of relativistic vortex beams for two sets of relativistic operators using circulation. One set includes new spin and orbital angular momentum (OAM) operators, which are derived from the parity-extended Poincaré group, and the other set consists of the (usual) Dirac spin and OAM operators. The first set predicts the same singularity in the circulation as in the case of nonrelativistic vortex beams. On the other hand, the second set anticipates that the singularity of the circulation is spin-orientation-dependent and can disappear, especially for a relativistic paraxial electron beam with spin parallel to the propagating direction. These contradistinctive predictions suggest that a relativistic electron beam experiment with spin-polarized electrons could for the first time answer a long-standing fundamental question, i.e., what are the proper relativistic observables, raised from the beginning of relativistic quantum mechanics following the discovery of the Dirac equation.

Orbital angular momentum constraints in the variational optimization of the two-electron reduced-density matrix

The direct variational determination of the two-electron reduced-density matrix (2-RDM) is usually carried out under the assumption that the 2-RDM is a real-valued quantity. However, in systems that possess orbital angular momentum symmetry, the description of states with a well-defined, non-zero z-projection of the orbital angular momentum requires a complex-valued 2-RDM. We consider a semidefinite program suitable for the direct optimization of a complex-valued 2-RDM and explore the role of orbital angular momentum constraints in systems that possess the relevant symmetries. For atomic systems, constraints on the expectation values of the square and z-projection of the orbital angular momentum operator allow one to optimize 2-RDMs for multiple orbital angular momentum states. Similarly, in linear molecules, orbital angular momentum projection constraints enable the description of multiple electronic states, and, moreover, for states with a non-zero z-projection of the orbital angular momentum, the use of complex-valued quantities is essential for a qualitatively correct description of the electronic structure. For example, in the case of molecular oxygen, we demonstrate that orbital angular momentum constraints are necessary to recover the correct energy ordering of the lowest-energy singlet and triplet states near the equilibrium geometry. However, care must still be taken in the description of the dissociation limit, as the 2-RDM-based approach is not size consistent, and the size-consistency error varies dramatically, depending on the z-projections of the spin and orbital angular momenta.

Magnetic anisotropy of ferromagnetic metals in low-symmetry systems

We have constructed an analytic formula to treat the perpendicular magnetic anisotropy energy in ferromagnetic metals with low symmetry, such as C4V and C3V. We find that the anisotropy energy is proportional to a part of the expectation values of the orbital angular momentum and magnetic dipole operator. Although the result is similar to the model proposed by Laan (1998) [2], we have derived a concrete expression for the spin-flip virtual excitation process term, which can be dominant in atoms with small magnetic moments and/or small exchange splitting. Pt monatomic layer with proximity-induced spin polarization grown on Fe is an example of this. Other multilayer systems such as Co/Pd and Co/Ni, and bilayer systems such as Fe(CoB)/MgO can be discussed similarly. Moreover, the relation between perpendicular magnetic anisotropy energy and measurable physical parameters is discussed based on X-ray magnetic circular dichroism spectroscopy.

Differential technique for the covariant orbital angular momentum operators

The orbital angular momentum operator expansion turns to be a powerful tool to construct the fully covariant partial wave amplitudes of hadron decay reactions and hadron photo- and electroproduction processes. In this paper we consider a useful development of the orbital angular momentum operator expansion method. We present the differential technique allowing the direct calculation of convolutions of two orbital angular momentum operators with an arbitrary number of open Lorentz indices. This differential technique greatly simplifies calculations when the reaction subject to the partial wave analysis involves high spin particles in the initial and/or final states. We also present a useful generalization of the orbital angular momentum operators.

Spin force and torque in non-relativistic Dirac oscillator on a sphere

The spin force operator on a non-relativistic Dirac oscillator (in the non-relativistic limit the Dirac oscillator is a spin one-half 3D harmonic oscillator with strong spin–orbit interaction) is derived using the Heisenberg equations of motion and is seen to be formally similar to the force by the electromagnetic field on a moving charged particle. When confined to a sphere of radius R, it is shown that the Hamiltonian of this non-relativistic oscillator can be expressed as a mere kinetic energy operator with an anomalous part. As a result, the power by the spin force and torque operators in this case are seen to vanish. The spin force operator on the sphere is calculated explicitly and its torque is shown to be equal to the rate of change of the kinetic orbital angular momentum operator, again with an anomalous part. This, along with the conservation of the total angular momentum, suggests that the spin force exerts a spin-dependent torque on the kinetic orbital angular momentum operator in order to conserve total angular momentum. The presence of an anomalous spin part in the kinetic orbital angular momentum operator gives rise to an oscillatory behavior similar to the Zitterbewegung. It is suggested that the underlying physics that gives rise to the spin force and the Zitterbewegung is one and the same in NRDO and in systems that manifest spin Hall effect.

Angular momentum of a relativistic wave packet

Quantum-mechanical spin is often thought of in terms of classical angular momentum. In fact spin is defined by its commutation relations and the spin and orbital angular momentum operators are very different. Here we solve the Dirac equation in a rotating frame of reference and create a localized wave packet from the resulting wave functions to examine the consequences of the rotational motion. We highlight some unexpected effects in the properties of the wave packet and show that the spin operator and orbital angular momentum operator describe different aspects of the rotational properties of the wave packet. It is also observed that neither quantum-mechanical spin nor orbital angular momentum can be fully understood within an inertial frame.

Weak magnetism correction to allowed β decay for reactor antineutrino spectra

The weak magnetism correction and its uncertainty to nuclear beta-decay play a major role in determining the significance of the reactor neutrino anomaly. Here we examine the common approximation used for one-body weak magnetism in the calculation of fission antineutrino spectra, wherein matrix elements of the orbital angular momentum operator contribution to the magnetic dipole current are assumed to be proportional to those of the spin operator. Although we find this approximation invalid for a large set of nuclear structure situations, we conclude that it is valid for the relevant allowed beta-decays between fission fragments. In particular, the uncertainty in the fission antineutrino due to the uncertainty in the one-body weak magnetism correction is found to be less than 1%. Thus, the dominant uncertainty from weak magnetism for reactor neutrino fluxes lies in the uncertainty in the two-body meson-exchange magnetic dipole current.

Hidden orbital polarization in diamond, silicon, germanium, gallium arsenide and layered materials

It was previously believed that the Bloch electronic states of non-magnetic materials with inversion symmetry cannot have finite spin polarizations. However, since the seminal work by Zhang et al. (Nat. Phys. 10, 387–393 (2014)) on local spin polarizations of Bloch states in non-magnetic, centrosymmetric materials, the scope of spintronics has been significantly broadened. Here, we show, using a framework that is universally applicable independent of whether hidden spin polarizations are small (e.g., diamond, Si, Ge and GaAs) or large (e.g., MoS2 and WSe2), that the corresponding quantity arising from orbital—instead of spin—degrees of freedom, the hidden orbital polarization is (i) much more abundant in nature since it exists even without spin–orbit coupling and (ii) more fundamental since the interband matrix elements of the site-dependent orbital angular momentum operator determine the hidden spin polarization. We predict that the hidden spin polarization of transition metal dichalcogenides is reduced significantly upon compression. We suggest experimental signatures of hidden orbital polarization from photoemission spectroscopies and demonstrate that the current-induced hidden orbital polarization may play a far more important role than its spin counterpart in antiferromagnetic information technology by calculating the current-driven antiferromagnetism in compressed silicon.