Linear Response Tensor Research Articles

Linear and nonlinear spin current response of anisotropic spin-orbit coupled systems

We calculate the linear and the second harmonic (SH) spin current response of two anisotropic systems with spin–orbit (SO) interaction. General expressions of wide applicability for the these response functions are first derived for a generic two-band Hamiltonian. The first system is a two-dimensional (2D) electron gas in the presence of Rashba and k-linear Dresselhaus SO couplings. The calculations show how narrow or wide the response spectra can be, what is their overall shape and size, and frequency shiftings, depending on which crystal orientation is selected. The quantitative knowing of this makes possible a comparative study for several orientations, which would allow to select a spectrum with particular characteristic. We find that vanishing linear and second order response tensors are achievable under SU(2) symmetry conditions, characterized by a collinear SO vector field. Additional conditions under which specific tensor components vanish are possible, without having such collinearity. Thus, a proper choice of the growth direction and SO strengths allows to select the polarization of the linear and SH spin currents according to the direction of flowing. The second system is an anisotropic 2D free electron gas with anisotropic Rashba interaction, which has been employed to study the optical conductivity of 2D puckered structures with anisotropic energy bands. The presence of mass anisotropy and an energy gap open several distinct scenarios for the allowed optical interband transitions, which manifest in the linear and SH response contrastingly. The linear response displays only out-of-plane spin polarized currents, while the SH spin currents flow with spin orientation lying parallel to the plane of the system strictly. The models illustrate the possibility of the nonlinear spin Hall effect in systems with SO interaction, under the presence or absence of time-reversal symmetry. The results suggest different ways to manipulate the linear and nonlinear optical generation of spin currents which could find spintronic applications.

Light propagation in (2+1) -dimensional electrodynamics: The case of linear constitutive laws

In this paper, we turn our attention to light propagation in three-dimensional electrodynamics. More specifically, we investigate the behavior of light rays in a continuous bi-dimensional hypothetical medium living in a three-dimensional ambient spacetime. Relying on a fully covariant approach, we assume that the medium is endowed with a local and linear response tensor which maps field strengths into excitations. In the geometric optics limit, we then obtain the corresponding Fresnel equation and, using well-known results from algebraic geometry, we derive the effective optical metric.

Deep neural network compression by Tucker decomposition with nonlinear response

Deep neural networks have shown impressive performance in many areas, including computer vision and natural language processing. Millions of parameters in deep neural network limit its deployment in low-end devices due to intensive memory and expensive computational cost. In the literature, several network compression techniques based on tensor decompositions have been proposed to compress deep neural networks. Existing techniques are designed in each network unit by approximating linear response or kernel tensor using various tensor decomposition methods. What is more, research has shown that there exists significant redundancy between different filters and feature channels of kernel tensor in each convolution layer. In this paper, we propose a new algorithm to compress deep neural network by considering both nonlinear response and the multilinear low-rank constraint in the kernel tensor. To overcome the resulted difficulty of nonconvex optimization, we propose a convex relaxation scheme such that it can be solved by alternating direction method of multipliers (ADMM) directly. Thus, the Tucker-2 rank and the feature matrix of Tucker decomposition can be determined simultaneously. The effectiveness of the proposed method is evaluated on CIFAR-10 and large-scale ILSVRC12 datasets for CNNs including ResNet-18, AlexNet and GoogleNet. According to our numerical computation, the proposed method is able to obtain highly reduction in model size with a small loss in accuracy. The compression performance of the proposed method is better than existing methods.

Systematic Analysis Method for Nonlinear Response Tensors

We propose a systematic analysis method for identifying essential parameters in various linear and nonlinear response tensors without which they vanish. By using the Keldysh formalism and the Chebyshev polynomial expansion method, the response tensors are decomposed into the model-independent and dependent parts, in which the latter is utilized to extract the essential parameters. An application of the method is demonstrated by analyzing the nonlinear Hall effect in the ferroelectric SnTe monolayer for example. It is shown that in this example the second-neighbor hopping is essential for the nonlinear Hall effect whereas the spin–orbit coupling is unnecessary. Moreover, by analyzing terms contributing to the essential parameters in the lowest order, the appearance of the nonlinear Hall effect can be interpreted by the subsequent two processes: the orbital magneto-current effect and the linear anomalous Hall effect by the induced orbital magnetization. In this way, the present method provides a microscopic picture of responses. By combining with computational analysis, it stimulates further discoveries of anomalous responses by filling in a missing link among hidden degrees of freedom in a wide variety of materials.

Symmetry of linear dielectric response tensors: Dispersion models fulfilling three fundamental conditions

Physically correct dispersion models must fulfill three fundamental conditions (time-reversal symmetry, Kramers–Kronig consistency, and conformity with sum rules). The application of these conditions on systems exhibiting low crystal symmetry, spatial dispersion, and/or magneto-optic effects is a non-trivial task. The aim of this contribution is to present an approach using decomposition of dielectric tensors into a set of independent spectral functions. For the derivation, the most general case of anisotropic dielectric response with optical activity is considered. The contribution discusses both the natural optical activity exhibiting spatial dispersion and the local magneto-optic effect of rotation of the plane of polarization induced by the external magnetic field. If the response tensor is expressed up to the term linear in the direction of the wave vector, then its symmetry can be classified into 16 types. Formulas expressing each type of the dielectric tensor using independent spectral functions are presented (the most complex case with the lowest symmetry requires 15 spectral functions). The symmetry for different internal and external conditions is demonstrated with the help of several simple models based on solving the classical equations of motion. It is shown that interpreting free particles in the magnetic field as bound particles is not correct. Instead, the Landau levels in a non-dissipative system must be interpreted as splitting of diamagnetic part of the dielectric response, rather than energy of bound states.

Quantum kinetic theory for unmagnetized and magnetized plasmas

Three different approaches to the treatment of quantum effects in plasmas are reviewed: quantum fluid theory (QFT), phase-space kinetic theory (PKT) and quantum plasmadynamics (QPD). The simplest form of QFT is analogous to a nonrelativistic fluid model for an unmagnetized plasma with a potential electric field, $$\phi $$ . The wave nature of the electron is included through the so-called Bohm term, as in the Madelung equations. The simplest form of PKT is based on the Wigner function and the kinetic equation that it satisfies in a quasi-classical $$\mathbf{p}$$ – $$\mathbf{x}$$ phase space. Further development of PKT involves including additional effects piecemeal. The electron spin is included through a classical model for a spin vector, $$\mathbf{s}$$ , with the phase space extended to include $$\mathbf{s}$$ . The inclusion of degeneracy is straightforward. Electromagnetic effects are described by including the vector potential, $$\mathbf{A}$$ , in Schrodinger’s equation. It is argued that further extensions, to a magnetized plasma and to include relativistic effects, raise conceptual difficulties concerning the phase-space approach. In a magnetic field, the electron (Landau) states are discrete, whereas the kinetic equation in PKT involves a derivative with respect to $$\mathbf{p}$$ . The relativistic case is based on Dirac’s equation, and to derive a Vlasov-like equation for electrons requires excluding the positron and virtual-pair contributions to the Dirac wavefunction, which cannot be achieved exactly. Moreover, specific spin states require a specific choice of spin operator in Dirac’s theory, whereas the Pauli matrices define the only (vectorial) spin operator in PKT. The Dirac wavefunctions for two spin operators are derived and shown to approximate the eigenstates of the Pauli theory only in the nonrelativistic limit. QPD is an exact theory, based on quantum electrodynamics, in which kinetic processes are described using Feynman diagrams. The presence of the plasma is taken into account through a statistical average of the electron propagator, analogous to the use of thermal Green functions. QPD is introduced for the unmagnetized case, and the generalization to include a background magnetic field is presented. It is shown how QPD is used to derive the linear response tensor for an electron gas in the relativistic quantum case for both unmagnetized and magnetized plasmas. A magnetized vacuum is shown to have response tensors analogous to a plasma, allowing processes (such as one-photon pair creation and photon splitting) that are forbidden in an unmagnetized vacuum. The various quantum effects that may be relevant to a plasma are summarized, and their possible application to laboratory and astrophysical plasmas are discussed briefly.

Vibrational (resonance) Raman optical activity with real time time dependent density functional theory.

We present a novel approach for the calculation of vibrational (resonance) Raman optical activity (ROA) spectra based on real time propagation. The ROA linear electronic response tensors are formulated in a propagator formalism in order to treat linear response (LR-) and real time time dependent density functional theory (RT-TDDFT) on equal footing. The length, mixed, and velocity representations of these tensors are discussed with respect to the potential origin dependence of the ROA invariants in the calculations. The propagator formalism allows a straight forward extension of the optical LR tensors in a mixed or velocity representation to a coupling with nonlocal potentials, where an extra term appears in the definition of the momentum operator, in order to maintain the gauge invariance. Using RT-TDDFT paves the way for an innovative, efficient calculation of both on- and off-resonance ROA spectra. Exemplary results are given for the off-resonance and (pre-)resonance spectra of (R)-methyloxirane, considering the resonance effects due to one or more electronically excited states. Moreover, the developed real time propagation approach allows us to obtain entire excitation profiles in a computationally efficient way.

Atomic-Scale Magnetic Toroidal Dipole under Odd-Parity Hybridization

Magnetic toroidal dipole (MTD) is one of a fundamental constituent to induce magneto-electric effects in the absence of both spatial inversion and time-reversal symmetries. We report on a microscopic investigation of the atomic-scale MTD in solids by taking into account the orbital degree of freedom with a different parity. We construct an effective two-orbital $d$-$f$ tight-binding model on a polar tetragonal system for describing the atomic-scale MTD, which are obtained by incorporating the atomic spin-orbit coupling and odd-parity hybridization. The effective model exhibits two types of the MTDs: in-plane $x, y$ components activated through spontaneous ferromagnetic ordering or external magnetic field and an out-of-plane $z$ component by a spontaneous odd-parity hybridization without spin moments. We show that the intra-orbital (inter-orbital) Coulomb interaction in multi-orbital systems plays an important role in stabilizing the in-plane (out-of-plane) MTD orderings. We also examine the magneto-electric effect under each MTD ordering by calculating a linear response tensor. We show that the odd-parity hybridization enhances the magneto-electric effect for the in-plane MTDs, while it suppresses that for the out-of-plane MTD.

Charge-spin response and collective excitations in Weyl semimetals

Weyl semimetals are characterized by unconventional electromagnetic response. We present analytical expressions for all components of the frequency- and wave-vector-dependent charge-spin linear-response tensor of Weyl fermions. The spin-momentum locking of the Weyl Hamiltonian leads to a coupling between charge and longitudinal spin fluctuations, while transverse spin fluctuations remain decoupled from the charge. A real Weyl semimetal with multiple Weyl nodes can show this charge-spin coupling in equilibrium if its crystal symmetry is sufficiently low. All Weyl semimetals are expected to show this coupling if they are driven into a non-equilibrium stationary state with different occupations of Weyl nodes, for example by exploiting the chiral anomaly. Based on the response tensor, we investigate the low-energy collective excitations of interacting Weyl fermions. For a local Hubbard interaction, the charge-spin coupling leads to a dramatic change of the zero-sound dispersion: its velocity becomes independent of the interaction strength and the chemical potential and is given solely by the Fermi velocity. In the presence of long-range Coulomb interactions, the coupling transforms the plasmon modes into spin plasmons. For real Weyl semimetals with multiple Weyl nodes, the collective modes are strongly affected by the presence of parallel static electric and magnetic fields, due to the chiral anomaly. In particular, the zero-sound frequency at fixed momentum and the spin content of the spin plasmons go through cusp singularities as the chemical potential of one of the Weyl cones is tuned through the Weyl node. We discuss possible experiments that could provide smoking-gun evidence for Weyl physics.

Classification of atomic-scale multipoles under crystallographic point groups and application to linear response tensors

Four types of atomic-scale multipoles, electric, magnetic, magnetic toroidal, and electric toroidal multipoles, give a complete set to describe arbitrary degrees of freedom for coupled charge, spin, and orbital of electrons. We here present a systematic classification of these multipole degrees of freedom towards the application in condensed matter physics. Starting from the multipole description under the rotation group in real space, we generalize the concept of multipoles in momentum space with the spin degree of freedom. We show how multipoles affect the electronic band structures and linear responses, such as the magneto-electric effect, magneto-current (magneto-gyrotropic) effect, spin conductivity, Piezo-electric effect, and so on. Moreover, we exhibit a complete table to represent the active multipoles under 32 crystallographic point groups. Our comprehensive and systematic analyses will give a foundation to identify enigmatic electronic order parameters and a guide to evaluate peculiar cross-correlated phenomena in condensed matter physics from microscopic point of view.

Symmetry-imposed shape of linear response tensors

A scheme suggested in the literature to determine the symmetry-imposed shape of linear response tensors is revised and extended to allow for the treatment of more complex situations. The extended scheme is applied to discuss the shape of the spin conductivity tensor for all magnetic space groups. This allows in particular investigating the character of longitudinal as well as transverse spin transport for arbitrary crystal structure and magnetic order that give rise e.g.\ to the spin Hall, Nernst and the spin-dependent Seebeck effects. In addition we draw attention to a new longitudinal spin transport phenomenon occurring in certain nonmagnetic solids.

Spin-orbit-induced longitudinal spin-polarized currents in nonmagnetic solids

For certain non-magnetic solids with low symmetry the occurrence of spin-polarized longitudinal currents is predicted. These arise due to an interplay of spin-orbit interaction and the particular crystal symmetry. This result is derived using a group-theoretical scheme that allows investigating the symmetry properties of any linear response tensor relevant to the field of spintronics. For the spin conductivity tensor it is shown that only the magnetic Laue group has to be considered in this context. Within the introduced general scheme also the spin Hall- and additional related transverse effects emerge without making reference to the two-current model. Numerical studies confirm these findings and demonstrate for (Au$_{1-x}$Pt$_{\rm x}$)$_4$Sc that the longitudinal spin conductivity may be in the same order of magnitude as the conventional transverse one. The presented formalism only relies on the magnetic space group and therefore is universally applicable to any type of magnetic order.

Propagation Effects in Magnetized Transrelativistic Plasmas

The transfer of polarized radiation in magnetized and nonmagnetized relativistic plasmas is an area of research with numerous flaws and gaps. The present paper is aimed at filling some gaps and eliminating the flaws. Starting from a Trubnikov's linear response tensor for a vacuum wave with |k| = ω/c in thermal plasma, the analytic expression for the dielectric tensor is found in the limit of high frequencies. The Faraday rotation and Faraday conversion measures are computed in their first orders in the ratio of the cyclotron frequency Ω0 to the observed frequency ω. The computed temperature dependencies of propagation effects bridge the known nonrelativistic and ultrarelativistic limiting formulae. The fitting expressions are found for high temperatures, where the higher orders in Ω0/ω cannot be neglected. The plasma eigenmodes are found to become linearly polarized at much larger temperatures than thought before. The results are applied to the diagnostics of the hot interstellar medium, hot accretion flows, and jets.

Dispersive properties of a magnetized degenerate electron gas

Accurate dispersion relations are fundamental in the evaluation of many astrophysical processes such as, for example, heat loss by neutrino emission. In this paper, the parallel propagating wave modes of a magnetized, degenerate, electron gas are evaluated exactly using the S-matrix form of the linear response tensor (for a relativistic quantum magnetized electron gas) and the vertex formalism. The modes are then discussed in terms of the dissipative regions in which they occur. In addition to analogous modes to the unmagnetized case, there also exist doublet modes due to the logarithmic singularites that occur in the dissipative regimes of both the longitudinal and transverse dispersion relations and modes above the gyromagnetic thresholds for one of the transverse components. These additional modes only appear over a restricted range of wave numbers and are unique to the magnetized case.

Local-field study of phase conjugation in metallic quantum wells with probe fields of both propagating and evanescent character

The phase conjugated response from nonmagnetic multi-level metallic quantum wells is analyzed and an essentially complete analytical solution is presented and discussed. The description is based on a semi-classical local-field theory for degenerate four-wave mixing in mesoscopic interaction volumes of condensed media developed by the present authors [T. Andersen and O. Keller, Phys. Scripta 58, 132 (1998)]. The analytical solution is supplemented by a numerical analysis of the phase conjugated response from a two-level quantum well in the case where one level is below the Fermi level and the other level is above. This is the simplest configuration of a quantum well phase conjugator in which the light-matter interaction can be tuned to resonance. The phase conjugated response is examined in the case where all the scattering takes place in one plane, and linearly polarized light is used in the mixing. In the numerical work we study a two-monolayer thick copper quantum well using the infinite barrier model potential. Our results show that the phase conjugated response from such a quantum-well system is highly dependent on the spatial dispersion of the matter response. The resonances showing up in the numerical results are analytically identified from the expressions for the linear and nonlinear response tensors. In addition to the general discussion of the phase conjugated response with varying frequency and parallel component of the wavevector, we present the phase conjugated response in the special case where the light is in resonance with the interband transition.

Response of a relativistic anisotropic thermal plasma. Part 1. Unmagnetized particles

Expressions for the linear response tensor are derived using a covariant forward-scattering method for unmagnetized relativistic thermal distributions restricted to one-dimensional (strictly parallel) and two-dimensional (strictly perpendicular) motions. The freedom to make Lorentz transformations is used to generalize the strictly perpendicular distribution to a counterpart of the nonrelativistic Dory–Guest–Harris (DGH) distribution.

The response tensor for a highly relativistic magnetized thermal plasma

The transverse part of the linear response tensor is evaluated for a highly relativistic thermal electron gas, starting from Trubnikov's response tensor for an arbitrary temperature. Three contributions to the response tensor are important. The diagonal components are dominated by an unmagnetized term, which gives the familiar dispersion k2=w2p0, where wp0 is the proper plasma frequency. The difference between the diagonal components is larger in magnitude than the off-diagonal components, implying that the natural modes are nearly linearly polarized. This leads to a generalized form of Faraday rotation in which linear polarization is partially converted into circular polarization at a rate per unit path length ∝λ3 (λ=wavelength).

Response of a relativistic anisotropic thermal plasma. Part 2. Magnetized particles

The linear response tensor for a relativistic counterpart of a bi-Maxwellian distribution is calculated in closed form for a magnetized plasma. The method used involves the following steps: (1) a covariant form of the response tensor for an arbitrary distribution of particles is derived using both a forward-scattering method and a covariant version of Vlasov theory; (2) the response tensor is evaluated for a strictly perpendicular, thermal distribution using a method due to Trubnikov; (3) the parallel distribution is built up by applying a Lorentz transformation to sum over a weighted, continuous distribution of parallel speeds. A convenient starting point for a detailed investigation of waves in such a plasma is the small-gyroradius approximation to the general expression for the response tensor, and the relevant approximation is derived and discussed briefly. The expansion of the general expression for the response tensor in Bessel functions in given in an appendix.

A novel construction of the response tensors for a collisionless unmagnetized plasma

The linear and nonlinear response tensors describe the response of a plasma to an electromagnetic field disturbance. A novel construction of the linear and quadratic response tensors for an unmagnetized collisionless plasma is presented here using a convenient 4-tensor notation. The construction uses the equations of charge continuity and gauge invariance, together with symmetry properties and dimensional arguments, to constrain the coefficients of an arbitrary linear combination of tensors. A new interpretation of the response tensors follows as expressions uniquely determined by fundamental physical constraints, and not simply as coefficients in a perturbation expansion of induced plasma response.

Characterization of the linear optical properties of a multiple quantum well structure in the sheet-model approximation

The possibility of determining the linear optical response tensor of a multiple quantum well (MQW) structure from optical reflection spectra is discussed. Taking into account local-field effects, the optical parameters of the MQW system are derived directly from the sheet conductivity tensor of the individual quantum well. For an amorphous Si/SiO 2 MQW structure deposited on a crystalline Si substrate, both s- and p-polarized reflection spectra were measured at different angles of incidence in the near-infrared and visible frequency regions. It was shown that the s-polarized spectra can be characterized by means of only one (complex) parameter, whereas in general it is impossible to account for the p-polarized spectra parametrically in a full range of angles of incidence even with four angular independent parameters. Finally, it was demonstrated that Feibelman's d-parameters can only be used to describe the experimental spectra at near-normal and grazing angles of incidence.