Third-rank Tensor Research Articles

Magnetoelectric polarizability and optical activity: Spin and frequency dependence

We extend a microscopic theory of polarization and magnetization to include the spin degree of freedom of the electrons, introducing a general spin-orbit coupling and Zeeman interaction term in the Hamiltonian. At finite frequencies and including spin, the magnetoelectric polarizability tensor is replaced by two separate tensors, one that relates the polarization P to the magnetic field B and a separate tensor that relates the magnetization M to the electric field E. When combined with other relevant response tensors a third-rank tensor that relates the induced current density to gradients in the electric field can be introduced; it is gauge invariant, in a form suitable for numerical calculations, and describes optical activity---including spin effects---even in materials that may lack time-reversal symmetry.

Hydrodynamics of quantum corrections to the Coulomb interaction via the third rank tensor evolution equation: application to Langmuir waves and spin-electron acoustic waves

The quantum effects in plasmas can be described by the hydrodynamics containing the continuity and Euler equations. However, novel quantum phenomena are found via the extended set of hydrodynamic equations, where the pressure evolution equation and the pressure flux third-rank tensor evolution equation are included. These give the quantum corrections to the Coulomb interaction. The spectra of the Langmuir waves and the spin-electron acoustic waves are calculated. The application of the pressure evolution equation ensures that the contribution of pressure in the Langmuir wave spectrum is proportional to $(3/5)v_{\textrm {Fe}}^{2}$ rather than $(1/3)v_{\textrm {Fe}}^{2}$ , where $v_{\textrm {Fe}}$ is the Fermi velocity.

Spatial dispersion contribution to second harmonic generation in inversion-symmetric materials

It is well known that second harmonic generation (SHG) from dipoles within the bulk vanishes in inversion-symmetric semiconductor materials as a consequence of parity symmetry. Hence SHG is then ascribed in the form of either surface dipole effects or bulk-related electric quadrupole or magnetic dipole effects. By incorporating the redefined spatial dispersion into the simplified bond-hyperpolarizability model, we show that the SHG spatial dispersion contribution for Si(001) and Si(111) facet orientations can be reformulated by a third-rank tensor containing one independent parameter, namely, the complex SHG spatial dispersion hyperpolarizability. Our results show that certain unexplained rotational anisotropy SHG intensity features for different incoming light wavelengths or optical penetration depths can be well reproduced only if the contribution from spatial dispersion is incorporated.

Positive definiteness in coupled strain gradient elasticity

The linear theory of coupled gradient elasticity has been considered for hemitropic second gradient materials, specifically the positive definiteness of the strain and strain gradient energy density, which is assumed to be a quadratic form of the strain and of the second gradient of the displacement. The existence of the mixed, fifth-rank coupling term significantly complicates the problem. To obtain inequalities for the positive definiteness including the coupling term, a diagonalization in terms of block matrices is given, such that the potential energy density is obtained in an uncoupled quadratic form of a modified strain and the second gradient of displacement. Using orthonormal bases for the second-rank strain tensor and third-rank strain gradient tensor results in matrix representations for the modified fourth-rank and the sixth-rank tensors, such that Sylvester’s formula and eigenvalue criteria can be applied to yield conditions for positive definiteness. Both criteria result in the same constraints on the constitutive parameters. A comparison with results available in the literature was possible only for the special case that the coupling term vanishes. These coincide with our results.

Liquid crystal distortions revealed by an octupolar tensor.

The classical theory of liquid crystal elasticity as formulated by Oseen and Frank describes the (orientable) optic axis of these soft materials by a director n. The ground state is attained when n is uniform in space; all other states, which have a nonvanishing gradient ∇n, are distorted. This paper proposes an algebraic (and geometric) way to describe the local distortion of a liquid crystal by constructing from n and ∇n a third-rank, symmetric, and traceless tensor A (the octupolar tensor). The (nonlinear) eigenvectors of A associated with the local maxima of its cubic form Φ on the unit sphere (its octupolar potential) designate the directions of distortion concentration. The octupolar potential is illustrated geometrically and its symmetries are charted in the space of distortion characteristics, so as to educate the eye to capture the dominating elastic modes. Special distortions are studied, which have everywhere either the same octupolar potential or one with the same shape but differently inflated.

Effect of magnetic anisotropy on direct chiral discrimination in paramagnetic NMR spectroscopy.

We present and analyse a complete analytic expression for the generalized shielding polarizability third-rank tensor Φαβγ, whose isotropic average, the pseudoscalar [capital Phi, Greek, macron], is proportional to a chiral macroscopic electric polarization P measurable in a pulsed nuclear magnetic resonance (NMR) experiment probing molecules with an arbitrarily degenerate ground state, in solution. We have recently predicted that [capital Phi, Greek, macron] can be measurable at room temperature in chiral paramagnetic molecules, and identified strong magnetic anistropy (as featured e.g. in lanthanide complexes) as a crucial molecular property to achieve room temperature chiral discrimination using NMR spectroscopy. As previously proposed, the components of Φαβγ are obtained as analytical third derivatives of the electronic free energy. Here we present the explicit calculation of these derivatives, which provide working expressions for the explicit accurate ab initio calculation of Φαβγ. We apply our theory by performing ab initio multiconfigurational calculations of all contributions to Φαβγ, for a set of ten DyIII complexes, characterized by a strongly axial ground Kramers doublet, but also by thermally accessible excited Kramers doublets at room temperature. The results show that the thermally populated excited state contributions, while generally reducing the value of [capital Phi, Greek, macron] calculated on the assumption of a thermally isolated ground state, still confirm the room temperature detectability of this property for all ten studied complexes. Trends on the relative sign of dominant contributions are then discussed on the basis of a crystal field model electrostatic potential splitting a ground spin-orbit multiplet, which provides an insight into the properties of the generalized shielding polarizability tensor for open shell species.

Quantization of spin Hall conductivity in two-dimensional topological insulators versus symmetry and spin-orbit interaction

The third-rank tensor of the static spin Hall conductivity is investigated for two-dimensional (2D) topological insulators by electronic structure calculations. Its seeming quantization is numerically demonstrated for highly symmetric systems independent of the gap size. 2D crystals with hexagonal and square Bravais lattice show similar effects, while true rectangular translational symmetry yields conductivity values much below the quantum $e^2/h$. Field-induced lifting the inversion symmetry does not influence the quantum spin Hall state up to band inversion but the conductivity quantization. Weak symmetry-conserving biaxial but also uniaxial strain has a minor influence as long as inverted gaps dictate the topological character. The results are discussed in terms of the atomic geometry and the Rashba contribution to the spin-orbit interaction (SOI). Translational and point-group symmetry as well as SOI rule the deviation from the quantization of the spin Hall conductance.

Octupolar tensors for liquid crystals

A third-rank three-dimensional symmetric traceless tensor, called the octupolar tensor, has been introduced to study tetrahedratic nematic phases in liquid crystals. The octupolar potential, a scalar-valued function generated on the unit sphere by that tensor, should ideally have four maxima (on the vertices of a tetrahedron), but it was recently found to possess an equally generic variant with three maxima instead of four. It was also shown that the irreducible admissible region for the octupolar tensor in a three-dimensional parameter space is bounded by a dome-shaped surface, beneath which is a separatrix surface connecting the two generic octupolar states. The latter surface, which was obtained through numerical continuation, may be physically interpreted as marking a possible intra-octupolar transition. In this paper, by using the resultant theory of algebraic geometry and the E-characteristic polynomial of spectral theory of tensors, we give a closed-form, algebraic expression for both the dome-shaped surface and the separatrix surface. This turns the envisaged intra-octupolar transition into a quantitative, possibly observable prediction.

Curved Witten-Dijkgraaf-Verlinde-Verlinde equation and N=4 mechanics

We propose a generalization of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation from ${\mathbb R}^n$ to an arbitrary Riemannian manifold. Its form is obtained by extending the relation of the WDVV equation with ${\cal N}{=}\,4$ supersymmetric $n$-dimensional mechanics from flat to curved space. The resulting `curved WDVV equation' is written in terms of a third-rank Codazzi tensor. For every flat-space WDVV solution subject to a simple constraint we provide a curved-space solution on any isotropic space, in terms of the rotationally invariant conformal factor of the metric.

Next point-of-interest recommendation via a category-aware Listwise Bayesian Personalized Ranking

In recent years, location-based social networks (LBSNs) have attracted much attention, and next point-of-interest (POI) recommendation has become an important task for LBSNs. However, previous efforts suffer from the high computational complexity, besides the transition pattern between POIs has not been well studied. In this paper, we proposed a two-fold approach for next POI recommendation. First, we predict the next category by using a third-rank tensor optimized by a Listwise Bayesian Personalized Ranking (LBPR) approach. Specifically, we introduce two functions, namely Plackett–Luce model and cross entropy, to generate the likelihood of a ranking list for posterior computation. The optimization criteria of LBPR are derived from a perspective of Bayesian analysis for ranking problems, and we formulate the listwise loss functions by two probability model. Then POI candidates filtered by the predicated category are ranked based on the spatial influence and category ranking influence. By explicit usage of category information to infer the user transition pattern in category-level, the proposed models are ideally better with the next new POI recommendation problem. The experiments on two real-world datasets demonstrate the significant improvements of our methods over several state-of-the-art methods.

New Boundary-Driven Twist States in Systems with Broken Spatial Inversion Symmetry

A full description of a magnetic sample includes a correct treatment of the boundary conditions (BCs). This is in particular important in thin film systems, where even bulk properties might be modified by the properties of the boundary of the sample. We study generic ferromagnets with broken spatial inversion symmetry and derive the general micromagnetic BCs of a system with Dzyaloshinskii-Moriya interaction (DMI). We demonstrate that the BCs require the full tensorial structure of the third-rank DMI tensor and not just the antisymmetric part, which is usually taken into account. Specifically, we study systems with C_{∞v} symmetry and explore the consequences of the DMI. Interestingly, we find that the DMI already in the simplest case of a ferromagnetic thin film leads to a purely boundary-driven magnetic twist state at the edges of the sample. The twist state represents a new type of DMI-induced spin structure, which is completely independent of the internal DMI field. We estimate the size of the texture-induced magnetoresistance effect being in the range of that of domain walls.

Two-measure theory with third-rank antisymmetric tensor for local scale symmetry breaking

Two-measure theory with third-rank antisymmetric tensor for local scale symmetry breaking

Octupolar order in three dimensions.

Octupolar order in three space dimensions is described by a real-valued, fully symmetric and traceless, third-rank tensor A. The real generalized eigenvalues of A are also the critical values of a real-valued potential [Formula: see text] defined on the unit sphere [Formula: see text] by A. Generalized eigenvalues of A and critical points of [Formula: see text] are equivalent means to describe octupolar order in a molecular assembly according to Buckingham's formula for the probability density distribution. Intuition suggests that [Formula: see text] would generically have four maxima, corresponding to the most probable molecular orientations, so that a (possibly distorted) tetrahedron would effectively describe A. This paper shows that another generic octupolar state flanks the expected one, featuring three maxima of [Formula: see text]. The two generic states are divided by a separatrix manifold, which may physically represent an intra-octupolar transition.

Effective successive POI recommendation inferred with individual behavior and group preference

The prevalence of smartphones and mobile social networks allow the users to share their location-based life experience much easier. The large amount of data generated in related location-based social networks provides informative cues on user's behaviors and preferences to support personalized location-based services, like point-of-interest (POI) recommendation. Yet achieving accurate personalized POI recommendation is challenging as the data available for each user is highly sparse. In addition, the computational complexity is high due to the large number of users. In this paper, a novel methodology for personalized successive POI recommendation is proposed. First, the preferred successive category of location is predicted using a third-rank tensor computed based on the partially observed transitions between the categories of user's successive locations where the missing transitions are uncovered by inferring the group preference. The group is achieved according to users' demographics and frequently visited locations. Then, a bipartite graph is constructed based on the recommended categories for each user. To obtain the personalized ranking of locations, a distance weighted HITS algorithm is proposed so that the location authority score is updated iteratively according to the visiting frequency of the group and some distance constraints. The proposed two-step approach with the category prediction incorporated aims to boost the location prediction performance via the smoothing and at the same time reduce the complexity. Experimental results obtained based on the real-world location-based social network data show that the proposed approach outperforms the existing state-of-the-art methods by a large margin.

Inferring a Personalized Next Point-of-Interest Recommendation Model with Latent Behavior Patterns

In this paper, we address the problem of personalized next Point-of-interest (POI) recommendation which has become an important and very challenging task in location-based social networks (LBSNs), but not well studied yet. With the conjecture that, under different contextual scenario, human exhibits distinct mobility patterns, we attempt here to jointly model the next POI recommendation under the influence of user's latent behavior pattern. We propose to adopt a third-rank tensor to model the successive check-in behaviors. By incorporating softmax function to fuse the personalized Markov chain with latent pattern, we furnish a Bayesian Personalized Ranking (BPR) approach and derive the optimization criterion accordingly. Expectation Maximization (EM) is then used to estimate the model parameters. Extensive experiments on two large-scale LBSNs datasets demonstrate the significant improvements of our model over several state-of-the-art methods.

The internal-strain tensor of crystals for nuclear-relaxed elastic and piezoelectric constants: on the full exploitation of its symmetry features.

Symmetry features of the internal-strain tensor of crystals (whose components are mixed second-energy derivatives with respect to atomic displacements and lattice strains) are formally presented, which originate from translational-invariance, atomic equivalences, and atomic invariances. A general computational scheme is devised, and implemented into the public Crystal program, for the quantum-mechanical evaluation of the internal-strain tensor of crystals belonging to any space-group, which takes full-advantage of the exploitation of these symmetry-features. The gain in computing time due to the full symmetry exploitation is documented to be rather significant not just for high-symmetry crystalline systems such as cubic, hexagonal or trigonal, but also for low-symmetry ones such as monoclinic and orthorhombic. The internal-strain tensor is used for the evaluation of the nuclear relaxation term of the fourth-rank elastic and third-rank piezoelectric tensors of crystals, where, apart from a reduction of the computing time, the exploitation of symmetry is documented to remarkably increase the numerical precision of computed coefficients.

On Preventing Location Attacks for Urban Vehicular Networks

The prevalence of global positioning system (GPS) equipped in vehicular networks exposes users’ location information to the location-based services. We argue that such data contains rich informative cues on drivers’ private behaviors and preferences, which will lead to the location privacy attacks. In this paper, we proposed a sophisticated prediction model to predict driver’s next location by using ak-order Markov chain-based third-rank tensor representing the partially observed transfer information of vehicles. Then Bayesian Personalized Ranking (BPR) is used to learn the unobserved transitions within the tensor for transition predication. Experimental results manifest the efficacy of the proposed model in terms of location predication accuracy, compared with several state-of-the-art predication methods. We also point out that the precision achieved by such advanced predication model is restricted to the order of the Markov chaink. Accordingly, we propose a schema to decrease the risks of such attacks by preventing the conformation of higher order Markov chain. Experimental results obtained based on the real-world vehicular network data demonstrated the effectiveness of our proposed schema.

Octupolar order in two dimensions.

Octupolar order is described in two space dimensions in terms of the maxima (and conjugated minima) of the probability density associated with a third-rank, fully symmetric and traceless tensor. Such a representation is shown to be equivalent to diagonalizing the relevant third-rank tensor, an equivalence which however is only valid in the two-dimensional case.

Single crystal plasticity with bend–twist modes

In this work a formulation is proposed and computationally implemented for rate dependent single crystal plasticity, which incorporates plastic bend–twist modes that arise from dislocation density based poly-slip mechanisms. The formulation makes use of higher order continuum theory and may be viewed as a generalized micromechanics model. The formulation is then linked to the burgers and Nye tensors, showing how their material rates are derivable from a newly proposed third-rank tensor Λp, which incorporates a crystallographic description of bend–twist plasticity through selectable slip-system level constitutive laws. A simple three-dimensional explicit finite element implementation is outlined and employed in three simulations: (a) bi-crystal bending; (b) tension on a notched single crystal; and (c) the large compression of a microstructure to induce the plastic buckling of secondary phases. All simulation are transient, for computational expediency. The results shed light on the physics resulting from dynamic inhomogeneous plastic deformation.

Simplified bond-hyperpolarizability model of second harmonic generation, group theory, and Neumann’s principle

We discuss the susceptibility third-rank tensor for second harmonic and sum-frequency generation, associated with low index surfaces of silicon (Si(001), Si(011), and Si(111)), from two different approaches: the simplified bond-hyperpolarizability model (SBHM) and group theory (GT). We show that the SBHM agrees very well with the experimental results for simple surfaces because the definitions of the bond vectors implicitly include the geometry of the crystal and therefore the symmetry group. However, for more complex surfaces it is shown that one can derive from GT the SBHM tensor, if Kleinman symmetry is allowed.