Topological Aspect and Bifurcation of Disclination Lines in Two—Dimensional Liquid Crystals

Using φ-mapping method and topological current theory,the topological structure and bifurcation of disclination lines in two-dimensional liquid crystals are studied.By introducing the strength density and the topological current of many disclination lines,the total disclination strength is topologically quantized by the Hopf indices and Brouwer degrees at the singularities of the director field when the Jacobian determinant of director field does not vanish.When the Jacobian determinant vanishes,the origin,annihilation and bifurcation processes of disclination lines are studied in the neighborhoods of the limit points and bifurcation points,respectively.The branch solutions at the limit point and the different directions of all branch curves at the bifurcation point are calculated with the conservation law of the topological quantum numbers.It is pointed out that a disclination line with a higher strength is unstable and it will evolve to the lower strength state through the bifurcation process.     

Contribution of Disclination Lines to Free Energy of Liquid Crystals in Single-Elastic Constant Approximation

In the light of C-mapping method and topological current theory, the contribution of disclination lines to free energy density of liquid crystals is studied in the single-elastic constant approximation. It is pointed out that the total free energy density can be divided into two parts. One is the usual distorted energy density of director field around the disclination lines. The other is the free energy density of disclination lines themselves, which is shown to be centralized at the disclination lines and to be topologically quantized in the unit of kn/2. The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of the director l~eld at the disclination lines, i.e. the disclination strengths. From the Lagrange‘s method of multipliers, the equilibrium equation and the molecular field ofliquid crystals are also obtained. The physical meaning of the Lagrangian multiplier is just the distorted energy density.     

Topological Aspects of Liquid Crystals

Using -mapping method and topological current theory, the properties and behaviors of disclination points in three-dimensional liquid crystals are studied. By introducing the strength density and the topological current of many disclination points, the total disclination strength is topologically quantized by the Hopf indices and Brouwer degrees at the singularities of the general director field when the Jacobian determinant of the general director field does not vanish. When the Jacobian determinant vanishes, the origin, annihilation, and bifurcation of disclination points are detailed in the neighborhoods of the limit point and bifurcation point, respectively. The branch solutions at the limit point and the different directions of all branch curves at the first- and second-order degenerated points are calculated. It is pointed out that a disclination point with a higher strength is unstable and will evolve to the lower strength state through the bifurcation process. An original disclination point can split into at most four disclination points at one time.     

Stability of Disclinations in Nematic Liquid Crystals

In the light of φ-mapping method and topological current theory, the stability of disclinations around a spherical particle in nematic liquid crystals is studied. We consider two different defect structures around a spherical particle: disclination ring and point defect at the north or south pole of the particle. We calculate the free energy of these different defects in the elastic theory. It is pointed out that the total Frank free energy density can be divided into two parts. One is the distorted energy density of director field around the disclinations. The other is the free energy density of disclinations themselves, which is shown to be concentrated at the defect and to be topologically quantized in the unit of (k-k₂₄)π/2. It is shown that in the presence of saddle-splay elasticity a dipole (radial and hyperbolic hedgehog) configuration that accompanies a particle with strong homeotropic anchoring takes the structure of a small disclination ring, not a point defect.     

Topological quantization of disclination points in three—dimensional liquid crystals

Using the φ-mapping method and topological current theory, we study the inner structure of disclination points in three-dimensional liquid crystals. By introducing the strength density and the topological current of many disclination points, it is pointed out that the disclination points are determined by the singularities of the general director field and they are topologically quantized by the Hopf indices and Brouwer degrees.     

Cascade “Melting” of a Linear Disclination in Chiral Nematic Droplets

A new structure of the elastic field of a chiral nematic liquid crystal in a cylindrical capillary with a singularity on the axis of the capillary in the form of a “melted” helical disclination has been obtained experimentally. It has been shown theoretically that the number of cylindrical surfaces, cascades at which a jump of the direction of the director orientation field n by π occurs, can be arbitrary. The particular number of cascades depends on the radius R of the cylinder and boundary conditions. An expression for the free energy of the elastic field has been obtained in the one-constant approximation of the continuum theory of liquid crystals.     

Topological and non inertial effects on the interband light absorption

In this work, we investigate the combined influence of the nontrivial topology introduced by a disclination and non inertial effects due to rotation, in the energy levels and the wave functions of a noninteracting electron gas confined to a two-dimensional pseudoharmonic quantum dot, under the influence of an external uniform magnetic field. The exact solutions for energy eigenvalues and wave functions are computed as functions of the applied magnetic field strength, the disclination topological charge, magnetic quantum number and the rotation speed of the sample. We investigate the modifications on the light interband absorption coefficient and absorption threshold frequency. We observe novel features in the system, including a range of magnetic field without corresponding absorption phenomena, which is due to a tripartite term of the Hamiltonian, involving magnetic field, the topological charge of the defect and the rotation frequency.     

Effect of Saddle-Splay Elasticity on Stability of Disclination Rings in Nematic Liquid Crystals

In this paper, the stability of disclination ring in nematic liquid crystals is studied. In the presence of saddle-splay elasticity (characterized by k₂₄) the disclination ring has a universal equilibrium radius. Depending on the values of the saddle-splay constant k₂₄, the universal equilibrium radius is altered. When k₂₄>0.92k (m=1/2) and k₂₄>0.88k (m=-1/2), the disclination will be a point rather than a ring, where $k$ is the Frank elastic constant in the one-constant approximation.     

Theory of Faraday rotation beats in quantum wells with large spin splitting

Spin dynamics of conduction electrons in a quantum well with a zinc blende structure is considered theoretically for the case where spin splitting exceeds the collisional broadening of energy levels. It is shown that, under certain conditions, the spin density component normal to the quantum well plane may oscillate with time even in the absence of an external magnetic field. These oscillations can be excited and detected using nonlinear two-pulse spectroscopy. Contrary to the case of small spin splitting, the external transverse magnetic field strongly affects spin dynamics in this regime.     

Excitons and excitonic complexes in GaAs/AlGaAs quantum wells with a low-density quasi-two-dimensional electron and hole channel

The recombination spectra of excitons and excitonic complexes in un-doped GaAs/AlGaAs single quantum wells are investigated. It is shown on the basis of a study of the magnetic-field dependence of the emission spectra and the degree of optical orientation in zero magnetic field and on the basis of electrooptic measurements that not only the density but also the sign of the charge carriers in a well depend strongly on the photoexcitation energy. It is shown on the basis of a comparative analysis of the spin splitting of the recombination lines of free and bound excitons that the recombination line which was attributed earlier to a positively charged exciton corresponds to the recombination of an exciton bound on a neutral acceptor. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 9, 707–713 (10 May 1998)     

“Aharonov-Bohm antiferromagnetism” and compensation points in the lattice of quantum rings

We investigate the magnetic properties of the lattice of non-interacting quantum rings using the 2D rotator model. The exact analytic expressions for the free energy as well as for the magnetization and magnetic susceptibility are found and analyzed. It is shown that such a system can be considered as a system with antiferromagnetic-like properties. We have shown also that all observable quantities in this case (free energy, entropy, magnetization) are periodic functions of the magnetic flux through the ring's area (as well known, such a behavior is typical for the Aharonov-Bohm effect). For the lattice of quantum rings with two different geometric parameters we investigate the ordinary compensation points (“temperature compensation points”, i.e. points at which the magnetization vanishes at fixed values of the magnetic field strength). It is shown that the positions of compensation points in the temperature scale are very sensitive to small changes in the magnetic field strength.     

Topology and geometry of nematic braids

Topological analysis of disclinations in nematic liquid crystals is an interesting and diverse topic that goes from strict mathematical theorems to applications in elaborate systems found in experiments and numerical simulations. The theory of nematic disclinations is shown from both the geometric and topological perspectives. Entangled disclination line networks are analyzed based on their shape and the behavior of their cross section. Methods of differential geometry are applied to derive topological results from reduced geometric information. For nematic braids, systems of −1/2 disclination loops, created by inclusion of homeotropic colloidal particles, a formalism of rewiring is constructed, allowing comparison and construction of an entire set of different conformations. The disclination lines are described as ribbons and a new topological invariant, the self-linking number, is introduced. The analysis is generalized from a constant −1/2 profile to general profile variations, while retaining the geometric treatment. The workings of presented topological statements are demonstrated on simple models of entangled nematic colloids, estimating the margins of theoretical assumptions made in the formal derivations, and reviewing the behavior of the disclinations not only under topological, but also under free-energy driven constraints.     

Topological approach to examine the singularity of the axial-vector current in an Abelian gauge field theory （QED）

A topological way to distinguish divergences of the Abelian axial-vector current in quantum field theory is proposed. By using the properties of the Atiyah-Singer index theorem, the nomtrivial Jacobian factor of the integration measure in the path-integral formulation of the theory is connected with the topological properties of the gauge field. The singularity of the fermion current related to the topological character can be correctly examined in a gauge background.     

Topological approach to examine the singularity of the axial-vector current in an Abelian gauge field theory (QED)

A topological way to distinguish divergences of the Abelian axial-vector current in quantum field theory is proposed. By usirg the properties of the Atiyah-Singer index theorem, the non-trivial Jacobian factor of the integration measure in the path-integral formulation of the theory is connected with the topological properties of the gauge field. The singularity of the fermion current related to the topological character can be correctly examined in a gauge background.     

Duality in the Azbel-Hofstadter Problem and Two-Dimensional d-Wave Superconductivity with a Magnetic Field

A single-parameter family of the lattice-fermion model is constructed. It is a deformation of the Azbel-Hofstadter problem by a parameter h = Delta/t (quantum parameter). A topological number is attached to each energy band. A duality between the classical limit ( h = +0) and the quantum limit ( h = 1) is revealed in the energy spectrum and the topological number. The model has a close relation to two-dimensional d-wave superconductivity with a magnetic field. Making use of the duality and a topological argument, we shed light on how quasiparticles with a magnetic field behave, especially in the quantum limit.     

The defect character of interface junction lines

Junction lines, where three or more interfaces meet in polycrystalline materials, are analysed from a topological point of view. Using circuit mapping methods, it is shown that, in contiguous polyerystals, the dislocations constituting the interfaces always react at junctions according to topological conservation principles. This conclusion is at variance with recent suggestions in the literature. In addition, it is shown that, in certain circumstances, junction lines can themselves exhibit defect character, i.e., dislocation and/or disclination character. Such defects arise in order to accommodate the coexistence of the abutting crystals. Simple examples are illustrated.     

The Origin and Bifurcation of the Space-Time Defects in the Early Universe

In the early universe, a new topological invariant is interpreted as the space-time dislocation flux and is quantized in the topological level. By extending to a topological current of dislocations, the dynamic form of the defects is obtained under the condition that the Jacobian determinant D(/u) 0. When D(/u) = 0, it is shown that there exists the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, the origin and bifurcation of the space-time dislocations are detailed in the neighborhoods of the limit points and bifurcation points of -mapping, respectively. It is pointed out that, since the dislocation current is identically conserved, the total topological quantum numbers of the branched dislocation fluxes will remain constant during their origin and bifurcation processes, which are important in the early universe because of spontaneous symmetry breaking.     

On mechanisms of the helix pitch variation in a thin cholesteric layer confined between two surfaces

Two possible mechanisms of the temperature-induced variation and jump of the helix pitch in a spatially bounded planar layer of a cholesteric liquid crystal (LC) was considered within the framework of the continuum theory of elasticity. These mechanisms are related to the existence of two configuration curves of the system free energy. The states with local free energy minima on each of the configuration curves correspond to topologically equivalent configurations of the LC director distribution and are quasi-equivalent in this sense. The transitions between such quasi-equivalent states are especially important in the first mechanism of the helix pitch jump proceeding without participation of defects. The second mechanism is related to transitions between the ground states of different configuration curves corresponding to topologically nonequivalent configurations. This mechanism requires either participation of disclination lines or the formation of defects.     

Arrays of Charged (±2π)-Twist Disclinations in Ferroelectric Liquid Crystals

The notion of an electrostatic charge of (±2)-twist disclinations is used to approximate the evaluation of the electrostatic interaction energy among disclinations forming arrays in finite samples of ferroelectric chiral smectic C liquid crystals. Screening effects of free charges in a material surrounding the disclination are taken into account by introducing a phenomenological depolarisation factor.The electrostatic interaction energy is important in chiral smectic C materials with high values of the spontaneous polarisation when screening effects of free charges are small. Then the electrostatic interaction leads to elimination of disclinations from the sample. When there is a high concentration of free charges in the sample (smaller value of depolarisation factor), the electrostatic interaction energy is of the order of the elastic interaction energy of disclinations what influences the equilibrium of disclination arrays in the sample. Two disclination configurations are considered. In the Brunet-Williams configuration the disclinations of opposite topological charge have also the opposite electrostatic charge so their attraction is augmented. This attraction can be balanced by the helical structure in the central part of the sample when the sample thickness is rather high.On the contrary, in the Glogarová-Pavel configuration the disclinations of opposite topological charge have the electrostatic charge of the same sign. The equilibrium in this configuration is either a balance of elastic attraction and electrostatic repulsion if elastic and Coulomb forces are of the same order or it is governed by the value of the anchoring energy when electrostatic interaction prevails over the elastic one.