Yang Guo-Hong

Stability of Disclinations in Nematic Liquid Crystals

In the light of -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 structure of disclination lines in 2-dimensional liquid crystals

Using φ-mapping method and topological current theory, the topological structure of disclination lines in 2-dimensional liquid crystals is studied. By introducing the strength density and the topological current of many disclination lines, it is pointed out that the disclination lines are determined by the singulaities of the director field, and topologically quantized by the Hopf indices and Brouwer degrees. Due to the equivalence in physics of the director fields n(x) and −n(x), the Hopf indices can be integers or half-integers, representing a generalization of our previous studies of integer Hopf indices.

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

Using -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.

Reduced Properties and Applications of Y(sl(2)) Algebra for a Two-Spin System

By taking a special constraint for a general realization of Y(sl(2)), two sets of sl(2) algebras are presented, in which a u(1) algebra is hidden. With the help of this constraint, the block-diagonal form can be written to the generator J of Yangian algebras, and especially it is a rotational transformation of a spin in the elementary quantum mechanics. This sheds new light on the physical meaning of Y(sl(2)).

Topological Aspects of Entropy and Phase Transition of Kerr Black Holes

In the light of topological current and the relationship between the entropy and the Euler characteristic, the topological aspects of entropy and phase transition of Kerr black holes are studied. From Gauss–Bonnet–Chern theorem, it is shown that the entropy of Kerr black holes is determined by the singularities of the Killing vector field of spacetime. By calculating the Hopf indices and Brouwer degrees of the Killing vector field at the singularities, the entropy S = A/4 for nonextreme Kerr black holes and S = 0 for extreme ones are obtained, respectively. It is also discussed that, with the change of the ratio of mass to angular momentum for unit mass, the Euler characteristic and the entropy of Kerr black holes will change discontinuously when the singularities on Cauchy horizon merge with the singularities on event horizon, which will lead to the first-order phase transition of Kerr black holes.

Effect of Disclination Lines on Free Energy of Nematic Liquid Crystals

In the light of -mapping method and topological current theory, the effect of disclination lines on the free energy density of nematic liquid crystals is studied. 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 disclination lines. The other is the saddle-splay energy density, which is shown to be centralized at the disclination lines and to be topologically quantized in the unit of kπ/2 when the Jacobian determinant of the director field does not vanish at the singularities of the director field. The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of the director field at the disclination lines, i.e., the disclination strengthes. When the Jacobian determinant vanishes, the generation, annihilation, intersection, splitting and merging processes of the saddle-splay energy density are detailed in the neighborhoods of the limit points and bifurcation points, respectively. It is shown that the disclination line with high topological quantum number is unstable and will evolve to the low topological quantum number states through the splitting process.

Rank-based deactivation model for networks with age

We study the impact of age on network evolution which couples addition of new nodes and deactivation of old ones. During evolution, each node experiences two stages: active and inactive. The transition from the active state to the inactive one is based on the rank of the node. In this paper, we adopt age as a criterion of ranking, and propose two deactivation models that generalize previous research. In model A, the older active node possesses the higher rank, whereas in model B, the younger active node takes the higher rank. We make a comparative study between the two models through the node-degree distribution.

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.

Euler Characteristic and Topological Phase Transition of NUT-Kerr-Newman Black Hole

From the Gauss–Bonnet–Chern theorem, the Euler characteristic of NUT-Kerr-Newman black hole is calculated to be some discrete numbers from 0 to 2. We find that the Bekenstein-Hawking entropy is the largest entropy in topology by taking into account of the relationship between the entropy and the Euler characteristic. The NUT-Kerr-Newman black hole evolves from the torus-like topological structure to the spherical structure with the changes of mass, angular momentum, electric and NUT charges. In this process, the Euler characteristic and the entropy are changed discontinuously, which give the topological aspect of the first-order phase transition of NUT-Kerr-Newman black hole. The corresponding latent heat of the topological phase transition is also obtained. The estimated latent heat of the black hole evolving from the star just lies in the range of the energy of gamma ray bursts.

Intrinsic topological structure of entropy of Kerr black holes

In the light of φ-mapping method and the relationship between entropy and the Euler characteristic, the intrinsic topological structure of entropy of Kerr black holes is studied. From the Gauss-Bonnet-Chem theorem, it is shown that the entropy of Kerr black hole is determined by singularities of the Killing vector field of spacetime. These singularities naturally carry topological numbers, Hopf indices and Brouwer degrees, which can also be viewed as topological quantization of entropy of Kerr black holes. Specific results S=A/4 for non-extreme Kerr black holes and S=0 for extreme ones are calculated independently by using the bove-mentioned methods.