Topological Structure of Vortex Solution in Jackiw-Pi Model

By using φ-mapping method, we discuss the topological structure of the self-duality solution in Jackiw-Pi model in terms of gauge potential decomposition. We set up relationship between Chern-Simons vortex solution and topological number, which is determined by Hopf index and Brouwer degree. We also give the quantization of flux in this case. Then, we study the angular momentum of the vortex, which can be expressed in terms of the flux.     

非相对论Chern-Simons多涡旋解的拓扑结构(英文)

运用规范势分解理论研究了Jackiw Pi模型中的自对偶方程, 得到一个新的自对偶方程, 发现了Chern Simons多涡旋解与拓扑荷之间的联系。为了研究Jackiw Pi模型多涡旋解的拓扑性质, 构造了一个新的静态自对偶Chern Simons多涡旋解,每个涡旋由5个实参数描述。 2个实参量用来描述涡旋的位置, 2个实参量用来描述涡旋的尺度和相位, 还有一个实参量描述涡旋的荷。 为了研究拓扑数对涡旋形状的影响, 给出了具有不同拓扑数的多涡旋解。 另外还研究了该涡旋解的磁通量的拓扑量子化。     

涡旋洛伦兹-高斯光束的远场矢量结构特征

利用矢量角谱法和稳相法,研究了涡旋洛伦兹-高斯光束的远场矢量结构特征,导出了横电项(TE项)和横磁项(TM项)远场电磁场和相应能流的解析表达式。通过相应的数值计算,分析了拓扑电荷数对涡旋洛伦兹-高斯光束及其矢量结构项远场能流分布的影响。TE项由位于竖直方向的2瓣或3瓣组成,TM项可由TE项旋转90得到。涡旋洛伦兹-高斯光束在拓扑电荷数小时内部中空,外部亮环均匀分布。增大拓扑电荷数,涡旋洛伦兹-高斯光束外部亮环上的能流呈起伏分布,内部变化相对复杂。涡旋洛伦兹-高斯光束及其矢量结构项的光斑尺寸随拓扑电荷数的增大而增大,但会饱和。研究显示,涡旋洛伦兹-高斯光束在实际应用时拓扑电荷数不宜过大。     

Vortex lines in a ferromagnetic spin-triplet superconductor

Based on Duan's topological current theory, we show that in a ferromagnetic spin-triplet superconductor there is a topological defect of string structures which can be interpreted as vortex lines. Such defects are different from the Abrikosov vortices in one-component condensate systems. We investigate the inner topological structure of the vortex lines. The topological charge density, velocity, and topological current of the vortex lines can all be expressed in terms of δ function, which indicates that the vortices can only arise from the zero points of an order parameter field. The topological charges of vortex lines are quantized in terms of the Hopf indices and Brouwer degrees of φ-mapping. The divergence of the self-induced magnetic field can be rigorously determined by the corresponding order parameter fields and its expression also takes the form of a δ-like function. Finally, based on the implicit function theorem and the Taylor expansion, we conduct detailed studies on the bifurcation of vortex topological current and find different directions of the bifurcation.     

Jackiw-Pi模型的新涡旋解

运用φ映射拓扑流理论研究了Jackiw-Pi模型中的自对偶方程，得到一个静态的自对偶解满足带有δ函数项的刘维尔方程, 从而得到了一个完整的带有拓扑信息的涡旋解,自然给出了磁通量子化.     

Direct observation of geometrical phase transitions in mesoscopic superconductors by scanning tunneling microscopy

Using scanning tunneling microscopy, we mapped the distribution of the local density of states in a single crystal superconductor heterostructure with an array of submicron normal metal islands. We observe the coexistence of strongly interacting multiquanta vortex lattice with interstitial Abrikosov vortices. The newly formed composite magnetic flux structure undergoes a series of phase transitions between different topological configuration states. The vortex configuration states are strongly dependent on the number of flux quanta and the nanoscale confinement architecture of the mesoscopic superconductor. Here, we present images of vortex phase transitions due to confinement effects when the number of magnetic flux quanta in the system changes. The vortex dynamics in these systems could serve as a model for behavior of confined many-body systems when the number of particles changes.     

SU(2)Chern-Simons涡旋解的拓扑结构

运用规范势分解理论研究了 Dunne Jackiw Pi Trugenberger 模型中的自对偶方程, 得到一个静态的自对偶Chern Simons多涡旋解, 每个涡旋由5个参数描述。 发现了自对偶解与拓扑数之间的关系, 而拓扑数由Brouwer度与Hopf指标确定。 同时, 也研究了该涡旋解的磁通量的拓扑量子化。The self dual equation and its solution in SU(2) Dunne Jackiw Pi Trugenberger model has Been discussed with special ansatz for the Lie algebraic structures of su(2) and gauge potential decomposition. We obtainer a new concrete self dual equation and found the relationship between SU(2) Chern Simons vortices and topological number which is determined by Hopf indices and Brouwer degrees of  mapping. (two positions, one scale, one phase per vortex and one charge of each vortex) m vortices solutions are discribed by using 5m parameters. The quantization of flux is also studied in this case.     

A note on Chern–Simons solitons—A type III vortex from the wall vortex

We study some properties of topological Chern–Simons vortices in 2+1 dimensions. As has already been understood in the past, in the large magnetic flux limit, they are well described by a Chern–Simons domain wall, which has been compactified on a circle with the symmetric phase inside and the asymmetric phase on the outside.Our goal is two-fold. First we want to explore how the tension depends on the magnetic flux discretized by the integer n. The BPS case is already known, but not much has been explored about the non-BPS potentials. A generic renormalizable potential has two dimensionless parameters that can be varied. Variation of only one of them leads to a type I and type II vortex, very similar to the Abrikosov–Nielsen–Olesen (ANO) case. Variation of both the parameters leads to a much richer structure. In particular we have found a new type of vortex, which is type I-like for small flux and then turns type II-like for larger flux. We could tentatively denote it a type III vortex. This results in a stable vortex with number of fluxes which can be greater than one.Our second objective is to study the Maxwell–Chern–Simons theory and understand how the large n limit of the CS vortex is smoothly connected with the large n limit of the ANO vortex.     

Separation of spin angular momentum in space-variant linearly polarized beam

We show that the spin angular momentum (SAM) flux in a space-variant linearly polarized beam can be separated in the focal plane. Such a beam carries only orbital angular momentum (OAM) and develops a net SAM flux upon focusing. The radial splitting of the SAM flux density is mediated by the phase vortex (or OAM) and can be controlled by the topological charge of the phase vortex. Optical trapping experiments verify the separation of the SAM flux density. The proposed approach enriches the manipulation of the angular momentum of light fields and inspires more designs of focus engineering, which would benefit optical micromanipulation of microscopic particles.     

Topological aspect of vortex lines in two-dimensional Gross—Pitaevskii theory

Using the -mapping topological theory, we study the topological structure of vortex lines in a two-dimensional generalized Gross-Pitaevskii theory in (3+1)-dimensional space-time. We obtain the reduced dynamic equation in the framework of the two-dimensional Gross-Pitaevskii theory, from which a conserved dynamic quantity is derived on the stable vortex lines. Such equations can also be used to discuss Bose-Einstein condensates in heterogeneous and highly nonlinear systems. We obtain an exact dynamic equation with a topological term, which is ignored in traditional hydrodynamic equations. The explicit expression of vorticity as a function of the order parameter is derived, where the δ function indicates that the vortices can only be generated from the zero points of Φ and are quantized in terms of the Hopf indices and Brouwer degrees. The -mapping topological current theory also provides a reasonable way to study the bifurcation theory of vortex lines in the two-dimensional Gross-Pitaevskii theory.     

Induced quantum numbers of a magnetic vortex at non-zero temperature

The phenomenon of the finite-temperature induced quantum numbers in fermionic systems with topological defects is analyzed. We consider an ideal gas of two-dimensional relativistic massive electrons in the background of a defect in the form of a pointlike magnetic vortex with arbitrary flux. This system is found to acquire, in addition to fermion number, also orbital angular momentum, spin, and induced magnetic flux, and we determine the functional dependence of the appropriate thermal averages and correlations on the temperature, the vortex flux, and the continuous parameter of the boundary condition at the location of the defect. We find that non-negativeness of thermal quadratic fluctuations imposes a restriction on the admissible range of values of the boundary parameter. The long-standing problem of the adequate definition of total angular momentum for the system considered is resolved.     

Dissociation of vortex stacks into fractional-flux vortices

We discuss the zero field superconducting phase transition in a finite system of magnetically coupled superconducting layers. Transverse screening is modified by the presence of other layers resulting in topological excitations with fractional flux. Vortex stacks trapping a full flux and present at any finite temperature undergo a dissociation transition which corresponds to the depairing of fractional-flux vortices in individual layers. We propose an experiment with a bilayer system allowing us to identify the dissociation of bound vortex molecules.     

Electronic properties of graphene with a topological defect

Various types of topological defects in graphene are considered in the framework of the continuum model for long-wavelength electronic excitations, which is based on the Dirac–Weyl equation. The condition for the electronic wave function is specified, and we show that a topological defect can be presented as a pseudomagnetic vortex at the apex of a graphitic nanocone; the flux of the vortex is related to the deficit angle of the cone. The cases of all possible types of pentagonal defects, as well as several types of heptagonal defects (with the numbers of heptagons up to three, and six) are analyzed. The density of states and the ground state charge are determined.     

Topological Majorana and Dirac zero modes in superconducting vortex cores

We provide an argument based on flux insertion to show that certain superconductors with a nontrivial topological invariant have protected zero modes in their vortex cores. This argument has the flavor of a two-dimensional index theorem and applies to disordered systems as well. It also provides a new way of understanding the zero modes in the vortex cores of a spinless px+ipy superconductor. Applying this approach to superconductors with and without time-reversal and spin-rotational symmetry, we predict the necessary and sufficient conditions for protected zero modes to exist in their vortices.     

Effect of an alternating field on the relaxation of effect of an alternating field on the relaxation of magnetic flux in a Josephson medium

We establish how trapped magnetic flux depends on the frequency and amplitude of an alternating field and how such a field affects the relaxation rate of the flux. We find that the nature of the flux creep changes in the process and that relaxation of the flux stops after the external field is switched off. We examine the dynamics of flux relaxation in a ring in the approximation in which the current density is assumed homogeneous, for various density dependences of the effective vortex activation energy. The critical current density and the vortex activation energy are obtained as functions of the external field strength. Finally, we explain the observed behavior in terms of the different field profiles emerging in the rings. Zh. éksp. Teor. Fiz. 111, 1047–1056 (March 1997)     

Microscale vortex laser with controlled topological charge

A microscale vortex laser is a new type of coherent light source with small footprint that can directly generate vector vortex beams. However, a microscale laser with controlled topological charge, which is crucial for virtually any of its application, is still unrevealed. Here we present a microscale vortex laser with controlled topological charge. The vortex laser eigenmode was synthesized in a metamaterial engineered non-Hermitian micro-ring cavity system at exceptional point. We also show that the vortex laser cavity can operate at exceptional point stably to lase under optical pumping.The microscale vortex laser with controlled topological charge can serve as a unique and general building block for nextgeneration photonic integrated circuits and coherent vortex beam sources. The method we used here can be employed to generate lasing eigenmode with other complex functionalities.     

Vortex condensation in the dual Chern–Simons–Higgs model

The contribution of nontrivial vacuum (topological) excitations, more specifically vortex configurations of the self-dual Chern–Simons–Higgs model, to the functional partition function is considered. By using a duality transformation, we arrive at a representation of the partition function in terms of which explicit vortex degrees of freedom are coupled to a dual gauge field. By matching the obtained action to a field theory for the vortices, the physical properties of the model in the presence of vortex excitations are then studied. In terms of this field theory for vortices in the self-dual Chern–Simons–Higgs model, we determine the location of the critical value for the Chern–Simons parameter below which vortex condensation can happen in the system. The effects of self-energy quantum corrections to the vortex field are also considered.     

推广的Gross-Pitaevskii理论中的涡旋问题(英文)

利用推广 Gross- Pitaevskii方程 ,分别研究了 (2 +1 )维时空和 3维空间的 Bose- Einstein凝聚体中涡旋的拓扑结构 .这一推广的方程能够被用于非均匀并且高度非线形的 Bose- Einstein凝聚系统 .利用Φ映射拓扑流理论 ,给出了基于序参数的涡旋速度场,以及该速度场的拓扑结构 .最后 ,仔细地探讨了这两种 Bose- Einstein系统中涡旋的各种分支条件.We studied the topological structure of vortex in the Bose-Einstein condensation with a generalized Gross-Pitaevskii equation in (2+1)-dimensional space-time and 3-dimensional space, respectively. Such equation can be used in discussing Bose-Einstein condensates in heterogeneous and highly nonlinear systems. An explicit expression for the vortex velocity field as a function of the order parameter field is derived in terms of the Φ -mapping theory, and the topological structure of ...     

Topological objects in two-gap superconductor

We discuss the non-Abelian topological objects, in particular the non-Abrikosov vortex and the magnetic knot made of the twisted non-Abrikosov vortex, in two-gap superconductor. We show that there are two types of non-Abrikosov vortex in Ginzburg-Landau theory of two-gap superconductor, the D-type which has no concentration of the condensate at the core and the N-type which has a non-trivial profile of the condensate at the core, under a wide class of realistic interaction potential. We prove that these non-Abrikosov vortices can have either integral or fractional magnetic flux, depending on the interaction potential. We show that they are described by the non-Abelian topology π₂(S ²) and π₁(S ¹), in addition to the well-known Abelian topology π₁(S ¹). Furthermore, we discuss the possibility to construct a stable magnetic knot in two-gap superconductor by twisting the non-Abrikosov vortex and connecting two periodic ends together, whose knot topology π₃(S ²) is described by the Chern-Simon index of the electromagnetic potential. We argue that similar topological objects may exist in multi-gap or multi-layer superconductors and multi-component Bose-Einstein condensates and superfluids, and discuss how these topological objects can be constructed in MgB₂, Sr₂RuO₄, ³He, and liquid metallic hydrogen.     

Geometric K-Homology of Flat D-Branes

We use the Baum-Douglas construction of K-homology to explicitly describe various aspects of D-branes in Type II superstring theory in the absence of background supergravity form fields. We rigorously derive various stability criteria for states of D-branes and show how standard bound state constructions are naturally realized directly in terms of topological K-cycles. We formulate the mechanism of flux stabilization in terms of the K-homology of non-trivial fibre bundles. Along the way we derive a number of new mathematical results in topological K-homology of independent interest.