Integer Winding Number Research Articles

Domain wall trajectory determined by its fractional topological edge defects

A domain wall (DW) in a ferromagnetic nanowire is composed of elementary topological bulk and edge defects with integer and fractional winding numbers, respectively, whose relative spatial arrangement determines the chirality of the DW. Here we show how we can understand and control the trajectory of DWs in magnetic branched networks, composed of connected nanowires, by considering their fractional elementary topological defects and how they interact with those innate to the network. We first develop a highly reliable mechanism for the injection of a DW of a given chirality into a nanowire and show that its chirality determines which branch the DW follows at a symmetric Y-shaped magnetic junction—the fundamental building block of the network. Using these concepts, we unravel the origin of the one-dimensional nature of magnetization reversal of connected artificial spin ice systems that have been observed in the form of Dirac strings. When a domain wall of a given chirality is injected into a magnetic nanowire, its trajectory through a branched network of Y-shaped nanowire junctions—such as a honeycomb lattice, for instance—can be pre-determined. This property has implications for data storage and processing.

Synchronization of quasiperiodic oscillations to a periodic force studied with semiconductor lasers

We experimentally study synchronization processes in a system of two different multisection semiconductor lasers. Periodic self-pulsations of laser 1 are injected into laser 2, which is operating in a regime with two-frequency quasiperiodic self-pulsations. The experimental system demonstrates the new type of transitions to synchrony between three frequencies which has been recently revealed using generic coupled phase and van der Pol oscillator models. In particular, resonances of quasiperiodic oscillations at integer winding numbers three and five are shown to break up before locking to the injected periodic signal. Moreover, carefully determining the coherence of the noisy oscillations, we reveal so far unexplored processes of coherence transfer to nonsynchronized oscillations.

Three-dimensional topological phases in a layered honeycomb spin-orbital model

We present an exactly solvable spin-orbital model based on the Gamma-matrix generalization of a Kitaev-type Hamiltonian. In the presence of small magnetic fields, the model exhibits a critical phase with a spectrum characterized by topologically protected Fermi points. Upon increasing the magnetic field, Fermi points carrying opposite topological charges move toward each other and annihilate at a critical field, signaling a phase transition into a gapped phase with trivial topology in three dimensions. On the other hand, by subjecting the system to a staggered magnetic field, an effective time-reversal symmetry essential to the existence of three-dimensional topological insulators is restored in the auxiliary free fermion problem. The nontrivial topology of the gapped ground state is characterized by an integer winding number and manifests itself through the appearance of gapless Majorana fermions confined to the two-dimensional surface of a finite system.

Classification of topological insulators and superconductors in three spatial dimensions

We systematically study topological phases of insulators and superconductors (or superfluids) in three spatial dimensions. We find that there exist three-dimensional (3D) topologically nontrivial insulators or superconductors in five out of ten symmetry classes introduced in seminal work by Altland and Zirnbauer within the context of random matrix theory, more than a decade ago. One of these is the recently introduced ${\mathbb{Z}}_{2}$ topological insulator in the symplectic (or spin-orbit) symmetry class. We show that there exist precisely four more topological insulators. For these systems, all of which are time-reversal invariant in three dimensions, the space of insulating ground states satisfying certain discrete symmetry properties is partitioned into topological sectors that are separated by quantum phase transitions. Three of the above five topologically nontrivial phases can be realized as time-reversal invariant superconductors. In these the different topological sectors are characterized by an integer winding number defined in momentum space. When such 3D topological insulators are terminated by a two-dimensional surface, they support a number (which may be an arbitrary nonvanishing even number for singlet pairing) of Dirac fermion (Majorana fermion when spin-rotation symmetry is completely broken) surface modes which remain gapless under arbitrary perturbations of the Hamiltonian that preserve the characteristic discrete symmetries, including disorder. In particular, these surface modes completely evade Anderson localization from random impurities. These topological phases can be thought of as three-dimensional analogs of well-known paired topological phases in two spatial dimensions such as the spinless chiral $({p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y})$-wave superconductor (or Moore-Read Pfaffian state). In the corresponding topologically nontrivial (analogous to ``weak pairing'') and topologically trivial (analogous to ``strong pairing'') 3D phases, the wave functions exhibit markedly distinct behavior. When an electromagnetic U(1) gauge field and fluctuations of the gap functions are included in the dynamics, the superconducting phases with nonvanishing winding number possess nontrivial topological ground-state degeneracies.

EDGE AND BULK MERONS IN DOUBLE QUANTUM DOTS WITH SPONTANEOUS INTERLAYER PHASE COHERENCE

We have investigated nucleation of merons in double quantum dots when a lateral distortion with a reflection symmetry is present in the confinement potential. We find that merons can nucleate both inside and at the edge of the dots. In addition to these merons, our results show that electron density modulations can be also present inside the dots. An edge meron appears to have approximately a half integer winding number.

Experimental umbilic diabolos in random optical fields

The intensity of a random optical field consists of bright speckle spots (maxima) separated from dark areas (minima and optical vortices) by saddle points. We show that hidden in this complicated landscape are umbilic points--singular points at which the eigenvalues Lambda (+/-) of the Hessian matrix that measure the curvature of the landscape become degenerate. Although not observed previously in random optical fields, umbilic points are the most numerous of all special points, outnumbering maxima, minima, saddle points, and vortices. We show experimentally that the directions of principal curvature, the eigenvectors Psi (+/-), rotate about intensity umbilic points with positive or negative half-integer winding number, in accord with theory, and that Lambda (+) and Lambda (-) generate a double cone known as a diabolo. At optical vortices the curvature of the amplitude is singular, and we show from both theory and experiment that for this landscape Psi (+/-) rotate about vortex centers with a positive integer winding number. Diabolos can be classified as elliptic or hyperbolic, and we present initial results for the measured fractions of these two different types of umbilic diabolos.

The size of the sync basin

We suggest a new line of research that we hope will appeal to the nonlinear dynamics community, especially the readers of this Focus Issue. Consider a network of identical oscillators. Suppose the synchronous state is locally stable but not globally stable; it competes with other attractors for the available phase space. How likely is the system to synchronize, starting from a random initial condition? And how does the probability of synchronization depend on the way the network is connected? On the one hand, such questions are inherently difficult because they require calculation of a global geometric quantity, the size of the "sync basin" (or, more formally, the measure of the basin of attraction for the synchronous state). On the other hand, these questions are wide open, important in many real-world settings, and approachable by numerical experiments on various combinations of dynamical systems and network topologies. To give a case study in this direction, we report results on the sync basin for a ring of n >> 1 identical phase oscillators with sinusoidal coupling. Each oscillator interacts equally with its k nearest neighbors on either side. For k/n greater than a critical value (approximately 0.34, obtained analytically), we show that the sync basin is the whole phase space, except for a set of measure zero. As k/n passes below this critical value, coexisting attractors are born in a well-defined sequence. These take the form of uniformly twisted waves, each characterized by an integer winding number q, the number of complete phase twists in one circuit around the ring. The maximum stable twist is proportional to n/k; the constant of proportionality is also obtained analytically. For large values of n/k, corresponding to large rings or short-range coupling, many different twisted states compete for their share of phase space. Our simulations reveal that their basin sizes obey a tantalizingly simple statistical law: the probability that the final state has q twists follows a Gaussian distribution with respect to q. Furthermore, as n/k increases, the standard deviation of this distribution grows linearly with square root of n/k. We have been unable to explain either of these last two results by anything beyond a hand-waving argument.

Polarization singularities in optical lattices.

Polarization singularities are shown to be unavoidable features of three-dimensional optical lattices. These singularities take the form of lines of circular polarization, C lines, and lines of linear polarization, L lines. The polarization figures surrounding a C line (L line) rotate about the line with winding number +/-1/2 (+/-1). C and L lines permeate the lattice, meander throughout the unit cell, and form closed loops. Surprisingly, every point in a linearly polarized optical lattice is found to be a singularity about which the surrounding polarization vectors rotate with an integer winding number.

Emergent polarization singularities.

Polarization singularities are shown to emerge spontaneously from the incoherent addition of uncorrelated optical fields that individually need not contain singularities. Examples of this phenomenon are given for both vector and ellipse fields. The incoherent addition of vector fields whose singularities have integer winding numbers is shown to yield fields whose singularities have half-integer winding numbers. These findings are used to make predictions about the singularities of the polarized component of the cosmic microwave background.

Vortex Configurations in Mesoscopic Superconducting Nanowires

Beyond the well-known Abrikosov and giant vortex configurations, new solutions to the Ginzburg–Landau model corresponding to vortices of integer and half-integer winding number are described. Phase diagrams (Bext, Energy) and magnetization curves have been determined, aiming towards an understanding of the magnetic properties of lead nanowires and the possible consequences of such solutions with respect to the switching mechanism between vortex states in mesoscopic superconductors.

Intersecting branes flip SU(5)

Within a toroidal orbifold framework, we exhibit intersecting brane-world constructions of flipped SU(5)× U(1) GUT models with various numbers of generations, other chiral matter representations and Higgs representations. We exhibit orientifold constructions with integer winding numbers that yield 8 or more conventional SU(5) generations, and orbifold constructions with fractional winding numbers that yield flipped SU(5)× U(1) models with just 3 conventional generations. Some of these models have candidates for the 5 and 5 Higgs representations needed for electroweak symmetry breaking, but not for the 10 and 10 representations needed for GUT symmetry breaking. We have also derived models with complete GUT and electroweak Higgs sectors, but these have undesirable extra chiral matter.

Non-Abelian thermal large gauge transformations in 2+1 dimensions

We discuss several different constructions of non-Abelian large gauge transformations at finite temperature. Pisarski's Ansatz with even winding number is related to Hopf mappings, and we present a simple new Ansatz that has any integer winding number at finite temperature.

Half-integer winding number solutions to the Ginzburg-Landau-Higgs equations

New solutions to the Abelian U(1) Higgs model, corresponding to vortices of integer and half-integer winding number bound onto the edges of domain walls and possibly surrounded by annular current flows, are described. Independently of their stability issue, the existence of these states could have interesting consequences in different physical contexts.

A Pivotal Model for the (1 + 1)-Dimensional Heisenberg and Sigma Models

An integrable interpolative (Pivotal) model for the (1 + 1)-dimensional Hyperbolic Heisenberg and Hyperbolic sigma models is proposed and some solutions classifiable by an integer winding number examined.

A Pivotal Model for the (1+1)-Dimensional Heisenberg and Sigma Models

An integrable interpolative (Pivotal) model for the (1 + 1)-dimensional Hyperbolic Heisenberg and Hyperbolic sigma models is proposed and some solutions classifiable by an integer winding number examined.

Winding Numbers, Complex Currents, and Non-Hermitian Localization

The nature of extended states in disordered tight binding models with a constant imaginary vector potential is explored. Such models, relevant to vortex physics in superconductors and to population biology, exhibit a delocalization transition and a band of extended states even for a one dimensional ring. Using an analysis of eigenvalue trajectories in the complex plane, we demonstrate that each delocalized state is characterized by an (integer) winding number, and evaluate the associated complex current. Winding numbers in higher dimensions are also discussed.

Gauge transformations with fractional winding numbers.

The role which gauge transformations of noninteger winding numbers might play in non-Abelian gauge theories is studied. The phase factor acquired by the semiclassical physical states in an arbitrary background gauge field when they undergo a gauge transformation of an arbitrary real winding number is calculated in the path integral formalism assuming that a $\ensuremath{\theta}F\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}$ term added to the Lagrangian plays the same role as in the case of integer winding numbers. Requiring that these states provide a representation of the group of "large" gauge transformations, a condition on the allowed backgrounds is obtained. It is shown that this representability condition is only satisfied in the monopole sector of a spontaneously broken gauge theory, but not in the vacuum sector of an unbroken or a spontaneously broken non-Abelian gauge theory. It is further shown that the recent proof of the vanishing of the $\ensuremath{\theta}$ parameter when gauge transformations of arbitrary fractional winding numbers are allowed breaks down in precisely those cases where the representability condition is obeyed because certain gauge transformations needed for the proof, and whose existence is assumed, are either spontaneously broken or cannot be globally defined as a result of a topological obstruction.

Analytical results for giant Shapiro steps in Josephson arrays.

We study two-dimensional Josephson arrays driven by a combined dc plus ac current, and with an applied transverse magnetic field of f flux quanta per plaquette. We present ansatz solutions for sufficiently large frequencies, which are a generalization of the traveling wave solutions found by Marino and Halsey for the case of dc current driving. For f=1/2 and 1/3, we compute the widths of the first few Shapiro steps for both integer and fractional winding numbers. These expressions consist of products of Bessel functions of (${\mathit{i}}_{\mathrm{ac}}$/${\mathrm{\ensuremath{\omega}}}_{\mathrm{ac}}$), where ${\mathit{i}}_{\mathrm{ac}}$ and ${\mathrm{\ensuremath{\omega}}}_{\mathrm{ac}}$ are the amplitude and the frequency of the driving ac current, respectively, times a frequency-dependent factor for fractional steps. In the limit of large frequencies, we find that the fractional steps are suppressed, whereas the maximum integer step widths saturate to a frequency-independent value. We show that the suppression of the fractional steps is due to decrease of the vertical (i.e., perpendicular to the direction of flow of the injected current) supercurrent relative to the normal current, whereas the persistence of the integer steps is due to the existence of zero-frequency (though spatially varying) terms in the expansion for the gauge-invariant phase differences, for which the normal current vanishes. These results are in reasonable agreement with the numerical computations carried out by other groups.

Vortices in complex scalar fields

Consider a spatially extended field which evolves in time according to a PDE. The solutions contain particle-like defects whose motions are parts of the full dynamics. We inquire into the formulation of an asymptotic “particle + field” defect dynamics which derives from the full PDE. We carry out this program for two complex scalar field equations in two space dimensions, the nonlinear Schrödinger equation and the nonlinear heat equation. The topological defects are zeros of the complex scalar field with nonzero integer winding numbers, called vortices. Vortices evolving under the nonlinear Schrödinger equation behave like point vortices in ideal fluid. Pairs of vortices evolving under the nonlinear heat equation with like (opposite) winding numbers undergo a repulsive (attractive) interaction.

Topologically nontrivial solutions to Yang-Mills equations with axisymmetric external sources.

We present a new set of solutions to Yang-Mills equations with axially symmetric external charge sources. Our solutions for the gauge fields are not explicitly axisymmetric, but the noninvariance of the fields under a rotation about the symmetry axis can be compensated by a gauge transformation about a symmetry axis in gauge space. All gauge-invariant quantities are therefore axisymmetric. Our solutions are characterized by a gauge-invariant integer winding number $n$, and all winding numbers are allowed. We prove that the total gauge-invariant charge of the system (source plus gauge fields) vanishes identically in our solutions for $n\ensuremath{\ne}0$, even if the source has net charge. We explicitly solve the equations of motion for a spherical shell of charge. The solution depends on the gauge coupling $g$, the total charge of the shell ${Q}_{S}$, and the topological number $n$. We use perturbative methods to obtain the solution in closed form for $\overline{\ensuremath{\alpha}}\ensuremath{\equiv}\frac{{g}^{2}{Q}_{S}}{(4\ensuremath{\pi})}\ensuremath{\ll}1$. We show analytically that in this limit the energy ${\mathcal{E}}_{n}$ of the system satisfies the bound ${\mathcal{E}}_{n}\ensuremath{\le}[\frac{{g}^{2}{Q}_{S}^{2}}{(8\ensuremath{\pi}a)}]\ifmmode\times\else\texttimes\fi{}\frac{1}{(2n+1)}$, where $a$ is the radius of the shell. Using relaxation methods to find the exact solution to the equations of motion numerically for arbitrary $\overline{\ensuremath{\alpha}}$, we establish that this bound is satisfied for all $g$, ${Q}_{S}$, and $n$.