Winding Number Research Articles

The stick number of rail arcs

Consider two parallel lines [Formula: see text] and [Formula: see text] in [Formula: see text]. A rail arc is an embedding of an arc in [Formula: see text] such that one endpoint is on [Formula: see text], the other is on [Formula: see text], and its interior is disjoint from [Formula: see text]. Rail arcs are considered up to rail isotopies, ambient isotopies of [Formula: see text] with each self-homeomorphism mapping [Formula: see text] and [Formula: see text] onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs are called stick rail arcs. The stick number of a rail arc class is the minimum number of sticks, line segments in a p.l. arc, needed to create a representative. This paper calculates the stick number of rail arcs classes with a crossing number at most 2 and uses a winding number invariant to calculate the stick numbers of infinitely many rail arc classes. Each rail arc class has two canonically associated knot classes, its under and over companions. This paper also introduces the rail stick number of knot classes, the minimum number of sticks needed to create a rail arcs whose under or over companion is the knot class. The rail stick number is calculated for 29 knot classes with crossing number at most 9. The stick number of multi-component rail arcs classes is considered as well as the lattice stick number of rail arcs.

Angular momentum-carrying radially-polarised twisted light

It is well-known that the fields of pure radially-polarised optical vortex modes do not possess the usual phase function eiℓϕ of typical twisted light and so they do not carry angular momentum. However, recent work has confirmed that radially-polarised modes with helical wave fronts have been created in the laboratory by several groups and they can now be readily generated using commercially available devices. They are endowed with the phase function eiℓϕ for arbitrary values of the winding number ℓ. Crucially, the original theoretical treatments disregard the longitudinal electric and magnetic field components. We show here that the longitudinal components must be included, not just for the consistency of the treatment, but because only this can lead to angular momentum. We show that for such created modes the cycle-averaged angular momentum is αħℓ where α=(Pw02)/(ħc2) is a dimensionless constant, with P the power in the mode and w0 the mode width at focus.

Superfluid Bose-Fermi mixture with spin-orbit coupling

We investigate a fermionic superfluid with Raman-induced spin-orbit coupling immersed in a Bose-Einstein condensate. By minimizing the total free energy, we find that, with moderate repulsive interspecies interaction, a phase separation occurs where the otherwise nontopological uniform phase is divided into two parts: a purely fermionic one and a Bose-Fermi mix characterized by nontrivial topology with the winding number $W=1$. We verify that Majorana zero modes emerge at the phase interfaces by numerical simulations of the coupled Bogoliubov--de Gennes and Gross--Pitaevskii equations in real space. The tunability of the phase interfaces enables a direct manipulation of the predicted Majorana zero modes.

Majorana modes of giant vortices

We study Majorana zero modes bound to giant vortices in topological superconductors or topological insulator/normal superconductor heterostructures. By expanding in inverse powers of a large winding number $n$, we find an analytic solution for asymptotically all $n$ zero modes required by the index theorem. Contrary to the existing estimates, the solution is not pinned to the vortex boundary and is composed of the warped lowest Landau level states. While the dynamics which shapes the zero modes is a subtle interference of the magnetic effects and Andreev reflection, the solution is very robust and is determined by a single parameter, the vortex radius. The resulting local density of states has a number of features which give remarkable signatures for an experimental observation of the Majorana fermions in two dimensions.

Optimal optical Ferris wheel solitons in a nonlocal Rydberg medium.

We propose a scheme for the creation of stable optical Ferris wheel (OFW) solitons in a nonlocal Rydberg electromagnetically induced transparency (EIT) medium. Depending on a careful optimization of both the atomic density and the one-photon detuning, we obtain an appropriate nonlocal potential provided by the strong interatomic interaction in Rydberg states that can perfectly compensate for the diffraction of the probe OFW field. Numerical results show that the fidelity remains larger than 0.96, while the propagation distance has exceeded 160 diffraction lengths. Higher-order OFW solitons with arbitrary winding numbers are also discussed. Our study provides a straightforward route to generate spatial optical solitons in the nonlocal response region of cold Rydberg gases.

First-order topological phase transitions and disorder-induced Majorana modes in interacting fermion chains

Using a combination of the mean-field Bogoliubov--de Gennes approach and the density matrix renormalization group method, we discover first-order topological transitions between topological superconducting and trivial insulating phases in a sawtooth lattice of intersite attractive fermions. The topological characterization of different phases is achieved in terms of winding numbers, Majorana edge modes, and entanglement spectra. By studying the effect of disorder on first-order topological phase transitions, we establish disorder-induced topological phase coexistence as a mechanism for generating a finite density of Majorana particles.

A complete derived invariant for gentle algebras via winding numbers and Arf invariants

Gentle algebras are in bijection with admissible dissections of marked oriented surfaces. In this paper, we further study the properties of admissible dissections and we show that silting objects for gentle algebras are given by admissible dissections of the associated surface. We associate to each gentle algebra a line field on the corresponding surface and prove that the derived equivalence class of the algebra is completely determined by the homotopy class of the line field up to homeomorphism of the surface. Then, based on winding numbers and the Arf invariant of a certain quadratic form over {mathbb {Z}}_2, we translate this to a numerical complete derived invariant for gentle algebras.

Topological natures of the Gauss-Bonnet black hole in AdS space

In the recent proposal [S. W. Wei et al., Black Hole Solutions as Topological Thermodynamic Defects, Phys. Rev. Lett. 129, 191101 (2022).], the black holes were viewed as topological thermodynamic defects by using the generalized off shell free energy. In this paper, we follow such proposal to study the local and global topological natures of the Gauss-Bonnet black holes in anti--de Sitter (AdS) space. The local topological natures are reflected by the winding numbers, where the positive and negative winding numbers correspond to the stable and unstable black hole branches. The global topological natures are reflected by the topological numbers, which are defined as the sum of the winding numbers for all black hole branches and can be used to classify the black holes into different classes. When the charge is present, we find that the topological number is independent on the values of the parameters, and the charged Gauss-Bonnet AdS black holes can be divided into the same class of the Reissner-Nordstr\"om anti-de Sitter black hole black holes with the same topological number 1. However, when the charge is absent, we find that the topological number has certain dimensional dependence. This is different from the previous studies, where the topological number is found to be a universal number independent of the black hole parameters. Furthermore, the asymptotic behaviors of curve $\ensuremath{\tau}({r}_{h})$ in small and large radii limit can be a simple criterion to distinguish the different topological number. We find a new asymptotic behavior as $\ensuremath{\tau}({r}_{h}\ensuremath{\rightarrow}0)=0$ and $\ensuremath{\tau}({r}_{h}\ensuremath{\rightarrow}\ensuremath{\infty})=0$ in the black hole system, which shows topological equivalency with the asymptotic behaviors $\ensuremath{\tau}({r}_{h}\ensuremath{\rightarrow}0)=\ensuremath{\infty}$ and $\ensuremath{\tau}({r}_{h}\ensuremath{\rightarrow}\ensuremath{\infty})=\ensuremath{\infty}$. We also give an intuitional proof of why there are only three topological classes in the black hole system under the condition $({\ensuremath{\partial}}_{{r}_{h}}S{)}_{P}>0$.

Topology of equatorial timelike circular orbits around stationary black holes

A topological approach has been successfully used to study the properties of the light ring and the null circular orbit, in a generic black hole background. However, for the equatorial timelike circular orbit, quite different from the light ring case, its radius is closely dependent of the energy and angular momentum of a test particle. This fact seems to restrict the extension of the topological treatment to the timelike circular orbit. In this paper, we confirm that the angular momentum does not affect the asymptotic behavior of the constructed vector with its zero points denoting the equatorial timelike circular orbits. As a result, a well-behaved topology to characterize the equatorial timelike circular orbits can be constructed. Our study shows that the total topological number of the timelike circular orbits vanishes for a generic black hole, which is dependent of the energy of the particle. Significantly, it reveals that if there the timelike circular orbits exist, they always come in pairs for fixed angular momentum. Meanwhile, the stable and unstable timelike circular orbits have positive or negative winding number. Of particular interest is that the marginally stable circular orbit corresponds to the bifurcation point of the zero point of the constructed vector. Moreover, we also examine the case when the particle energy acts as the control parameter. It is shown that there will be topological phase transition when the value of the particle energy is one. Below this value, the timelike circular orbits always come in pairs for fixed energy. Otherwise, we will have one more unstable timelike circular orbit. We further apply the treatment to the Kerr black hole. All the results given in a generic black hole background are exactly reproduced. These strongly indicate that our topological approach can be generalized to the equatorial timelike circular orbits.

Hydrodynamic Enhancement of p-atic Defect Dynamics.

We investigate numerically and analytically the effects of hydrodynamics on the dynamics of topological defects in p-atic liquid crystals, i.e., two-dimensional liquid crystals with p-fold rotational symmetry. Importantly, we find that hydrodynamics fuels a generic passive self-propulsion mechanism for defects of winding number s=(p-1)/p and arbitrary p. Strikingly, we discover that hydrodynamics always accelerates the annihilation dynamics of pairs of ±1/p defects and that, contrary to expectations, this effect increases with p. Our Letter paves the way toward understanding cell intercalation and other remodeling events in epithelial layers.

Robust formation of metachronal waves in directional chains of phase oscillators

Biological systems can rely on collective formation of a metachronal wave in an ensemble of oscillators for locomotion and for fluid transport. We consider one-dimensional chains of phase oscillators with nearest-neighbor interactions, connected in a loop and with rotational symmetry, so each oscillator resembles every other oscillator in the chain. Numerical integrations of the discrete phase oscillator systems and a continuum approximation show that directional models (those that do not obey reversal symmetry), can exhibit instability to short wavelength perturbations but only in regions where the slope in phase has a particular sign. This causes short wavelength perturbations to develop that can vary the winding number that describes the sum of phase differences across the loop and the resulting metachronal wave speed. Numerical integrations of stochastic directional phase oscillator models show that even a weak level of noise can seed instabilities that resolve into metachronal wave states.

Tailoring the phase transition from topological superconductor to trivial superconductor induced by magnetic textures of a spin chain on a p -wave superconductor

We theoretically investigate the phase transition from a nontrivial topological $p$-wave superconductor to a trivial $s$-wave-like superconducting phase through a gapless phase, driven by different magnetic textures as a one-dimensional spin-chain impurity, e.g., Bloch-type, in-plane, and out-of-plane N\'eel-type spin chains, etc. In our proposal, the chain of magnetic impurities is placed on a spin-triplet $p$-wave superconductor where we obtain numerically as well as analytically an effective $s$-wave-like pairing due to spin rotation, resulting in gradual destruction of the Majorana zero modes present in the topological superconducting phase. In particular, when the impurity spins are antiferromagnetically aligned, i.e., spiral wave vector ${G}_{\mathrm{s}}=\ensuremath{\pi}$, the system becomes an effective $s$-wave superconductor without Majorana zero modes in the local density of states. The Shiba bands, on the other hand, formed due to the overlapping of Yu-Shiba-Rusinov states play a crucial role in this topological to trivial superconductor phase transition, confirmed by the sign change in the minigap within the Shiba bands. We also characterize this topological phase transition via gap closing and winding number analysis. Moreover, interference of the Shiba bands exhibiting oscillatory behavior within the superconducting gap $\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{p}$ to ${\mathrm{\ensuremath{\Delta}}}_{p}$, as a function of ${G}_{s}$, also reflects an important evidence for the formation of an effective $s$-wave pairing. Such oscillation is absent in the $p$-wave regime. We also analyze the case of two-dimensional $p$-wave superconductor hosting Majorana edge modes (in absence of any magnetic chain) and show that initially Majorana zero modes (in presence of one-dimensional magnetic chain) can hybridize with such Majorana edge modes. Interestingly, in the topological regime with a fixed ${G}_{s}$ value, the Mazorana zero modes survive at the ends of the magnetic chain even when the Majorana edge states disappear at some critical value of the chemical potential and exchange coupling strength.

Statistical Topology—Distribution and Density Correlations of Winding Numbers in Chiral Systems

Statistical Topology emerged as topological aspects continue to gain importance in many areas of physics. It is most desirable to study topological invariants and their statistics in schematic models that facilitate the identification of universalities. Here, the statistics of winding numbers and of winding number densities are addressed. An introduction is given for readers with little background knowledge. Results that my collaborators and I obtained in two recent works on proper random matrix models for the chiral unitary and symplectic cases are reviewed, avoiding a technically detailed discussion. There is a special focus on the mapping of topological problems to spectral ones as well as on the first glimpse of universality.

One-dimensional noninteracting topological insulators with chiral symmetry

We construct microscopical models of one-dimensional non-interacting topological insulators in all of the chiral universality classes. Specifically, we start with a deformation of the Su-Schrieffer-Heeger (SSH) model that breaks time-reversal symmetry, which is in the AIII class. We then couple this model to its time-reversal counterpart in order to build models in the classes BDI, CII, DIII and CI. We find that the $\mathbb{Z}$ topological index (the winding number) in individual chains is defined only up to a sign. This comes from noticing that changing the sign of the chiral symmetry operator changes the sign of the winding number. The freedom to choose the sign of the chiral symmetry operator on each chain independently allows us to construct two distinct possible chiral symmetry operators when the chains are weakly coupled -- in one case, the total winding number is given by the sum of the winding number of individual chains while in the second case, the difference is taken. We find that the chiral models that belong to $\mathbb{Z}$ classes, AIII, BDI and CII are topologically equivalent, so they can be adiabatically deformed into one another so long as the chiral symmetry is preserved. We study the properties of the edge states in the constructed models and prove that topologically protected edge states must all be localised on the same sublattice (on any given edge). We also discuss the role of particle-hole symmetry on the protection of edge states and explain how it manages to protect edge states in $\mathbb{Z}_2$ classes, where the integer invariant vanishes and chiral symmetry alone does not protect the edge states anymore. We discuss applications of our results to the case of an arbitrary number of coupled chains, construct possible chiral symmetry operators for the multiple chain case, and briefly discuss the generalisation to any odd number of dimensions.

Topological interpretation for phase transitions of black holes

In this work, we develop a universal picture from topology in the thermodynamic parameters space to describe first order phase transitions of black holes. By employing an off-shell internal energy, we find two types of topological defects. The first is normal, describing black holes which move with a nonzero velocity and acceleration, in a central force field. The second type of defects are exotic: they are static in the space, not describing black holes but encoding information about first order transitions. For each defect, we assign a winding number and an inertial mass. By studying neutral and charged black holes in asymptotically anti-de Sitter space, we show that first order transitions can be viewed as once or twice interchange of winding numbers between black holes and the exotic defects, through wired action at a distance. This corresponds to the usual notion: a smaller black hole grows into a larger black hole or vice versa. However, our topological analysis illustrates that the transition can also be locally interpreted as virtual collisions between black holes and the exotic defects. In this interpretation, a smaller black hole first grows into a new exotic defect whereas an original exotic defect grows into a larger black hole. All the defects simply change their positions and momentums, rather than interchanging the winding numbers. Critical point of the transition can be extracted when all the defects meet in the parameters space. Certain quantities, such as the Jacobians and the velocities of normal defects show universal behaviors near the critical point.

Wannier topology and quadrupole moments for a generalized Benalcazar-Bernevig-Hughes model

We analyze a special separable and chiral-symmetric model with a quantized quadrupole moment, extending the Benalcazar-Bernevig-Hughes model [Science 357, 61 (2017)]. Using nested-Wilson loop formalism, we give an exact expression for Wannier centers, sector polarizations, and quadrupole moments. These are connected to the winding numbers of the constitutive one-dimensional chains. We prove that these winding numbers can characterize the model's Wannier topology as a $\mathbb{Z}\times\mathbb{Z}$ set. These results clearly show that the quantization of the quadrupole moment can arise without additional spatial symmetry (except for translation symmetry) for the bulk. By switching from the Wannier representation to the Bloch representation, we derive an alternative expression for the bulk quadrupole moment and obtain its exact value. Combining the bulk quadrupole and edge polarizations, we analytically calculate the corner charge in a large square system and make the bulk-boundary correspondence explicit in an edge-consistent gauge. Our work reveals the relationship between zero-energy states at the boundary, charge localization, and the bulk quadrupole of the extended model.

Homomorphism obstructions for satellite maps

A knot in a solid torus defines a map on the set of (smooth or topological) concordance classes of knots in S 3 S^3 . This set admits a group structure, but a conjecture of Hedden suggests that satellite maps never induce interesting homomorphisms: we give new evidence for this conjecture in both categories. First, we use Casson-Gordon signatures to give the first obstruction to a slice pattern inducing a homomorphism on the topological concordance group, constructing examples with every winding number besides ± 1 \pm 1 . We then provide subtle examples of satellite maps which map arbitrarily deep into the n n -solvable filtration of Cochran, Orr, and Teichner [Ann. of Math. (2) 157 (2003), pp. 433–519], act like homomorphisms on arbitrary finite sets of knots, and yet which still do not induce homomorphisms. Finally, we verify Hedden’s conjecture in the smooth category for all small crossing number satellite operators but one.

Many-body Aharonov-Bohm caging in a lattice of rings

We study a system of a few ultracold bosons loaded into the states with orbital angular momentum $l=1$ of a one-dimensional staggered lattice of rings. Local eigenstates with winding numbers $+l$ and $-l$ form a Creutz ladder with a real dimension and a synthetic one. States with opposite winding numbers in adjacent rings are coupled through complex tunnelings, which can be tuned by modifying the central angle $\phi$ of the lattice. We analyze both the single-particle case and the few boson bound-state subspaces for the regime of strong interactions using perturbation theory, showing how the geometry of the system can be engineered to produce an effective $\pi$-flux through the plaquettes. We find non-trivial topological band structures and many-body Aharonov-Bohm caging in the $N$-particle subspaces even in the presence of a dispersive single-particle spectrum. Additionally, we study the family of models where the angle $\phi$ is introduced at an arbitrary lattice periodicity $\Gamma$. For $\Gamma>2$, the $\pi$-flux becomes non-uniform, which enlarges the spatial extent of the Aharonov-Bohm caging as the number of flat bands in the spectrum increases. All the analytical results are benchmarked through exact diagonalization.

Topological Properties of Floquet Winding Bands in a Photonic Lattice.

The engineering of synthetic materials characterized by more than one class of topological invariants is one of the current challenges of solid-state based and synthetic materials. Using a synthetic photonic lattice implemented in a two-coupled ring system we engineer an anomalous Floquet metal that is gapless in the bulk and shows simultaneously two different topological properties. On the one hand, this synthetic lattice presents bands characterized by a winding number. The winding emerges from the breakup of inversion symmetry, and it directly relates to the appearance of Bloch suboscillations within its bulk. On the other hand, the Floquet nature of the lattice results in well-known anomalous insulating phases with topological edge states. The combination of broken inversion symmetry and periodic time modulation studied here enriches the variety of topological phases available in lattices subject to Floquet driving and suggests the possible emergence of novel phases when periodic modulation is combined with the breakup of spatial symmetries.

Topological Unidirectional Guided Resonances Emerged from Interband Coupling.

Unidirectional guided resonances (UGRs) are optical modes in photonic crystal slabs that radiate toward one side without the need for mirrors on the other. In this Letter, we report a mechanism to realize UGRs by tuning the interband coupling effect originating from up-down symmetry breaking. We theoretically find that UGRs that reside along high-symmetric lines correspond to phase singularities of far-field radiation, depicted by phase winding numbers as a type of topological indices. We investigate the phase dislocation lines in three-dimensional parameter space and elaborate on the interplay between UGRs and non-Hermitian degeneracies accordingly. Our findings reveal the topological nature of UGRs about their generation, evolution, and annihilation in general parameter spaces, thus paving the way to new possibilities of light manipulation.