Winding Number Research Articles

Toward realistic models in T2/ℤ2 flux compactification

Abstract We consider a six dimensional gauge theory compactified on $T^2/\mathbb {Z}_2$ with magnetic flux. The configurations of models are classified by winding numbers at the fixed points. Requiring the existence of generation numbers and Yukawa coupling, we see that allowed and forbidden configurations are described by geometry of winding numbers.

Multiple skin transitions in two-band non-Hermitian systems with long-range nonreciprocal hopping

Abstract Non-Hermitian skin effect is a prominent feature in non-Hermitian physics, leading to novel topological properties and expanding the traditional energy band theories. In this paper, we investigate a two-band non-Hermitian system in which multiple skin transitions are induced by long-range nonreciprocal hopping. The spectral winding number under periodic boundary conditions reveals the localization directions of skin states. Further, we present the analytical solution of transition points by tracing the self-intersecting points on the complex plane. Interestingly, the current system exhibits the abundant non-Hermitian skin effects, including the normal, W-shaped, and bipolar localization properties, which the eigenstate distributions and the generalized Brillouin zone can clearly illustrate. We also provide a phase diagram to represent the skin transition properties of the system comprehensively. Further, we demonstrate that the multimer non-Hermitian lattices also present the anomalous skin effect and multiple transitions, which occur in the region of the bulk band touching, the same as the two-band lattice. Moreover, a feasible scheme is proposed to realize the current non-Hermitian system with long-range nonreciprocal hopping by a topoelectrical circuit. This work further supplies the content of skin transitions and may help us explore more plentiful localization features in the two-band non-Hermitian systems.

Optimization Design of Variable Reluctance Resolver Based on Three-phase Symmetrical Winding

Optimization Design of Variable Reluctance Resolver Based on Three-phase Symmetrical Winding

Anomalous Hall effects in chiral superconductors

We report theoretical results for the electronic contribution to thermal and electrical transport for chiral superconductors belonging to even or odd-parity E1 and E2 representations of the tetragonal and hexagonal point groups. Chiral superconductors exhibit novel properties that depend on the topology of the order parameter and Fermi surface, and—as we highlight—the structure of the impurity potential. An anomalous thermal Hall effect is predicted and shown to be sensitive to the winding number, ν, of the chiral order parameter via Andreev scattering that transfers angular momentum from the chiral condensate to excitations that scatter off the random potential. For heat transport in a chiral superconductor with isotropic impurity scattering, i.e., point-like impurities, a transverse heat current is obtained for ν = ±1, but vanishes for |ν| > 1. This is not a universal result. For finite-size impurities with radii of order or greater than the Fermi wavelength, R ≥ ℏ/pf, the thermal Hall conductivity is finite for chiral order with |ν|≥ 2, and determined by a specific Fermi-surface average of the differential cross-section for electron-impurity scattering. Our results also provide quantitative formulae for analyzing and interpreting thermal transport measurements for superconductors predicted to exhibit broken time-reversal and mirror symmetries.

Topological phase transitions in the non-Hermitian SSH model with long-range hopping terms induced by gains and losses

The latest progress in the one-dimensional chain SSH model provide the basic understanding of non-Hermitian topological phases. By periodically introducing the gains and losses of imaginary potential ig,-ig,-ig,ig with anti-PT symmetry, we have investigated the topological property of non-Hermitian SSH model with long-range hopping terms. We find that the gains and losses of imaginary potential causes the model to exhibit non-Hermiticity, and the non-Hermiticity induces system to undergo topological phase transitions from topologically trivial phase with non-Hermitian winding number 0 to topologically nontrivial phase with non-Hermitian winding number 1 and −1. The introduction of gains and losses greatly broadens the topological non-trivial region and play a non-trivial role in the model. We fully characterize the topological phase transitions by the band gap closing-reopening, the non-Hermitian winding numbers v and the hidden Chern number C.

Restoration of non-Hermitian bulk-boundary correspondence by counterbalancing skin effect

In systems exhibiting the non-Hermitian skin effect (NHSE), the bulk spectrum under open boundary conditions (OBC) significantly differs from that of its periodic counterpart. This disparity renders the conventional bulk-boundary correspondence (BBC) inapplicable. Here we propose an intuitive approach called doubling and swapping to restore the BBC, using the non-Hermitian Su-Schrieffer-Heeger model as an example. Explicitly, we construct a modified system free of NHSE by swapping the asymmetric intracell hoppings in every second primitive unit cell. Importantly, this change does not alter the OBC spectrum. As a result, the modified periodic system can serve as the bulk for defining topological invariants that accurately predict edge states and topological phase transitions. The basic principle is applicable to many other systems. By extending the study to disordered systems in which the asymmetric hoppings are randomly swapped, we show that two types of winding numbers can also be defined to account for the NHSE and topological edge states, respectively.

Multifrequency topological interface modes in a nonlocal acoustic system with large winding numbers

Topological interface states have wide applications for their robustness and energy localization. Multiple topological interface states in 1D acoustic systems are preferred for signal enhancement at specific frequencies. However, previous 1D topological systems with local interactions can realize only one topological interface state in a bandgap. Here, we find that connected acoustic Su–Schrieffer–Heeger (SSH) chains with nonlocal interactions can achieve two or more topological interface states at different frequencies in a single bandgap, even in the subwavelength region. The difference in frequency can be modulated by merely adjusting the structural design at the interface. A mechanical metamaterial with a large winding number is designed to realize multiple nondegenerate topological interface states. This metamaterial can enhance weak acoustic signals at multiple specific frequencies simultaneously and has applications in the fields of acoustic detecting and color imaging.

The winding number of coronal flux ropes. I. Data-driven time-dependent magnetofrictional modelling

Magnetic flux ropes are key structures in solar and solar-terrestrial research. Their magnetic twist is an important quantity for understanding their eruptivity, their evolution in interplanetary space, and their consequences for planetary space environments. The magnetic twist is expressed in terms of a winding number that describes how many times a field line winds about the axis of the flux rope (FR). Due to the complexity of calculating the winding number, current methods rely largely on its approximation. We use a data-driven simulated FR to investigate the winding number $T_g$ in comparison to the commonly used twist proxy $T_w$, which describes a winding of two infinitesimally close field lines. We also estimate the magnetic flux enclosed in the resultant FR(s). We use the magnetic field analysis tools (MAFIAT) software to compute $T_g$ and $T_w$ for data-driven time-dependent magnetofrictional modelling of AR12473. We find that the FR boundaries can significantly differ depending on whether they are defined using the twist approximation $T_w$ or the winding number $T_g$. This also significantly affects the FR structure and the estimates of the enclosed magnetic flux. For the event analysed here, $T_g$ also reveals that the twisted flux system consists of two separate intertwined FRs. The results of this study suggest that the computation of the winding number ($T_g$) is important for investigations of solar FRs.

Topological interpretation of black hole phase transition in Gauss–Bonnet gravity

Phase transitions of Einstein–Gauss–Bonnet black holes are studied using Duan’s [Formula: see text]-field topological current theory, where black holes are treated as topological defects in the thermodynamic parameter space. The kinetics of thermodynamic defects are studied using Duan’s bifurcation theory. In this picture, a first-order phase transition between small/large black hole phases is interpreted as the interchange of winding numbers between the defects as a result of some action at a distance. We observe a first-order phase transition between small/large black holes for [Formula: see text] Einstein–Gauss–Bonnet theory similar to Reissner–Nordström black holes in AdS space. This implies that these black hole solutions share the same topology and phase structure. We have also studied the phase transition of neutral black holes in [Formula: see text] and found a transition between unstable small and large stable black hole phases similar to the case of neutral black holes in AdS space. Recently, it has been conjectured that black holes with similar topological structure exhibit the same thermodynamic properties. Our results strengthen the conjecture by connecting the topological nature of black holes to phase transitions.

Azimuthal modulation of light propagation through 1DPCs via efficient nonlinear frequency mixing

The present study aims to investigate the transmitted and reflected light beams from a one-dimensional defect photonic crystal (1DPC) composed of a three-level ladder-type quantum system. The lower leg of the ladder scheme is driven by a weaker probe beam, while the upper leg is driven by a stronger control beam. Unlike natural atoms, this type of model with broken symmetry permits the generation of a sum-frequency signal beam between the majority of higher and lower quantum states, resulting in the formation of a cyclic closed-loop arrangement for the interaction between light and matter. The Laguerre Gaussian (LG) field with strong coupling and a weak probe light interact with the quantum system. A new weak signal light is produced as a result of the system’s symmetric breakdown, and the medium becomes azimuthal dependent. We investigate the spatial dependence of the transmitted and reflected light from a defect 1DPC by using azimuthal modulation of the LG light. Additionally, we will discuss how the absorption spectrum’s spatial modulation varies with winding numbers. We discover that in certain regions, the gain can be realized without population inversion under certain specified parametric parameters.

Topological phases and non-Hermitian topology in tunable nonreciprocal cyclic three-mode optical systems

We propose a method for simulating a 1D non-Hermitian Su-Schrieffer-Heeger model with modulated nonreciprocal hopping using a cyclic three-mode optical system. The current system exhibits different localization of topologically nontrivial phases, which can be characterized by the winding number. We find that the eigenenergies of such a system undergo a real-complex transition as the nonreciprocal hopping changes, accompanied by a non-Bloch parity-time symmetry breaking. We explain this phase transition by considering the evolution of saddle points on the complex energy plan and the ratio of complex eigenenergies. Additionally, we demonstrate that the skin states resulting from the non-Hermitian skin effect possess higher-order exceptional points under the critical point of the non-Bloch parity-time phase transition. Furthermore, we investigate the non-Hermitian skin phase transition by the directional mean inverse participation ratio and the generalized Brillouin zone. This work provides an alternative way to investigate the novel topological and non-Hermitian effects in nonreciprocal optical systems.

Miniature atom bottle traps enabled by chiral doughnut light

We highlight what we believe to be a novel optical set-up which enables the confinement of cold atoms in a finite set of sub-wavelength bottle traps. This involves two counter-propagating vortex beams with the same winding number ℓ = ±1 and the same circular polarization (. Strong focusing generates significant longitudinal field components which become responsible for an on-axis standing wave enabling the axial confinement of far blue-detuned atoms. The off-axis radial confinement is provided by the optical potential due to the transverse components of the light. The trap characteristics are illustrated using experimentally accessible parameters and are tunable by changing the power, focusing and ellipticity of the light. Atoms trapped in such a set-up are useful for applications, including quantum simulation and quantum information processing.

Joule Thomson expansion, Maxwell equal area law and topological interpretation of Phantom RN AdS black holes

This paper is devoted to study the thermodynamic topological defects, Joule–Thomson (J−T) and Maxwell’s equal area law of Phantom RN AdS black holes (BHs). The inversion temperatures and inversion curves are obtained and by using the isenthalpic curves in the temperature–pressure (T−P) plane to locate the cooling and heating regions. Using Maxwell’s equal-area law, we select different independent conjugate variables to study the phase transitions of Phantom RN AdS BHs. We find that phase transition rely on the electric potential and horizon radius of the BH when its charge is constant. Phase transition of Phantom RN AdS BH are proportional to the ratio of event horizon to its cosmological constant, where the latter is assumed to be constant. Moreover, we consider the Phantom RN AdS BH as defects with a topological nature within thermodynamic domain and examine the local and global topology by computing the winding numbers at the defects, which concludes that the overall topological charge is either equal to 0 or 1.

Identifying gap-closings in open non-Hermitian systems by biorthogonal polarization

We investigate gap-closings in one- and two-dimensional tight-binding models with two bands, containing non-Hermitian hopping terms, and open boundary conditions (OBCs) imposed along one direction. We compare the bulk OBC spectra with the periodic boundary condition (PBC) spectra, pointing out that they do not coincide, which is an intrinsic characteristic of non-Hermitian systems. The non-Hermiticity, thus, results in the failure of the familiar notions of bulk-boundary correspondence found for Hermitian systems. This necessitates the search for topological invariants which can characterize gap-closings in open non-Hermitian systems correctly and unambiguously. We elucidate the behavior of two possible candidates applicable for one-dimensional slices—(1) the sum of winding numbers for the two bands defined on a generalized Brillouin zone and (2) the biorthogonal polarization (BP). While the former shows jumps/discontinuities for some of the non-Hermitian systems studied here, at points when an edge mode enters the bulk states and becomes delocalized, it does not maintain quantized values in a given topological phase. On the contrary, BP shows jumps at phase transitions, and the quantized value of one or zero, which corresponds to whether an actual edge mode exists or whether that mode is delocalized and absorbed within the bulk (not being an edge mode anymore).

Thermodynamic Topology of Black Holes in f(R) Gravity

Abstract In this work, we study the thermodynamic topology of a static, a charged static, and a charged rotating black hole in f(R) gravity. For charged static black holes, we work in two different ensembles: the fixed charge (q) ensemble and fixed potential (ϕ) ensemble. For charged rotating black holes, four different types of ensembles are considered: fixed (q, J), fixed (ϕ, J), fixed (q, Ω), and fixed (ϕ, Ω) ensemble, where J and Ω denote the angular momentum and the angular frequency, respectively. Using the generalized off-shell free energy method, where the black holes are treated as topological defects in their thermodynamic spaces, we investigate the local and global topologies of these black holes via the computation of winding numbers at these defects. For the static black hole we work in three models. We find that the topological charge for a static black hole is always −1 regardless of the values of the thermodynamic parameters and the choice of f(R) model. For a charged static black hole, in the fixed charge ensemble, the topological charge is found to be zero. Contrastingly, in the fixed ϕ ensemble, the topological charge is found to be −1. For charged static black holes, in both the ensembles, the topological charge is observed to be independent of the thermodynamic parameters. For charged rotating black holes, in the fixed (q, J) ensemble, the topological charge is found to be 1. In the fixed (ϕ, J) ensemble, we find the topological charge to be 1. In the case of the fixed (q, Ω) ensemble, the topological charge is 1 or 0 depending on the value of the scalar curvature (R). In the fixed (Ω, ϕ) ensemble, the topological charge is −1, 0, or 1 depending on the values of R, Ω, and ϕ. Therefore, we conclude that the thermodynamic topologies of the charged static black hole and charged rotating black hole are influenced by the choice of ensemble. In addition, the thermodynamic topology of the charged rotating black hole also depends on the thermodynamic parameters.

Winding Number, Density of States, and Acceleration

Abstract Winding number and density of states are two fundamental physical quantities for non-self-adjoint quasi-periodic Schrödinger operators, which reflect the asymptotic distribution of zeros of the characteristic determinants of the truncated operators under Dirichlet boundary condition, with respect to complexified phase and the energy, respectively. We will prove that the winding number is in fact Avila’s acceleration, it is also closely related to the density of states by a generalized Thouless formula for non-self-adjoint Schrödinger operators and Avila’s global theory. Communicating Editor: Prof Svetlana Jitomirskaya

Fabrication of Zigzag Parylene Nanofibers in Liquid Crystals with Electric Field-Induced Defect Structures.

Liquid crystals (LCs) have been adopted to induce tunable physical properties that dynamically originated from their unique intrinsic properties responding to external stimuli, such as surface anchoring condition and applied electric field, which enables them to be the template for aligning functional guest materials. We fabricate the fiber array from the electrically modulated (in-plain) nematic LC template using the chemical vapor polymerization (CVP) method. Under an electric field, an induced defect structure with a winding number of -1/2 contains a periodic zigzag disclination line. It is known that LC defect structures can trap the guest materials, such as particles and chemicals. However, the resulting fibers grow along the LC directors, not trapped in the defects. To show the versatility of our platform, nanofibers are fabricated on patterned electrodes representing the alphabets 'CVP.' In addition, the semifluorinated moieties are added to fibers to provide a hydrophobic surface. The resultant orientation-controlled fibers will be used in controllable smart surfaces that can be used in sensors, electronics, photonics, and biomimetic surfaces.

Quantum phases of spin-1/2 extended XY model in transverse magnetic field

In this study, a spin-1/2 extended anisotropic XY chain has been introduced in which both time reversal and SU(2) symmetries are broken but Z 2 symmetry is preserved. Magnetic and topological phase diagrams in the parameter space have been drawn in the presence of transverse magnetic field. Entanglement measures like mutual information and quantum discord are also evaluated and it indicates that these transitions are second order in nature. Quantum phase transition is noted at zero magnetic field, as well as magnetic long range order is found to withstand magnetic field of any strength. Exact analytic results for spin-spin correlation functions have been obtained in terms of Jordan-Wigner fermionization. Existence of long range magnetic order has been investigated numerically by finding correlation functions as well as the Binder cumulant in the ground state. Dispersion relation, ground state energy, and energy gap are obtained analytically. In order to find the topologically nontrivial phase, sign of Pfaffian invariant and value of winding number have been evaluated. Both magnetic and topological phases are robust against the magnetic field and found to move coercively in the parameter space with the variation of its strength. Long range orders along two orthogonal directions and two different topological phases are found and their one-to-one correspondence has been found. Finally casting the spinless fermions onto Majorana fermions, properties of zero energy edge states are studied. Three different kinds of Majorana pairings are noted. In the trivial phase, next-nearest-neighbor Majorana pairing is found, whereas two different types of nearest-neighbor Majorana pairings are identified in the topological superconducting phase.

Ground states of a distinct spin–orbit-coupled spin-1 Bose–Einstein Condensate in a toroidal trap

This study delves into the influence of a new type of spin–orbit coupling (SOC) on spin-1 Bose–Einstein condensates (BECs) in quasi-two-dimensional annular regions. The unconventional characteristics of the annular potential trap coupled with innovative artificial gauge field bring about a diverse array of phase diagrams for the ground state behavior of the BECs. In the antiferromagnetic state, we observe a distinct petal-like phase structure with components exhibiting novel density and phase distributions. In the ferromagnetic state, the spin-1 BECs exhibit rich asymmetric states which showcase a new type of vortex-molecule structure with multiple winding numbers in the huge hole. Many other novel breaking structures have also been discovered by modulating the strength of the nonlinear interaction. We contribute the different ground states of distinct spin–orbit couplings to the different symmetries.

On a continuation of quaternionic and octonionic logarithm along curves and the winding number

This paper deals with the problem of finding a continuous extension of the hypercomplex (quaternionic or octonionic) logarithm along (quaternionic or octonionic) paths which avoid the origin. The main difficulty depends upon this fact: while a branch of the complex logarithm can be defined in a small open neighbourhood of a strictly negative real point, no continuous branch of the hypercomplex logarithm can be defined in any open set which contains a strictly negative real point. To overcome this difficulty, we use the logarithmic manifold introduced in [4]: in general, the existence of a lift of a path to this manifold is not guaranteed and, indeed, the problem of lifting a path to the logarithmic manifold is completely equivalent to the problem of finding a continuation of the hypercomplex logarithm along this path.The second part of the paper scrutinizes the existence of a notion of winding number (with respect to the origin) for hypercomplex loops that avoid the origin, even though it is known that the definition of winding number for such loops is not natural in Rn when n is greater than 2. The surprise is that, in the hypercomplex setting, the new definition of winding number introduced in this paper can be given and has full meaning for a large class of hypercomplex loops (untwisted loops with companion that avoid the origin).Finally an original but rather natural notion of homotopy for these hypercomplex loops (the c-homotopy) is presented and it is proved to be suitable to comply with the intrinsic geometrical meaning of the winding number for this class of loops, namely, two such hypercomplex loops are c-homotopic if, and only if, they have the same winding number.