Total Winding Number Research Articles

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.

Defect patterns of two-dimensional nematic liquid crystals in confinement.

A two-dimensional or quasi-two-dimensional nematic liquid crystal refers to a surface-confined system. When such a system is further confined by external line boundaries or excluded from internal line boundaries, the nematic directors form a deformed texture that may display defect points or defect lines, for which winding numbers can be clearly defined. Here, a particular attention is paid to the case when the liquid crystal molecules prefer to form a boundary nematic texture in parallel to the wall surface (i.e., following the homogeneous boundary condition). A general theory, based on geometric argument, is presented for the relationship between the sum of all winding numbers in the system (the total winding number) and the type of confinement angles and curved segments. The conclusion is validated by comparing the theoretical defect rule with existing nematic textures observed experimentally and theoretically in recent years.

Euler-obstructed Cooper pairing: Nodal superconductivity and hinge Majorana zero modes

Since the proposal of monopole Cooper pairing in [Phys. Rev. Lett. 120, 067003 (2018)], considerable research efforts have been dedicated to the study of Cooper pairing order parameters constrained (or obstructed) by the nontrivial normal-state band topology at Fermi surfaces in 3D systems. In the current work, we generalize the topologically obstructed pairings between Chern states (like the monopole Cooper pairing) by proposing Euler obstructed Cooper pairing in 3D systems. The Euler obstructed Cooper pairing widely exists between two Fermi surfaces with nontrivial band topology characterized by nonzero Euler numbers; such Fermi surfaces can exist in 3D PT-protected spinless-Dirac/nodal-line semimetals with negligible spin-orbit coupling, where PT is the space-time inversion symmetry. An Euler obstructed pairing channel must have pairing nodes on the pairing-relevant Fermi surfaces, and the total winding number of the pairing nodes is determined by the sum or difference of the Euler numbers on the Fermi surfaces. In particular, we find that when the normal state is time-reversal invariant and the pairing is weak, a sufficiently-dominant Euler obstructed pairing channel with zero total momentum leads to nodal superconductivity. If the Fermi surface splitting is small, the resultant nodal superconductor hosts hinge Majorana zero modes. The possible dominance of the Euler obstructed pairing channel near the superconducting transition and the robustness of the hinge Majorana zero modes against disorder are explicitly demonstrated using effective or tight-binding models. Our work presents the first class of higher-order nodal superconductivity originating from the topologically obstructed Cooper pairing.

Topological Defects in Nematic Liquid Crystals: Laboratory of Fundamental Physics

Topological Defects The curvature of a closed manifold exhibiting orientational order has a strong impact on positions and the number of topological defects (TDs). The surface integrated Gaussian curvature determines the total winding number of TDs within the manifold. If a region possessing a large enough negative Gaussian curvature is introduced it can trigger pairs {defect, antidefect}, possessing opposite signs of the winding number. More details can be found in article number 2000752 by Mitja Kralj, Marko Kralj, and Samo Kralj.

Effective Floquet model for minimally twisted bilayer graphene

We construct an effective Floquet lattice model for the triangular network that emerges in interlayer-biased minimally twisted bilayer graphene and which supports two chiral channels per link for a given valley and spin. We introduce the Floquet scheme with the one-channel triangular network and subsequently extend it to the two-channel case. From the bulk topological index (winding number) and finite system calculations, we find that both cases host anomalous Floquet insulators (AFIs) with a different gap-opening mechanism. In the one-channel network, either time-reversal or in-plane inversion symmetry has to be broken to open a gap. In contrast, in the two-channel network, interchannel coupling can open a gap without breaking these symmetries yielding a valley AFI with counterpropagating edge states. This phase is topologically trivial with respect to the total winding number but robust in the absence of intervalley scattering. Finally, we demonstrate the applicability of the Floquet model with magnetotransport calculations.

Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional Systems with Space-Time Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle

We show that the Wannier obstruction and the fragile topology of the nearly flat bands in twisted bilayer graphene at magic angle are manifestations of the nontrivial topology of two-dimensional real wave functions characterized by the Euler class. To prove this, we examine the generic band topology of two dimensional real fermions in systems with space-time inversion $I_{ST}$ symmetry. The Euler class is an integer topological invariant classifying real two band systems. We show that a two-band system with a nonzero Euler class cannot have an $I_{ST}$-symmetric Wannier representation. Moreover, a two-band system with the Euler class $e_{2}$ has band crossing points whose total winding number is equal to $-2e_2$. Thus the conventional Nielsen-Ninomiya theorem fails in systems with a nonzero Euler class. We propose that the topological phase transition between two insulators carrying distinct Euler classes can be described in terms of the pair creation and annihilation of vortices accompanied by winding number changes across Dirac strings. When the number of bands is bigger than two, there is a $Z_{2}$ topological invariant classifying the band topology, that is, the second Stiefel Whitney class ($w_2$). Two bands with an even (odd) Euler class turn into a system with $w_2=0$ ($w_2=1$) when additional trivial bands are added. Although the nontrivial second Stiefel-Whitney class remains robust against adding trivial bands, it does not impose a Wannier obstruction when the number of bands is bigger than two. However, when the resulting multi-band system with the nontrivial second Stiefel-Whitney class is supplemented by additional chiral symmetry, a nontrivial second-order topology and the associated corner charges are guaranteed.

Winding numbers of nodal points in Fe-based superconductors

We analyze the nodal points in multi-orbital Fe-based superconductors from a topological perspective. We consider the $s^{+-}$ gap structure with accidental nodes, and the $d$-wave gap with nodes along the symmetry directions. In both cases, the nodal points can be moved by varying an external parameter, e.g., a degree of inter-pocket pairing. Eventually, the nodes merge and annihilate via a Lifshitz-type transition. We discuss the Lifshitz transition in Fe-based superconductors from a topological point of view. We show, both analytically and numerically, that the merging nodal points have winding numbers of opposite sign. This is consistent with the general reasoning that the total winding number is a conserved quantity in the Lifshitz transition.

Configurable topological textures in strain graded ferroelectric nanoplates

Topological defects in matter behave collectively to form highly non-trivial structures called topological textures that are characterised by conserved quantities such as the winding number. Here we show that an epitaxial ferroelectric square nanoplate of bismuth ferrite subjected to a large strain gradient (as much as 105 m−1) associated with misfit strain relaxation enables five discrete levels for the ferroelectric topological invariant of the entire system because of its peculiar radial quadrant domain texture and its inherent domain wall chirality. The total winding number of the topological texture can be configured from − 1 to 3 by selective non-local electric switching of the quadrant domains. By using angle-resolved piezoresponse force microscopy in conjunction with local winding number analysis, we directly identify the existence of vortices and anti-vortices, observe pair creation and annihilation and manipulate the net number of vortices. Our findings offer a useful concept for multi-level topological defect memory.

Nonintersecting Brownian motions on the unit circle

We consider an ensemble of $n$ nonintersecting Brownian particles on the unit circle with diffusion parameter $n^{-1/2}$, which are conditioned to begin at the same point and to return to that point after time $T$, but otherwise not to intersect. There is a critical value of $T$ which separates the subcritical case, in which it is vanishingly unlikely that the particles wrap around the circle, and the supercritical case, in which particles may wrap around the circle. In this paper, we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as $n\to\infty$, and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution. We also give a streamlined approach to identifying the Pearcey and tacnode processes in scaling limits. The formula of the tacnode correlation kernel is new and involves a solution to a Lax system for the Painlev\'{e} II equation of size 2 $\times$ 2. The proofs are based on the determinantal structure of the ensemble, asymptotic results for the related system of discrete Gaussian orthogonal polynomials, and a formulation of the correlation kernel in terms of a double contour integral.

Speed Regulation Technique of One Bearingless 8/6 Switched Reluctance Motor With Simpler Single Winding Structure

Some types of bearingless switched reluctance motors (BSRMs) have been developed up to now. Their differences mainly focus on different stator-rotor tooth number and winding structures: two layers of windings and single layer of windings for 12/8 or 8/6 BSRM. Their driving principles are just the same: driving current and radial force current are regulated to generate torque and radial force separately. The main difference is the loading of two currents on one single layer of windings or on two separate layers of windings. Further analysis proves that single layer winding structure has more advantages, for example simpler motor and winding structure. Based on the above researches, one 8/6 BSRM with simpler single layer of winding structure has been developed furthermore. Its main characteristic is that the total winding number of motor is decreased from eight to six, which is only two more than four windings of a normal 8/6 SRM. Only six converters are demanded to achieve levitation operation. Its special driving theory and speed regulation technique of the motor by applying three phase windings have been introduced. To testify the new idea, theoretical analysis and practical test have been finished for one test motor. One initial speed of 5000 r/min of levitation operation has been achieved.

Disoriented chiral condensate dynamics in the(2+1)DO(3)model

The dynamics of symmetry breaking after a quench is numerically simulated on a lattice for the (2+1)-dimensional $O(3)$ model. In addition to the standard sigma model with a temperature-dependent ${\ensuremath{\Phi}}^{4}$ potential the energy functional includes a four-derivative current-current coupling to stabilize the size of the emerging extended topological textures. The total winding number can be conserved by constraint. As a model for the chiral phase transition during the cooling phase after a hadronic collision this allows us to investigate the interference of ``baryon-antibaryon'' production with the developing disoriented aligned domains. The growth of angular correlations, condensate, and average orientation is studied in dependence of texture size, quench rate, and symmetry breaking. The classical dissipative dynamics determines the rate of energy emitted from the relaxing source for each component of the 3-vector field which provides a possible signature for the domains of disoriented chiral condensate (DCC). We find that the ``pions'' are emitted in two distinct pulses; for sufficiently small lattice size the second one carries the DCC signal, but it is strongly suppressed as compared to simultaneous ``sigma''-meson emission. We compare the resulting anomalies in the distributions of DCC pions with probabilities derived within the commonly used coherent state formalism.

The probability for formation of collapsing textures and texture knots

Global textures are interesting because they are promising candidates for seeds in the structure formation of the universe. The important configurations are those which will be able to collapse. The type of configurations that I will consider are half knots and true knots. I define a half knot as one that covers more than a half-S 3, and a true knot as a configuration that covers the whole of S 3. Configurations that are half knots will be able to collapse, a new argument for this collapse criterion is given in this paper. I will also give some critical remarks on the method of using the total winding number as a criterion for identifying a knot. I propose another more direct method and make use of it when I estimate the probability for formation of both true and half knots by numerical simulations. In three dimensions the probability for formation of a true knot is found to be p knot = 0.003 and the probability for formation of a half knot is found to be p half = 0.066. Taking into account the chance of annihilation the probability for collapse is found to be in the interval p coll ϵ [0.01, 0.04]. In the skyrmion picture of hadrons the investigations of formation of true knots are important, and some implications of my result on the predictions of a “topological” theory of baryon production in jets are discussed.

Third-order braid invariants

The author analyses the topological invariants of three-braided curves a(t), b(t) and c(t). 3-braids are represented as a single phase curve gamma (t) in a two-dimensional configuration space. This configuration space consists of a set of triangular regions connected at their vertices. The curve gamma (t) passes through a vertex whenever a(t), b(t) and c(t) are collinear. The sequence of vertices completely describes the braid (up to uniform twists). The length T of this sequence can be employed as a measure of topological complexity. The energy of a set of braided magnetic flux tubes is expected to be proportional to T2+W2, where W is the total winding number (or signed crossing number) of the braid. Second-order winding numbers are integrals of closed 1-forms like d theta ab. The author presents a third-order winding number Psi ( gamma ) which is also an integral of a closed 1-form, but which depends on relations between all three curves.

On the origin of tertiary turns in covalently closed double-stranded cyclic DNA.

At certain stages of the life cycle, many viral DNA's occur as duplexes (for a review, see ref. 1). Physicochemical studies by Vinograd and co-workers have shown that such molecules contain tertiary turns topologically locked into the molecules,l 2 as a consequence of which the molecules are more compact hydrodynamically and hence have sedimentation coefficients higher than those for molecules without tertiary turns. When examined under the electron microscope by Kleinschmidt method, such molecules appear in agreement with the notion that the molecules contain tertiary turns; hence the name twisted DNA or supercoil is frequently used. If one single-chain scission is introduced in a DNA, a swivel is provided. By rotating around the single bond opposite to the point of the single-chain scission, the molecule may release its tertiary turns. This is accompanied by a decrease in sedimentation coefficient, as well as other changes in the physicochemical properties of the molecule. For the purpose of simplicity, we shall refer to double-stranded molecules as covalently DNA or closed circles, and double-stranded molecules with one or more single-chain scissions as free-swiveling DNA or simply as cyclic Inasmuch as all the DNA's thus far isolated from nature appear to be twisted, the origin of the tertiary turns in such molecules is of interest. Several hypotheses have been suggested: (1) The molecule was partially unwound (i.e., a segment of denatured region was present) at the time of final closure. Since the total winding number of a is a topological invariant, I the winding of the unwound region associated with the base-pairing after closure necessarily introduces tertiary turns of the opposite sense into the molecule. One reason for such an unwound region might be to accommodate the closing enzyme.1 (2) The number of base-pairs per Watson-Crick helical turn of a helix in vivo is different from the number of base-pairs per turn of the helix in the purified DNA. The change of the number of Watson-Crick helical turns of the results in tertiary turns in the molecule. (3) The molecule in vivo at the time of final closure is wound on a protein core; the removal of the core during the purification of the produces tertiary turns.2 4 The work of Gellert with Escherichia coli extracts showed that it was possible to form X from X in vitro.5 It was later shown that the active ingredients in the extracts were a polynucleotide-joining enzyme and its cofactor, diphosphopyridine nucleotide (DPN).1-9 In the presence of the cofactor, the enzyme can link two polynucleotide chains through a 3'5' phosphodiester bond, provided that the 3' hydroxyl and the 5' phosphate ends of