Winding number

The same classification into winding numbers can be used in a system with N nuclear degrees of freedom, in which the Cl seam is an N — 2)-dimensional hyperline as in Fig. 1. For example, if we take N = 3, then the seam is a line the... 

Figure 5. Examples of Feynman paths belonging to different homotopy classes, illustrating how the winding number n is defined.

Retracing the argument used to justify point (2), it is clear that, in a multiply connected space, a given path is only coupled to those paths into which it can be continuously deformed. By definition, these are all the paths that belong to the same homotopy class. Paths belonging to different homotopy classes are thus decoupled from one another [41 5]. For a reactive system with a Cl that has the space of Fig. 1, this means that a path with a given winding number n is coupled to all paths with the same n, but is decoupled from paths with different n. As a result, the Kernel separates into [41-45]... 

Figure 6. Diagram showing how the winding number n of the Feynman paths should be defined with respect to the cut line. In (a), the cut line (chains) is placed between (() = — and 2n — in (b), between (() = ti/4 and —In/A. In (c), the wave function describes a unimolecular reaction, in which the initial state occupies the (gray shaded) area shown. Feynman paths originate from all points within this area (inset) their winding number n is defined with respect to the common cut line.

We can represent this function in the single space, provided we use a common cut line for all three components. This is shown schematically in Fig. 17. Use of the common cut line is equivalent to taking the linear combinations in the double space, then cutting a 27i-wide section out of the entire (4)). The winding numbers n of the Feynman paths that enter the three equivalent reagent channels must all be dehned with respect to the common cut line, since they are analogous to paths starting at different points in the initial state of a unimolecular reaction (Section 11.D). 

Figure 18. Complete unwinding of an encircling nuclear wave function >0 by mapping onto higher cover spaces, (a) The function in the single space (b) e in the double space (c) 4 in the quadruple space (d) schematic picture of in a 2hn cover space. In each case, will be completely unwound if it contains contributions from Feynman paths belonging to (b) 2, (c) 4, and (d) h different winding-number classes.

Figure 4 Distributions for separations between the nearest distances nearest NPs, saddle points, NPs with the same (++) and opposite winding numbers (+-) in a chaotic Sinai billiard. The radial distribution of nearest distances for completely random points (26) is shown by the dashed curve in (a). The corresponding distribution for the Berry model function for a chaotic state (2) and random superposition of 16 eigen functions for a rectangular box with the same size and energy are shown by dots and thin curves, respectively.

Since under large gauge transformations Ai => WAiU+iWdiU, Ncs => Ncs + Nw where Nw is the winding number of the gauge transformation ... 

This is in contradistinction to requiring the appending of yet another parameter, the "winding number" a, as is done for the IUPAC name [2]-[cyclotricontane]-[cycloheptatricontane]-catenane (a = 2). 

There is, however, another type of transition possible in two dimensions, a transition between states without LRO. This is the Kosterlitz-Thouless transition [8] mentioned in Sections II and V.B.l. It is relevant to superconductivity, commensurate-incommensurate transitions [61], planar magnetism, the electron gas system, and to many other systems in two dimensions. It involves vortices (thus the requirement of a two-component order parameter) characterized by a winding number q = (1/2-rr) dr V0, in which 0 is the phase of the order parameter (see also Ref. 4), the amplitude being fixed. These free vortices have an energy [see Eq. (28)] given by... 

The corresponding trajectories can be best visualized as motion restricted to a two-dimensional torus, as shown in Fig. 1. If the frequency ratio, or the winding number ( i/( 2, is a rational number, the two DOFs are in resonance and an individual trajectory will close on itself on the torus. By contrast, if coi/a)2 is an irrational number, then as time evolves a single trajectory will eventually cover the torus. The motion in the latter case is called conditionally periodic. 

Figure 47. Quantum transport suppressed by the cantorus. Shown here is the logarithm of P I, li) versus I, with the initial state given by /, = 4. The winding number of the cantorus is given by 2 — g. From top to bottom the dotted lines have the slopes 3.2, 18, and 6.5, respectively. [From N. T. Maitra and E. J. Heller, Phys. Rev. E 61, 3620 (2000).]...

In order to show the -dependence of the flow on the TCM for large we numerically calculate the stable and unstable manifolds on the TCM. Figure 5 depicts them for (Z, ) = (1,1). We confirmed the following things. When is increased, the winding number JV of the stable and unstable manifolds of c and d on the TCM around the body of the TCM is monotonically decreased. From numerical observation, JV is saturated to certain value in the limit oo. For even relatively small value of JV is almost saturated—for instance, < 100 for Z = 1,2,3,4,5. 

The connection between the observation in Fig. 5 and the observation in Figs. 6b and 7b is unknown here. Now we elucidate this connection. Figure 6b for (Z, ) = (1,1) shows one branch of Wtcm c) and the triple collision orbits on the Poincare section. A remarkable point is that the triple collision curves tc and accumulate at 10 points on 0. As shown in Fig. 9a, these points are the points at which if tcm c) and if TCM d) cross the plane x = 0 and which are, of course, just on 0. We denote these points by PTCM,r=o- It is clear that the number of points of Ptcm,t=o is related to the existence of tori in the Poincare section 3>. If the tori exist, its outer most torus has periodic points. These periodic points have the stable and unstable manifolds. Branches of these stable and unstable manifolds run toward precisely if tcm c) and if TCM d) on 0. This situation was observed in Fig. 8. Therefore, the number of the points of Ptcm,t=o is related to the existence of the tori. At the same time, the number of the points of PrcM,r=Q just corresponds to the winding number xF of f tcm c) or ifTcmid) around the body of the TCM as mentioned in the previous subsection. In Figs. 7b and 9b, the case of (Z, = (1,7) is shown. As... 

Fig. 21 Plots obtained by mean-field calculations for an EHFMI [24]. Calculations are performed for a two-dimensional 16x16 square lattice with open boundary conditions. Parameters used are U = St and t = —0.2t t denotes the second nearest neighbor transfer integrals tjk)- The number of doped holes is 8 half of them are centers of merons and the rest are centers of antimerons. (a) Plot for spin configuration. Centers of spin vortices are indicated as M for a meron (winding number -H spin vortex) and A for an antimeron (winding number —1 spin vortex), respectively, (b) Plot for current density j (short black arrows) and V x (long orange arrows). M and A here indicate centers of counterclockwise and clockwise loop currents, respectively (c) Plot for D(x), which connects j(x) and V/(x) as j(x) = D(x) V/(x) (d) Plot for 2j (thick orange line arrows are not attached but directions are the same as those of the black arrows) and 2Z)(x) V/(x) (black arrows)...

If the magnetic field is absent, loop currents with winding number 1 are created with the total sum of them being zero. We consider this situation below. For simplicity, we only retain adjacent pairs in the second sum in (59) we replace rij by its average value given by 1 /-Jnx, and consider a square lattice of a lattice constant 1 /. As a result, the following very simple interaction potential for... 

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