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Analytic Property of Vortex-Type Spin Structures in Itinerant Electron System with Inversion Symmetry

Analytic properties of the multiple spin density waves showing vortex structures, which have recently been found in the fcc Hubbard model using the generalized Hartree-Fock (GHF) approximation, have been investigated. It is shown that the 2Q multiple helical spin density waves (SDW) consist of two types of half-skyrmion vortices and two types of antivortices. In the 3Q multiple helical SDW, the half-skyrmion vortex structures with the"westerlies"are shown to be twisted along the axis perpendicular to the vortex plane. The 12Q multiple SDW (12QMSDW) are analyzed as a superposition of the 2-4QMSDW on the (001) plane and the remaining 4QMSDW. It is shown analytically that the 2-4QMSDW forms the planar vortices and antivortices with zero core polarization, so that the 12QMSDW forms the same types of vortices with additional SDW polarization in the [001] direction. These results are in good agreement with the visualization image data based on the GHF calculations.


Introduction
The B20-type transition metal compounds such as MnSi 1 and FeGe 2 are wellknown to show a vortex-type structure called the magnetic skyrmions [3][4][5][6][7][8] in which the magnetic moments being perpendicular to the vortex plane at the core center incline spirally away from the center and point in the opposite direction on the periphery. Their magnetic structures and properties have been much investigated in the past decade, 9,10 since the magnetic skyrmions are a new type of topologically protected system and are expected to be applicable to the nano-size magnetic devices. More recently, new kind of vortex structure called the half skyrmion, in which the inclined magnetic moments are in the vortex plane on the periphery, has been found in the β-Mn type Co 8 Zn 9 Mn 3 compound. 11 In the MnGe compound, 12 the 3 dimensional type of skyrmions have been discovered. Moreover a new kind of 4Q multiple spin density waves (MSDW) has been found in MnSi 1−x Ge x (0.25 < x < 0.7) alloys. 13 These experimental results indicate that a variety of the vortex-type magnetic structures and related MSDW are possible in itinerant electron systems.
Although the Dzyaloshinski-Moriya (DM) interactions 14,15 play an important role in the formation of skyrmions mentioned above, [16][17][18] recent theoretical investigations based on the Heisenberg model have shown that the skyrmion-type lattice structures can also be stabilized by the competition between the ferro-and antiferro-magnetic interactions. 19 The skyrmion type structure without the DM interactions should be stabilized also in itinerant electron system because more complex competitions between the long-range ferro-and antiferro-magnetic interactions are expected via electron hopping process.
We recently investigated the real-space magnetic structures of the multiple spin density waves (MSDW) on the fcc lattice using the application visualization system (AVS) image technique to understand the skyrmion-type magnetic structure in itinerant electron system, 20 and found that the skyrmion-type vortex structures are described by the multiple helical SDW.
Using the phenomenological Ginzburg-Landau theory, [21][22][23] we showed that the skyrmion type vortex structures are possible even in the itinerant electron system with inversion symmetry.
In the subsequent paper, 24 which we refer to I hereafter, we performed the self-consistent magnetic-structure calculations for the Hubbard model 25-31 on the fcc lattice using the Gen- Although we have shown that the vortex-type skyrmion structures are possible even in the itinerant electron system with inversion symmetry, it is based on the observation of the magnetic structures with use of the AVS image technique. Therefore, the details of their vortex structures as well as the type of the skyrmion structures have not yet been well understood.
We need alternative approach to clarify the magnetic structures obtained from the numerical calculations.
In this paper, we examine the analytic property of the vortex structures obtained by the self-consistent GHF calculations for the Hubbard model on the fcc lattice. We start from the principal MSDW obtained by the Fourier analysis of the calculated magnetic structures, and determine the vortex core positions assuming the vortex magnetic structure. Expanding the MSDW with respect to the position vector around the vortex cores, we determine analytically the local structure of the vortices.
We determined the local magnetic structures of the 2QH, the 3QH, and the 12QMSDW obtained in the self-consistent GHF calculations. We found that the 2QH consists of the two types of vortices and antivortices. The former vortices are the half skyrmions, 32, 33 and form 2/20 the antiferromagnetic (AF) lattice. The 3QH forms the half-skyrmion-base vortex structures being twisted along the axis perpendicular to the vortex plane. The 12QMSDW also form the vortex lattice structure, but these vortices have no core polarization and are not the skyrmion system. Moreover we found that the two types of antivortices with no core polarization appear in common to the 2QH and the 12QMSDW, while the core centers in the 3QH have the magnetic moments which helically rotate along the axis perpendicular to the vortex plane.
In the following section, we summarize the magnetic-structure diagram on the fcc lattice obtained in the self-consistent GHF calculations. 24 New data points are added to the diagram, and a new type of the AF-base half-skyrmion structure is reported there. In Sect. 3.1, the local structures of the 2QH is analyzed, and the boundaries of the half-skyrmions are analytically determined. In Sect. 3.2, the 3QH structure is analyzed on the basis of the 2QH local structures. Analytic expressions of the half skyrmions with the "westerlies" are given. Section 3.3 is devoted to the analysis of the 12QMSDW. We regard the 12QMSDW as a superposition of the 2-4QMSDW (or the 8QMSDW) and the remaining 4QMSDW, and analyze first the former, taking the same steps as in the 2QH analysis. We will show that the 2-4QMSDW consists of the vortex and antivortex structures with zero core polarization, so that the 12QMSDW forms the vortex structures which are different from the half skyrmions. In the last section 4, we summarize the present work and discuss the remaining problems.

Magnetic-Structure Diagram
We considered in paper I 24 the Hubbard model 25-31 on the fcc lattice with nearest neighbor transfer integral t and Coulomb interaction energy U , and performed the magnetic structure calculations as a function of U and n e on the basis of the GHF approximation. We adopted a large cluster consisting of the 10 × 10 × 10 fcc unit cells with periodic boundary condition and applied the recursion method 34,35 to calculate the electronic states of magnetic structures.
We considered, as the input magnetic structures for self-consistent calculations, the single-Q helical state (1QH), the 2QH, the 3QH, as well as the paramagnetic state (P) and the ferromagnetic state (F). Starting from these structures and varying the wave numbers, we performed the self-consistent GHF calculations at zero temperature and determined the stable magnetic structure comparing their total energies. Since the output magnetic structures do not necessarily agree with the starting ones, we also obtained the other magnetic structures in these calculations.
We summarize the magnetic structures obtained in paper I in Fig. 1. Note that we increased the number of points in the figure, making further numerical calculations, in order to make clearer the stability regions in the diagram. In the upper-right and the lower-left regions, we find the F and P states, respectively. Between the F and P, the conical structure (C), the antiferromagnetic structure of the first and second kinds (AF), the 1QH, the 2QH, the 4Q multiple spin density waves (4QMSDW), the 12Q multiple SDW (12QMSDW), and 3/20  the magnetic moment on site R l = (x l , y l , z l ) is given by a 4QMSDW as follows.
The magnetic moments (1) on the nth layer ( , i.e., z l = na/2) can be folded as follows.

2Q multiple helical SDW
We found in I the half-skyrmion type vortex structure on the xy plane at (n e , U ) = (1.70 − 1.73, 5.5) using the AVS image analysis. The principal terms of the Fourier analysis for the calculated magnetic moments m l are given by two elliptical-helical waves as follows.
Here m 1 = 0.075, m 2 = 0.044 for (n e , U ) = (1.7, 5.5), and the wave vectors are given by , and q = 0.2. We neglect in the following analysis additional 4Q-MSDW satellite terms found in the Fourier analysis in I.
The vortices are accompanied by the vortex cores. Assuming the existence of the skyrmiontype vortex structure on the xy plane, the condition for the core center R c is given by Solving the above equations, we obtain R c as Here n and n ′ are integer, L is the wave length defined by L = a/q, and z l is the z coordinate 5/20 of the lattice points on a xy plane. We note that R c are not necessarily located on the lattice points in general, though it is not essential in the following analysis.
Introducing the lattice point r l = (x l ,ŷ l ,ẑ l ) measured from the core center R c (R l = r l + R c ), Eq. (3) is written as When n and n ′ are even, i.e., n = 2k and n ′ = 2k ′ , we have Expanding m l with respect to Q 1 ·r l and Q 2 ·r l up to the second order, we obtain With use of the polar coordinates (r l , φ l ) on the xy plane, we obtain This is a clockwise vortex with the winding number w = 1, helicity α = −π/2, and core polarization p c = 2m 2 which is perpendicular to the vortex plane. The vortex structure may be schematically expressed by (I) in Fig. 2.
When n is odd (n = 2k + 1) and n ′ is even (n ′ = 2k ′ ) in R c , Eq. (6) is expressed as Taking the same steps as before, we obtain This is an antivortex with w = −1, α = π/2, and p c = 0. The antivortex structure may be expressed by (II) in Fig. 2.
In the case of n = 2k and n ′ = 2k ′ + 1, we obtain This is the antivortex with w = −1, α = −π/2, p c = 0, and may be expressed by (IV) in Fig.   2. The 4 types of vortex structures mentioned above are summarized in Fig. 2.
This yields two types of linear functions which are orthogonal each other.
Here integers l and l ′ are parameters to be determined. One can determine the boundary for each half skyrmion from these lines. For example, for the half-skyrmion (III) centered at 7/20 Figure 3 shows the AVS image on the xy plane for (n e , U ) = (1.7, 5.5) and the analytic result for k = k ′ = 1 and q = 0.2. We verify that the analytic results describe the half-skyrmion half-antiskyrmion pairs as well as their boundaries. In addition, we find the antivortex structures (II) and (IV) between the half-skyrmion particles with the same helicity. The antivortices were overlooked in the previous AVS-image observation in paper I.

8/20
The 3QH is regarded as a superposition of the 2QH given by Eq. (3) with m 1 = m 2 = m and the single-Q helical wave me(z l ), where e(z l ) = i cos Q 3 ·R l + j sin Q 3 ·R l . Thus the 3QH forms the half-skyrmion vortex structures with a "westerlies" me(z l ). 24 The local structure, for example, around R c = ((k + 1/4)L, k ′ L, 0) is obtained from Eq. (9) as follows.
The AVS image obtained for (n e , U ) = (1.68, 6) with q = 0.2 is shown in Fig. 4. Analytic results of the half-skyrmion vortex structures with the "westerlies" mi, which are centered at R c = (25a/4, 5a, 0) and R c = (35a/4, 15a/2, 0), respectively, are indicated by the curved arrow circles I and III on the xy plane (z l = 0). The "westerlies" me(z l ) rotates on the xy plane with the translation along the z axis. Therefore the vortices with "westerlies", I and III, also rotate along the z axis with the period L = 5a. Their magnetic moments at R c would conically rotate along the z axis; m l = me(z l ) ± 2mk. The antivortices with "westerlies", II and IV, obtained by the present analysis, also rotate along the z axis, but the magnetic moments at R c would show the helical rotation along the z axis. The same local structures appear on the yz and zx planes because Eq. (17) is invariant for the cyclic transformation of the x, y, and z coordinates.
The 12QMSDW are regarded as a superposition of three 4QMSDW's with x, y, z polarizations, respectively. In order to understand the magnetic structure of the 12QMSDW, we consider the first two 4QMSDW in Eq. (19) (2-4QMSDW) in the following section.  Half-skyrmion vortices with the "westerlies" mi (I and III) and antivortices (II and IV) found by the present analysis are shown by the curved arrow circles with yellow (left) or red (right) color, and the blue circles, respectively. Use a zoom-in tool to see more detailed structure.
The 2-4QMSDW contains vortex structures. In order to clarify this feature, we assume that the MSDW forms the planar vortex structures. The conditions for their core centers R c = (x 0 , y 0 , z 0 ) are given by For z 0 such that q ′ z 0 = nπ, the above equations reduce to We find 4 possible sets of solutions for Eqs. (24) and (25); R c = (l 1 a/2q, (l 2 + 1/2)a/2q, na/2q ′ ), (l 1 a/2q, y 0 , na/2q ′ ), (x 0 , l 2 a/2q ′ , na/2q ′ ), and (l 1 a/2q ′ , l 2 a/2q ′ , na/2q ′ ), where l 1 , l 2 , and n are integer. Among them, the second and third solutions are not suitable as the core centers because they contain arbitrary parameters, x 0 and y 0 . In the first case, we do not find the vortex structure around R c . In what follows, we consider the last case: As in Sect. 3.1, we introduce the position vector r l such that R l = r l + R c , and expand Eq. (20) with respect to Q · r l . When q ′ = kq, we obtain m l in the lowest order as follows.
Here c When k = 2 as in the case of (n e , U ) = (1.75, 5), we have R c as Here L = a/q denotes the longest wave length of the system. We then examine the local magnetic structures m l around 16 independent R l 1 l 2 n on a (001) plane with n. We obtained around R 00n , R 02n , R 20n , and R 22n , the shear-flow type SDW structures with a node on a line through R c : Here the signs should be taken as (+, +) for R 00n , (+, −) for R 02n , (−, +) for R 20n , and (−, −) for R 22n . Around R 11n , R 13n , R 31n , and R 33n , we have Here the signs should be taken as (+, +) for R 11n , (+, −) for R 13n , (−, +) for R 31n , and We obtained the SDW local structures with a node on the (001) plane around R 01n , R 03n , R 21n , and R 23n .
On the other hand, we obtained the clockwise planar vortex structure around R 12n : the anticlockwise planar vortex structure around R 30n : and the antivortices around R 10n and R 32n : respectively.
Moreover, we can verify that m l = 0 on the lines (l 1 L/2, y l , nL/4) and (x l , (2l 2 + 1)L/4, nL/4). The planar vortices and antivortices in the 2-4QMSDW are separated each other by these node lines with no magnetic moment. The local structures of the 2-4QMSDW around 16 points on the xy plane are summalized in Fig. 5. We find the vortices around R 120 , R 300 , and R 340 , and the antivortices around R 100 , R 140 , and R 320 . They are separated by the node lines mentioned above.

Magnetic structure of 12QMSDW
We can analyze the 12QMSDW given by Eq. (19) as a superposition of the 2-4QMSDW on the xy plane (the first two terms in Eq. (19)) and the remaining 1-4QMSDW in the z direction (the last term in Eq. (19)). For example, around R 12n = (1/4, 1/2, n/4)L, we have a planar vortex structure of 2-4QMSDW given by Eq. (32) for q ′ = 2q. The magnetic moments of the remaining 1-4QMSDW, m l (1-4Q), can be expressed after transformation R l = r l + R 12n as Expanding m l (1-4Q) with respect to Q · r l , we obtain for even n, and for odd n.
This is a vortex structure in which the z components of the magnetic moments are polarized in opposite directions across the line y = L/2 or x = L/4 through R 12n = (1/4, 1/2, n/4)L.
Consequently, there is no core polarization in the vortex, and thus it is not the skyrmion.
These antivortices also have no core polarization at R 32n and R 10n .
It should be noted that Eq. (19) for the 12QMSDW has the cyclic symmetry for x, y, The vortex structure around R n12 = (n/4, 1/4, 1/2)L on the yz plane, for example, is expressed as and the vortex structure around R 2n1 = (1/2, n/4, 1/4)L on the zx plane is given by Here the polar coordinates (r l , φ l ) are defined on the yz (zx) plane in Eq. (42) (Eq. (43)).
In Figs  direction, while those at the lower half are polarized in the −z direction. In particular, there is no polarization at their core centers. Thus these vortices are not half skyrmions, though we suggested the latter in paper I. We also verify in the AVS image the antivortices around R 320 and R 100 , as found in the present analysis. The vortices on the xy plane changes their sign alternatively with increasing the layer number n, and the positive (negative) regions of the z polarization in these vortices rotate anticlockwise with increasing n.

Summary
We have examined the analytic property of the multiple spin density waves (MSDW) showing the vortex structures, which were found in the AVS image analysis for the magnetic structures of the Hubbard model on the fcc lattice obtained by the generalized Hartree-Fock (GHF) approximation, 24 in order to understand the vortex structures and their relations to In the 2Q multiple helical SDW (2QH) with wave number q = 0.2, which was found at (n e , U ) = (1.7, 5.5) in the GHF calculations, we determined the two types of vortex structures with different helicity (±π/2) and core polarization (±2m) on the xy plane. These are the half-skyrmion and half-antiskyrmion particles, forming the antiferromagnetic (AF) structure on a giant face-centered square lattice. We also found the antivortex structures with zero core polarization between the half skyrmions with the same helicity, which were overlooked in our previous AVS image analyses.
The half-skyrmion lattice structure has recently been reported in the Co 8 Zn 9 Mn 3 compound. 11 The AF half-skyrmion vortex lattice structure agrees with that was observed in 16/20 Co 8 Zn 9 Mn 3 . However the antivortices with no core polarization between the half-skyrmions have not yet been verified experimentally. In the experimental analysis, the half-antiskyrmions with positive core polarization are allocated from the viewpoint of the local-moment model. 11 Experimental data are obtained under the external field 20mT being perpendicular to the