#  Phase degree of freedom and topology in multiple-$Q$ spin textures

Kotaro Shimizu  Department of Applied Physics, The University of Tokyo, Tokyo
113-8656, Japan  Shun Okumura  The Institute for Solid State Physics, The
University of Tokyo, Kashiwa 277-8581, Japan  Yasuyuki Kato  Yukitoshi Motome
Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan

###### Abstract

A periodic array of topological spin textures, such as skyrmions and
hedgehogs, is called the multiple-$Q$ spin texture, as it is represented by a
superposition of multiple spin density waves. Depending on the way of
superposition, not only the magnetic but also the topological properties are
modified, leading to a variety of quantum transport and optical phenomena
caused by the emergent electromagnetic fields through the Berry phase. Among
others, the phase degree of freedom of the superposed waves is potentially
important for such modifications, but its effect has not been fully
investigated thus far. Here we perform systematic theoretical analyses of
magnetic and topological properties of the multiple-$Q$ spin textures with the
phase degree of freedom in two and three dimensions. By introducing a
hyperspace with an additional dimension corresponding to the phase degree of
freedom, we establish a generic framework to deal with the phase shift in the
multiple-$Q$ spin textures. Using the hyperspace representation, we elaborate
the complete topological phase diagrams for the superpositions of three proper
screws or sinusoidal waves in two dimensions and those of four in three
dimensions. In the two-dimensional case, we find that the phase shift as well
as the magnetization change can yield the skyrmion lattices with the skymion
number of $-2$, $-1$, $1$, and $2$, corresponding to the evolution of the
Dirac strings connecting hedgehogs and antihedgehogs in the three-dimensional
hyperspace. We show that the high skyrmion numbers $\pm 2$ appear in wider
parameter regions for the sinusoidal superpositions than the screw ones.
Meanwhile, in the three-dimensional case, we clarify that the topological
phase diagrams include various types of the hedgehog lattices whose total
number of hedgehogs and antihedgehogs ranges up to $48$ in a cubic unit.
Interestingly, the phase shift can generate unusual Dirac strings running on
the horizontal planes perpendicular to the magnetization direction, which
gives rise to unconventional pair creation of hedgehogs and antihedgehogs
while increasing the magnetization in the case of the screw superpositions. We
also show that the amplitude of the emergent magnetic field is maximized by
fusion of hedgehogs and antihedgehogs on the horizontal Dirac strings in both
proper screw and sinusoidal cases. In addition, by analyzing the numerical
data in the previous studies, we demonstrate that phase shifts are indeed
caused by an external magnetic field, associated with the topological
transitions in the multiple-$Q$ spin textures. Our results illuminate the
topological aspects of the skyrmion and hedgehog lattices with the phase
degree of freedom, which would be extended to other multiple-$Q$ textures and
useful for the exploration of topologically nontrivial magnetic phases and
exotic quantum phenomena.

##  I Introduction

Topology is originally a mathematical concept to discuss the properties of a
geometric object, but has been extended to a variety of research fields in
these decades [1, 2, 3, 4]. In condensed matter physics, the electronic states
of solids have been discussed by topology of the electronic band structures in
momentum space, which led to the discovery of topological states of matter,
e.g., the quantum Hall state [5, 6, 7, 8] and the topological insulator [9,
10, 11, 12, 13, 14, 15, 16]. Topology appears also in geometric structures of
the spin textures in magnets. The typical examples are swirling noncoplanar
spin textures, such as magnetic skyrmions [17, 18, 19, 20] and Bloch points
[21, 22, 23] (or equivalently, magnetic hedgehogs [24, 25, 26]). These
topological spin textures are characterized by an integer called the
topological invariant: for instance, the skyrmion number for skyrmions [27, 3,
28] and the monopole charge for hedgehogs [24, 3]. The topological invariant
is robust against perturbation, which ensures the topological protection of
the spin textures. Moreover, the noncoplanar spin structures can generate the
so-called emergent electromagnetic fields through the Berry phase mechanism
[29, 24, 4, 30, 31, 28]. They are fictitious electromagnetic fields acting on
electrons coupled to the spin textures, and thus, give rise to unusual quantum
transport and optical phenomena, such as the topological Hall effect [32, 33,
34, 35, 36, 37], the Nernst effect [38, 39, 40], the magneto-optical Kerr
effect [41, 42], and the emergent inductance [43, 44, 45, 46, 47]. Owing to
these distinguishing properties, the topological spin textures have attracted
a lot of attention for not only fundamental physics but also applications to
next-generation electronic devices.

In magnetic materials, the topological spin textures often appear in the form
of a periodic array of the topological objects. For instance, the magnetic
skyrmions appear by forming a periodic lattice called the skyrmion lattice
(SkL) [48, 49, 50, 51, 52, 53], and the magnetic hedgehogs (and the
antihedgehogs) appear as the hedgehog lattice (HL) [54, 25, 55, 26, 56, 57,
58]. These periodic structures can be represented by superpositions of
multiple spin density waves, and hence, called multiple-$Q$ spin textures. An
example is shown in Fig. 1(a), where a SkL is given by a superposition of
three proper screws and called the $3Q$-SkL. As such superpositions yield
superstructures as the interference patterns, the topological spin textures
can be viewed as “spin moiré” [59]. Analogous to moiré fringes in optics,
there are many ways to modulate the spin moiré, such as the number of
superposed waves [60, 61, 62], the amplitudes of each spin density waves [63],
and the angles between the propagating directions of the superposed waves
[59]. Such modulations bring about various topological phases with different
topological invariants and topological phase transitions between them.

Figure 1:  Variations of spin textures while changing the phase in the
superpositions of three proper screws with the wave vectors ${\bf q}_{1}$,
${\bf q}_{2}$, and ${\bf q}_{3}$. (b) and (c) are obtained from (a) by the
phase shift of $\frac{\pi}{2}$ and $\pi$, respectively. The left panels
display the schematic pictures of the superposed waves and the right panels
show the spin textures obtained by the superpositions. The color of the arrows
in the left panels represents the out-of-plane component of spins, as
indicated in the inset of (c). The skyrmion number changes from (a) $N_{\rm
sk}=1$ to (b) $N_{\rm sk}=0$, and to (c) $N_{\rm sk}=-1$, and the symmetry of
the spin texture changes from (a) sixfold, (b) threefold, and (c) sixfold
rotational symmetry. The black rhombus represents the magnetic unit cell. See
Sec. III.2 for the details.

Among such parameters in spin moiré, it was recently pointed out that the
phase degree of freedom in the superposed waves is an important parameter to
control not only the spin textures but also their symmetry and topological
properties [64, 65]. The situation is illustrated for superpositions of three
proper screws in Fig. 1. Figure 1(a) shows a SkL by a superposition of three
proper screws running in the $120^{\circ}$ directions of ${\bf q}_{1}$, ${\bf
q}_{2}$, and ${\bf q}_{3}$. The spin texture comprises a hexagonal array of
skyrmions with the skyrmion number $N_{\rm sk}=1$ per magnetic unit cell and
has sixfold rotational symmetry; see Sec. III.2 for the details. Let us
consider a phase shift in the ${\bf q}_{1}$ component from this state. The
results obtained by $\frac{\pi}{2}$ and $\pi$ shifts are shown in Figs. 1(b)
and 1(c), respectively. The symmetry is reduced to threefold for the
$\frac{\pi}{2}$ shift, but recovered to sixfold for the $\pi$ shift.
Accordingly, the topological property is also changed: The $\frac{\pi}{2}$
shift gives a periodic array of half skyrmions called merons and antimerons,
leading to $N_{\rm sk}=0$, while the $\pi$ shift leads to a SkL with $N_{\rm
sk}=-1$. Thus, the phases of the superposed waves are relevant degrees of
freedom, but their impact has not been fully elucidated thus far, for not only
SkLs but also the other topological spin textures like HLs.

In this paper, we systematically clarify the effect of phase shifts on the
typical multiple-$Q$ spin textures, two-dimensional (2D) SkLs and three-
dimensional (3D) HLs, focusing on their topological properties and the
emergent magnetic fields. We first establish a generic framework to deal with
the phase shift by introducing the hyperspace with an additional dimension
corresponding to the phase degree of freedom, inspired by the description of
the phason degree of freedom in quasicrystals [66, 67, 68, 69]. In the
hyperspace representation, the 2D SkLs composed of the three spin density
waves with the phase degree of freedom are mapped to 3D HLs in which the Dirac
strings connecting the hedgehogs and antihedgehogs correspond to the skymion
and antiskyrmion cores in the original 2D SkLs. Similarly, the 3D HLs composed
of four spin density waves are mapped to four-dimensional (4D) loop lattices
in which intersections of the membranes defined by the loops, which we call
“the Dirac planes”, by 3D hyperplanes give hedgehog-antihedgehog pairs
connected by the Dirac strings in the original 3D HLs. Analyzing the
topological objects in the hyperspace representation, we systematically
elucidate the evolution of the multiple-$Q$ spin structures for the phase
shift as well as the magnetization change.

In the 2D case, considering the superpositions of three proper screws or
sinusoidal waves, we obtain various $3Q$-SkLs with $N_{\rm sk}$ ranging from
$-2$ to $2$ depending on the phase and magnetization. We find that the phase
diagram is dominated by the SkLs with $N_{\rm sk}=\pm 1$ in the case of the
proper screw superpositions, whereas the $N_{\rm sk}=\pm 2$ regions become
wider in the sinusoidal case. Interestingly, at zero magnetization, we always
obtain the $N_{\rm sk}=\pm 1$ ($\pm 2$) SkLs for any phase shifts in the screw
(sinusoidal) case; namely, the $N_{\rm sk}=\pm 2$ ($\pm 1$) SkLs are obtained
only with nonzero magnetization in the screw (sinusoidal) case.

On the other hand, in the 3D case, we clarify the topological phase diagrams
for the superpositions of four proper screws or sinusoidal waves. We find
various $4Q$-HLs classified by the number of the hedgehogs and antihedgehogs
per unit cube, $N_{\rm m}$: $N_{\rm m}=8$, $16$, $32$, and $48$ for the screw
case, and $N_{\rm m}=8$ $16$, $24$, $32$, and $48$ for the sinusoidal case.
For the former case, the emergent magnetic field is always negative, while for
the latter, it takes both positive and negative values. Notably, we find
unusual Dirac strings running on the horizontal planes perpendicular to the
magnetization direction. In the case of the screw superpositions, they give
rise to pair creation of the hedgehogs and antihedgehogs and accordingly the
increase of $N_{\rm m}$ from $16$ to $48$ while increasing the magnetization.
This is highly unusual since the increase of the magnetization usually results
in pair annihilation and the reduction of $N_{\rm m}$. In this screw case,
$N_{\rm m}$ is always $16$ for any phases at zero magnetization, and the HLs
with larger $N_{\rm m}$ appear only for nonzero magnetization. In contrast, in
the case of the sinusoidal superpositions, the zero magnetization state has
always the largest $N_{\rm m}=48$, and $N_{\rm m}$ decreases monotonically
while increasing the magnetization. We also show that, in both cases, the
amplitude of the emergent magnetic field is maximally enhanced by fusion of
three hedgehogs and antihedgehogs on the horizontal Dirac strings where
$N_{\rm m}$ changes from $48$ to $16$.

Finally, we study how the phases evolve in the actual multiple-$Q$ spin
textures in microscopic models. Specifically, analyzing the numerical data for
the 2D Kondo lattice model [70] and the 3D effective spin model [57], we
extract the sum of phases in the superposed waves by fitting the spin
configurations obtained by the numerical simulations. We show that phase
shifts indeed take place in both cases around the topological phase
transitions caused by an external magnetic field: For the SkL, the sum of
phases jumps from $\sim 0$ to $\sim\frac{\pi}{4}$ accompanied by the reduction
of $|N_{\rm sk}|$ from $2$ to $1$, while for the HLs, it rapidly decreases
from $\sim\frac{\pi}{3}$ to $\sim 0$ accompanied by the reduction of $N_{\rm
m}$ from $16$ to $8$.

Our results establish the generic and systematic way to investigate the effect
of phase shifts in the multiple-$Q$ spin textures. Moreover, they open a way
for unexplored topological magnetic states and phase transitions, which may
bring about nontrivial electronic structures and quantum transport properties
through the emergent electromagnetic fields. Thus, our findings would shed
light on the engineering of the multiple-$Q$ spin textures and related physics
through the phase degree of freedom which has been overlooked thus far.

The rest of the paper is organized as follows. In Sec. II, we introduce the
hyperspace representation for general multiple-$Q$ spin textures. In Sec. III,
applying the framework to 2D $3Q$ states (Sec. III.1), we elucidate the effect
of phase shifts and magnetization changes on the spin textures, the symmetry,
and the topological properties of the $3Q$ states composed of three proper
screws (Sec. III.2) and three sinusoidal waves (Sec. III.3). In Sec. IV, we
present the results for the 3D $4Q$ states: the hyperspace representation
(Sec. IV.1), and the effect of phase shifts and magnetization changes on the
$4Q$ states composed four proper screws (Sec. IV.2) and four sinusoidal waves
(Sec. IV.3). In Sec. V, we present the analysis of the actual numerical data
for the 2D Kondo lattice model (Sec. V.1) and the 3D effective spin model
(Sec. V.2). We discuss the results in Sec. VI. Section VII is devoted to the
summary of this paper.

##  II Phase degree of freedom and hyperspace representation

In this section, we propose a theoretical framework to systematically analyze
the phase degree of freedom in multiple-$Q$ spin structures. We consider a
generic form of the multiple-$Q$ spin structures in $d$-dimensional continuous
space, which is given by the function of the real-space position ${\bf
r}=(x_{1},\ldots,x_{d})$ as

$\displaystyle{\bf S}({\bf
r})\propto\sum_{\eta=1}^{N_{Q}}~{}\left(\psi_{\eta}^{\rm c}{\bf
e}_{\eta}^{1}\cos\mathcal{Q}_{\eta}+\psi_{\eta}^{\rm s}{\bf
e}_{\eta}^{2}\sin\mathcal{Q}_{\eta}\right)+m\hat{\bf z},$ (1)

where $N_{Q}$ is the number of superposed waves, $\psi_{\eta}^{\rm c}$ and
$\psi_{\eta}^{\rm s}$ are the amplitudes of cosinusoidal and sinusoidal waves
with the wave vector ${\bf q}_{\eta}=(q_{\eta}^{1},\ldots,q_{\eta}^{d})$,
respectively, ${\bf e}_{\eta}^{1}$ and ${\bf e}_{\eta}^{2}$ are unit vectors,
$\mathcal{Q}_{\eta}={\bf q}_{\eta}\cdot{\bf r}+\varphi_{\eta}$, and $m$
represents the uniform magnetization and $\hat{\bf z}$ is the unit vector
along the $z$ direction. The spin length is normalized as $|{\bf S}({\bf
r})|=1$ for any ${\bf r}$. Note that $m$ is not the net magnetization because
of the normalization.

###  II.1 Phase degree of freedom

Let us consider how the spin structures are modulated by changing the phases
$\varphi_{\eta}$. When $N_{Q}\leq d$ and ${\bf q}_{\eta}$ are linearly
independent, the periodic spin textures are described by the set of $N_{Q}$
linearly-independent magnetic translation vectors ${\bf a}_{\eta}$, which
satisfy

$\displaystyle{\bf a}_{\eta}\cdot{\bf
q}_{\eta^{\prime}}=2\pi\delta_{\eta\eta^{\prime}},$ (2)

where $\delta_{\eta\eta^{\prime}}$ is the Kronecker delta. In this case, a
phase shift from $\varphi_{\eta}$ to $\varphi_{\eta}+\Delta\varphi_{\eta}$ is
reduced to a spatial translation from ${\bf r}$ to ${\bf r}+\Delta{\bf r}$
with

$\displaystyle\Delta{\bf
r}=\sum_{\eta=1}^{N_{Q}}\frac{\Delta\varphi_{\eta}}{2\pi}{\bf a}_{\eta},$ (3)

since the following relation holds:

$\displaystyle{\bf q}_{\eta}\cdot\Delta{\bf
r}=\sum_{\eta^{\prime}}\frac{\Delta\varphi_{\eta^{\prime}}}{2\pi}{\bf
q}_{\eta}\cdot{\bf a}_{\eta^{\prime}}=\Delta\varphi_{\eta}.$ (4)

Hence, the phase degree of freedom is irrelevant when $N_{Q}\leq d$ [71].

The situation is, however, different for $N_{Q}>d$. In this case, the wave
vectors ${\bf q}_{\eta}$ are not linearly independent of each other, as
exemplified for $N_{Q}=3$ and $d=2$ in Fig. 1. This means that a phase shift
cannot be reduced to a spatial translation, as ${\bf a}_{\eta}$ defined by Eq.
(2) with $d$ out of $N_{Q}$ wave vectors ${\bf q}_{\eta}$ do not satisfy the
relation in Eq. (3).

###  II.2 Hyperspace representation

To discuss the effect of the phase shift in the case of $N_{Q}>d$
systematically, it is convenient to introduce $N_{Q}$-dimensional ($N_{Q}$D)
hyperspace. In the hyperspace, we can introduce a position vector ${\bf
R}=(X_{1},X_{2},\ldots,X_{N_{Q}})$ and $N_{Q}$ linearly-independent wave
vectors ${\bf Q}_{\eta}=(Q_{\eta}^{1},Q_{\eta}^{2},\ldots,Q_{\eta}^{N_{Q}})$
($\eta=1,2,\ldots,N_{Q}$) so that they satisfy the relations [72]

$\displaystyle\mathcal{Q}_{\eta}={\bf q}_{\eta}\cdot{\bf
r}+\varphi_{\eta}={\bf Q}_{\eta}\cdot{\bf R}.$ (5)

In this hyperspace, we can define the $N_{Q}$ linearly-independent magnetic
translation vectors ${\bf A}_{\eta}$ ($\eta=1,2,\ldots,N_{Q}$) satisfying

$\displaystyle{\bf A}_{\eta}\cdot{\bf
Q}_{\eta^{\prime}}=2\pi\delta_{\eta\eta^{\prime}}.$ (6)

By using Eqs. (5) and (6), the hyperspace position ${\bf R}$ and the real-
space one ${\bf r}$ are related with each other as

$\displaystyle\left(\begin{array}[]{c}X_{1}\\\ X_{2}\\\ \vdots\\\
X_{N_{Q}}\end{array}\right)=M_{Q}^{-1}M_{q}\left(\begin{array}[]{c}x_{1}\\\
x_{2}\\\ \vdots\\\ x_{d}\\\ 1\end{array}\right),$ (16)

where

$\displaystyle M_{Q}$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}{\bf Q}_{1}\\\ {\bf Q}_{2}\\\ \vdots\\\ {\bf Q}_{N_{Q}}\end{array}\right)=\left(\begin{array}[]{cccc}Q_{1}^{1}&Q_{1}^{2}&\ldots&Q_{1}^{N_{Q}}\\\ Q_{2}^{1}&Q_{2}^{2}&\ldots&Q_{2}^{N_{Q}}\\\ \vdots&\vdots&\ddots&\vdots\\\ Q_{N_{Q}}^{1}&Q_{N_{Q}}^{2}&\ldots&Q_{N_{Q}}^{N_{Q}}\end{array}\right),$ (25) $\displaystyle M_{q}$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{c | c}{\bf q}_{1}&\varphi_{1}\\\ {\bf q}_{2}&\varphi_{2}\\\ \vdots&\vdots\\\ {\bf q}_{N_{Q}}&\varphi_{N_{Q}}\end{array}\right)=\left(\begin{array}[]{cccc}q_{1}^{1}&\ldots&q_{1}^{d}&\varphi_{1}\\\ q_{2}^{1}&\ldots&q_{2}^{d}&\varphi_{2}\\\ \vdots&\ddots&\vdots&\vdots\\\ q_{N_{Q}}^{1}&\ldots&q_{N_{Q}}^{d}&\varphi_{N_{Q}}\\\ \end{array}\right).$ (34)

Note that $M_{Q}^{-1}$ is given as

$\displaystyle M_{Q}^{-1}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\left(\begin{array}[]{cccc}{\bf A}_{1}&{\bf
A}_{2}&\ldots&{\bf A}_{N_{Q}}\end{array}\right)$ (36) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\left(\begin{array}[]{cccc}A_{1}^{1}&A_{2}^{1}&\ldots&A_{N_{Q}}^{1}\\\
A_{1}^{2}&A_{2}^{2}&\ldots&A_{N_{Q}}^{2}\\\ \vdots&\vdots&\ddots&\vdots\\\
A_{1}^{N_{Q}}&A_{2}^{{N_{Q}}}&\ldots&A_{N_{Q}}^{{N_{Q}}}\end{array}\right).$
(41)

The relation in Eq. (16) can be regarded as a surjective mapping from the set
of the real-space position ${\bf r}$ and the phases $\varphi_{\eta}$ onto the
hyperspace position ${\bf R}$ [73]. In this setting, Eq. (5) gives one-to-one
correspondence between a multiple-$Q$ spin configuration with the phase
variables $\varphi_{\eta}$ in the original $d$-dimensional real space and that
in the $N_{Q}$D hyperspace with fixed phases (in other words, without the
phase degrees of freedom). In this representation, a phase shift by
$\Delta\varphi_{\eta}$ in the original real space corresponds to a translation
in the hyperspace by

$\displaystyle\Delta{\bf
R}=\sum_{\eta=1}^{N_{Q}}\frac{\Delta\varphi_{\eta}}{2\pi}{\bf A}_{\eta}.$ (42)

Consequently, the hyperspace representation enables us to treat the phase
degree of freedom as additional coordinates in the hyperspace. We note that
the situation is analogous to the hyperspace introduced to understand the
structures of quasiperiodic crystals, where the number of translation vectors
are in general larger than the system dimension and the quasycrystals are
obtained by a “slice" of a periodic structure in the hyperspace with
additional dimensions spanned by the same number of vectors [66, 67, 68, 69].
In the quasicrystals, the additional variables in the hyperspace are called
phasons [74, 75, 76, 77], which also supports the analogy.

##  III $3Q$ skyrmion lattices

In this section, we discuss the effect of phase shifts on the spin structures
composed of three wave vectors in two dimensions, i.e., $N_{Q}=3$ and $d=2$,
by using the hyperspace representation introduced in Sec. II. Specifically, we
consider Eq. (1) with three ${\bf q}_{\eta}$ given by

$\displaystyle{\bf q}_{1}=\left(q,0\right),\ {\bf
q}_{2}=\left(-\frac{q}{2},\frac{\sqrt{3}q}{2}\right),\ {\bf
q}_{3}=\left(-\frac{q}{2},-\frac{\sqrt{3}q}{2}\right),$ (43)

where $|{\bf q}_{\eta}|=q$. For this $3Q$ spin structure, the 2D magnetic
translation vectors are defined as

$\displaystyle{\bf a}_{1}=\frac{2\pi}{q}\left(1,\frac{1}{\sqrt{3}}\right),\ \
{\bf a}_{2}=\frac{2\pi}{q}\left(0,\frac{2}{\sqrt{3}}\right).$ (44)

In the following, we focus on the two types of the $3Q$ spin structures. One
is the superposition of proper screws, which includes a $3Q$-SkL found in a
wide range of materials, as introduced in Sec. I. We call this type the screw
$3Q$ state; see Sec. III.2. The other type is the superposition of sinusoidal
waves, which includes a $3Q$-SkL with the skyrmion number of two found in the
Kondo lattice system on the triangular lattice [70] and its effective spin
model [78]. We call this type the sinusoidal $3Q$ state; see Sec. III.3.
Before going into the analyses, we present the hyperspace representation in
Sec. III.1, which is commonly used for these two types.

###  III.1 Hyperspace representation of the $3Q$ states

Figure 2:  (a) Schematic pictures of the relations between the wave vectors in
the 3D reciprocal hyperspace, ${\bf Q}_{\eta}$, and those in the 2D reciprocal
space, ${\bf q}_{\eta}$. (b) Corresponding relations between the magnetic
translation vectors in the 3D hyperspace, ${\bf A}_{\eta}$, and those in the
2D real space, ${\bf a}_{\eta}$. The projections of ${\bf A}_{\eta}$ onto the
$xy$ plane denoted by $\tilde{\bf a}_{\eta}$ are also shown.

Using the framework introduced in Sec. II.2, we construct the hyperspace
representation of the $3Q$ spin structures. First, we set the 3D wave vectors
${\bf Q}_{\eta}=(Q_{\eta}^{X},Q_{\eta}^{Y},Q_{\eta}^{Z})$ in Eq. (5) as

$\displaystyle{\bf Q}_{1}=q\left(1,0,\frac{1}{\sqrt{2}}\right),$ (45)
$\displaystyle{\bf
Q}_{2}=q\left(-\frac{1}{2},\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right),$ (46)
$\displaystyle{\bf
Q}_{3}=q\left(-\frac{1}{2},-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right),$
(47)

so that the projection of ${\bf Q}_{\eta}$ onto the $q^{x}q^{y}$ plane is
${\bf q}_{\eta}$, and ${\bf Q}_{1}$, ${\bf Q}_{2}$, and ${\bf Q}_{3}$ are
orthogonal to each other as shown in Fig. 2(a), without loss of generality.
Then, we obtain the corresponding magnetic translation vectors ${\bf
A}_{\eta}$ in the 3D hyperspace as

$\displaystyle{\bf A}_{1}=\frac{4\pi}{3q}\left(1,0,\frac{1}{\sqrt{2}}\right),$
(48) $\displaystyle{\bf
A}_{2}=\frac{4\pi}{3q}\left(-\frac{1}{2},\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right),$
(49) $\displaystyle{\bf
A}_{3}=\frac{4\pi}{3q}\left(-\frac{1}{2},-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right).$
(50)

Figure 2(b) illustrates the relation between ${\bf A}_{\eta}$ and ${\bf
a}_{\eta}$. Note that ${\bf a}_{\eta}$ is given by ${\bf a}_{\eta}=\tilde{\bf
a}_{\eta}-\tilde{\bf a}_{3}$, where $\tilde{\bf a}_{\eta}$ is the projection
of ${\bf A}_{\eta}$ onto the $xy$ plane.

By using Eq. (16), the hyperspace positions are related with the real-space
positions as

$\displaystyle\left(\begin{array}[]{c}X\\\ Y\\\
Z\end{array}\right)=V_{3}\left(\begin{array}[]{c}x\\\ y\\\
1\end{array}\right),$ (57)

where $V$ includes the phases as

$\displaystyle V_{3}=\left(\begin{array}[]{ccc}{\bf v}_{1}&{\bf v}_{2}&{\bf
v}_{3}\end{array}\right)=\left(\begin{array}[]{ccc}1&0&\frac{2}{3q}\left(\varphi_{1}-\frac{1}{2}(\varphi_{2}+\varphi_{3})\right)\\\
0&1&\frac{1}{\sqrt{3}q}\left(\varphi_{2}-\varphi_{3}\right)\\\
0&0&\frac{\sqrt{2}}{3q}\left(\varphi_{1}+\varphi_{2}+\varphi_{3}\right)\end{array}\right).$
(62)

It is worth noting that Eq. (57) can be rewritten as

${\bf R}=x{\bf v}_{1}+y{\bf v}_{2}+{\bf v}_{3}.$ (63)

This means that the spin configuration on the original 2D $xy$ plane is the
same as the one on a slice of the hyperspace spin configuration spanned by
${\bf v}_{1}$ and ${\bf v}_{2}$ including the point at ${\bf v}_{3}$, namely,
the horizontal plane with

$Z=\frac{\sqrt{2}}{3q}\tilde{\varphi},$ (64)

where

$\displaystyle\tilde{\varphi}=\sum_{\eta}\varphi_{\eta}=\varphi_{1}+\varphi_{2}+\varphi_{3}.$
(65)

Thus, only the summation of the phases $\varphi_{\eta}$ is relevant for the
present $3Q$ spin textures, instead of each value of $\varphi_{\eta}$. Note
that $\tilde{\varphi}$ has $2\pi$ periodicity; namely, the spin configuration
in the 3D hyperspace becomes equivalent with period of
$\frac{2\sqrt{2}\pi}{3q}$ in the $Z$ direction. In the original 2D space, the
$2\pi$ phase shift in $\tilde{\varphi}$
($\Delta\tilde{\varphi}=\sum_{\eta}\Delta\varphi_{\eta}=2\pi$) corresponds to
a spatial translation by

$\displaystyle\Delta{\bf r}_{\eta}=\tilde{\bf
a}_{\eta}-\sum_{\eta^{\prime}=1}^{3}\frac{\Delta\varphi_{\eta^{\prime}}}{2\pi}\tilde{\bf
a}_{\eta^{\prime}},$ (66)

where $\eta$ may take any of $1$, $2$, and $3$.

In the following, we apply the above hyperspace representation to analyze the
effect of phase shifts on the magnetic and topological properties of two types
of 2D $3Q$ states.

###  III.2 Screw $3Q$ state

In this subsection, we analyze the effect of phase shifts on the $3Q$ state
composed of three proper screws. The spin texture is given by Eq. (1) with
$N_{Q}=3$ and

$\displaystyle\psi_{\eta}^{\rm c}=\psi_{\eta}^{\rm s}=\frac{1}{\sqrt{3}},$
(67) $\displaystyle{\bf e}_{\eta}^{1}=\hat{\bf z},\ {\bf e}_{\eta}^{2}={\bf
e}_{\eta}^{0}\times{\bf e}_{\eta}^{1},\ {\bf
e}_{\eta}^{0}=\left(\frac{q^{x}_{\eta}}{q},\frac{q^{y}_{\eta}}{q},0\right)^{\mathsf{T}},$
(68)

where $\mathsf{T}$ denotes the transpose of the vector. The explicit form is
given as

$\displaystyle{\bf S}({\bf r})$ $\displaystyle\propto$
$\displaystyle\left(\begin{array}[]{c}\frac{\sqrt{3}}{2}(\sin\mathcal{Q}_{2}-\sin\mathcal{Q}_{3})\\\
-\sin\mathcal{Q}_{1}+\frac{1}{2}(\sin\mathcal{Q}_{2}+\sin\mathcal{Q}_{3})\\\
\cos\mathcal{Q}_{1}+\cos\mathcal{Q}_{2}+\cos\mathcal{Q}_{3}+\sqrt{3}m\end{array}\right).$
(72)

Before going into the analyses of the topological properties, let us discuss
the symmetry of the screw $3Q$ state in Eq. (72). Table 1 summarizes the
symmetry operations on the spin texture, together with the decomposition of
each operation into the change of the sum of phases $\tilde{\varphi}$, spatial
translation, and magnetization change. We here consider the point group
operations and time-reversal operation that do not change the wave vectors of
the proper screws. Other symmetry operations are expressed by the combinations
of those in Table 1, e.g., $C_{2y}=C_{2z}C_{2x}=C_{6z}^{3}C_{2x}$. From Table
1, we can obtain the symmetry operations which do not change both
$\tilde{\varphi}$ and $m$, i.e., the spin texture. Specifically, we find that
the screw $3Q$ state is symmetric under $C_{6z}$, $\mathcal{T}C_{2x}$, and
their combinations for $\tilde{\varphi}=0$ and $\pi$, otherwise $C_{3z}$,
$\mathcal{T}C_{2x}$, and their combinations. We note that spatial translation
combined with the $C_{6z}$ rotation can be represented by shifting the
rotation axis.

Table 1:  Symmetry operations for the screw $3Q$ state in Eq. (72) and their decompositions into the change in the sum of phases, spatial translation in the original 2D space, and the magnetization change: $C_{n\alpha}$ represents an $n$-fold rotation about the axis in the $\alpha$ direction and $\mathcal{T}$ represents the time-reversal operation.  operation | sum of phases | translation | magnetization
---|---|---|---
$C_{6z}$ | $\tilde{\varphi}\rightarrow 2\pi-\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}+\tilde{\bf a}_{\eta^{\prime}}$ | $m\rightarrow m$
$C_{2x}$ | $\tilde{\varphi}\rightarrow\pi+\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}$ | $m\rightarrow-m$
$\mathcal{T}$ | $\tilde{\varphi}\rightarrow\pi+\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}$ | $m\rightarrow-m$
Figure 3:  (a) Schematic of the three proper screws in the 3D reciprocal
hyperspace, whose wave vectors ${\bf Q}_{\eta}$ are defined by Eqs. (45),
(46), and (47) for the screw $3Q$ state in Eq. (72). (b) Corresponding cubic
MUC in the hyperspace and spin structures on the three horizontal planes with
$Z=\frac{\sqrt{2}}{3q}(2n+1)\pi$ ($n=0,1,2$). The right panel in (b) shows the
2D spin texture on the three slices, which corresponds to the spin structure
in Eq. (72) with
$\tilde{\varphi}=\varphi_{1}+\varphi_{2}+\varphi_{3}=(2n+1)\pi$, where $n$ is
an integer. The rhombus consisting of the two triangles and one hexagon
denotes the 2D MUC in the original 2D plane.

####  III.2.1 Hedgehogs in hyperspace

To discuss the effect of phase shifts on the screw $3Q$ state in Eq. (72), we
study the corresponding spin texture in the 3D hyperspace whose reciprocal
space is spanned by the wave vectors ${\bf Q}_{\eta}$ in Eqs. (45), (46), and
(47). The schematic picture is shown in Fig. 3(a). As discussed in the
previous subsection, a real-space spin configuration for a given phase
summation $\tilde{\varphi}$ on the original 2D plane corresponds to a
hyperspace spin configuration on the horizontal plane with
$Z=\frac{\sqrt{2}}{3q}\tilde{\varphi}$ [see Eq. (64)]. Such a correspondence
is exemplified in Fig. 3(b) for $\tilde{\varphi}=(2n+1)\pi$. In this case, the
intersections of the 3D cubic magnetic unit cell (MUC) and the horizontal
planes with $Z=\frac{\sqrt{2}}{3q}(2n+1)\pi$ $(n=0,1,2)$ give the hexagon and
the two triangles, which comprise the rhombus MUC in the original 2D plane.

In the 3D hyperspace, the spin structure composed of three proper screws may
comprise a 3D topological spin structure called $3Q$-HL [25, 79, 57, 59]. It
has a periodic array of topological defects called the hedgehogs and
antihedgehogs, whose cores are the singular points where the spin length
vanishes (see Sec. III.2.2). Indeed, by solving the equation ${\bf S}({\bf
r})=0$ for Eq. (72), we obtain the following eight solutions:

$\displaystyle(\mathcal{Q}_{1}^{*},\mathcal{Q}_{2}^{*},\mathcal{Q}_{3}^{*})$
$\displaystyle=$ $\displaystyle\left(\pi+p_{1}^{\rm scr}(m),\pi+p_{1}^{\rm
scr}(m),\pi+p_{1}^{\rm scr}(m)\right),$ (73) $\displaystyle\left(\pi-
p_{1}^{\rm scr}(m),\pi-p_{1}^{\rm scr}(m),\pi-p_{1}^{\rm scr}(m)\right),$
$\displaystyle\left(\pi-p_{2}^{\rm scr}(m),\pi-p_{2}^{\rm scr}(m),p_{2}^{\rm
scr}(m)\right)$ $\displaystyle\qquad\qquad\quad\mbox{and cyclic
permutations},$ $\displaystyle\left(\pi+p_{2}^{\rm scr}(m),\pi+p_{2}^{\rm
scr}(m),2\pi-p_{2}^{\rm scr}(m)\right)$
$\displaystyle\qquad\qquad\quad\mbox{and cyclic permutations},$

where

$\displaystyle p_{1}^{\rm scr}(m)=\arccos\left(\frac{m}{\sqrt{3}}\right),\quad
p_{2}^{\rm scr}(m)=\arccos\left(\sqrt{3}m\right).$ (74)

By using the relation

$\displaystyle{\bf R}^{*}=\sum_{\eta}\frac{\mathcal{Q}_{\eta}^{*}}{2\pi}{\bf
A}_{\eta},$ (75)

we obtain the positions of the eight singular points in the hyperspace as

$\displaystyle(X^{*},Y^{*},Z^{*})=$
$\displaystyle\quad\frac{\sqrt{2}}{3q}\left(0,0,3(\pi+p_{1}^{\rm
scr}(m))\right),\ \frac{\sqrt{2}}{3q}\left(0,0,3(\pi-p_{1}^{\rm
scr}(m))\right),$ $\displaystyle\quad\left(\frac{\pi-2p_{2}^{\rm
scr}(m)}{3q},\frac{\pi-2p_{2}^{\rm scr}(m)}{\sqrt{3}q},\frac{\sqrt{2}(2\pi-
p_{2}^{\rm scr}(m))}{3q}\right)$
$\displaystyle\quad\qquad\qquad\qquad\qquad\mbox{ and $C_{3}^{Z}$ symmetric
points},$ $\displaystyle\quad\left(-\frac{\pi-2p_{2}^{\rm
scr}(m)}{3q},-\frac{\pi-2p_{2}^{\rm
scr}(m)}{\sqrt{3}q},\frac{\sqrt{2}(4\pi+p_{2}^{\rm scr}(m))}{3q}\right)$
$\displaystyle\quad\qquad\qquad\qquad\qquad\mbox{ and $C_{3}^{Z}$ symmetric
points},$ (76)

where the $C_{3}^{Z}$ symmetric points are obtained by $2\pi/3$ and $4\pi/3$
rotations about the $Z$ axis.

####  III.2.2 Topological transition in 3D hyperspace

Figure 4:  Change of the positions of the hedgehogs and antihedgehogs, and the
Dirac strings connecting them in the spin structure in the 3D hyperspace
corresponding to the screw $3Q$ state in Eq. (72) while changing $m$: (a)
$m=0$, (b) $m=0.4$, (c) $m=0.8$, and (d) $m=1.6$. The red, blue, magenta, and
cyan spheres are the topological defects with $Q_{\rm m}=2$, $-2$, $1$, and
$-1$, respectively. The green and purple lines denote the Dirac strings with
the vorticity $\zeta=+1$ and $-1$, respectively.

Figure 4 illustrates the systematic change of the topological defects while
changing the magnetization along the $Z$ direction, $m$. When $m=0$, four out
of the eight solutions in Eq. (76) become identical to
$\frac{\sqrt{2}}{3q}\left(0,0,\frac{9\pi}{2}\right)$ and the rest four become
$\frac{\sqrt{2}}{3q}\left(0,0,\frac{3\pi}{2}\right)$, since $p_{1}^{\rm
scr}(0)=\frac{\pi}{2}$ and $p_{2}^{\rm scr}(0)=\frac{\pi}{2}$. Hence, there
are only two topological defects located on the $Z$ axis, as shown in Fig.
4(a). Following the arguments in Refs. [62, 79, 25, 57, 59], we compute the
monopole charge for these defects, which is defined by

$\displaystyle Q_{\rm m}=\frac{1}{4\pi}\int d\tilde{{\bf S}}\cdot{\bf b}({\bf
R}),$ (77)

where ${\bf b}({\bf R})=(b_{X}({\bf R}),b_{Y}({\bf R}),b_{Z}({\bf R}))$ is the
scalar spin chirality defined in the hyperspace as

$\displaystyle b_{i}({\bf R})=\frac{1}{2}\varepsilon^{ijk}{\bf S}({\bf
R})\cdot\left(\frac{\partial{\bf S}({\bf R})}{\partial
X_{j}}\times\frac{\partial{\bf S}({\bf R})}{\partial X_{k}}\right),$ (78)

where $\varepsilon^{ijk}$ is the Levi-Civita Symbol and ${\bf S}({\bf R})$ is
the spin at ${\bf R}$ in the 3D hyperspace; the integral in Eq. (77) is taken
on a closed surface surrounding the defect in the hyperspace. Here, $Q_{\rm
m}$ takes an integer, which defines the topological nature of the defects; a
defect with positive (negative) $Q_{\rm m}$ is a hedgehog (an antihedghog)
which is regarded as a source (sink) of the emergent magnetic fields [79, 25,
57, 59]. We find that the topological defect at
$\frac{\sqrt{2}}{3q}\left(0,0,\frac{3\pi}{2}\right)$ in Fig. 4(a) is a
hedgehog with $Q_{\rm m}=+2$ (red sphere) and the other one at
$\frac{\sqrt{2}}{3q}\left(0,0,\frac{9\pi}{2}\right)$ is an antihedgehog with
$Q_{\rm m}=-2$ (blue sphere). Following the procedures in Ref. [59], we also
identify four Dirac strings, which is the lines connecting the hedgehog and
antihedgehog by the spins pointing downward (antiparallel to the direction of
the magnetization). The Dirac strings are distinguished by their vorticity
given by

$\displaystyle\zeta=\frac{1}{2\pi}\oint d{\bf l}\cdot\bm{\nabla}\phi({\bf
R}),$ (79)

where $\phi({\bf r})$ is the azimuthal angle of ${\bf S}({\bf R})$ and the
integral is taken along a closed path surrounding the string at ${\bf R}$ on a
plane perpendicular to the $Z$ axis [80, 59]. We find four Dirac strings, as
shown in Fig. 4(a): One is the line along the $Z$ axis (green line) and the
other three run through the MUC boundaries and connect the topological defects
in the neighboring MUCs (purple lines). The former has the vorticity of
$\zeta=+1$, while the latter three have $\zeta=-1$.

When introducing $m$, each topological defect splits into four; the hedgehog
with $Q_{\rm m}=+2$ splits into three hedgehogs with $Q_{\rm m}=+1$ (magenta
spheres) and one antihedgehog with $Q_{\rm m}=-1$ (cyan sphere), while the
antihedgehog with $Q_{\rm m}=-2$ splits into three antihedgehogs with $Q_{\rm
m}=-1$ and one hedgehog with $Q_{\rm m}=+1$, as shown in Fig. 4(b). Note that
the total monopole charge is conserved in each splitting. Thus, we have
totally four hedgehogs with $Q_{\rm m}=+1$ and four antihedgehogs with $Q_{\rm
m}=-1$ by introducing $m$. With a further increase of $m$, three pairs
connected by the Dirac strings with $\zeta=-1$ disappear with pair
annihilation at $m=1/\sqrt{3}$, leaving one pair connected by the Dirac string
with $\zeta=+1$, as shown in Fig. 4(c). The remaining hedgehog and
antihedgehog move toward each other while further increasing $m$, as shown in
Fig. 4(d), and they also vanish with pair annihilation at $m=\sqrt{3}$.
Consequently, while increasing $m$, we have two topological transitions caused
by pair annihilation of the hedgehogs and antihedgehogs at $m=1/\sqrt{3}$ and
$m=\sqrt{3}$.

####  III.2.3 Topological transition on 2D plane

Figure 5:  (a) Hedgehogs and antihedgehogs, and Dirac strings for $m=0$ in the
3D hyperspace. The pale colors are used for the objects in the neighboring
MUCs denoted by the dashed cubes. The gray plane represents the 2D MUC on the
intersection of the plane with $Z=\frac{\sqrt{2}}{3q}\frac{9}{4}\pi$. (b) The
spin configuration on the gray plane in (a), which corresponds to the spin
structure in Eq. (72) with $\tilde{\varphi}=\frac{9}{4}\pi$. The green and
purple circles denote the intersections of the Dirac strings with $\zeta=+1$
and $-1$, respectively. The right panel displays the 2D real-space spin
configuration by repeating the MUC. Similar figures for (c),(d) $m=0$ and
$\tilde{\varphi}=3\pi$, and (e),(f) $m=0.9$ and
$\tilde{\varphi}=\frac{9}{4}\pi$.

Using the results on the topological defects and the Dirac strings in the 3D
hyperspace, we can discuss in a systematic way the topological properties of
the 2D spin texture in Eq. (72) while changing $\tilde{\varphi}$ and $m$.
Figure 5 displays the relation between the horizontal slices in the 3D
hyperspace at $Z=\frac{\sqrt{2}}{3q}\tilde{\varphi}$ and the 2D spin textures
with $\tilde{\varphi}$. In Fig. 5(a), we show the configurations of the
hedgehogs and antihedgehogs, and the Dirac strings in the hyperspace at $m=0$,
and the horizontal plane with $Z=\frac{\sqrt{2}}{3q}\frac{9}{4}\pi$. The spin
configuration on the plane is shown in Fig. 5(b), which corresponds to the 2D
spin texture in Eq. (72) with $\tilde{\varphi}=\frac{9}{4}\pi$. The
topological property of the 2D spin texture is characterized by the skyrmion
number [27, 3, 28]

$\displaystyle N_{\rm sk}=\frac{1}{4\pi}\int_{\rm 2D~{}MUC}dXdY~{}b_{Z}({\bf
R}),$ (80)

where the integral is taken within the 2D rhombic MUC. The skyrmion number is
also obtained by counting the vorticities of the Dirac strings as [59]

$\displaystyle N_{\rm sk}=-\sum_{k}\zeta_{k}(Z),$ (81)

where $\zeta_{k}(Z)$ denotes the vorticity of the $k$th Dirac string
intersecting the 2D MUC [81]. For example, in the case of Fig. 5(b), there are
two intersections by the Dirac strings with $\zeta=+1$ and three with
$\zeta=-1$, and hence, $N_{\rm sk}=+1$. Thus, this simple counting in the
hyperspace representation enables us to identify the 2D spin texture at $m=0$
with $\tilde{\varphi}=\frac{9}{4}\pi$ as the $3Q$-SkL with $N_{\rm sk}=+1$,
while the direct integration by Eq. (80) gives the same conclusion.

Figures 5(c) and 5(d) illustrate the situations with $\tilde{\varphi}=3\pi$ at
$m=0$. In this case, the 2D slice has a single intersection by the Dirac
string with $\zeta=+1$, and hence, the spin texture with
$\tilde{\varphi}=3\pi$ is the $3Q$-SkL with $N_{\rm sk}=-1$. This demonstrates
a switching of the topological property by the phase shift. In the hyperspace
representation, such topological transitions occur when the gray horizontal
plane crosses the hedgehogs or antihedgehogs at the end points of the Dirac
strings.

Since the topological defects change their positions with $m$ as shown in Fig.
4, the topological properties of the 2D spin structures change also with $m$.
A demonstration is shown in Figs. 5(e) and 5(f) for $m=0.9$. At this value of
$m$, three pairs of the hedgehogs and antihedgehogs already vanish by pair
annihilation, and only a single pair remains on the $Z$ axis, as shown in Fig.
5(e). In this case, when the 2D slice intersects the Dirac string connecting
the hedgehog-antihedgehog pair, the 2D spin structures becomes a $3Q$-SkL with
$N_{\rm sk}=-1$, as exemplified in Fig. 5(f) for
$\tilde{\varphi}=\frac{9}{4}\pi$.

In this way, we can systematically investigate the changes of the magnetic
textures and the topological properties of the 2D $3Q$ spin structures while
changing $\tilde{\varphi}$ and $m$. The procedure is summarized as follows:
(i) Define the 3D spin texture in the hyperspace by using Eqs. (5) and (6), in
the present case, Eqs. (45)-(50), (ii) identify the hedgehogs and
antihedgehogs, and the Dirac strings connecting them for the 3D spin structure
in the hyperspace, (iii) consider the 2D slice of the 3D spin structure at the
horizontal plane with $Z=\frac{\sqrt{2}}{3q}\tilde{\varphi}$, which gives the
2D spin texture with the phase summation $\tilde{\varphi}$, and (iv) take the
sum of the vorticities of the Dirac strings intersecting the plane, which
gives the skyrmion number $N_{\rm sk}$ of the 2D spin structure through Eq.
(81).

####  III.2.4 Topological phase diagram

Figure 6:  Topological phase diagram for the screw $3Q$ state in Eq. (72)
determined by the skyrmion number $N_{\rm sk}$ while changing $m$ and
$\tilde{\varphi}$. The colored regions are topologically nontrivial phases
with nonzero $N_{\rm sk}$, and the white areas in the left and right hand
sides denote topologically trivial phases with $N_{\rm sk}=0$. The black dots
represent the pair annihilation of the hedgehogs and antihedgehogs in the 3D
hyperspace, which are located at $(m,\tilde{\varphi})=(-\sqrt{3},0)$,
$(-1/\sqrt{3},\pi)$, $(1/\sqrt{3},0)$, and $(\sqrt{3},\pi)$. The phase diagram
is periodic in the $\tilde{\varphi}$ direction with period of $2\pi$.  Figure
7:  Real-space spin configurations of the screw $3Q$ states: (a) $m=0.7$ and
$\tilde{\varphi}=0$, (b) $m=0.7$ and $\tilde{\varphi}=\pi$, (c) $m=-0.7$ and
$\tilde{\varphi}=0$, and (d) $m=-0.7$ and $\tilde{\varphi}=\pi$. Each spin
configuration is topologically nontrivial with (a) $N_{\rm sk}=-2$, (b)
$N_{\rm sk}=-1$, (c) $N_{\rm sk}=1$, and (d) $N_{\rm sk}=2$. The notations are
common to those in Fig. 5.

Figure 6 summarizes the topological phase diagram on the plane of $m$ and
$\tilde{\varphi}$ for Eq. (72) obtained by the above procedure. The result is
periodic in the $\tilde{\varphi}$ direction with period of $2\pi$ and
symmetric with respect to $\tilde{\varphi}=\pi$. Note that the phase diagram
for $m<0$ is obtained by mirroring that for $m>0$ with $\pi$ shift of
$\tilde{\varphi}$ and the sign inversion of $N_{\rm sk}$ since the spin
texture with ($\varphi_{\eta},m$) is obtained by time-reversal operation on
that with ($\varphi_{\eta}+\pi,-m$). We find four topologically nontrivial
phases with $N_{\rm sk}=-2$, $-1$, $1$, and $2$. The major portions of the
phase diagram are occupied by the SkLs with $N_{\rm sk}=\pm 1$, while the SkL
with $N_{\rm sk}=\pm 2$ appear in the small areas in between them only for
$m\neq 0$; the SkL at $m=0$ always has $N_{\rm sk}=\pm 1$. The black dots
appearing at the ends of the $N_{\rm sk}=\pm 1$ domes at $m=\pm 1/\sqrt{3}$
and $m=\pm\sqrt{3}$ correspond to the topological transitions by the pair
annihilation of the hedgehogs and antihedgehogs in the hyperspace; see Sec.
III.2.2.

Typical spin configurations of the screw $3Q$-SkLs with different $N_{\rm sk}$
are shown in Fig. 7. Figure 7(a) shows the spin configuration of the SkL with
$N_{\rm sk}=-2$ at $m=0.7$ and $\tilde{\varphi}=0$. In this state, small two
skyrmions exist in the MUC and constitute a honeycomb lattice structure. There
are two Dirac strings with $\zeta=1$ crossing the 2D plane, resulting in
$N_{\rm sk}=-2$. Figure 7(b) is for the SkL with $N_{\rm sk}=-1$ at $m=0.7$
and $\tilde{\varphi}=\pi$. In this case, a single skyrmion exists at the
center of the MUC. The Dirac string with $\zeta=1$ through the skyrmion core
leads to $N_{\rm sk}=-1$. Figures 7(c) and 7(d) show the spin configurations
for $m=-0.7$ at $\tilde{\varphi}=0$ and $\tilde{\varphi}=\pi$, respectively.
These are obtained by flipping all the spins in Figs. 7(b) and 7(a), and
hence, $N_{\rm sk}=1$ and $2$, respectively. We note that the spin
configurations with $\tilde{\varphi}=n\pi$ ($n$ is an integer) have sixfold
rotational symmetry, while the others with $\tilde{\varphi}\neq n\pi$ are
threefold, consistent with the symmetry arguments in Table 1.

Let us conclude this section by discussing some implications of our
topological phase diagram to the phase control. In the previous experimental
and theoretical studies for the chiral magnets [60, 48, 49, 50, 82, 83], the
SkLs with $N_{\rm sk}=-1$ ($+1$) were observed in an external magnetic field
applied to the ($-$)$\hat{\bf z}$ direction. This corresponds to the state
with $\tilde{\varphi}\sim\pi$ for $m>0$ and $\tilde{\varphi}\sim 0$ for $m<0$
in the phase diagram in Fig. 6. The other SkLs with $N_{\rm sk}=\pm 2$,
however, have not been reported thus far. Our topological phase diagram
indicates that it is necessary to cause the phase shift by $\simeq\pi$ in a
magnetic field for reaching the $N_{\rm sk}=\pm 2$ states. This is an
interesting issue to be addressed since the emergent magnetic field in the
$N_{\rm sk}=\pm 2$ states becomes twice as large as that in the $N_{\rm
sk}=\pm 1$ ones. In addition, in the previous studies, the SkLs with $N_{\rm
sk}=\pm 1$ turn into a $1Q$ conical state or a uniformly polarized state while
increasing the magnetic field, not into the topologically trivial $3Q$ state
with $N_{\rm sk}=0$ shown in the phase diagram in Fig. 6. This suggests that
it is difficult to access the points where the hedgehogs and antihedgehogs
cause pair annihilation in the hyperspace (the black dots in Fig. 6). Once one
can avoid the transition to the $1Q$ conical state, it might be possible to
find novel topological phenomena arising from the singularity in the emergent
electromagnetic fields due to the pair annihilation. It is worth noting that
some possible ways to control the phase degree of freedom were recently
proposed [65]. We will discuss this issue in Sec. VI.

###  III.3  Sinusoidal 3$Q$ state

We next discuss the phase degree of freedom for the sinusoidal $3Q$ state
given by

$\displaystyle{\bf S}({\bf
r})\propto\left(\begin{array}[]{c}\frac{\sqrt{3}}{2}\sin\theta(-\cos\mathcal{Q}_{2}+\cos\mathcal{Q}_{3})\\\
(-1)^{\Gamma}\frac{1}{2}\sin\theta\left(2\cos\mathcal{Q}_{1}-\cos\mathcal{Q}_{2}-\cos\mathcal{Q}_{3}\right)\\\
\cos\theta(\cos\mathcal{Q}_{1}+\cos\mathcal{Q}_{2}+\cos\mathcal{Q}_{3}+3\tilde{m})\end{array}\right),$
(85)

which is obtained from Eq. (1) by taking $N_{Q}=3$ and

$\displaystyle\psi_{\eta}^{\rm c}=\frac{1}{\sqrt{3}},\ \ \psi_{\eta}^{\rm
s}=0,$ (86) $\displaystyle{\bf
e}_{1}^{1}=\left(0,(-1)^{\Gamma}\sin\theta,\cos\theta\right)^{\mathsf{T}},\ \
{\bf e}_{2}^{1}=R_{\Gamma}{\bf e}_{1}^{1},\ \ {\bf
e}_{3}^{1}=R_{\Gamma}^{2}{\bf e}_{1}^{1},$ (87)
$\displaystyle\tilde{m}=\frac{m}{\sqrt{3}\cos\theta},$ (88)

where $\Gamma$ takes 0 or 1, $0<\theta<\frac{\pi}{2}$, and $R_{\Gamma}$
represents a $(-1)^{\Gamma}\frac{2\pi}{3}$ rotation about the $z$ axis given
by

$\displaystyle
R_{\Gamma}=\left(\begin{array}[]{ccc}\cos\left((-1)^{\Gamma}\frac{2\pi}{3}\right)&-\sin\left((-1)^{\Gamma}\frac{2\pi}{3}\right)&0\\\
\sin\left((-1)^{\Gamma}\frac{2\pi}{3}\right)&\cos\left((-1)^{\Gamma}\frac{2\pi}{3}\right)&0\\\
0&0&1\end{array}\right).$ (92)

In Eq. (85), $\Gamma$ is a parameter to describe the chirality of the spin
texture; the spin texture with $\Gamma=1$ is obtained by flipping the $S_{y}$
component of that with $\Gamma=0$. Note that the spin texture with $\Gamma=0$
has threefold rotational symmetry, whereas that with $\Gamma=1$ does not.
Meanwhile, $\theta$ describes the angle of the sinusoidal plane in the
constituent waves. In the following, we mainly focus on the spin textures with
$\Gamma=0$, while we touch on those with $\Gamma=1$ in Sec. V.1.

Following the arguments in Sec. III.2, we summarize the symmetry operations
and their decompositions for the sinusoidal $3Q$ state in Eq. (85) with
$\Gamma=0$ and $1$ in Tables 2 and 3, respectively. Other symmetry operations
are expressed by the combinations of those in the tables. From Table 2, we
find that the sinusoidal $3Q$ state with $\Gamma=0$ is symmetric under
$C_{3z}$, $\mathcal{I}$, $\mathcal{T}C_{2x}$, and their combinations for
$\tilde{\varphi}=0$ and $\pi$, otherwise $C_{3z}$, $\mathcal{T}C_{2x}$, and
their combinations. Meanwhile, from Table 3, we find that the state with
$\Gamma=1$ is symmetric under $\mathcal{I}$, $\mathcal{T}C_{2x}$, and their
combinations for $\tilde{\varphi}=0$, $\pi$, otherwise $\mathcal{T}C_{2x}$.

Table 2:  Similar table to Table 1 for the sinusoidal $3Q$ state with $\Gamma=0$ in Eq. (85): $\mathcal{I}$ represents the spatial-inversion operation and the other notations are common to those in Table 1.  operation | sum of phases | translation | magnetization
---|---|---|---
$C_{3z}$ | $\tilde{\varphi}\rightarrow\tilde{\varphi}$ | $0$ | $m\rightarrow m$
$C_{2x}$ | $\tilde{\varphi}\rightarrow\pi+\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}$ | $m\rightarrow-m$
$\mathcal{I}$ | $\tilde{\varphi}\rightarrow 2\pi-\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}+\tilde{\bf a}_{\eta^{\prime}}$ | $m\rightarrow m$
$\mathcal{T}$ | $\tilde{\varphi}\rightarrow\pi+\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}$ | $m\rightarrow-m$
Table 3:  Similar table for the sinusoidal $3Q$ state with $\Gamma=1$ in Eq. (85).  operation | sum of phases | translation | magnetization
---|---|---|---
$C_{2x}$ | $\tilde{\varphi}\rightarrow\pi+\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}$ | $m\rightarrow-m$
$\mathcal{I}$ | $\tilde{\varphi}\rightarrow 2\pi-\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}+\tilde{\bf a}_{\eta^{\prime}}$ | $m\rightarrow m$
$\mathcal{T}$ | $\tilde{\varphi}\rightarrow\pi+\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}$ | $m\rightarrow-m$

####  III.3.1 Hedgehogs in hyperspace

Following the procedure in Sec. III.2.1, we can identify the hyperspace
positions of the topological defects in the 3D spin texture corresponding to
Eq. (85). Solving ${\bf S}({\bf r})=0$, we obtain the following eight
solutions:

$\displaystyle(\mathcal{Q}_{1}^{*},\mathcal{Q}_{2}^{*},\mathcal{Q}_{3}^{*})$
$\displaystyle=$ $\displaystyle\left(\pi+p^{\rm sin}(\tilde{m}),\pi+p^{\rm
sin}(\tilde{m}),\pi+p^{\rm sin}(\tilde{m})\right),$ (93)
$\displaystyle\left(\pi-p^{\rm sin}(\tilde{m}),\pi-p^{\rm
sin}(\tilde{m}),\pi-p^{\rm sin}(\tilde{m})\right),$
$\displaystyle\left(\pi+p^{\rm sin}(\tilde{m}),\pi-p^{\rm
sin}(\tilde{m}),\pi-p^{\rm sin}(\tilde{m})\right)$
$\displaystyle\qquad\qquad\quad\mbox{and cyclic permutations},$
$\displaystyle\left(\pi-p^{\rm sin}(\tilde{m}),\pi+p^{\rm
sin}(\tilde{m}),\pi+p^{\rm sin}(\tilde{m})\right)$
$\displaystyle\qquad\qquad\quad\mbox{and cyclic permutations},$

where

$\displaystyle p^{\rm sin}(\tilde{m})=\arccos(\tilde{m}).$ (94)

By using the relation in Eq. (75), we obtain the positions of the eight
singular points as

$\displaystyle(X^{*},Y^{*},Z^{*})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{3q}\left(0,0,3(\pi+p^{\rm
sin}(\tilde{m}))\right),$ (95)
$\displaystyle\frac{\sqrt{2}}{3q}\left(0,0,3(\pi-p^{\rm
sin}(\tilde{m}))\right),$ $\displaystyle\left(\frac{4p^{\rm
sin}(\tilde{m})}{3q},0,\frac{\sqrt{2}}{3q}(3\pi-p^{\rm
sin}(\tilde{m}))\right)$ $\displaystyle\qquad\qquad\quad\mbox{and $C_{3}^{Z}$
symmetric points},$ $\displaystyle\left(-\frac{4p^{\rm
sin}(\tilde{m})}{3q},0,\frac{\sqrt{2}}{3q}(3\pi+p^{\rm
sin}(\tilde{m}))\right)$ $\displaystyle\qquad\qquad\quad\mbox{and $C_{3}^{Z}$
symmetric points}.$

Figure 8:  Hedgehogs and antihedgehogs, and Dirac strings in the spin
structure in the 3D hyperspace corresponding to the sinusoidal $3Q$ state in
Eq. (85) for (a) $\tilde{m}=0$ and (b) $\tilde{m}=0.8$. The notations are
common to those in Fig. 4. The results are obtained for $\Gamma=0$ in Eq.
(85); in the case of $\Gamma=1$, all the monopole charges of the hedgehogs and
antihedgehogs, and the vorticities of the Dirac strings reverse their signs.

####  III.3.2 Topological transition in 3D hyperspace

Figure 8 illustrates the evolution of the topological defects and the Dirac
strings in the hyperspace while changing $\tilde{m}$ in Eq. (85) with
$\Gamma=0$. The monopole charges, the Dirac strings, and their vorticities are
obtained by the same procedure as in Sec. III.2.2; see Eqs. (77) and (79).
Note that $\theta$ in Eq. (85) is irrelevant to the positions of the
topological objects in the hyperspace. In the absence of the magnetization
($\tilde{m}=0$), four out of the eight defects in Eq. (95) are classified into
the hedgehogs with $Q_{\rm m}=1$ and the other fours are antihedgehogs with
$Q_{\rm m}=-1$. The eight topological defects form the NaCl-like structure, as
shown in Fig. 8(a). We find that four pairs of the hedgehogs and antihedgehogs
are connected by four Dirac strings with $\zeta=\pm 1$, which cross each other
at the center of the MUC. When introducing $\tilde{m}$, the hedgehog and
antihedgehog pairs move toward each other along the Dirac strings, and the
cube defined by the eight defects shrinks, as shown in Fig. 8(b). All of the
eight defects come to the center of the MUC and vanish with pair
annihilatation simultaneously at $\tilde{m}=1$. In the case of $\Gamma=1$, the
positions of the topological defects remain the same, but the signs of all
$Q_{\rm m}$ and $\zeta$ are reversed.

Figure 9:  (a) Hedgehogs and antihedgehogs, and Dirac strings for
$\tilde{m}=0$ in the 3D hyperspace for the sinusoidal $3Q$ state with
$\Gamma$=0. The gray plane represents the intersection of the plane with
$Z=\frac{\sqrt{2}}{3q}3\pi$ ($\tilde{\varphi}=3\pi$). (b) The spin
configuration on the gray plane in (a), which corresponds with the spin
structure in Eq. (85) with $\tilde{\varphi}=3\pi$ and $\Gamma=0$. We take
$\theta=\arccos\frac{1}{\sqrt{3}}$ in Eq. (87). The vorticity at the black
circle takes $\zeta=-2$ (see the text for details). (c) and (d) Similar
figures for $\tilde{m}=0$ and $\tilde{\varphi}=2\pi$. The notations are common
to those in Fig. 5.

####  III.3.3 Topological transition on 2D plane

As in Sec. III.2.3, the 2D spin structure with phase $\tilde{\varphi}$ in Eq.
(85) is obtained as the slice of the 3D hedgehog lattice at
$Z=\frac{\sqrt{2}}{3q}\tilde{\varphi}$, and the skyrmion number $N_{\rm sk}$
is given by the sum of the vorticity of the Dirac strings as Eq. (81). Figure
9(a) shows the configurations of the hedgehogs, antihedgehogs, and the Dirac
strings for the spin structure of Eq. (85) with $\Gamma=0$ in the hyperspace
at $\tilde{m}=0$, and the slice at $Z=\frac{\sqrt{2}}{3q}3\pi$. The spin
configuration on the slice is shown in Fig. 9(b), which corresponds to the 2D
spin texture in Eq. (85) with $\tilde{\varphi}=(2n+1)\pi$ ($n$ is an integer)
and $\Gamma=0$. While the topological objects in the hyperspace are
independent of the value of $\theta$ as mentioned in Sec. III.3.2, the spin
configuration in the original 2D plane depends on $\theta$; we take
$\theta=\arccos\frac{1}{\sqrt{3}}$ in Fig. 9(b). In the hyperspace, as the
four Dirac strings cross each other at the center of the MUC and the slice
includes the crossing point, the sum of the vorticities at the intersection is
given by $\zeta=3\times(-1)+1=-2$, as depicted by the black circles in the
figure. Thus, we can identify the 2D spin texture at $\tilde{m}=0$ and
$\tilde{\varphi}=3\pi$ as the $3Q$-SkL with $N_{\rm sk}=2$.

Figures 9(c) and 9(d) illustrate the situation with $\tilde{\varphi}=2\pi$ at
$\tilde{m}=0$. In this case, the 2D slice has two intersections of the Dirac
strings with $\zeta=+1$, and hence, the spin texture with
$\tilde{\varphi}=2\pi$ is the $3Q$-SkL with $N_{\rm sk}=-2$, which is a time-
reversal counterpart of the spin texture in Fig. 9(b). Thus, similar to the
screw $3Q$ spin structures in Sec. III.2.3, the phase shift can cause
topological transitions in the sinusoidal ones.

####  III.3.4 Topological phase diagram

Figure 10:  Topological phase diagram for the sinusoidal $3Q$ state in Eq.
(85) determined by the skyrmion number $N_{\rm sk}$ on the plane of
$\tilde{m}$ and $\tilde{\varphi}$. The upper (lower) signs of $N_{\rm sk}$ are
for $\Gamma=0$ ($1$). The notations are common to those in Fig. 6. The black
points represent the simultaneous annihilation of the four hedgehogs and the
four antihedgehogs in the hyperspace, which are located at
$(\tilde{m},\tilde{\varphi})=(-1,0)$ and $(1,\pi)$.  Figure 11:  Real-space
spin configurations of the sinusoidal $3Q$ state in Eq. (85) with (a)-(d)
$\Gamma=0$ and (e)-(h) $\Gamma=1$, and $\theta=\arccos\frac{1}{\sqrt{3}}$:
(a)(e) $\tilde{m}=0.5$ and $\tilde{\varphi}=\frac{\pi}{2}$, (b)(f)
$\tilde{m}=0.5$ and $\tilde{\varphi}=\pi$, (c)(g) $\tilde{m}=-0.5$ and
$\tilde{\varphi}=0$, (d)(h) $\tilde{m}=-0.5$ and
$\tilde{\varphi}=\frac{\pi}{2}$. Each spin configuration is topologically
nontrivial and has (a) $N_{\rm sk}=-1$, (b) $N_{\rm sk}=2$, (c) $N_{\rm
sk}=-2$, (d) $N_{\rm sk}=1$, (e) $N_{\rm sk}=1$, (f) $N_{\rm sk}=-2$, (g)
$N_{\rm sk}=2$, and (h) $N_{\rm sk}=-1$. The white circles in (f) denote the
vorticity $\zeta=2$. Other notations are common to those in Fig. 9. In (b) and
(f), the four Dirac strings cross at the center of the MUC; see Fig. 8 and the
text for details.

Figure 10 summarizes the topological phase diagram on the plane of $\tilde{m}$
and $\tilde{\varphi}$ for the 2D spin texture in Eq. (85). The result is
common to $\Gamma=0$ and $1$, while the sign of the skyrmion number $N_{\rm
sk}$ in each phase is opposite: The upper (lower) signs are for $\Gamma=0$
($1$). Similar to Fig. 6, the phase diagram has $2\pi$ periodicity and
symmetric with respect to $\tilde{\varphi}=\pi$, and the result for $m<0$ is
obtained by mirroring that for $m>0$ with $\pi$ shift of $\tilde{\varphi}$ and
the sign inversion of $N_{\rm sk}$. We find four topologically nontrivial
phases with $N_{\rm sk}=-2$, $-1$, $1$, and $2$ as the proper screw case in
Fig. 6, but with different distributions of each phase. In the present
sinusoidal case, the large portions of the phase diagram are occupied by the
SkL with $N_{\rm sk}=\pm 2$, while the SkL with $N_{\rm sk}=\pm 1$ appear in
between them only for $\tilde{m}\neq 0$; the state at $\tilde{m}=0$ always has
$N_{\rm sk}=\pm 2$, in contrast to Fig. 6. Both $N_{\rm sk}=\pm 2$ ($\mp 2$)
and $\mp 1$ ($\pm 1$) regions end at $\tilde{m}=1$ ($-1$) and
$\tilde{\varphi}=\pi$ ($0$) with the simultaneous pair annihilation of all the
hedgehogs and antihedgehogs in the hyperspace, which are denoted by the black
dots in Fig. 10.

Figure 11 showcases typical spin configurations of the sinusoidal $3Q$ state
with $\theta=\arccos\frac{1}{\sqrt{3}}$. Here, we take $\Gamma=0$ in Figs.
11(a)-11(d) and $\Gamma=1$ in Figs. 11(e)-11(h). Figure 11(a) shows the spin
configuration of the $N_{\rm sk}=-1$ state at $\tilde{m}=0.5$ and
$\tilde{\varphi}=\frac{\pi}{2}$. In this state, there is a single Bloch type
skyrmion with $N_{\rm sk}=-1$ per MUC. The Dirac string with $\zeta=1$ through
the skyrmion core contributes to $N_{\rm sk}=-1$. Figure 11(b) is for the
$N_{\rm sk}=2$ state at $\tilde{m}=0.5$ and $\tilde{\varphi}=\pi$. This spin
structure has a skyrmion with $N_{\rm sk}=2$ at the center of the MUC. In this
state, the 2D plane in the hyperspace intersects the crossing point of the
four Dirac strings, which gives the total vorticity as $-2$; see Sec. III.3.3.
Figures 11(c) and 11(d) show the spin configurations with $\tilde{m}=-0.5$ at
$\tilde{\varphi}=0$ and $\tilde{\varphi}=\frac{\pi}{2}$; the former is
obtained by time-reversal operation on Fig. 11(b), while the latter is
obtained by time-reversal operation combined with sixfold rotation operation
about the $z$ axis on Fig. 11(a). The corresponding results for $\Gamma=1$ are
shown in Figs. 11(e)-11(h). In contrast to the screw $3Q$ case in Sec.
III.2.4, all these sinusoidal $3Q$ cases with $\Gamma=0$ have threefold
rotational symmetry independent of $\tilde{\varphi}$ and $\tilde{m}$, while
those with $\Gamma=1$ do not; see Sec. III.3. In addition, the spin textures
with $\tilde{\varphi}=n\pi$ ($n$ is an integer) has inversion symmetry
independent of $\Gamma$, which is consistent with the symmetry arguments in
Tables 2 and 3.

The phase diagram in Fig. 10 indicates that the system undergoes a topological
phase transition from $N_{\rm sk}=\pm 2$ to $\pm 1$, and finally to $N_{\rm
sk}=0$ while increasing $\tilde{m}$. Such transitions were found in the
previous numerical study of the Kondo lattice model on a triangular lattice
while increasing the magnetic field [84]. Since the $N_{\rm sk}=\pm 2$ and
$\pm 1$ phases appear predominantly in the different $\tilde{\varphi}$ regions
in our phase diagram, the topological phase transition between them might be
accompanied by a phase shift, but the phase degree of freedom was not studied
in the previous study. We will discuss this issue by analyzing the phases in
the spin structures obtained by the previous study in Sec. V.1.

##  IV $4Q$ hedgehog lattices

Figure 12:  (a) Schematic picture of the wave vectors in 3D reciprocal space,
${\bf q}_{\eta}$. (b) Corresponding magnetic translation vectors in the 3D
real space, ${\bf a}_{\eta}$, represented by orange arrows. The brown arrows
represent the projections of ${\bf A}_{\eta}$ onto the $xyz$ space denoted by
$\tilde{\bf a}_{\eta}$. The gray rhombohedron is the MUC, and the dashed cube
with the side length of $L=\frac{2\sqrt{3}\pi}{q}$ includes four MUCs. The
small gray cube has the side length of $\frac{L}{2}$.

In this section, we elucidate the effect of phase shifts on the spin textures
composed of four wave vectors in three dimensions, i.e., $N_{Q}=4$ and $d=3$,
by using the hyperspace representation. Specifically, we consider Eq. (1) with
four ${\bf q}_{\eta}$ given by

$\displaystyle{\bf q}_{1}=\frac{q}{\sqrt{3}}\left(1,1,1\right),\ \ {\bf
q}_{2}=\frac{q}{\sqrt{3}}\left(-1,-1,1\right),$ $\displaystyle{\bf
q}_{3}=\frac{q}{\sqrt{3}}\left(-1,1,-1\right),\ \ {\bf
q}_{4}=\frac{q}{\sqrt{3}}\left(1,-1,-1\right).$ (96)

For this $4Q$ spin structure, the 3D magnetic translation vectors are defined
as

$\displaystyle{\bf a}_{1}=\frac{\sqrt{3}\pi}{q}\left(1,0,1\right),\ \ {\bf
a}_{2}=\frac{\sqrt{3}\pi}{q}\left(1,1,0\right),$ $\displaystyle{\bf
a}_{3}=\frac{\sqrt{3}\pi}{q}\left(0,1,1\right).$ (97)

The four wave vectors and the three magnetic translation vectors are depicted
in Figs. 12(a) and 12(b), respectively. While the 3D MUC is given by the gray
rhombohedron in Fig. 12(b), we compute the topological properties for the
dashed cube with the side length of $L=\frac{2\sqrt{3}\pi}{q}$, which includes
four MUCs, in the following analyses.

In parallel with the arguments in Sec. III, we focus on the two types of the
$4Q$ spin structures in the following: the superposition of four proper screws
and that of four sinusoidal waves. The former screw $4Q$ state breaks spatial-
inversion symmetry and possesses the chirality; see Sec. IV.2. This type of
spin structure is found in a noncentrosymmetric material MnSi1-xGex and a
centrosymmetric material SrFeO3, as introduced in Sec. I. On the other hand,
the latter sinusoidal $4Q$ state retains spatial-inversion symmetry or $S_{4}$
improper rotational symmetry about the $z$ axis; see Sec. IV.3. In the next
subsection, we present the hyperspace representation applicable to these two
types of spin configurations.

###  IV.1 Hyperspace representation of the $4Q$ states

Following the same procedure as for the $3Q$ states in Sec. III.1, we
construct the hyperspace representation of the $4Q$ spin structures.
Considering 4D reciprocal hyperspace, we set the wave vectors ${\bf
Q}_{\eta}=(Q_{\eta}^{X},Q_{\eta}^{Y},Q_{\eta}^{Z},Q_{\eta}^{W})$ in Eq. (5)
without loss of generality:

$\displaystyle{\bf
Q}_{1}=q\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{q}\right),$
(98) $\displaystyle{\bf
Q}_{2}=q\left(-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{q}\right),$
(99) $\displaystyle{\bf
Q}_{3}=q\left(-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{q}\right),$
(100) $\displaystyle{\bf
Q}_{4}=q\left(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{q}\right),$
(101)

where $Q_{\eta}^{X}$, $Q_{\eta}^{Y}$, and $Q_{\eta}^{Z}$ are taken to be
proportional to $q_{\eta}^{x}$, $q_{\eta}^{y}$, and $q_{\eta}^{z}$ in Eq.
(96), respectively. In this setting, ${\bf q}_{\eta}$ is a projection of ${\bf
Q}_{\eta}$ onto the $q^{x}q^{y}q^{z}$ space. Then, we obtain the corresponding
magnetic translation vectors ${\bf A}_{\eta}$ in the 4D hyperspace as

$\displaystyle{\bf
A}_{1}=\frac{\pi}{2q}\left(\sqrt{3},\sqrt{3},\sqrt{3},q\right),$ (102)
$\displaystyle{\bf
A}_{2}=\frac{\pi}{2q}\left(-\sqrt{3},-\sqrt{3},\sqrt{3},q\right),$ (103)
$\displaystyle{\bf
A}_{3}=\frac{\pi}{2q}\left(-\sqrt{3},\sqrt{3},-\sqrt{3},q\right),$ (104)
$\displaystyle{\bf
A}_{4}=\frac{\pi}{2q}\left(\sqrt{3},-\sqrt{3},-\sqrt{3},q\right).$ (105)

Note that ${\bf a}_{1}$, ${\bf a}_{2}$, and ${\bf a}_{3}$ are given by
$\tilde{\bf a}_{1}-\tilde{\bf a}_{3}$, $\tilde{\bf a}_{1}-\tilde{\bf a}_{2}$,
and $\tilde{\bf a}_{1}-\tilde{\bf a}_{4}$, respectively, where $\tilde{\bf
a}_{\eta}$ are the projections of ${\bf A}_{\eta}$ onto the $xyz$ space; see
Fig. 12(b).

By using Eq. (16), the hyperspace positions ${\bf R}=(X,Y,Z,W)$ are related
with the real-space positions ${\bf r}$ as

$\displaystyle\left(\begin{array}[]{c}X\\\ Y\\\ Z\\\
W\end{array}\right)=V_{4}\left(\begin{array}[]{c}x\\\ y\\\ z\\\
1\end{array}\right),$ (114)

where $V$ includes the phases as

$\displaystyle
V_{4}=\left(\begin{array}[]{cccc}1&0&0&\frac{\sqrt{3}}{4q}\left(\varphi_{1}-\varphi_{2}-\varphi_{3}+\varphi_{4}\right)\\\
0&1&0&\frac{\sqrt{3}}{4q}\left(\varphi_{1}-\varphi_{2}+\varphi_{3}-\varphi_{4}\right)\\\
0&0&1&\frac{\sqrt{3}}{4q}\left(\varphi_{1}+\varphi_{2}-\varphi_{3}-\varphi_{4}\right)\\\
0&0&0&\frac{1}{4}\left(\varphi_{1}+\varphi_{2}+\varphi_{3}+\varphi_{4}\right)\end{array}\right).$
(119)

Equation (114) tells that the spin configuration in the original 3D $xyz$
space is the same as the one on a hyperplane in the 4D hyperspace with

$\displaystyle W=\frac{1}{4}\tilde{\varphi},$ (120)

where

$\displaystyle\tilde{\varphi}=\sum_{\eta}\varphi_{\eta}=\varphi_{1}+\varphi_{2}+\varphi_{3}+\varphi_{4}.$
(121)

Similar to the $3Q$ case in Sec. III.1, only the summation of the phases
$\varphi_{\eta}$ is relevant, instead of each value of $\varphi_{\eta}$, and
$\tilde{\varphi}$ has $2\pi$ periodicity. Due to this periodicity, the $2\pi$
phase shift in $\tilde{\varphi}$
($\Delta\tilde{\varphi}=\sum_{\eta}\varphi_{\eta}=2\pi$) corresponds to a
spatial translation by

$\displaystyle\Delta{\bf r}_{\eta}=\tilde{\bf
a}_{\eta}-\sum_{\eta^{\prime}=1}^{4}\frac{\Delta\varphi_{\eta^{\prime}}}{2\pi}\tilde{\bf
a}_{\eta^{\prime}},$ (122)

where $\eta$ may take any of 1, 2, 3, and 4.

###  IV.2 Screw $4Q$ state

In this subsection, we analyze the effect of phase shifts on the screw $4Q$
state composed of four proper screws given by

$\displaystyle{\bf S}({\bf r})$ $\displaystyle\propto$
$\displaystyle\left(\begin{array}[]{c}\frac{1}{\sqrt{2}}\left((-\cos{\mathcal{\mathcal{Q}}}_{1}+\cos{\mathcal{Q}}_{2}-\cos{\mathcal{Q}}_{3}+\cos{\mathcal{Q}}_{4})+\frac{1}{\sqrt{3}}(-\sin{\mathcal{Q}}_{1}+\sin{\mathcal{Q}}_{2}-\sin{\mathcal{Q}}_{3}+\sin{\mathcal{Q}}_{4})\right)\\\
\frac{1}{\sqrt{2}}\left((\cos{\mathcal{Q}}_{1}-\cos{\mathcal{Q}}_{2}-\cos{\mathcal{Q}}_{3}+\cos{\mathcal{Q}}_{4})-\frac{1}{\sqrt{3}}(\sin{\mathcal{Q}}_{1}-\sin{\mathcal{Q}}_{2}-\sin{\mathcal{Q}}_{3}+\sin{\mathcal{Q}}_{4})\right)\\\
\sqrt{\frac{2}{3}}(\sin{\mathcal{Q}}_{1}+\sin{\mathcal{Q}}_{2}+\sin{\mathcal{Q}}_{3}+\sin{\mathcal{Q}}_{4})+2m\end{array}\right),$
(126)

which is obtained from Eq. (1) by taking $N_{Q}=4$ and

$\displaystyle\psi_{\eta}^{\rm c}=\psi_{\eta}^{\rm s}=\frac{1}{2},$ (127)
$\displaystyle{\bf e}_{\eta}^{1}=\frac{\hat{\bf z}\times{\bf
e}_{\eta}^{0}}{|\hat{\bf z}\times{\bf e}_{\eta}^{0}|},\ {\bf
e}_{\eta}^{2}={\bf e}_{\eta}^{0}\times{\bf e}_{\eta}^{1},\ {\bf
e}_{\eta}^{0}=\frac{{\bf q}_{\eta}}{q}.$ (128)

With the same manner to Tables 1–3, we summarize the symmetry operations and
their decompositions for the screw $4Q$ state in Table 4. Similar to the
previous arguments, from Table 4, we can obtain the symmetry operations which
do not change the spin texture. In the 3D system, however, some of the
symmetry operations are nonsymmorphic. For instance, the $C_{4z}$ operation at
$\tilde{\varphi}=\pi$ is reduced to the spatial translation by $\tilde{\bf
a}_{\eta}$ as shown in Table 4, and hence, the operations of
$\\{C_{4z}|-\tilde{\bf a}_{\eta}\\}$ does not change the spin texture, where
$\\{\mathcal{O}|{\bf t}\\}$ denotes the translation by ${\bf t}$ after
operating $\mathcal{O}$. This corresponds to a screw operation. Note that the
situation is different from the $3Q$ cases, where a translation combined with
rotation can be represented by a shift of the rotation axis. By considering
such relations, we find that the screw $4Q$ state is symmetric under $C_{4z}$,
$\mathcal{T}C_{2x}$, and their combinations for $\tilde{\varphi}=0$,
$\\{C_{4z}|-\tilde{\bf a}_{\eta}\\}$, $\mathcal{T}C_{2x}$, and their
combinations for $\tilde{\varphi}=\pi$, otherwise $C_{2z}$,
$\mathcal{T}C_{2x}$, and their combinations.

Table 4:  Similar table to Tables 1–3 for the screw $4Q$ state in Eq. (126). The notations are common to those in Table 1. We take $\varphi_{\eta}=\frac{\tilde{\varphi}}{4}$ without loss of generality.  operation | sum of phases | translation | magnetization
---|---|---|---
$C_{4z}$ | $\tilde{\varphi}\rightarrow 2\pi-\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}$ | $m\rightarrow m$
$C_{2x}$ | $\tilde{\varphi}\rightarrow\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}+\tilde{\bf a}_{\eta^{\prime}}$ | $m\rightarrow-m$
$\mathcal{T}$ | $\tilde{\varphi}\rightarrow\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}+\tilde{\bf a}_{\eta^{\prime}}$ | $m\rightarrow-m$

####  IV.2.1 Hedgehogs in hyperspace

Figure 13:  Real-space distribution of the hedgehogs and antihedgehogs, and
the Dirac strings within the $L^{3}$ cube [see Fig. 12(b)] for the screw $4Q$
state in Eq. (126) while changing $m$ and $\tilde{\varphi}$: (a) $m=0$, (b)
$m=0.3$, and (c) $m=0.9$ at $\tilde{\varphi}=\pi/3$, and (d) $m=0$, (e)
$m=0.7$, and (f) $m=0.9$ at $\tilde{\varphi}=\pi$. The notations are common to
those in Fig. 4, except for the Dirac strings running on the horizontal
planes, whose vorticities are ill-defined, denoted by the white lines.

To discuss the phase shift in the screw $4Q$ state in Eq. (126), we study the
corresponding spin texture in the 4D hyperspace whose reciprocal space is
spanned by the wave vectors ${\bf Q}_{\eta}$ in Eqs. (98)–(101). As discussed
in the previous subsection, a real-space spin configuration for a given phase
summation $\tilde{\varphi}$ in the original 3D space corresponds to a
hyperspace spin configuration on the hyperplane with
$W=\frac{\tilde{\varphi}}{4}$. Following the procedure in Sec. III.2.1, we
first compute the positions of the topological defects in the 4D hyperspace.
In the present case, however, the solutions for ${\bf S}({\bf R})=0$ are given
by lines rather than points in the 4D hyperspace. When
$|m|<\frac{4}{\sqrt{6}}\sin W$, we obtain two solutions analytically:

$\displaystyle(X^{*},Y^{*},Z^{*})=L\left(0,0,\frac{1}{2}\pm\frac{1}{2\pi}r_{1}^{\rm
scr}(m,W)\right),$ (129)

where

$\displaystyle r_{1}^{\rm scr}(m,W)=\arccos\left(\frac{\sqrt{6}m}{4\sin
W}\right).$ (130)

Here, we show the solutions within $0\leq X,Y<\frac{L}{2}$ and $0\leq Z<L$,
but the spatial translations of them with ${\bf a}_{\eta}$ in Eqs. (97) also
satisfy ${\bf S}({\bf R})=0$. Meanwhile, when $|m|<\frac{4}{\sqrt{6}}\cos W$,
we obtain other two solutions as

$\displaystyle(X^{*},Y^{*},Z^{*})$ $\displaystyle=$ $\displaystyle
L\left(\frac{1}{4},\frac{1}{4},\frac{1}{2\pi}r_{2}^{\rm scr}(m,W)\right),$
(131) $\displaystyle
L\left(\frac{1}{4},\frac{1}{4},\frac{1}{2}-\frac{1}{2\pi}r_{2}^{\rm
scr}(m,W)\right),$

where

$\displaystyle r_{2}^{\rm scr}(m,W)=\arcsin\left(\frac{\sqrt{6}m}{4\cos
W}\right).$ (132)

The spatial translations also apply to this case. These solutions do not
change their $XY$ coordinates with $m$ and $\tilde{\varphi}$. In addition to
the above analytical solutions, we also find the other solutions by
numerically solving the equation ${\bf S}({\bf R})=0$. The two conditions
$S_{x}({\bf R})=0$ and $S_{y}({\bf R})=0$ can be reduced to

$\displaystyle\tan\left(\frac{q}{\sqrt{3}}Z^{*}\right)=\pm\sqrt{\frac{3\tan^{2}W-1}{3-\tan^{2}W}}$
(133)

and

$\displaystyle\tan\left(\frac{q}{\sqrt{3}}Y^{*}\right)=\mp\sqrt{\frac{\left(\sqrt{3}\tan
W+1\right)\left(\sqrt{3}+\tan W\right)}{\left(\sqrt{3}\tan
W-1\right)\left(\sqrt{3}-\tan
W\right)}}\tan\left(\frac{q}{\sqrt{3}}X^{*}\right)$ (134)

Meanwhile, $S_{z}({\bf R})=0$ is reduced to

$\displaystyle\cos\left(\frac{q}{\sqrt{3}}X^{*}\right)\cos\left(\frac{q}{\sqrt{3}}Y^{*}\right)\cos\left(\frac{q}{\sqrt{3}}Z^{*}\right)\sin
W$
$\displaystyle-\sin\left(\frac{q}{\sqrt{3}}X^{*}\right)\sin\left(\frac{q}{\sqrt{3}}Y^{*}\right)\sin\left(\frac{q}{\sqrt{3}}Z^{*}\right)\cos
W$ $\displaystyle+\frac{\sqrt{6}}{4}m=0.$ (135)

We find that the numerical solutions exist when $\frac{1}{\sqrt{3}}<\tan
W<\sqrt{3}$, namely $\frac{2\pi}{3}<\tilde{\varphi}<\frac{4\pi}{3}$. We note
that the solutions in Eqs. (133)-(135) were not mentioned in the previous
study for the screw $4Q$ state with $\tilde{\varphi}=\pi$ [62]. While above
solutions are points in the original 3D space, the line solutions in original
3D space are also obtained for only $m=0$. When $W=0$, $\frac{\pi}{6}$ and
$\frac{\pi}{3}$, we find analytical solutions

$\displaystyle(X^{*},Y^{*},Z^{*})=\left(0,0,*\right),\left(*,\frac{L}{4},0\right),\left(0,*,\frac{L}{4}\right),$
(136)

where $*$ takes any value, respectively.

In the 4D hyperspace, the topological objects defined by the above solutions
of ${\bf S}({\bf R})=0$ except for Eq. (136) form closed loops, and the
intersection of the loops by the hyperplane with $W=\frac{\tilde{\varphi}}{4}$
gives topological point defects in the original 3D space, which correspond to
the hedgehogs and antihedgehogs discussed below. The hedgehog and antihedgehog
always appear in pairs for each closed loop. The Dirac string connecting the
hedgehog-antihedgehog pair in the 3D space is derived from the intersection of
a 2D membrane in the 4D hyperspace whose edge and surface are defined by ${\bf
S}({\bf R})=0$ and ${\bf S}({\bf R})=-\hat{\bf z}$, respectively. The 2D
membrane can be regarded as an extension of the Dirac string in the higher
dimension and hence, we may call it the Dirac plane. Thus, the hyperspace
representation of the $4Q$ state is given by a 4D lattice composed of such
closed loops, and the $4Q$ spin texture with the phase degree of freedom is
defined as a 3D intersection of the 4D loop lattice.

Since it is difficult to visualize the 4D hyperspace, we present the
topological objects in the original 3D space which are derived from the
hyperspace representation above. Figure 13 shows the systematic change of the
topological defects in the 3D space while changing $m$ in Eq. (126) with
$\tilde{\varphi}=\frac{\pi}{3}$ and $\pi$. Following the procedure in Sec.
III.2.2, we compute the monopole charge for the topological defects, $Q_{\rm
m}$ in Eq. (77), and the vorticity of the Dirac strings which connect the
hedgehogs and antihedgehogs, $\zeta$ in Eq. (79), by replacing ${\bf R}$ with
${\bf r}$. In order to distinguish the different topological phases, we also
compute the total number of the topological defects, the hedgehogs and
antihedgehogs, within the cube shown in Fig. 12(b), denoted by $N_{\rm m}$;
the total number per MUC is given by $\frac{N_{\rm m}}{4}$.

First, we discuss the case of $\tilde{\varphi}=\frac{\pi}{3}$ shown in Figs.
13(a), 13(b), and 13(c). For $m=0$, there are 16 topological defects in total
and half of them are hedgehogs with $Q_{\rm m}=+1$ and the others are
antihedgehogs with $Q_{\rm m}=-1$, as shown in Fig. 13(a). The hedgehogs and
antihedgehogs derived from Eq. (129) are connected by the Dirac strings with
$\zeta=-1$, while the other topological defects from Eq. (131) are connected
by the Dirac strings with $\zeta=+1$. All the Dirac strings run along the $z$
axis and have the same length of $\frac{L}{2}$. When introducing $m$, the
hedgehogs and antihedgehogs move toward their counterparts along the Dirac
strings, as exemplified in Fig. 13(b) for $m=0.3$. The Dirac strings with
$\zeta=-1$ become shorter than those with $\zeta=+1$: The lengths change as
$L\left(\frac{1}{2}-\frac{1}{\pi}r_{2}^{\rm
scr}\left(m,\frac{\tilde{\varphi}}{4}\right)\right)$ and
$\frac{L}{\pi}r_{1}^{\rm scr}\left(m,\frac{\tilde{\varphi}}{4}\right)$ for
$\zeta=+1$ and $-1$, respectively. This differentiation gives rise to a net
emergent magnetic field, as will be discussed in Sec. IV.2.2. By further
increasing $m$, the hedgehogs and antihedgehogs connected by the Dirac strings
with $\zeta=-1$ disappear with pair annihilation at
$m=\frac{4}{\sqrt{6}}\sin\frac{\tilde{\varphi}}{4}$; namely $N_{\rm m}$ is
reduced to 8, which defines a topological transition between the phases with
different $N_{\rm m}$. The remaining hedgehogs and antihedgehogs move toward
each other while further increasing $m$, as shown in Fig. 13(c). They also
vanish with pair annihilation at
$m=\frac{4}{\sqrt{6}}\cos\frac{\tilde{\varphi}}{4}$, which defines the other
topological transition into a topologically trivial state with $N_{\rm m}=0$.

Next, we discuss the case of $\tilde{\varphi}=\pi$ shown in Figs. 13(d),
13(e), and 13(f). When $m=0$, as shown in Fig. 13(d), the system has eight
pairs of the hedgehogs and antihedgehogs, similar to the case of
$\tilde{\varphi}=\frac{\pi}{3}$. However, while the positions of the
topological defects as well as their total number are same as those in Fig.
13(a), half of the hedgehog and antihedgehog pairs are exchanged; all the
Dirac strings have the hedgehogs at their lower edges in Fig. 13(d), whereas
only half of them do in Fig. 13(a). Moreover, we find additional Dirac strings
running on the planes perpendicular to the $z$ axis, which are denoted by the
white lines in the figure. Note that the vorticity in Eq. (79) is ill-defined
for these horizontal Dirac strings. As shown in Fig. 13(d), they intersect
with the Dirac strings running along the $z$ axis, whose vorticities $\zeta$
change their signs at the crossing points. By introducing $m$, the hedgehogs
and antihedgehogs move toward each other along the vertical Dirac strings as
in the case of $\tilde{\varphi}=\frac{\pi}{3}$, while the horizontal Dirac
strings are intact. When $m$ exceeds $\frac{2}{3}$, however, $N_{\rm m}$
increases from 16 to 48, as depicted in Fig. 13(e). This is caused by a
peculiar topological transition with the increase of $N_{\rm m}$ discussed in
detail below. The additional defects are obtained from the numerical solutions
with Eqs. (133), (134), and (135) for $m>\frac{2}{3}$, which appear in pair on
the horizontal Dirac strings; the horizontal Dirac strings are cut into
pieces, both ends of which form hedgehogs or antihedgehogs. In other words, a
cut results in pair creation of the hedgehog and antihedgehog. In this region,
three hedgehogs and three antihedgehogs (a pair of hedgehog and antihedgehog
with the vertical Dirac string, and hedgehog pair and antihedgehog pair with
the horizontal Dirac strings) form a cluster like a twisted two-barred cross.
While further increasing $m$, the hedgehogs and antihedgehogs move along the
Dirac strings, and $N_{\rm m}$ decreases from 48 to 16 at
$m=\sqrt{\frac{2}{3}}$. Here, two hedgehogs and one antihedgehog (or one
hedgehog and two antihedgehogs) collide with each other at the same time in
each cluster, leaving one (anti)hedgehog (see below). After this topological
transition, all the Dirac strings have the hedgehogs at the upper edges, and
hence, the vorticities are $\zeta=+1$, as shown in Fig. 13(f). Finally, the
remaining hedgehogs and antihedgehogs move toward their counterparts along the
Dirac strings, and cause pair annihilation at $m=\frac{2}{\sqrt{3}}$, by which
the system enters into a topologically trivial phase with $N_{\rm m}=0$.

Figure 14:  Enlarged figures for the systematic evolution of the hedgehogs and
antihedgehogs while changing $m$ with $\tilde{\varphi}=\pi$: (a) $m=0$, (b)
$m=0.7$, (c) $m=0.8$, and (d) $m=0.9$. The hedgehogs and antihedgehogs
enclosed with the black dotted lines in (b) are generated by pair creation on
the horizontal Dirac strings. On the other hand, the three topological defects
enclosed with the black dotted lines in (c) collide with each other
simultaneously and cause a fusion into a hedgehog or an antihedgehog in (d). A
part of the topological defects and the Dirac strings are depicted for better
visibility. The notations are same with those in Fig. 13.

The evolution of the topological objects for the case of $\tilde{\varphi}=\pi$
includes, at least, two striking features. One is the increase of the total
number of the hedgehogs and antihedgehogs while increasing $m$. This is highly
nontrivial since usually the hedgehogs and antihedgehogs move toward each
other along the Dirac string and cause pair annihilation, and hence, their
number does not increase while increasing $m$ [60, 62, 79, 25, 59]. It is
worthy noting that the increase is caused by pair creation on the horizontal
Dirac strings whose vorticities are ill-defined. The process is detailed in
Figs. 14(a) and 14(b). The other striking feature is the simultaneous fusion
of three defects, as depicted in Figs. 14(c) and 14(d). In this process, two
hedgehogs (antihedgehogs) on the horizontal Dirac string and one antihedgehog
(hedgehog) on the vertical Dirac string move toward the crossing point of the
vertical and horizontal Dirac strings, and cause the fusion into a single
hedgehog (antihedgehog). Note that the total monopole charge is conserved
through the fusion. Hence, both striking features originate from the peculiar
horizontal Dirac strings.

Thus far, we show the results only for $\tilde{\varphi}=\pi/3$ and $\pi$, but
we note that the other cases fall into either behavior qualitatively.
Specifically, the cases for $\frac{2\pi}{3}<\tilde{\varphi}<\frac{4\pi}{3}$
belong to the latter, while the others to the former. In the latter class, we
find the additional numerical solutions as described above, which lead to the
peculiar pair creation and fusion of the topological defects. We note that the
fusion occurs simultaneously for all the created pairs when
$\tilde{\varphi}=\pi$, while it takes place successively half by half for
other values in $\frac{2\pi}{3}<\tilde{\varphi}<\frac{4\pi}{3}$, as discussed
below.

####  IV.2.2 Topological phase diagram

Figure 15:  Topological phase diagram for the screw $4Q$ state on the plane of
$m$ and $\tilde{\varphi}$, determined by the number of hedgehogs and
antihedgehogs within the cube, $N_{\rm m}$ [see Fig. 12(b)]. The contour plot
indicates the emergent magnetic field $-\bar{b}_{z}$; the white lines denote
the contours drawn every $0.1$, and the black dashed lines denote the boundary
to $\bar{b}_{z}=0$. The gray solid lines denote pair annihilation of hedgehogs
and antihedgehogs. The yellow line and the orange lines denote pair creation
of hedgehogs and antihedgehogs and fusion of three topological defects,
respectively, while increasing $m$.  Figure 16:  Real-space spin
configurations on the isosurfaces with $S_{z}({\bf r})=-0.9$ within the
$L^{3}$ cube for the screw 4$Q$ state in Eq. (126) for different topological
phases in Fig. 15: (a) $m=0$ and $\tilde{\varphi}=\frac{\pi}{3}$ ($N_{\rm
m}=16$), (b) $m=0$ and $\tilde{\varphi}=\pi$ ($N_{\rm m}=16$), (c) $m=0.4$ and
$\tilde{\varphi}=0$ ($N_{\rm m}=8$), (d) $m=0.7$ and
$\tilde{\varphi}=\frac{5\pi}{6}$ ($N_{\rm m}=32$), (e) $m=0.7$ and
$\tilde{\varphi}=\pi$ ($N_{\rm m}=48$), and (f) $m=0.9$ and
$\tilde{\varphi}=\pi$ ($N_{\rm m}=16$). The spin configurations are also shown
on the bottom plane of the cube. The color of the arrows and the isosurfaces
denote the $xyz$ and $xy$ components of ${\bf S}({\bf r})$, respectively; see
the inset in (a). In each figure, the right panel shows the top view.

Figure 15 summarizes the topological phase diagram on the plane of $m$ and
$\tilde{\varphi}$ for the screw $4Q$ state in Eq. (126), determined by $N_{\rm
m}$. The result is periodic in the $\tilde{\varphi}$ direction with period of
$2\pi$ and symmetric with respect to $\tilde{\varphi}=\pi$. We also plot the
emergent magnetic field $\bar{b}_{z}$, which is defined as

$\displaystyle\bar{b}_{z}=\frac{1}{4\pi L}\int d\mathcal{V}b_{z}({\bf r}),$
(137)

where the volume integration is taken within the $L^{3}$ cube in Fig. 12(b).
Following the arguments in Refs. [62, 79, 25, 59], $\bar{b}_{z}$ is rewritten
into

$\displaystyle\bar{b}_{z}=-\frac{1}{L}\sum_{k}\left(l_{k}^{+}-l_{k}^{-}\right),$
(138)

where $l_{k}^{+}$ and $l_{k}^{-}$ denote the length of the $k$th Dirac strings
with $\zeta=+1$ and $\zeta=-1$ projected onto the $z$ axis, respectively, and
the sum is taken for all the Dirac strings involved in the cube. Note that the
values of $-\bar{b}_{z}$ are plotted by contour in Fig. 15. The phase diagram
for $m<0$ is obtained in the same form with sign reversal of $-\bar{b}_{z}$
since the spin texture with $(\varphi_{\eta},m)$ is obtained by time-reversal
operation on that with $(\varphi_{\eta}+\pi,-m)$.

In the phase diagram, we find the topological phases with $N_{\rm m}=8$, $16$,
$32$, and $48$, in addition to the trivial phase with $N_{\rm m}=0$ in the
large $m$ region. When $0<\tilde{\varphi}<\frac{2\pi}{3}$ or
$\frac{4\pi}{3}<\tilde{\varphi}<2\pi$, the phases with $N_{\rm m}=8$ and 16
appear. In the phase with $N_{\rm m}=16$ for small $m$, 8 hedgehogs and 8
antihedgehogs are connected by the Dirac strings with $\zeta=+1$ and $-1$, and
the Dirac strings with $\zeta=-1$ are shorter than those with $\zeta=+1$, as
exemplified in Fig. 13(b). While increasing $m$, the length difference between
the long and short Dirac strings increases, leading to the increase of
$-\bar{b}_{z}$. Specifically, the value of $-\bar{b}_{z}$ is given by

$-\bar{b}_{z}=\pm 2\left[1-\frac{2}{\pi}\left(r_{1}^{\rm
scr}\left(m,\frac{\tilde{\varphi}}{4}\right)+r_{2}^{\rm
scr}\left(m,\frac{\tilde{\varphi}}{4}\right)\right)\right],$ (139)

where the upper (lower) sign is for $0<\tilde{\varphi}<\frac{2\pi}{3}$
($\frac{2\pi}{3}<\tilde{\varphi}<2\pi$). At
$m=\frac{4}{\sqrt{6}}\sin\frac{\tilde{\varphi}}{4}$ for
$0<\tilde{\varphi}<\frac{2\pi}{3}$ and
$m=\frac{4}{\sqrt{6}}\cos\frac{\tilde{\varphi}}{4}$ for
$\frac{4\pi}{3}<\tilde{\varphi}<2\pi$, half of the topological defects
connected by the Dirac strings with $\zeta=-1$ disappear with pair
annihilation, leaving the defects connected by the Dirac stings with
$\zeta=+1$, as exemplified in Fig. 13(c). The pair annihilation occurs at
smaller $m$ when $\tilde{\varphi}$ approaches $0$ or $2\pi$, and
$-\bar{b}_{z}$ is enhanced to $-\bar{b}_{z}\to 2$ when $m\to 0$ at
$\tilde{\varphi}=0$ or $2\pi$ (in the limit, there are only four Dirac string
with $\zeta=+1$ and length $\frac{L}{2}$). Meanwhile, in the phase with
$N_{\rm m}=8$, the increase of $m$ reduces the length of the Dirac strings
with $\zeta=+1$, leading to the decrease of $-\bar{b}_{z}$ as

$-\bar{b}_{z}=\begin{cases}2\left[1-\frac{2}{\pi}r_{2}^{\rm
scr}\left(m,\frac{\tilde{\varphi}}{4}\right)\right]&{\rm for}\ \
0<\tilde{\varphi}<\frac{2\pi}{3}\\\ \frac{4}{\pi}r_{1}^{\rm
scr}\left(m,\frac{\tilde{\varphi}}{4}\right)&{\rm for}\ \
\frac{4\pi}{3}<\tilde{\varphi}<2\pi.\end{cases}$ (140)

Finally, $-\bar{b}_{z}$ vanishes by the pair annihilation of the remaining
hedgehogs and antihedgehogs at
$m=\frac{4}{\sqrt{6}}\cos\frac{\tilde{\varphi}}{4}$ for
$0<\tilde{\varphi}<\frac{2\pi}{3}$ and
$m=\frac{4}{\sqrt{6}}\sin\frac{\tilde{\varphi}}{4}$ for
$\frac{4\pi}{3}<\tilde{\varphi}<2\pi$, and the system enters into the
topologically trivial phase with $N_{\rm m}=0$ for larger $m$.

On the other hand, when $\frac{2\pi}{3}<\tilde{\varphi}<\frac{4\pi}{3}$, the
topological phases with $N_{\rm m}=32$ and 48 appear additionally in the
intermediate region of $m$, as shown in Fig. 15. In this range of
$\tilde{\varphi}$, while increasing $m$ from the phase with $N_{\rm m}=16$,
pair creation of the hedgehogs and antihedgehogs occurs on the phase boundary
to the phase with $N_{\rm m}=48$, which is denoted by the yellow line in the
figure. No anomaly is found in $-\bar{b}_{z}$ at the topological transition
since the pair-created topological objects move along the horizontal Dirac
strings and this evolution does not contribute to $l_{k}^{\pm}$ in Eq. (138);
see Fig. 13(e). In these phases, however, the increase of $m$ reduces the
length of the Dirac strings with $\zeta=-1$, leading to the increase of
$-\bar{b}_{z}$ as

$-\bar{b}_{z}=2\left[1-\frac{2}{\pi}\left(r_{1}^{\rm
scr}\left(m,\frac{\tilde{\varphi}}{4}\right)-r_{2}^{\rm
scr}\left(m,\frac{\tilde{\varphi}}{4}\right)\right)\right].$ (141)

While increasing $m$, half of the pair-created topological defects disappear
through the fusion, which causes the topological transition from $N_{\rm
m}=48$ to $32$ (orange lines in the figure). In the $N_{\rm m}=32$ region, the
value of $-\bar{b}_{z}$ is given by

$\displaystyle-\bar{b}_{z}=$ $\displaystyle
2\left[1\mp\frac{2}{\pi}\left(r_{1}^{\rm
scr}\left(m,\frac{\tilde{\varphi}}{4}\right)+r_{2}^{\rm
scr}\left(m,\frac{\tilde{\varphi}}{4}\right)\right)\right.$ (142)
$\displaystyle\qquad\left.\pm\frac{4}{\pi}\arccos\left(\sqrt{2\cos^{2}\frac{\tilde{\varphi}}{4}-\frac{1}{2}}\right)\right],$

where the upper (lower) signs are for $\frac{2\pi}{3}<\tilde{\varphi}\leq\pi$
($\pi\leq\tilde{\varphi}<\frac{4\pi}{3}$). With a further increase of $m$, the
rest half of the pair-created topological defects cause the fusion and $N_{\rm
m}$ is reduced from $32$ to $16$. In the $N_{\rm m}=16$ phase, all of the
topological defects are connected by the Dirac strings with $\zeta=+1$, as
exemplified in Fig. 13(f), where $-\bar{b}_{z}$ is given as

$-\bar{b}_{z}=2\left[1+\frac{2}{\pi}\left(r_{1}^{\rm
scr}\left(m,\frac{\tilde{\varphi}}{4}\right)-r_{2}^{\rm
scr}\left(m,\frac{\tilde{\varphi}}{4}\right)\right)\right].$ (143)

Note that $-\bar{b}_{z}$ takes the maximum value of $-\bar{b}_{z}=2$ at
$(m,\tilde{\varphi})=\left(\sqrt{\frac{2}{3}},\pi\right)$, where all the pair-
created defects cause the fusion simultaneously.

Figure 16 showcases typical spin configurations of the screw $4Q$ states for
all the topological phases in Fig. 15, together with the hedgehogs and
antihedgehogs, and the Dirac strings. The spin configurations are shown on the
isosurfaces with $S_{z}({\bf r})=-0.9$ as well as the bottom plane of the
$L^{3}$ cube. The isosurfaces, by definition, extend from the hedgehogs and
antihedgehogs, and involve the Dirac strings inside. Figure 16(a) is for the
$N_{\rm m}=16$ state at $m=0$ and $\tilde{\varphi}=\frac{\pi}{3}$. From the
spin configurations on the isosurfaces, it is observed that the helicity of
the spin texture gradually increases or decreases with the $z$ coordinates,
which leads to $\pm\pi$ difference between the top and bottom of each Dirac
string. Figure 16(b) is for the different $N_{\rm m}=16$ state at $m=0$ and
$\tilde{\varphi}=\pi$. The result demonstrates that the phase shift
drastically changes the spin configurations as well as the real-space
distributions of the topological objects, while $N_{\rm m}$ is same as in Fig.
16(a). In this state, the isosurfaces have complicated 3D networks due to the
existence of the Dirac strings running on the horizontal planes. Figure 16(c)
is for the $N_{\rm m}=8$ phase at $m=0.4$ and $\tilde{\varphi}=0$. In this
state, the isosurfaces become much simpler. Figures 16(d) and 16(e) are for
the $N_{\rm m}=32$ state at $m=0.7$ and $\tilde{\varphi}=\frac{5\pi}{6}$ and
$N_{\rm m}=48$ states at $m=0.7$ and $\tilde{\varphi}=\pi$, respectively. In
both states, the spin configuration on a horizontal $xy$ plane comprise a SkL
with Bloch type skyrmions, as exemplified on the bottom plane of the cube in
the figure. We note that the skyrmions are deformed and elongated along the
direction of the horizontal Dirac strings. Figure 16(f) is for the $N_{\rm
m}=16$ state at $m=0.9$ and $\tilde{\varphi}=\pi$. In this state, while
$N_{\rm m}$ takes the same value as that in Figs. 16(a) and 16(b), the spin
configuration is completely different from them. In all the cases, the spin
configurations have twofold rotational symmetry about each vertical Dirac
string. We note that the spin configurations with $\tilde{\varphi}=0$ have
fourfold rotational symmetry about each Dirac string, as exemplified in Fig.
16(c), while those with $\tilde{\varphi}=\pi$ are symmetric for the screw
operation $\\{C_{4z}|-\tilde{\bf a}_{\eta}\\}$, as exemplified in Figs. 16(b),
16(e), and 16(f). These results are consistent with the symmetry arguments in
Table 4.

Let us briefly discuss the present results in comparison with the previous
studies. A $4Q$ HL was experimentally discovered in MnSi1-xGex [85]. In this
study, the spin texture at zero magnetic field was interpreted as the screw
$4Q$ state with four hedgehogs and four antihedgehogs ($N_{\rm m}=8$), which
corresponds to $\tilde{\varphi}=0$ in our results, although the value of
$\tilde{\varphi}$ was not examined experimentally. Meanwhile, the topological
properties and the emergent magnetic field in the screw $4Q$ state were
theoretically studied at $\tilde{\varphi}=\pi$ while changing the external
magnetic field [62], but the study was limited to the $N_{\rm m}=16$ state as
the solutions in Eqs. (133)-(135) were not included. Recently, some of the
authors studied the evolution of the screw $4Q$ state on a 3D simple cubic
lattice by variational calculations and simulated annealing, and found various
types of topological transitions depending on the direction of the magnetic
field [57]. We will discuss one of them, paying attention to the phase shift
in Sec. V.2.

###  IV.3 Sinusoidal $4Q$ state

Next, we analyze the phase shift in the sinusoidal $4Q$ state given by

$\displaystyle{\bf S}({\bf
r})\propto\left(\begin{array}[]{c}\cos\mathcal{Q}_{1}-\cos\mathcal{Q}_{2}-\cos\mathcal{Q}_{3}+\cos\mathcal{Q}_{4}\\\
\cos\mathcal{Q}_{1}-\cos\mathcal{Q}_{2}+\cos\mathcal{Q}_{3}-\cos\mathcal{Q}_{4}\\\
\cos\mathcal{Q}_{1}+\cos\mathcal{Q}_{2}-\cos\mathcal{Q}_{3}-\cos\mathcal{Q}_{4}+2\sqrt{3}m\end{array}\right),$
(147) (148)

which is obtained from Eq. (1) by taking $N_{Q}=4$ and

$\displaystyle\psi_{\eta}^{\rm c}=\frac{1}{2},\ \ \psi_{\eta}^{\rm s}=0,\ \
{\bf e}_{\eta}^{1}=\frac{{\bf q}_{\eta}}{q}.$ (149)

Following the arguments in Sec. IV.2, we summarize the symmetry operations for
the sinusoidal $4Q$ state in Table 5. We find that the spin texture is
unchanged for $C_{4z}$, $\mathcal{I}$, $\mathcal{T}C_{2x}$, and their
combinations for $\tilde{\varphi}=0$, $\\{C_{4z}|-\tilde{\bf a}_{\eta}\\}$,
$\mathcal{I}$, $\mathcal{T}C_{2x}$, and their combinations for
$\tilde{\varphi}=\pi$, otherwise $C_{2z}$, $S_{4z}$, $\mathcal{T}C_{2x}$, and
their combinations, where $S_{4z}$ represents the fourfold improper rotation
operation about the $z$ axis.

Table 5:  Similar table to Tables 1–4 for the sinusoidal $4Q$ state in Eq. (148).  operation | sum of phases | translation | magnetization
---|---|---|---
$C_{4z}$ | $\tilde{\varphi}\rightarrow 2\pi-\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}$ | $m\rightarrow m$
$C_{2x}$ | $\tilde{\varphi}\rightarrow\tilde{\varphi}$ | $0$ | $m\rightarrow-m$
$\mathcal{I}$ | $\tilde{\varphi}\rightarrow 2\pi-\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}+\tilde{\bf a}_{\eta^{\prime}}$ | $m\rightarrow m$
$\mathcal{T}$ | $\tilde{\varphi}\rightarrow\tilde{\varphi}$ | $\tilde{\bf a}_{\eta}+\tilde{\bf a}_{\eta^{\prime}}$ | $m\rightarrow-m$

####  IV.3.1 Hedgehogs in hyperspace

Figure 17:  Real-space distribution of the hedgehogs and antihedgehogs, and
the Dirac strings within the $L^{3}$ cube [see Fig. 12(b)] for the sinusoidal
$4Q$ state in Eq. (148) while changing $m$ and $\tilde{\varphi}$: (a) $m=0$,
(b) $m=0.1$, and (c) $m=0.7$ at $\tilde{\varphi}=\pi/3$, and (d) $m=0$, (e)
$m=0.5$, and (g) $m=0.7$ at $\tilde{\varphi}=\pi$. The notations are common to
those in Fig. 13.

Following the same procedure in Sec. IV.2.1, we compute the positions of the
topological defects in the 4D hyperspace for the spin texture corresponding to
Eq. (148). Here, we show the solutions ${\bf S}({\bf R})=0$ within $0\leq
X,Y<\frac{L}{2}$ and $0\leq Z<L$, but the spatial translations of them with
${\bf a}_{\eta}$ in Eq. (97) also satisfy ${\bf S}({\bf R})=0$. When
$|m|<\frac{2}{\sqrt{3}}\sin W$, we obtain two solutions analytically:

$\displaystyle(X^{*},Y^{*},Z^{*})=L\left(0,0,\frac{1}{4}\pm\frac{1}{2\pi}r_{1}^{\rm
sin}(m,W)\right),$ (150)

where

$\displaystyle r_{1}^{\rm sin}(m,W)=\arccos\left(\frac{\sqrt{3}m}{2\sin
W}\right).$ (151)

Meanwhile, when $|m|<\frac{2}{\sqrt{3}}\cos W$, we obtain other two analytical
solutions as

$\displaystyle(X^{*},Y^{*},Z^{*})$ $\displaystyle=$ $\displaystyle
L\left(\frac{1}{4},\frac{1}{4},\frac{3}{4}+\frac{1}{2\pi}r_{2}^{\rm
sin}(m,W)\right),$ (152) $\displaystyle
L\left(\frac{1}{4},\frac{1}{4},\frac{1}{4}-\frac{1}{2\pi}r_{2}^{\rm
sin}(m,W)\right),$

where

$\displaystyle r_{2}^{\rm sin}(m,W)=\arcsin\left(\frac{\sqrt{3}m}{2\cos
W}\right).$ (153)

The solutions in Eqs. (150) and (152) do not change their $XY$ coordinates
with $m$ and $\tilde{\varphi}$, as Eqs. (129) and (131) for the screw $4Q$
case. On the other hand, when
$-\frac{2}{\sqrt{3}}\cos^{2}W<m<\frac{2}{\sqrt{3}}\sin^{2}W$, we obtain four
solutions:

$\displaystyle(X^{*},Y^{*},Z^{*})$ $\displaystyle=$ $\displaystyle
L\left(\frac{1}{2\pi}r_{3+}^{\rm
sin}(m,W),\frac{1}{2}-\frac{1}{2\pi}r_{3+}^{\rm
sin}(m,W),\frac{1}{2}+\frac{W}{2\pi}\right),$ $\displaystyle
L\left(\frac{1}{2}-\frac{1}{2\pi}r_{3+}^{\rm
sin}(m,W),\frac{1}{2\pi}r_{3+}^{\rm
sin}(m,W),\frac{1}{2}+\frac{W}{2\pi}\right),$ $\displaystyle
L\left(\frac{1}{2\pi}r_{3+}^{\rm sin}(m,W),\frac{1}{2\pi}r_{3+}^{\rm
sin}(m,W),\frac{1}{2}-\frac{W}{2\pi}\right),$ $\displaystyle
L\left(\frac{1}{2}-\frac{1}{2\pi}r_{3+}^{\rm
sin}(m,W),\frac{1}{2}-\frac{1}{2\pi}r_{3+}^{\rm
sin}(m,W),\frac{1}{2}-\frac{W}{2\pi}\right),$

where

$\displaystyle r_{3+}^{\rm
sin}(m,W)=\frac{1}{2}\arccos\left(\cos\left(2W\right)+\sqrt{3}m\right).$ (155)

For $-\frac{2}{\sqrt{3}}\sin^{2}W<m<\frac{2}{\sqrt{3}}\cos^{2}W$, we obtain
four solutions:

$\displaystyle(X^{*},Y^{*},Z^{*})$ $\displaystyle=$ $\displaystyle
L\left(\frac{1}{2\pi}r_{3-}^{\rm
sin}(m,W),\frac{1}{2}-\frac{1}{2\pi}r_{3-}^{\rm
sin}(m,W),\frac{W}{2\pi}\right),$ $\displaystyle
L\left(\frac{1}{2}-\frac{1}{2\pi}r_{3-}^{\rm
sin}(m,W),\frac{1}{2\pi}r_{3-}^{\rm sin}(m,W),\frac{W}{2\pi}\right),$
$\displaystyle L\left(\frac{1}{2\pi}r_{3-}^{\rm
sin}(m,W),\frac{1}{2\pi}r_{3-}^{\rm sin}(m,W),1-\frac{W}{2\pi}\right),$
$\displaystyle L\left(\frac{1}{2}-\frac{1}{2\pi}r_{3-}^{\rm
sin}(m,W),\frac{1}{2}-\frac{1}{2\pi}r_{3-}^{\rm
sin}(m,W),1-\frac{W}{2\pi}\right),$

where

$\displaystyle r_{3-}^{\rm
sin}(m,W)=\frac{1}{2}\arccos\left(\cos\left(2W\right)-\sqrt{3}m\right).$ (157)

In contrast to Eqs. (150) and (152), the solutions in Eqs.
(LABEL:eq:4qnc_sol3) and (LABEL:eq:4qnc_sol4) change their $XY$ coordinates
with $m$ and $\tilde{\varphi}$, while the $Z$ coordinates do not change. These
solutions correspond to those obtained numerically in the screw $4Q$ case.

Figure 17 showcases the systematic change of the topological defects in the
original 3D space while changing $m$ in Eq. (148) with
$\tilde{\varphi}=\frac{\pi}{3}$ and $\pi$. The monopole charge of the
hedgehogs and antihedgehogs, $Q_{\rm m}$, and the vorticity of the Dirac
strings, $\zeta$, are calculated by the same procedure in Sec. IV.2.1.

First, we discuss the case of $\tilde{\varphi}=\frac{\pi}{3}$ shown in Figs.
17(a), 17(b), and 17(c). When $m=0$, there are 48 defects in total and half of
them are hedgehogs with $Q_{\rm m}=+1$ and the others are antihedgehogs with
$Q_{\rm m}=-1$, as shown in Fig. 17(a). These topological defects are derived
from Eqs. (150), (152), (LABEL:eq:4qnc_sol3), and (LABEL:eq:4qnc_sol4). There
are two types of the Dirac strings connecting the hedgehogs and antihedgehogs:
One is the vertical strings for the hedgehog-antihedgehog pairs, and the other
is the horizontal ones for the hedgehog-hedgehog pairs or the antihedgehog-
antihedgehog pairs. Similar to the screw $4Q$ case, the vorticity is ill-
defined for the latter horizontal ones, and they intersect the former vertical
ones whose vorticities change their signs at the crossing points; three
hedgehogs and three antihedgehogs form a cluster like a twisted two-barred
cross. However, the horizontal ones are straight, in contrast to the curved
ones in the screw $4Q$ case (see Fig. 13). Moreover, they are found for all
$\tilde{\varphi}$, while they appear only for
$\frac{2\pi}{3}<\tilde{\varphi}<\frac{4\pi}{3}$ in the screw $4Q$ case. All
the vertical Dirac strings have the length of $\frac{L}{2}$ at $m=0$, while
the horizontal ones have a long or short length, resulting in the two types of
clusters; the hedgehogs and antihedgehogs in the four clusters with longer
horizontal strings are given by Eqs. (152) and (LABEL:eq:4qnc_sol4), while
those in the rest four with shorter ones are by Eqs. (150) and
(LABEL:eq:4qnc_sol3). By introducing $m$, the topological defects move along
the Dirac strings, and two hedgehogs and one antihedgehog (or one hedgehog and
two antihedgehogs) collide with each other, leaving one (anti)hedgehog, at
$m=\frac{2}{\sqrt{3}}\sin^{2}\frac{\tilde{\varphi}}{4}$ in the four clusters
with shorter horizontal Dirac strings. The fusion process is similar to those
in Fig. 14. At the topological transition by the fusion, $N_{\rm m}$ decreases
from $48$ to $32$, and for larger $m$, the vertical Dirac strings left by the
fusion, which have $\zeta=-1$, coexist with the clusters remaining, as shown
in Fig. 17(b). By further increasing $m$, the hedgehogs and antihedgehogs
connected by the vertical Dirac strings cause the pair annihilation at
$m=\frac{2}{\sqrt{3}}\sin\frac{\tilde{\varphi}}{4}$, leaving four clusters
derived from Eqs. (152) and (LABEL:eq:4qnc_sol4), as shown in Fig. 17(c). At
the topological transition by the pair annihilation, $N_{\rm m}$ is further
reduced from $32$ to $24$. When
$m=\frac{2}{\sqrt{3}}\cos^{2}\frac{\tilde{\varphi}}{4}$, the fusion takes
place in the remaining four clusters, which reduces $N_{\rm m}$ from 24 to 8,
and leaves four vertical Dirac strings with $\zeta=-1$. Finally, the hedgehogs
and antihedgehogs on the Dirac strings pair annihilate at
$m=\frac{2}{\sqrt{3}}\cos\frac{\tilde{\varphi}}{4}$, where the system becomes
topologically trivial with $N_{\rm m}=0$.

Next, we discuss the case of $\tilde{\varphi}=\pi$ shown in Figs. 17(d),
17(e), and 17(f). When $m=0$, $N_{\rm m}$ is 48 as in the case of
$\tilde{\varphi}=\frac{\pi}{3}$; the positions of the hedgehogs and
antihedgehogs connected by the vertical Dirac strings are the same, but those
connected by the horizontal Dirac strings are different, as shown in Fig.
17(d). By introducing $m$, the topological defects move along the Dirac
strings as shown in Fig. 17(e), and the fusion occurs at
$m=\frac{1}{\sqrt{3}}$ simultaneously in all the clusters. This leaves eight
pairs of the hedgehogs and antihedgehogs connected by the vertical Dirac
strings, as shown in Fig. 17(f); $N_{\rm m}$ decreases from $48$ to $16$. In
this state, all the Dirac strings have the length of $\frac{L}{\pi}r_{1}^{\rm
sin}\left(m,\frac{\tilde{\varphi}}{4}\right)=L\left(\frac{1}{2}-\frac{1}{\pi}r_{2}^{\rm
sin}\left(m,\frac{\tilde{\varphi}}{4}\right)\right)$ and the vorticity
$\zeta=-1$. By further increasing $m$, the remaining pairs annihilate at
$m=\sqrt{\frac{2}{3}}$.

####  IV.3.2 Topological phase diagram

Figure 18:  Topological phase diagram for the sinusoidal $4Q$ state determined
by $N_{\rm m}$, with the contour plot of $-\bar{b}_{z}$, on the plane of $m$
and $\tilde{\varphi}$. $N_{\rm m}$ is indicated in each phase. The white lines
denote the contours drawn every 0.1, and the black dashed lines denote
$\bar{b}_{z}=0$. The gray and orange lines are the phase boundaries between
different $N_{\rm m}$ phases: The former denotes the pair annihilation of
hedgehogs and antihedgehogs, while the latter denotes the fusion of three
topological defects.

Performing similar calculations to those in Sec. IV.2.2 while changing
$\tilde{\varphi}$ and $m$, we elaborate the phase diagram shown in Fig. 18. We
also plot $-\bar{b}_{z}$ by the contour, as in Sec. IV.2.2. Similar to Fig.
15, the result is again symmetric with respect to $\tilde{\varphi}=\pi$ and
has $2\pi$ periodicity, and the phase diagram for $m<0$ is obtained in the
same form with sign inversion of $-\bar{b}_{z}$. We find the topological
phases with $N_{\rm m}=8$, 16, 24, 32, and 48 in the phase diagram. There are
two interesting features, in comparison with the result for the screw $4Q$
case in Fig. 15. One is that the topological phases with large values of
$N_{\rm m}$, such as $N_{\rm m}=24$, 32, and 48, appear in the large portions
of the phase diagram. In particular, the system has $N_{\rm m}=48$ for all
$\tilde{\varphi}$ in the small $m$ limit, and turns into the $N_{\rm m}=32$
state by the fusion on the orange lines in the phase diagram. Near
$\tilde{\varphi}=0$ or $2\pi$, the $N_{\rm m}=24$ regions extend widely in the
intermediate $m$ region, and the $N_{\rm m}=32$ states appear in between. The
value of $-\bar{b}_{z}$ in the phase with $N_{\rm m}=48$ is given by

$-\bar{b}_{z}=2\left[1+\frac{2}{\pi}\left(r_{1}^{\rm
sin}\left(m,\frac{\tilde{\varphi}}{4}\right)-r_{2}^{\rm
sin}\left(m,\frac{\tilde{\varphi}}{4}\right)\right)\right].$ (158)

Meanwhile, the values of $-\bar{b}_{z}$ in the $N_{\rm m}=32$ and $24$ phases
are given as

$-\bar{b}_{z}=2\left[1-\frac{2}{\pi}\left(r_{1}^{\rm
sin}\left(m,\frac{\tilde{\varphi}}{4}\right)+r_{2}^{\rm
sin}\left(m,\frac{\tilde{\varphi}}{4}\right)\right)-\frac{\tilde{\varphi}}{\pi}\right],$
(159)

and

$-\bar{b}_{z}=2\left[1-\frac{2}{\pi}r_{2}^{\rm
sin}\left(m,\frac{\tilde{\varphi}}{4}\right)-\frac{\tilde{\varphi}}{\pi}\right],$
(160)

respectively, when $0\leq\tilde{\varphi}\leq\pi$; $-\bar{b}_{z}$ for
$\pi\leq\tilde{\varphi}\leq 2\pi$ is obtained by the symmetry with respect to
$\tilde{\varphi}=\pi$. These are in stark contrast to the screw $4Q$ case
where the topological phases with $N_{\rm m}=32$ and 48 appear only in the
limited region for $\frac{2\pi}{3}<\tilde{\varphi}<\frac{4\pi}{3}$ and $N_{\rm
m}=24$ is not found. The other interesting feature is that the sign of
$-\bar{b}_{z}$ is not limited to positive unlike the screw $4Q$ case; it can
be negative while changing $m$ and $\tilde{\varphi}$ in the phases with
$N_{\rm m}=32$ and $24$. The sign changes are caused by the competition
between the lengths of the Dirac strings with $\zeta=+1$ and $-1$; see also
Eqs. (159) and (160).

We find that $-\bar{b}_{z}$ takes the minimum value of $-2$ at
$(m,\tilde{\varphi})=\left(\frac{1}{\sqrt{3}},\pi\right)$, where all the
defects connected by the horizontal Dirac strings cause the fusion
simultaneously. This state has the hedgehog-antihedgehog pairs given by the
analytical solutions in Eqs. (150) and (152) connected by the Dirac strings
whose vorticities and lengths are all $\zeta=-1$ and $\frac{L}{4}$,
respectively. On the other hand, $-\bar{b}_{z}$ takes the maximum value of $2$
in the limit of $m\to 0$ at $\tilde{\varphi}=0$ ($2\pi$). This is understood
as follows. When $\tilde{\varphi}=0$, the system has the hedgehogs and
antihedgehogs given by Eqs. (152) and (LABEL:eq:4qnc_sol4) connected by the
Dirac strings with the vorticities $\zeta=1$ and the lengths
$L\left(\frac{1}{2}-\frac{1}{\pi}r_{2}^{\rm
sin}\left(m,\frac{\tilde{\varphi}}{4}\right)\right)$. This leads to

$-\bar{b}_{z}=2\left[1-\frac{2}{\pi}r_{2}^{\rm
sin}\left(m,\frac{\tilde{\varphi}}{4}\right)\right].$ (161)

Meanwhile, when $\tilde{\varphi}=2\pi$, the hedgehogs and antihedgehogs are
given by Eqs. (150) and (LABEL:eq:4qnc_sol3), and $-\bar{b}_{z}$ is given by

$\displaystyle-\bar{b}_{z}=\frac{4}{\pi}r_{1}^{\rm
sin}\left(m,\frac{\tilde{\varphi}}{4}\right).$ (162)

Note that Eqs. (161) and (162) are related with each other by the symmetry
with respect to $\tilde{\varphi}=\pi$. Hence, $-\bar{b}_{z}$ goes to $2$ in
the limit of $m\rightarrow 0$ for both $\tilde{\varphi}=0$ and $2\pi$.

Figure 19:  Real-space spin configurations on the isosurfaces with $S_{z}({\bf
r})=-0.9$ for the sinusoidal 4$Q$ state in Eq. (148) for different topological
phases in Fig. 18: (a) $m=0$ and $\tilde{\varphi}=\pi$ ($N_{\rm m}=48$), (b)
$m=0.2$ and $\tilde{\varphi}=\frac{\pi}{3}$ ($N_{\rm m}=32$), (c) $m=0.6$ and
$\tilde{\varphi}=\frac{\pi}{3}$ ($N_{\rm m}=24$), (d) $m=0.6$ and
$\tilde{\varphi}=\pi$ ($N_{\rm m}=16$), and (e) $m=0.8$ and
$\tilde{\varphi}=\frac{5\pi}{6}$ ($N_{\rm m}=8$). The notations are common to
those in Fig. 16.

Figure 19 showcases typical spin configurations of the sinusoidal $4Q$ states
for all the topological phases in Fig. 18, in a similar manner to Fig. 16.
Figure 19(a) is for the $N_{\rm m}=48$ state at $m=0$ and
$\tilde{\varphi}=\pi$. In this state, all the isosurfaces have the same shape
and volume. Figure 19(b) is for the $N_{\rm m}=32$ state at $m=0.2$ and
$\tilde{\varphi}=\frac{\pi}{3}$. The isosurfaces including the pairs of the
hedgehog and antihedgehog connected by the Dirac strings with $\zeta=-1$ are
small (hardly seen in the figure) compared to those including the twisted two-
barred crosses, and they disappear by pair annihilation while increasing $m$,
as shown in Fig. 19(c) for the $N_{\rm m}=24$ state at $m=0.6$ and
$\tilde{\varphi}=\frac{\pi}{3}$. In these states, any horizontal $xy$ plane
intersecting the Dirac strings with $\zeta=-1$ gives a SkL with antiskyrmions,
as exemplified on the bottom planes of the cubes. Figure 19(d) is for the
$N_{\rm m}=16$ state at $m=0.6$ and $\tilde{\varphi}=\pi$. In this state, the
horizontal Dirac strings disappear, and all the isosurfaces become small,
while they retain knoblike features as the remnant of the horizontal Dirac
strings. In this case also, the horizontal $xy$ plane is a SkL with
antiskyrmions, and the antiskyrmions appear on the $xy$ planes with almost all
$z$ coordinates. Figure 19(e) is for the $N_{\rm m}=8$ state at $m=0.8$ and
$\tilde{\varphi}=\frac{5\pi}{6}$. In this state, the number of the Dirac
strings is halved and both Dirac strings and surrounding isosurfaces are
shrunk. Hence, the antiskyrmions are found on the horizontal planes with
limited $z$ coordinates, in contrast to the case in Fig. 19(d). In all the
above cases, the spin configurations have fourfold improper rotational
symmetry about each vertical Dirac string, as shown in Fig. 19. Moreover, the
spin configurations with $\tilde{\varphi}=\pi$ are symmetric for the screw
operation $\\{C_{4z}|-\tilde{\bf a}_{\eta}\\}$ and the spatial-inversion
operation, as exemplified in Figs. 19(a) and 19(d). These results are
consistent with the symmetry arguments in Table 5.

##  V Numerical analysis of phase shift

Thus far, we elucidated the topological properties of 2D $3Q$-SkLs and 3D
$4Q$-HLs by systematically changing the phases of their constituent waves as
well as the magnetization. In this section, we study how the actual phases of
the superposed waves evolve while increasing an external magnetic field based
on specific model Hamiltonians. For the 2D SkLs, we analyze the numerical data
of the real-space spin configurations obtained for the Kondo lattice model in
Ref. [70], and extract the phases for two types of SkLs with $|N_{\rm sk}|=1$
and $2$ in Sec. V.1. In Sec. V.2, we apply similar analysis to the 3D HLs
obtained for an effective spin model in Ref. [57] to extract the phases for
two types of HLs with $N_{\rm m}=8$ and $16$.

###  V.1 Phase shift in the sinusoidal 3$Q$ state

First, using the numerical data obtained for the Kondo lattice model on a 2D
triangular lattice in the previous study [70], we extract the phases for the
$3Q$ states, and discuss their magnetic field dependences in comparison with
our result in Sec. III.3. The Kondo lattice model is a fundamental model for
the systems in which itinerant electrons are coupled with localized spins,
whose Hamiltonian is given by

$\displaystyle\mathcal{H}$ $\displaystyle=$ $\displaystyle-\sum_{{\bf
r}_{l},{\bf r}_{l^{\prime}},\sigma}t_{{\bf r}_{l}{\bf
r}_{l^{\prime}}}\hat{c}^{{\dagger}}_{{\bf r}_{l}\sigma}\hat{c}_{{\bf
r}_{l^{\prime}}\sigma}-J\sum_{{\bf r}_{l},\sigma,\sigma^{\prime}}{\bf S}_{{\bf
r}_{l}}\cdot\hat{c}^{{\dagger}}_{{\bf
r}_{l}\sigma}\bm{\sigma}_{\sigma\sigma^{\prime}}\hat{c}_{{\bf
r}_{l}\sigma^{\prime}}$ (163) $\displaystyle-h\sum_{{\bf r}_{l}}S_{{\bf
r}_{l}}^{z},$

where the operator $\hat{c}^{{\dagger}}_{{\bf r}_{l}\sigma}$ $(\hat{c}_{{\bf
r}_{l}\sigma})$ creates (annihilates) an electron with spin index $\sigma=\pm$
at site ${\bf r}_{l}$. The first term represents the kinetic energy of the
itinerant electrons and $t_{{\bf r}_{l}{\bf r}_{l^{\prime}}}$ denotes the
hopping integral between the sites ${\bf r}_{l}$ and ${\bf r}_{l^{\prime}}$.
The second term represents the spin-charge coupling with the coefficient $J$,
where $\bm{\sigma}$ is the vector of Pauli matrices. The localized spins ${\bf
S}_{{\bf r}_{l}}$ are treated as classical vectors with $|{\bf S}_{{\bf
r}_{l}}|=1$. The last term denotes the Zeeman coupling to the external
magnetic field $h$, which is taken into account only for the localized spins
for simplicity. In the previous study, the ground state of the model in Eq.
(163) with the nearest-neighbor hopping $t_{1}=1$, the third-neighbor hopping
$t_{3}=-0.85$, and $J=0.5$ was studied by using the numerical method based on
the kernel polynomial method (KPM) and the Langevin dynamics (LD), which is
called the modified KPM-LD method [86, 70, 87]. The ground state was obtained
by minimizing the grand potential $\Omega$ given by
$\Omega=\braket{\mathcal{H}}/N-\mu n_{\rm e}$, where $N$ is the number of
sites ($N=96^{2}$), $\mu=-3.5$ is the chemical potential, and $n_{\rm e}$ is
the electron density defined by $n_{\rm e}=\sum_{{\bf
r}_{l},\sigma}\braket{\hat{c}^{{\dagger}}_{{\bf r}_{l}\sigma}\hat{c}_{{\bf
r}_{l}\sigma}}/N$. The model was found to stabilize $3Q$ states whose wave
vectors are dictated by the Fermi surfaces and given by Eq. (43) with
$q=\frac{\pi}{3}$. When $0\leq h\lesssim 0.00325$, the ground state obtained
by the numerical simulation is a $3Q$-SkL with $|N_{\rm sk}|=2$, which is well
described by a superposition of three sinusoidal waves like in Eq. (85). On
the other hand, it turns into a different $3Q$-SkL with $|N_{\rm sk}|=1$ for
$h\gtrsim 0.00325$, and finally, to yet another $3Q$ state with $N_{\rm sk}=0$
for $h\gtrsim 0.0065$. In the following, we focus on the region for $0\leq
h\leq 0.006$ since the $3Q$ state with $N_{\rm sk}=0$ for $h\gtrsim 0.0065$
cannot be well represented by Eq. (85).

From the spin configurations obtained in the previous study, we extract the
phases of the constituent waves in the two SkLs by assuming Eq. (85). For this
purpose, we estimate the cost function defined by

$\displaystyle U(\\{\varphi_{\eta}\\},m,\theta,\Gamma;\hat{\bf n},\xi)=$
$\displaystyle\quad\frac{1}{N}\sum_{{\bf r}_{l}}\left(1-{\bf S}^{\rm KPM-
LD}_{{\bf r}_{l}}\cdot\tilde{{\bf S}}^{{\rm sin}3Q}_{{\bf
r}_{l}}(\\{\varphi_{\eta}\\},m,\theta,\Gamma;\hat{\bf n},\xi)\right),$ (164)

where ${\bf S}^{\rm KPM-LD}_{{\bf r}_{l}}$ is the spin configuration obtained
by the modified KPM-LD simulation and $\tilde{{\bf S}}^{{\rm sin}3Q}_{{\bf
r}_{l}}$ is that generated from Eq. (85) as

$\displaystyle\tilde{{\bf S}}^{{\rm sin3Q}}_{{\bf
r}_{l}}(\\{\varphi_{\eta}\\},m,\theta,\Gamma;\hat{\bf n},\xi)=R\left(\hat{\bf
n},\xi\right){\bf S}^{{\rm sin}3Q}_{{\bf
r}_{l}}(\\{\varphi_{\eta}\\},m,\theta,\Gamma).$ (165)

Here, ${\bf S}^{{\rm sin}3Q}_{{\bf
r}_{l}}(\\{\varphi_{\eta}\\},m,\theta,\Gamma)$ is given by Eq. (85) with
$q=\frac{\pi}{3}$, and $R\left(\hat{\bf n},\xi\right)$ denotes the 3D rotation
matrix about the unit vector $\hat{\bf n}$ with the angle $\xi$. We note that
the model in Eq. (163) does not change the energy by flipping all the $y$
components of the localized spins, which corresponds to the change of $\Gamma$
between $0$ and $1$ in Eq. (85); hence, the ground states with $\Gamma=0$ and
$1$ are energetically degenerate. By minimizing $U$ in Eq. (164) for the
numerical data $\\{{\bf S}^{\rm KPM-LD}_{{\bf r}_{l}}\\}$ at each value of
$h$, we obtain the optimal values of the phases, $\varphi_{\eta}^{*}$, as well
as the other parameters. For comparison with $\tilde{\varphi}$ in Sec. III.3,
we define the sum of $\varphi_{\eta}^{*}$ in the form of

$\displaystyle\tilde{\varphi}^{*}=\pi-\left|\pi-{\rm
Mod}\left[\sum_{\eta=1}^{3}\varphi^{*}_{\eta},2\pi\right]\right|,$ (166)

paying attention to the sixfold rotational symmetry of the model and the
symmetry by the transformation from $\varphi_{\eta}$ to $-\varphi_{\eta}$, in
addition to the $2\pi$ periodicity.

Figure 20:  Magnetic field dependences of (top) the cost function $U$ in Eq.
(164) and (bottom) the grand potential calculated from $\\{{\bf S}_{{\bf
r}_{l}}^{\rm KPM-LD}\\}$ and $\\{\tilde{\bf S}_{{\bf r}_{l}}^{{\rm
sin}3Q}\\}^{*}$, and the difference of the grand potential, $\Delta\Omega$.

We plot the $h$ dependence of the optimal values of $U$ in the upper panel of
Fig. 20. We obtain sufficiently small values less than $4\times 10^{-4}$ for
$h<0.00325$. On the other hand, $U$ suddenly increases to $\simeq 4\times
10^{-2}$ at $h=0.0035$ and shows a gradual increase while increasing $h$. The
sudden increase of $U$ is related with the topological phase transition at
$h\simeq 0.00325$ as discussed below. The values for $h>0.00325$ are, however,
still very small, a few percent. The results indicate that the optimal spin
state $\\{\tilde{\bf S}_{{\bf r}_{l}}^{{\rm sin}3Q}\\}^{*}$ reproduces well
$\\{{\bf S}_{{\bf r}_{l}}^{\rm KPM-LD}\\}$ for all $h$.

In the lower panel of Fig. 20, we show the $h$ dependences of the grand
potential calculated from the numerically obtained ground state $\\{{\bf
S}_{{\bf r}_{l}}^{\rm KPM-LD}\\}$ and the optimal state $\\{\tilde{\bf
S}_{{\bf r}_{l}}^{{\rm sin}3Q}\\}^{*}$. We also plot the difference of the
grand potential $\Delta\Omega=\Omega^{{\rm sin}3Q}-\Omega^{{\rm KPM-LD}}$. The
results indicate that the optimal states $\\{\tilde{\bf S}_{{\bf r}_{l}}^{{\rm
sin}3Q}\\}^{*}$ almost perfectly reproduce $\\{{\bf S}_{{\bf r}_{l}}^{\rm KPM-
LD}\\}$ for $h<0.00325$ ($\Delta\Omega<9.0\times 10^{-6}$, whose relative
error is less than $3.2\times 10^{-3}$ %), while slight modifications could
improve the results for $h>0.00325$ ($2.0\times 10^{-4}<\Delta\Omega<4.0\times
10^{-4}$, whose relative error is less than $0.14$ %). We return to this point
in the end of this subsection.

Figure 21:  (a) Magnetic field dependences of $\tilde{\varphi}^{*}$ and
$|N_{\rm sk}|$ calculated from the optimal spin textures $\\{\tilde{{\bf
S}}_{{\bf r}_{l}}^{{\rm sin}3Q}\\}^{*}$. For comparison, $|N_{\rm sk}|$
obtained by the modified KPM-LD simulations is plotted by the orange points
connected by the dashed line. At $h=0$, we plot the results of
$\tilde{\varphi}^{*}$ at both $\tilde{\varphi}^{*}\simeq 0$ and $\pi$ because
of the time-reversal symmetry; the solid and dotted lines are the guides for
the eye (see the text). (b) Evolution of $\tilde{m}^{*}$ and
$\tilde{\varphi}^{*}$ obtained by the optimization. The background is a part
of the phase diagram in Fig. 10. The orange and green circles represent that
the data are for $|N_{\rm sk}|=2$ and $|N_{\rm sk}|=1$, respectively.

Figure 21(a) shows the $h$ dependences of $\tilde{\varphi}^{*}$ and $|N_{\rm
sk}|$ calculated from $\\{\tilde{{\bf S}}_{{\bf r}_{l}}^{{\rm sin}3Q}\\}^{*}$.
For comparison, we plot $|N_{\rm sk}|$ obtained by the modified KPM-LD
simulations. For the $|N_{\rm sk}|=2$ state at $h=0$, we obtain
$\tilde{\varphi}^{*}\simeq\pi$, but $\tilde{\varphi}^{*}$ is equivalent to
$\tilde{\varphi}^{*}+\pi$ at $h=0$ because of time-reversal symmetry; to show
this explicitly, we plot the data points at both $\tilde{\varphi}^{*}\simeq 0$
and $\pi$. By increasing $h$, we find that $\tilde{\varphi}^{*}\simeq 0$ tends
to be favored in the region with $|N_{\rm sk}|=2$, whereas
$\tilde{\varphi}^{*}$ becomes $\simeq\pi$ only at $h=0.001$. This is
presumably due to the failure of the modified KPM-LD simulation in the small
$h$ region where the states with $\tilde{\varphi}^{*}=0$ and $\pi$ have a
small energy difference. When $h$ is increased above $0.003$,
$\tilde{\varphi}^{*}$ is shifted to $\sim\frac{\pi}{4}$, and becomes almost
constant in the region with $|N_{\rm sk}|=1$. This clearly shows that the
topological transition with the change of $|N_{\rm sk}|$ caused by the
magnetic field is accompanied by the phase shift. We note that $|N_{\rm sk}|$
for the optimal spin configuration remains $2$ at $h=0.0035$ where the phase
is shifted already, while $|N_{\rm sk}|$ is reduced to $1$ for the modified
KPM-LD result; see below.

Figure 21(b) shows the evolution of $\tilde{m}^{*}$ and $\tilde{\varphi}^{*}$
obtained for $\\{\tilde{\bf S}_{{\bf r}_{l}}^{{\rm sin}3Q}\\}^{*}$, plotted on
the phase diagram in Fig. 10. Note that $\tilde{m}^{*}$ obtained by the
fitting does not correspond to the magnetization in the simulation data; see
also Eq. (88). The orange and green circles represent $|N_{\rm sk}|=2$ and 1,
respectively. The result indicates that $|N_{\rm sk}|$ calculated from
$\\{\tilde{\bf S}_{{\bf r}_{l}}^{{\rm sin}3Q}\\}^{*}$ is consistent with the
phase diagram calculated in Sec. III.3.2 and the topological transition from
$|N_{\rm sk}|=2$ to 1 is associated with the phase shift, except for
$h=0.0035$. When $h=0.0035$, $\\{\tilde{\bf S}_{{\bf r}_{l}}^{{\rm
sin}3Q}\\}^{*}$ gives $|N_{\rm sk}|=2$, although the values of $\tilde{m}^{*}$
and $\tilde{\varphi}^{*}$ are in the $|N_{\rm sk}|=1$ region, as shown in Fig.
21(b).

The discrepancy at $h=0.0035$ is presumably due to the insufficient
approximation by Eq. (165). Throughout the above analysis, we use Eq. (165)
for two different topological phases with different $|N_{\rm sk}|$. However,
it was recently pointed out that the spin state with $|N_{\rm sk}|=1$ is
better described by a different form of a superposition of sinusoidal and
cosinsoidal waves [88]. An extension including such a superposition reconciles
the discrepancy of $N_{\rm sk}$ at $h=0.0035$ in Fig. 21, and at the same
time, suppresses the increases of $U$ and $\Delta\Omega$ in the $|N_{\rm
sk}|=1$ region [89]. We here, however, restrict ourselves to Eq. (165) to
discuss the phase shift within the same form of the constituent waves as in
Sec. III.3.

###  V.2 Phase shift in the screw 4$Q$ state

Next, we perform similar analysis for the 3D $4Q$ states on a 3D simple cubic
lattice obtained in the previous study [57] for an effective spin model
derived from the Kondo lattice model with an antisymmetric spin-orbit coupling
[90, 57]. The Hamiltonian reads

$\displaystyle\mathcal{H}$ $\displaystyle=$ $\displaystyle
2\sum_{\eta}\left[-J{\bf S}_{{\bf q}_{\eta}}\cdot{\bf S}_{-{\bf
q}_{\eta}}+\frac{K}{N}({\bf S}_{{\bf q}_{\eta}}\cdot{\bf S}_{-{\bf
q}_{\eta}})^{2}\right.$ (167) $\displaystyle\qquad\ \left.-iD\frac{{\bf
q}_{\eta}}{|{\bf q}_{\eta}|}\cdot({\bf S}_{{\bf q}_{\eta}}\times{\bf S}_{-{\bf
q}_{\eta}})\right]-h\sum_{l}S_{{\bf r}_{l}}^{z},$

where ${\bf S}_{{\bf q}_{\eta}}=\frac{1}{\sqrt{N}}\sum_{l}{\bf S}_{{\bf
r}_{l}}e^{-i{\bf q}_{\eta}\cdot{\bf r}_{l}}$; the first, second, and third
terms represent the bilinear, biquadratic, and DM-type interactions in
momentum space, while the last term is the Zeeman coupling. The parameters are
set as $J=1$, $K=0.6$, $D=0.3$, and $N=16^{3}$. The first sum in Eq. (167) is
taken for a particular set of the wave vectors ${\bf q}_{\eta}$ shown in Fig.
12(a) and $q=|{\bf q}_{\eta}|=\frac{\sqrt{3}\pi}{4}$, i.e., $L=8$. The model
was found to stabilize $4Q$ states for $0\leq h\lesssim 1.395$, by using the
variational calculation for $h=0$ and the simulated annealing for $h\geq 0$
[57]. When $h=0$, the ground state is given by a superposition of four proper
screws, similar to the one discussed in Sec. IV.2. When $0\leq h\lesssim
0.575$, the ground state obtained by the simulated annealing is a $4Q$-HL with
$N_{\rm m}=16$. By increasing $h$, half of the hedgehog-antihedgehog pairs
cause pair annihilation at $h\simeq 0.575$, which changes the spin state to a
different $4Q$-HL with $N_{\rm m}=8$. The remaining hedgehogs and
antihedgehogs pair annihilate at $h\simeq 1.395$ and the system turns into a
topologically trivial $4Q$ state with $N_{\rm m}=0$, and finally, into the
forced ferromagnetic state for $h\gtrsim 1.395$. In the following, we focus on
the region for $0\leq h\leq 1.2$ where the spin configurations can be well
described by Eq. (126).

Following the procedure in Sec. V.1, we extract the phases of the constituent
waves in the two $4Q$-HLs by assuming that the ground state is well
approximated by a superposition of proper screws like in Eq. (126). In this
case, we estimate the cost function defined by

$\displaystyle U$
$\displaystyle(\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4},m)=$ (168)
$\displaystyle\frac{1}{N}\sum_{{\bf r}_{l}}\left(1-{\bf S}^{\rm SA}_{{\bf
r}_{l}}\cdot{\bf S}^{{\rm scr}4Q}_{{\bf
r}_{l}}(\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4},m)\right),$

where ${\bf S}^{\rm SA}_{{\bf r}_{l}}$ is the spin configuration obtained by
the simulated annealing and ${\bf S}^{{\rm scr}4Q}_{{\bf r}_{l}}$ is that
generated from Eq. (126) with ${\bf e}_{\eta}^{2}\rightarrow-{\bf
e}_{\eta}^{2}$ [in the previous study [57], $D$ was taken to be positive in
Eq. (167), which prefers left-handed screws]. We define the sum of
$\varphi_{\eta}^{*}$ in the same form as in Eq. (166), with the summation of
$\eta$ from 1 to 4.

Figure 22:  Magnetic field dependences of (top) the cost function $U$ in Eq.
(168) and (bottom) the energy per site calculated from $\\{{\bf S}_{{\bf
r}_{l}}^{\rm SA}\\}$ and $\\{{\bf S}_{{\bf r}_{l}}^{{\rm scr}4Q}\\}^{*}$, and
the difference of the energy, $\Delta E$.

The upper panel of Fig. 22 shows the $h$ dependence of the optimal values of
$U$. In the whole range of $h$, the optimal $U$ is less than $0.021$,
indicating that the optimal $\\{{\bf S}_{{\bf r}_{l}}^{{\rm scr}4Q}\\}^{*}$
reproduces well the numerically obtained ground state $\\{{\bf S}_{{\bf
r}_{l}}^{\rm SA}\\}$. The lower panel of Fig. 22 shows the $h$ dependences of
the energy per site calculated from $\\{{\bf S}_{{\bf r}_{l}}^{{\rm
scr}4Q}\\}^{*}$ and $\\{{\bf S}_{{\bf r}_{l}}^{\rm SA}\\}$, $E^{{\rm scr}4Q}$
and $E^{{\rm SA}}$, respectively, and their difference, $\Delta E=E^{{\rm
scr}4Q}-E^{{\rm SA}}$. We find that the energy is well reproduced for all $h$
(the relative error is less than $1.9$ %.) Both $U$ and $\Delta E$ show rapid