#  Hidden Topological Transitions in Emergent Magnetic Monopole Lattices

Yasuyuki Kato and Yukitoshi Motome  Department of Applied Physics, the
University of Tokyo, Tokyo 113-8656, Japan

###### Abstract

Topological defects, called magnetic hedgehogs, realize emergent magnetic
monopoles, which are not allowed in the ordinary electromagnetism described by
Maxwell’s equations. Such monopoles were experimentally discovered in magnets
in two different forms: tetrahedral $4Q$ and cubic $3Q$ hedgehog lattices. The
spin textures are modulated by the chemical composition, an applied magnetic
field, and temperature, leading to quantum transport and optical phenomena
through movement and pair annihilation of magnetic monopoles, but the
theoretical understanding remains elusive, especially in the regions where
different types of hedgehog lattices are competing. Here we propose a
theoretical model that can stabilize both tetrahedral and cubic hedgehog
lattices, and perform a thorough investigation of the phase diagram while
changing the interaction parameters, magnetic field, and temperature, by using
a recently developed method that delivers exact solutions in the thermodynamic
limit. We find that the model exhibits various types of topological
transitions with changes of the density of monopoles and antimonopoles, some
of which are accompanied by anomalies in the thermodynamic quantities, while
the others are hidden with less or no anomaly. We also find another hidden
topological transition with pair annihilation of two-dimensional vortices in
the three-dimensional system. These results not only provide useful
information for understanding the existing experimental data, but also
challenge the identification of hidden topological transitions and the
exploration of emergent electromagnetism in magnetic monopole lattices.

##  I Introduction

Figure 1:  Cubic $3Q$ (left) and tetrahedral $4Q$ (right) HLs, and the
possible transitions between them. Main panels show the position of hedgehogs
and antihedgehogs in the $3Q$ and $4Q$ HLs with surrounding spins. Enlarged
pictures of the spin configurations around the hedgehogs and antihedgehogs,
and the corresponding monopoles and antimonopoles defined by the sources and
sinks of the emergent magnetic fields denoted by the light green arrows are
also shown. Top panels show the characteristic wave numbers ${\bf Q}_{\eta}$
and the spin helices superposed to form the HLs. The labeled numbers indicate
$\eta$, and $\Lambda$ denotes the period of the helices. The gray cubes are
guides to the eye.

Topological defects are found ubiquitously, from cosmology, meteorology,
biology, and material science. In particular, in condensed matter physics,
various types of the topological defects have been discovered, which play an
important role in the properties of matter. Typical examples are found in
magnets, such as domain walls and vortices in swirling spin textures [1, 2, 3,
4]. Among such magnetic topological defects, magnetic hedgehog and
antihedgehog are unique since they realize magnetic monopole and antimonopole,
respectively [5], which are not allowed in the ordinary electromagnetism
described by Maxwell’s equations — these spin textures have spin vanishing
singularities at the cores, which can be viewed as source and sink of an
emergent magnetic flux quanta arising from the Berry phase mechanism [6]. They
are characterized by a topological index called monopole charge, which is
defined by how many times the spins around the core wraps a unit sphere. The
magnetic hedgehog and antihedgehog have attracted much interests since they
are expected to generate unique electromagnetic phenomena beyond the ordinary
electromagnetism and also carry information associated with their topological
indices.

Such magnetic hedgehogs and antihedgehogs were experimentally discovered in
MnSi_1-$x$Ge_x in a form of a periodic lattice structure called the magnetic
hedgehog lattice (HL) [7, 8, 9, 10, 11, 12, 13, 14, 15]. Two different types
of HLs have been reported depending on $x$: a tetrahedral HL for $0.3\lesssim
x\lesssim 0.7$ and a cubic HL for $0.7\lesssim x\leq 1.0$. Both spin
structures are regarded as a superposition of multiple spin helices. The
tetrahedral HL is composed of four spin helices propagating along the
directions from the center to four corners of a tetrahedron, and the cubic HL
is composed of three spin helices propagating along the three orthogonal
directions; hence, they are also called the $4Q$ and $3Q$ HLs (Fig. 1).
Similar tetrahedral $4Q$ HL was also found in SrFeO3 [16]. In these HLs,
unconventional quantum transport and optical phenomena, such as the
topological Hall effect [7, 8, 17], the topological Nernst effect [18], and
the magneto-Seebeck effect [19], were observed in an applied magnetic field,
presumably associated with the unique electromagnetism of magnetic monopoles
and antimonopoles. Although further exotic phenomena could be expected in the
competing region between the different types of HLs, no in-depth research has
been reported thus far.

The emergent electromagnetic phenomena in HLs have also been studied
theoretically. For instance, to explain the stability of HLs, different types
of lattice spin models were proposed, by including short-range [20, 21], long-
range multiple spin interactions [22, 23, 24, 25], and long-range anisotropic
spin interactions [26, 27]. Effects of the magnetic field were also studied in
detail, and interesting topological transitions were found to occur with pair
annihilation of magnetic monopoles and antimonopoles [22, 25]. However,
comprehensive understanding has been unreached yet, especially including the
competition between different types of HLs. The fundamental questions that we
address in this study are (i) how different types of HLs transform into each
other, (ii) how they respond to an external magnetic field, and (iii) how they
behave at finite temperature. Although all these questions are crucial for not
only understanding of the unique electromagnetism of magnetic monopoles but
also providing a guiding principle for further experimental exploration, these
remain challenging issues because of the lack of appropriate models and the
huge computational cost for the comprehensive study in three-dimensional
systems.

In this paper, we propose a model that stabilizes two types of HLs, the $3Q$
and $4Q$ HLs, at zero magnetic field, and study their competing region while
changing the magnetic field and temperature (Fig. 1). By using the exact
steepest descent method recently developed by the authors [27], we clarify the
phase diagram in the thermodynamic limit. Through the analysis, we unveil a
variety of the topological transitions with changes of the density of emergent
magnetic monopoles and antimonopoles. Notably, these topological transitions,
when not accompanied by magnetic phase transitions, are “hidden”, namely, they
show less or no anomaly in the thermodynamic quantities, such as the specific
heat and the magnetization. We also find another type of a hidden topological
transition not related to monopoles, caused by pair annihilation of two-
dimensional vortices in the three-dimensional system. It is worth noting that
these hidden topological transitions are hard to detect by numerical studies
such as the conventional brute-forced Monte Carlo simulation at finite
temperature, and their complete identifications are captured for the first
time by using the present exact method.

The structure of the paper is as follows. In Secs. II and III, we outline the
theoretical model and the method for the analysis of the magnetic and
topological transitions in the HLs. In Sec. IV, we present the results of the
ground-state phase diagrams including both $3Q$ and $4Q$ HLs, and the magnetic
field–temperature phase diagrams for three representative parameter sets in
their competing region. Finally, Sec. V is devoted for the summary.

##  II Model

Since we are interested in the competition between different types of HLs as
observed in MnSi1-xGex, we consider a model which can stabilize HLs even at
zero magnetic field. A candidate is found in spin lattice models with long-
range interactions mediated by itinerant electrons [22, 25], which are
variants of the models studied for various types of swirling spin textures
[28, 29, 30, 31, 32, 33, 34, 35, 36, 24, 37, 26, 38, 27]. In the previous
study [22], two models were independently studied for the cubic $3Q$ and
tetrahedral $4Q$ HLs. To study the competition between the two, we integrate
the two models by interpolating the interaction parameters. The Hamiltonian is
given by

$\displaystyle\mathcal{H}=2\sum_{\eta}\Bigl{[}$ $\displaystyle-J_{\eta}{\bf
S}_{{\bf Q}_{\eta}}\cdot{\bf S}_{-{\bf
Q}_{\eta}}+\frac{K_{\eta}}{N}\bigl{(}{\bf S}_{{\bf Q}_{\eta}}\cdot{\bf
S}_{-{\bf Q}_{\eta}}\bigr{)}^{2}$ $\displaystyle-i{\bf D}_{\eta}\cdot{\bf
S}_{{\bf Q}_{\eta}}\times{\bf S}_{-{\bf Q}_{\eta}}\Bigr{]}-\sum_{\bf r}{\bf
h}\cdot{\bf S}_{\bf r},$ (1)

where

$\displaystyle{\bf S}_{\bf Q}=\frac{1}{\sqrt{N}}\sum_{\bf r}{\bf S}_{\bf
r}e^{-i{\bf Q}\cdot{\bf r}},$ (2)

${\bf S}_{\bf r}=(S^{x}_{\bf r},S^{y}_{\bf r},S^{z}_{\bf r})$ denotes the spin
degree of freedom at site ${\bf r}$ on a simple cubic lattice, and $N$ is the
total number of spins; we consider the classical spin limit where ${\bf
S}_{\bf r}\in\mathbb{R}^{3}$ and $|{\bf S}_{\bf r}|=1$, for simplicity. The
first term of the Hamiltonian in Eq. (1) represents an effective long-range
spin interaction of the Ruderman-Kittel-Kasuya-Yosida type [39, 40, 41], where
${\bf Q}_{\eta}$ with $\eta=1,2,\ldots,7$ are the characteristic wave numbers
given by the nesting vectors of the Fermi surfaces of itinerant electrons in
the limit of weak spin-charge coupling [28]. The second term represents an
effective biquadratic interaction, which is most dominant in the higher-order
perturbation in terms of the spin-charge coupling [28]. The third term
describes an antisymmetric interaction of the Dzyaloshinskii-Moriya (DM) type
[42, 43], where the DM vectors are taken parallel to the corresponding
characteristic wave number as

$\displaystyle{\bf D}_{\eta}=D_{\eta}\frac{{\bf Q}_{\eta}}{|{\bf Q}_{\eta}|}.$
(3)

The last term in Eq. (1) represents the Zeeman coupling with an external
magnetic field ${\bf h}$. For the three interaction terms, to describe both
the cubic $3Q$ and tetrahedral $4Q$ HLs, we choose the characteristic wave
numbers as

$\displaystyle{\bf Q}_{1}=\begin{pmatrix}+Q,0,0\end{pmatrix},\ {\bf
Q}_{2}=\begin{pmatrix}0,+Q,0\end{pmatrix},\ {\bf
Q}_{3}=\begin{pmatrix}0,0,+Q\end{pmatrix},$ (4) $\displaystyle{\bf
Q}_{4}=\begin{pmatrix}+Q,-Q,-Q\end{pmatrix},\ {\bf
Q}_{5}=\begin{pmatrix}-Q,+Q,-Q\end{pmatrix},$ $\displaystyle{\bf
Q}_{6}=\begin{pmatrix}-Q,-Q,+Q\end{pmatrix},\ {\bf
Q}_{7}=\begin{pmatrix}+Q,+Q,+Q\end{pmatrix},$ (5)

where the former three in Eq. (4) [the latter four in Eq. (II)] prefer the
cubic $3Q$ (tetrahedral $4Q$) HL [22]; see Fig. 1. We parametrize the coupling
constants to interpolate the cubic and tetrahedral cases:

$\displaystyle\begin{pmatrix}J_{\eta},K_{\eta},D_{\eta}\end{pmatrix}=\begin{pmatrix}J(1-p),&K(1-p),&D(1-p)\end{pmatrix},$
(6)

for $\eta=1,2,3$, and

$\displaystyle\begin{pmatrix}J_{\eta},K_{\eta},D_{\eta}\end{pmatrix}=\begin{pmatrix}Jp,&Kp,&Dp\end{pmatrix},$
(7)

for $\eta=4,5,6,7$, with the mixing ratio $0\leq p\leq 1$. The model
stabilizes the cubic $3Q$ (tetrahedral $4Q$) HL at $p=0$ ($p=1$) when both $K$
and $D$ are sufficiently large [22, 24]. Thus, considering that the
interaction parameters are derived from the Fermi surface nesting, the
interpolation by $p$ implicitly assumes smooth deformation of the Fermi
surface with switching of the nesting vectors between the cubic and
tetrahedral types. In the following calculations, we take the energy unit as
$J=1$ and the lattice constant as unity. We set $K=0.6$, $D=0.3$, and
$Q=2\pi/\Lambda$ with $\Lambda=8$; $\Lambda$ corresponds to the magnetic
period of the stable spin textures [22].

##  III Method

Although the model in Eq. (1) is derived from the previous ones [22], the
systematic study of the phase diagram while changing the mixing ratio $p$, the
magnetic field ${\bf h}$, and temperature $T$ is not a simple task. Indeed,
the previous studies were limited to only the ground state for $p=0$ and $1$
by using simulated annealing. In the present study, we adopt a steepest
descent approach recently developed by the authors [27]. This method can
provide an exact solution for a class of models including Eq. (1) in the
thermodynamic limit, not only for the ground state but also at finite
temperature. Although the details of the method is found in Ref. [27], we
briefly describe the framework below to make the present paper to be self-
contained.

A key observation is that the Hamiltonian can be written in terms of the
averaged spins for each sublattice,

$\displaystyle\overline{\bf S}_{{\bf r}_{0}}=\frac{1}{N_{\rm MUC}}\sum_{\bf
R}{\bf S}_{{\bf R}+{\bf r}_{0}},$ (8)

where ${\bf R}$ and ${\bf r}_{0}$ are the position vectors of the magnetic
unit cell (MUC) and the internal sublattice site, respectively. This is
because, in the model in Eq. (1), all the magnetically ordered states have the
magnetic periods dictated by the wave numbers ${\bf Q}_{\eta}$ in Eqs. (4) and
(II), and hence, the MUC always fits into a cube of $\Lambda^{3}(\equiv
N_{0})$ spins with the translation vectors $(\Lambda,0,0)$, $(0,\Lambda,0)$,
and $(0,0,\Lambda)$. Thus, ${\bf R}=\Lambda(N^{x},N^{y},N^{z})$ and ${\bf
r}_{0}=(r^{x}_{0},r^{y}_{0},r^{z}_{0})$ with integers $N^{\mu}\in[0,L)$ and
$r^{\mu}_{0}\in[0,\Lambda)$; $N_{\rm MUC}$ is the number of MUC, i.e., $N_{\rm
MUC}=L^{3}$. Then, one can write the partition function by using
$\overline{\bf S}_{{\bf r}_{0}}$ as

$\displaystyle Z=\int\Bigl{(}\prod_{{\bf r}}d{\bf S}_{\bf
r}\Bigr{)}e^{-\beta\mathcal{H}}=\int\Bigr{[}\prod_{{\bf r}_{0}}d\overline{\bf
S}_{{\bf r}_{0}}\rho_{L^{d}}(\overline{\bf S}_{{\bf
r}_{0}})\Bigr{]}e^{-\beta\mathcal{H}},$ (9)

where $\beta$ is the inverse temperature and $\rho_{L^{d}}(\overline{\bf
S}_{{\bf r}_{0}})$ is the density of state for $\overline{\bf S}_{{\bf
r}_{0}}$. Note that the dimension of integration in Eq. (9) is reduced from
$2N_{\rm MUC}N_{0}$ to $3N_{0}$. After some algebra using the Pearson random
walk [44, 45], we obtain

$\displaystyle Z\to\int\Bigl{(}\prod_{{\bf r}_{0}}d\overline{\bf S}_{{\bf
r}_{0}}\Bigr{)}e^{N_{\rm MUC}G(\\{\overline{S}^{\alpha}_{{\bf r}_{0}}\\})},$
(10)

where

$\displaystyle G(\\{\overline{S}^{\alpha}_{{\bf r}_{0}}\\})$
$\displaystyle\quad=-\frac{\beta\mathcal{H}}{N_{\rm MUC}}+\sum_{{\bf
r}_{0}}\Bigl{[}\ln\Bigl{(}\frac{4\pi\sinh v_{0{\bf r}_{0}}}{v_{0{\bf
r}_{0}}}\Bigr{)}-v_{0{\bf r}_{0}}|\overline{\bf S}_{{\bf r}_{0}}|\Bigr{]},$
(11)

with

$\displaystyle|\overline{\bf S}_{{\bf r}_{0}}|=\coth v_{0{\bf
r}_{0}}-\frac{1}{v_{0{\bf r}_{0}}}\equiv S(v_{0{\bf r}_{0}}).$ (12)

From Eq. (10), the partition function in the thermodynamic limit ($N_{\rm
MUC}\to\infty$) is obtained by using the steepest descent method as

$\displaystyle Z\sim e^{N_{\rm MUC}G(\\{\overline{\overline{S}^{\alpha}_{{\bf
r}_{0}}}\\})},$ (13)

with $\\{\overline{\overline{S}^{\alpha}_{{\bf r}_{0}}}\\}=\mathop{\rm
arg~{}max}_{\\{\overline{S}^{\alpha}_{{\bf
r}_{0}}\\}}G(\\{\overline{S}^{\alpha}_{{\bf r}_{0}}\\})$. In other words, the
solution in this method is obtained by maximizing
$G(\\{\overline{S}^{\alpha}_{{\bf r}_{0}}\\})$. It is worth noting that Eq.
(12) is useful for the parametrization of $\overline{\bf S}_{{\bf r}_{0}}$ by
enabling us to write

$\displaystyle\overline{\bf S}_{{\bf r}_{0}}=$ $\displaystyle S(v_{0{\bf
r}_{0}})\bigl{(}\cos\varphi_{{\bf r}_{0}}\sin\theta_{{\bf
r}_{0}},\sin\varphi_{{\bf r}_{0}}\sin\theta_{{\bf r}_{0}},\cos\theta_{{\bf
r}_{0}}\bigr{)},$ (14)

with three real numbers $v_{0{\bf r}_{0}}$, $\theta_{{\bf r}_{0}}$, and
$\varphi_{{\bf r}_{0}}$. For the maximization of
$G(\\{\overline{S}^{\alpha}_{{\bf r}_{0}}\\})$, we use NVIDIA A100 GPU with a
JAX-based [46] library, Optax [47].

From the optimized values of $\overline{\overline{S}^{\alpha}_{{\bf r}_{0}}}$,
the free energy per spin is obtained as

$\displaystyle f=-\frac{G(\\{\overline{\overline{S}^{\alpha}_{{\bf
r}_{0}}}\\})}{\beta N_{0}}.$ (15)

In addition, the internal energy $\varepsilon$ and the specific heat $C$ per
spin are computed as

$\displaystyle\varepsilon=\frac{\langle\mathcal{H}\rangle}{N}=\lim_{\\{\overline{S}^{\alpha}_{{\bf
r}_{0}}\\}\to\\{\overline{\overline{S}^{\alpha}_{{\bf
r}_{0}}}\\}}\frac{\mathcal{H}}{N},$ (16) $\displaystyle
C=\frac{\partial{\varepsilon}}{\partial T},$ (17)

respectively, where $\langle\cdots\rangle$ denotes the thermal average. As the
real-space spin configuration is given by $\langle{\bf S}_{{\bf R}+{\bf
r}_{0}}\rangle=\overline{\overline{\bf S}_{{\bf r}_{0}}}$, the magnetization
$m$ and the spin scalar chirality are computed as

$\displaystyle m=$ $\displaystyle\frac{1}{N_{0}h}\sum_{{\bf
r}_{0}}\overline{\overline{\bf S}_{{\bf r}_{0}}}\cdot{\bf h},$ (18)
$\displaystyle\chi_{\rm sc}=$ $\displaystyle\frac{1}{N_{0}h}\sum_{{\bf
r}_{0}}{\bm{\chi}}_{{\bf r}_{0}}\cdot{\bf h},$ (19)

respectively, where $h=|{\bf h}|$; ${\bm{\chi}}_{{\bf r}_{0}}$ represents the
local scalar spin chirality defined as [22, 23]

$\displaystyle\chi^{\gamma}_{{\bf r}_{0}}=$
$\displaystyle\frac{1}{2}\sum_{\alpha,\beta,\nu_{\alpha},\nu_{\beta}}\varepsilon^{\alpha\beta\gamma}\nu_{\alpha}\nu_{\beta}\overline{\overline{\bf
S}_{{\bf r}_{0}}}\cdot\bigl{(}\overline{\overline{\bf S}_{{\bf
r}_{0}+\nu_{\alpha}\hat{\bm{\alpha}}}}\times\overline{\overline{\bf S}_{{\bf
r}_{0}+\nu_{\beta}\hat{\bm{\beta}}}}\bigr{)},$ (20)

where $\alpha,\beta,\gamma=x,y,z$, $\varepsilon^{\alpha\beta\gamma}$ is the
Levi-Civita symbol, $\nu_{\alpha(\beta)}=\pm 1$, and
$\hat{\bm{\alpha}}(\hat{\bm{\beta}})$ is the unit translation vector in the
$\alpha(\beta)$ direction. To identify the magnetically ordered phases, we
define the order parameter as

$\displaystyle m_{\eta}=\sqrt{\frac{1}{N_{0}}\overline{\overline{\bf S}_{{\bf
Q}_{\eta}}}\cdot\overline{\overline{\bf S}_{-{\bf Q}_{\eta}}}},$ (21)

which corresponds to the square root of the normalized spin structure factor,
with $\overline{\overline{\bf S}_{{\bf Q}}}=\frac{1}{\sqrt{N_{0}}}\sum_{{\bf
r}_{0}}\overline{\overline{\bf S}_{{\bf r}_{0}}}e^{-i{\bf Q}\cdot{\bf
r}_{0}}$. Moreover, to investigate the topological property of the
magnetically ordered phases, extending the previous studies [22, 23, 24, 26,
25], we identify the monopoles and antimonopoles by computing the monopole
charge in each cubic unit of the cubic lattice from configurations of
normalized spins $\langle{\bf S}_{{\bf R}+{\bf r}_{0}}\rangle/|\langle{\bf
S}_{{\bf R}+{\bf r}_{0}}\rangle|$ [27]. We also obtain the number of monopoles
and antimonopoles per MUC, $N_{\rm m}$, to distinguish different topological
phases.

##  IV Results

In this section, we show the results obtained for the model in Eq. (1) by
using the steepest descent method described in the previous section. In Sec.
IV.1, we present the $p$ dependences of the magnetic order parameters at zero
magnetic field and zero temperature (Sec. IV.1.1), and the $p$–$h$ phase
diagrams at zero temperature with three different magnetic field directions,
${\bf h}\parallel[100]$, $[110]$, and $[111]$ (Sec. IV.1.2). In Sec. IV.2, we
present the $h$–$T$ phase diagrams for the three directions of the magnetic
field, focusing on the competing regime between the $3Q$ and $4Q$ HLs, at
$p=0.4$ (Sec. IV.2.1), $p=0.5$ (Sec. IV.2.2), and $p=0.6$ (Sec. IV.2.3).

###  IV.1 Ground-state phase diagrams

####  IV.1.1 Zero magnetic field

Figure 2:  Order parameters $m_{\eta_{\ell}}$ at zero temperature as a function of the mixing ratio $p$ for the coupling constants in Eqs. (6) and (7): (a) the entire view for $0\leq p\leq 1$ and (b) an enlarged view for $0.4\leq p\leq 0.6$. $m_{\eta_{\ell}}$ are grouped into two as $\\{m_{1},m_{2},m_{3}\\}$ and $\\{m_{4},m_{5},m_{6},m_{7}\\}$, and sorted in each group in descending order of the values of $m_{\eta_{\ell}}$. The numbers in parentheses in (b) indicate the numbers of monopoles and antimonopoles per MUC, $N_{\rm m}$.  Figure 3:  Ground-state spin configurations and positions of monopoles and antimonopoles for (a) the $3Q$ state with $N_{\rm m}=8$ at $p=0.4$, (b) the $3Q$ state with $N_{\rm m}=4$ at $p=0.5$, (c) the $5Q$ state with $N_{\rm m}=16$ at $p=0.52$, and (d) the $4Q$ state with $N_{\rm m}=16$ at $p=0.6$. The color of arrows represents the $[111]$ component of the spins, $S^{[111]}_{\bf r}$, as indicated in the inset of (a). The magenta and cyan spheres represent the monopoles (hedgehogs) and antimonopoles (antihedgehogs), respectively. The right panels show the top view of the HLs (top) and the distribution of the order parameters $m_{\eta}$ (bottom).  Table 1:  Magnetic phases, possible numbers of monopoles and antimonopoles per MUC $N_{\rm m}$, and conditions for the order parameters found in the ground-state and finite-temperature phase diagrams for the model in Eq. (1) with $J=1$, $K=0.6$, $D=0.3$, and $\Lambda=8$. The list of the related figures is also shown. The starred $N_{\rm m}$ are found only at finite temperature. The possible sets of $\eta$ for nonzero $m_{\eta}$ are shown in the form of $\\{\eta_{1},\eta_{2},\cdots\\}$ if necessary.  | Phase | $N_{\rm m}$ | Conditions | Related figures
---|---|---|---|---
${\bf h}=0$ | $3Q$ | 0∗, 4, 8 |  $m_{\eta}\neq 0$ for all $\eta=1\text{--}3$. |  2, 3(a), 3(b), 4, 6, 9
| $4Q$ | 16 |  $m_{\eta}=0$ for all $\eta=1\text{--}3$ with $m_{\eta^{\prime}}\neq 0$ for all $\eta^{\prime}=4\text{--}7$. |  2, 3(d), 4
| $5Q$ | 0∗, 16 | Five of $m_{\eta}$ are nonzero: $\\{1,4\text{--}7\\}$, $\\{2,4\text{--}7\\}$, and $\\{3,4\text{--}7\\}$. |  2, 3(c), 9
${\bf h}\parallel[100]$ | $1Q$ | 0 | One of $m_{\eta}$ is nonzero: $\\{1\\}$. |  4(a), 5(a), 8(a)
| $3Q$ | 0, 4, 8 | (same as $3Q$ for ${\bf h}=0$) |  4(a), 5(a), 8(a)
| $3Q^{\prime}$ | 0 | Three of $m_{\eta}$ are nonzero: $\\{1,4,7\\}$ and $\\{1,5,6\\}$. |  4(a), 8(a)
| $4Q$ | 0, 8, 16 | (same as $4Q$ for ${\bf h}=0$). |  4(a), 11(a)
| $5Q$ | 0∗, 16 | Five of $m_{\eta}$ are nonzero: $\\{1,4\text{--}7\\}$. |  8(a)
${\bf h}\parallel[110]$ | $1Q$ | 0 | One of $m_{\eta}$ is nonzero: $\\{1\\}$ and $\\{2\\}$. |  4(b), 5(b), 7
| $1Q^{\prime}$ | 0 | One of $m_{\eta}$ is nonzero: $\\{6\\}$ and $\\{7\\}$. |  4(b), 8(b), 10, 11(b)
| $2Q$ | 0∗ | Two of $m_{\eta}$ are nonzero: $\\{1,2\\}$. |  5(b), 7
| $2Q^{\prime}$ | 0∗ | Two of $m_{\eta}$ are nonzero: $\\{6,7\\}$. |  8(b), 11(b)
| $3Q$ | 0, 2, 4, 6, 8 | (same as $3Q$ for ${\bf h}=0$) |  4(b), 5(b), 7, 8(b), 10
| $3Q^{\prime}$ | 0∗ | Three of $m_{\eta}$ are nonzero: $\\{1,4,7\\}$, $\\{1,5,6\\}$, $\\{2,4,6\\}$, and $\\{2,5,7\\}$. |  8(b)
| $4Q$ | 0, 8, 16, 24∗ | (same as $4Q$ for ${\bf h}=0$) |  4(b), 11(b)
| $5Q$ | 0∗, 4∗, 16 | Five of $m_{\eta}$ are nonzero: $\\{1,4\text{--}7\\}$ and $\\{2,4\text{--}7\\}$. |  8(b)
${\bf h}\parallel[111]$ | $1Q$ | 0 | One of $m_{\eta}$ is nonzero: $\\{1\\}$, $\\{2\\}$, and $\\{3\\}$. |  4(c), 5(c)
| $1Q^{\prime}$ | 0 | One of $m_{\eta}$ is nonzero: $\\{7\\}$. |  4(c), 8(c), 11(c)
| $3Q$ | 0, 2, 4, 6∗, 8 | (same as $3Q$ for ${\bf h}=0$) |  4(c), 5(c), 8(c)
| $3Q^{\prime}$ | 0∗ | Three of $m_{\eta}$ are nonzero: $\\{1,4,7\\}$, $\\{2,5,7\\}$, and $\\{3,6,7\\}$. |  8(c)
| $4Q$ | 8, 16 | (same as $4Q$ for ${\bf h}=0$) |  4(c), 11(c), 12
| $5Q$ | 0∗ | (same as $5Q$ for ${\bf h}=0$) |  8(c)

Figure 2 shows the mixing ratio $p$ dependences of the order parameters
$m_{\eta_{\ell}}$ [Eq. (21)] for the ground state. The label of wave numbers
$\eta$ are grouped into $1$–$3$ and $4$–$7$, sorted within each group, and
assigned $\ell$ in descending order as $m_{\eta_{1}}\geq m_{\eta_{2}}\geq
m_{\eta_{3}}$ and $m_{\eta_{4}}\geq m_{\eta_{5}}\geq m_{\eta_{6}}\geq
m_{\eta_{7}}$.

With an increase of $p$, the ground state changes from the $3Q$ phase to the
$5Q$ phase, and then to the $4Q$ phase, as shown in Fig. 2(a). In the $3Q$
phase, not only $m_{1\text{--}3}$ but also $m_{4\text{--}7}$ are nonzero,
since $m_{4\text{--}7}$ correspond to higher harmonics of $m_{1\text{--}3}$,
e.g., ${\bf Q}_{4}={\bf Q}_{1}-{\bf Q}_{2}-{\bf Q}_{3}$. Meanwhile, in the
$4Q$ phase, $m_{1\text{--}3}=0$, and only $m_{4\text{--}7}$ are nonzero. In
the intermediate $5Q$ phase, one of the $3Q$ components, $m_{1}$, remains
nonzero in addition to all the $4Q$ components $m_{4\text{--}7}$. As shown in
the enlarged view in Fig. 2(b), the $3Q$–$5Q$ transition at $p\simeq 0.517$ is
discontinuous with jumps of the order parameters, while the $5Q$–$4Q$
transition at $p\simeq 0.529$ is continuous.

In the entire range of $p$, the ground-state spin configuration contains
topological defects, monopoles and antimonopoles, indicating that all the
$3Q$, $4Q$, and $5Q$ states are HLs. The number of monopoles and antimonopoles
per MUC, $N_{\rm m}$, is indicated in the parentheses in Fig. 2(b), and the
ground-state spin configurations, the positions of the monopole and
antimonopoles, and the distributions of $m_{\eta}$ for representative
parameters at $p=0.4$, $0.5$, $0.52$, and $0.6$ are shown in Fig. 3. A closer
look reveals additional phase transitions within the $3Q$ phase: two
discontinuous transitions at $p\simeq 0.499$ and $p\simeq 0.509$, where
$N_{\rm m}$ is reduced from $8$ to $4$ and recovered from $4$ to $8$,
respectively, as shown in Fig. 2(b). This means that an intermediate $3Q$
phase with $N_{\rm m}=4$ intervenes in the $3Q$ phase with $N_{\rm m}=8$. At
these discontinuous transitions, $m_{\eta_{l}}$ also changes discontinuously:
$m_{\eta_{2}}=m_{\eta_{3}}$ for the $N_{\rm m}=8$ phase, while
$m_{\eta_{2}}>m_{\eta_{3}}$ for the $N_{\rm m}=4$ phase. Thus, these
transitions are magnetic phase transitions with topological changes. In
addition, we find phase transitions within the $N_{\rm m}=8$ state: a
continuous one from $m_{\eta_{1}}=m_{\eta_{2}}=m_{\eta_{3}}$ to
$m_{\eta_{1}}>m_{\eta_{2}}=m_{\eta_{3}}$ at $p\simeq 0.467$ and a
discontinuous one at $p\simeq 0.475$. We do not discuss these transitions in
this paper because they are not relevant in the topological point of view. All
the phases with different $N_{\rm m}$ are summarized in the top row for ${\bf
h}=0$ of Table 1.

Let us make two remarks. One is on the competition between the $3Q$ and $4Q$
phases. In Fig. 2, the $3Q$ region is slightly wider than the $4Q$ one. This
is presumably because the characteristic wave numbers of the $4Q$ components
coincide with the higher harmonics of those of the $3Q$ components in the
present model, which may work in favor of stabilizing the $3Q$ phase over
$4Q$. The other remark is on the intermediate $5Q$ phase. This is a peculiar
phase with spontaneous symmetry breaking by selecting one of the $3Q$ wave
numbers, which has never been reported in experiments to our knowledge. While
further theoretical studies are necessary on the stability of this phase in a
wider parameter range, our result would stimulate the experimental exploration
in the competing regime between the $3Q$ and $4Q$ HLs.

####  IV.1.2 Magnetic field dependence

Figure 4:  Ground-state phase diagrams on the $p$–$h$ plane for (a) ${\bf
h}\parallel[100]$, (b) ${\bf h}\parallel[110]$, and (c) ${\bf
h}\parallel[111]$. The calculations are done for every $0.05$ ($0.1$) with
respect to $p$ ($h$). The colored regimes indicate the magnetically ordered
phases while the white regimes indicate the forced ferromagnetic (FFM) phase.
The numbers in parentheses represent the number of monopoles in a MUC, $N_{\rm
m}$.

Figure 4 shows the $p$–$h$ phase diagram for the cases of ${\bf
h}\parallel[100]$, ${\bf h}\parallel[110]$, and ${\bf h}\parallel[111]$. At
$h=0$, only the $3Q$ and $4Q$ phases appear and the intermediate $5Q$ phase
found in the previous section is absent because of the low resolution in the
parameter setting; here, the calculations are done for every $0.05$ with
respect to $p$. When the magnetic field is applied, both $3Q$ and $4Q$ phases
remain stable, but in most cases, they turn into a single-$Q$ phase ($1Q$ or
$1Q^{\prime}$) before the saturation to the forced ferromagnetic (FFM) phase,
which is connected to the paramagnetic (PM) state at finite temperature. The
stability of the $3Q$ and $4Q$ phases depends on the relative directions of
${\bf h}$ and ${\bf Q}_{\eta}$: When ${\bf h}$ is applied in parallel to one
of ${\bf Q}_{\eta}$, the $3Q$ and $4Q$ phases turn into the single-$Q$ phase
characterized by ${\bf Q}_{\eta}\parallel{\bf h}$ at a relatively weak
magnetic field, and the single-$Q$ phase is stable in a relatively wider field
range; see Fig. 4(a) for the $3Q$ case where ${\bf Q}_{1}\parallel{\bf h}$ and
Fig. 4(c) for the $4Q$ case where ${\bf Q}_{7}\parallel{\bf h}$. In the other
field directions, the $3Q$ and $4Q$ phases remain more stable, and the
single-$Q$ phases become narrower; the extreme case is the $4Q$ case under
${\bf h}\parallel[100]$, which shows direct saturation to FFM without any
single-$Q$ phase.

It is worth noting that the relative stability between the $3Q$ and $4Q$
states does not change so much under the magnetic field, namely, the phase
boundaries between them are almost independent of $h$ within the present
resolution. This is similar to the experimental results for MnSi_1-$x$Ge_x
[13]. In the competing region near $p=0.5$ in $h\parallel[100]$, however, we
find complex successive phase transitions from $3Q$ to $3Q^{\prime}$ and to
$1Q$ before the saturation. The $3Q^{\prime}$ phase is found only in the
finite field region, and the magnetic order is composed of a mixture of the
$3Q$ and $4Q$ components similar to the $5Q$ phase discussed in the previous
section: two out of the $4Q$ components $\\{m_{4},m_{7}\\}$ or
$\\{m_{5},m_{6}\\}$ and one of the $3Q$ component $m_{1}$. Interestingly, the
spin texture in the $3Q^{\prime}$ phase exhibits two-dimensional modulation on
the [$\bar{1}\bar{1}1$] or [$\bar{1}11$] plane since the three wave numbers
are coplanar. We will discuss an interesting field direction dependence of the
spin scalar chirality in this two-dimensional $3Q^{\prime}$ phase in Sec.
IV.2.2.

Both $3Q$ and $4Q$ phases exhibit various types of topological transitions
accompanied by changes of $N_{\rm m}$ under the magnetic field. In the $4Q$
case, $N_{\rm m}$ decreases monotonically, from $16$ to $8$ for all the field
directions, and then from $8$ to $0$ before the transition to $1Q^{\prime}$ or
the saturation to FFM for the cases of ${\bf h}\parallel[100]$ and $[110]$. In
contrast, in the $3Q$ case, $N_{\rm m}$ does not always decrease monotonically
and shows more complex field dependence. For example, in the case of ${\bf
h}\parallel[110]$, $N_{\rm m}$ changes as $8\to 6\to 2\to 6\to 2$ before the
transition to the $1Q$ phase, as shown in Fig. 4(b). We will study how these
topological transitions evolve with temperature and how thermodynamic
quantities behave at these transitions in the next section.

###  IV.2 Magnetic field–temperature phase diagrams

In this section, we present the results of the magnetic field–temperature
phase diagrams. Besides magnetic phase transitions, we discuss topological
transitions associated with changes of $N_{\rm m}$ caused by varying the
temperature and the magnetic field. Focusing on the competing region between
the $3Q$ and $4Q$ phases, we show the results for three values of $p$:
$p=0.4$, $0.5$, and $0.6$ in Secs. IV.2.1, IV.2.2, and IV.2.3, respectively.

####  IV.2.1 $p=0.4$

Figure 5:  Magnetic field–temperature phase diagrams (left) and $h$–$T$ maps
of the spin scalar chirality $\chi_{{\rm sc}}$ (right) for (a) ${\bf
h}\parallel[100]$, (b) ${\bf h}\parallel[110]$, and (c) ${\bf
h}\parallel[111]$ at $p=0.4$. The calculations are done for every $0.05$
($0.02$) with respect to $h$ ($T$). The magenta dashed line in the left panel
of (b) indicates the parameter range for Fig. 7. The gray dashed lines in the
right panels represent the phase boundaries for guides to the eye.  Figure 6:
Temperature dependences of (a) the order parameters $m_{\eta_{\ell}}$ and (b)
the specific heat $C$ at $h=0$ and $p=0.4$. The black solid and dashed lines
in (a) indicate functions proportional to $|T-T_{c}|^{1/2}$ and
$|T-T_{c}|^{3/2}$, respectively.

Let us begin with the case of $p=0.4$, which is close to the competing region
but still in the $3Q$ phase; see Figs. 2 and 4. Figure 5 shows the $h$–$T$
phase diagrams at $p=0.4$ for three different directions of the magnetic
field, ${\bf h}\parallel[100]$, $[110]$, and $[111]$ (left panels). At $h=0$,
the ground state is the $3Q$ HL with $N_{\rm m}=8$, namely, four
monopole–antimonopole pairs within the MUC; see Fig. 3(a). The $3Q$ phase
remains stable against raising temperature, and shows a phase transition to
the PM phase at $T\simeq 0.52$. Figure 6 shows the temperature dependences of
the order parameters $m_{\eta}$ and the specific heat per spin $C$ at $h=0$.
Near the critical temperature $T_{c}$, $m_{1\text{--}3}$ exhibit the critical
behavior of the mean-field universality class as
$m_{1\text{--}3}\propto|T-T_{c}|^{1/2}$, while $m_{4\text{--}7}$ behave
differently as $m_{4\text{--}7}\propto|T-T_{c}|^{3/2}$. This indicates that
$m_{1\text{--}3}$ are the primary order parameters, and $m_{4\text{--}7}$,
which correspond to higher harmonics of $m_{1\text{--}3}$ as discussed in Sec.
IV.1.1, are induced as the secondary order parameters. We note that the mean-
field universality class is consistently understood from the fact that the
range of the interactions in Eq. (1) is infinite. We find that $C$ shows a
jump at $T_{c}$ and becomes zero for $T>T_{c}$, which is also consistent with
the mean-field universality class.

Figure 7:  Magnetic field dependences of (a) $m_{\eta}$, (b) $C$ and $dm/dh$,
and (c) $N_{\rm m}$ and $\chi_{\rm sc}$. The calculations are done for ${\bf
h}\parallel[110]$ at $T=0.45$ and $p=0.4$; see the magenta dashed line in the
left panel of Fig. 5(b). The data for $m_{4\text{--}7}$ in (a) and $\chi_{\rm
sc}$ in (c) are multiplied by factors of $10$ and $10^{4}$, respectively, for
better visibility. The solid lines in (a) indicate functions proportional to
$|T-T_{c}|^{1/2}$. The green and blue arrows in (b) indicate the hidden
topological transitions associated with monopoles and vortices, respectively.
(d) Spin configurations and positions of monopoles and antimonopoles at
$h=0.3$. The inset shows distribution of $m_{\eta}$. Two-dimensional spin
configurations on a $(001)$ slice at (e) $h=0.305$, (f) $h=0.350$, and (g)
$h=0.400$ in the $2Q$ phase. The orange and blue plaquettes represent the
cores of vortices and antivortices.

In an applied magnetic field, the $3Q$ phase remains stable with changes of
$N_{\rm m}$, but it undergoes phase transitions to the $1Q$ phase and then to
the PM phase at low temperature, as seen in the ground state in Sec. IV.1.2.
In contrast, we find different behaviors in the high-temperature region; the
system undergoes a phase transition to the $2Q$ phase before entering into PM
in ${\bf h}\parallel[110]$ [Fig. 5(b)] and a direct transition from the $3Q$
phase to PM in ${\bf h}\parallel[111]$ [Fig. 5(c)]. In addition, within the
$3Q$ phase we find topological transitions that are not seen in the low-
temperature region: the transition from $N_{\rm m}=8$ to $0$ in ${\bf
h}\parallel[100]$ [Fig. 5(a)] and $N_{\rm m}=6$ to $4$ in ${\bf
h}\parallel[110]$ [Fig. 5(b)]. As shown in the right panels of Fig. 5,
$\chi_{\rm sc}$ has nonzero values in the entire regions of the $3Q$ phases in
the magnetic field, including the topologically trivial one with $N_{\rm
m}=0$. We find that $\chi_{\rm sc}$ is drastically reduced at the topological
transitions where $N_{\rm m}$ becomes zero, and it vanishes at the transitions
to the other $1Q$, $2Q$, or PM phases either continuously or discontinuously.

Let us closely look at the phase transitions under the magnetic field at
finite temperature, focusing on the case of $T=0.45$ in ${\bf h}||[110]$,
where the intermediate $2Q$ phase appears in addition to the multiple $3Q$
phases with different $N_{\rm m}$. Figures 7(a)–7(c) show $h$ dependences of
$m_{\eta}$, $C$, the field derivative of the magnetization $dm/dh$, $N_{\rm
m}$, and $\chi_{\rm sc}$. When $h$ is applied, the spiral component $m_{3}$,
which is perpendicular to the magnetic field (${\bf Q}_{3}\perp{\bf h}$),
deviates from the other $3Q$ components ($m_{1}=m_{2}$) due to the symmetry
breaking by $h$ and decreases as increasing $h$, as shown in Fig. 7(a). With
further increasing $h$, $m_{3}$ continuously vanishes at $h\simeq 0.305$ as
$m_{3}\propto|h-h_{c}|^{1/2}$, and the system turns into the $2Q$ phase. Due
to the disappearance of one of the primary order parameters, the secondary
order parameters $m_{4\text{--}7}$ also vanish as
$m_{4\text{--}7}\propto|h-h_{c}|^{1/2}$. At the transition, both $C$ and
$dm/dh$ show a jump, as shown in Fig. 7(b).

In addition to this magnetic phase transition, the system undergoes successive
topological transitions within the $3Q$ phase at $h\simeq 0.238$ and $0.270$,
where $N_{\rm m}$ changes stepwise as $N_{\rm m}=8$ to $6$ and to $4$ as shown
in Fig. 7(c). They are associated with annihilations of a pair of monopole and
antimonopole per MUC. Interestingly, these transitions are not accompanied by
any anomalies in $C$ and $dm/dh$ [green arrows in Fig. 7(b)], in stark
contrast to the magnetic phase transition at $h\simeq 0.305$ where the
topology also changes as $N_{\rm m}$ vanishes. We also note that $\chi_{\rm
sc}$ does not show any anomalies at these topological transitions, while it
decreases rapidly along with the decrease of $N_{\rm m}$ after showing a broad
maximum in the $N_{\rm m}=8$ phase, as shown in Fig. 7(c). Thus, the
topological transitions with changes of $N_{\rm m}$, when they are not
accompany by the magnetic phase transition, are “hidden” transitions that do
not show any anomalies in the thermodynamic quantities. We note that similar
hidden transitions were found in the previous study of the ground state [22],
but their features, especially at finite temperature, have not been analyzed
because of the high computational cost and less accuracy in the numerical
simulations.

Through the magnetic phase transition from $3Q$ to $2Q$ at $h\simeq 0.305$,
the spin texture changes from the three- to two-dimensional one. Figure 7(d)
shows the spin configuration and the positions of monopoles and antimonopoles
in the $3Q$ phase close to the critical field. The spin structure is three-
dimensional, but the modulation in the direction of ${\bf Q}_{3}$ ($z$
direction) is weak as $m_{3}$ is small compared to $m_{1}$ and $m_{2}$ as
shown in the inset. When entering into the $2Q$ phase, the modulation in the
$z$ direction is completely eliminated as $m_{3}$ becomes zero, and the spin
texture becomes two-dimensional. Figures 7(e)–7(g) show the spin
configurations on a $(001)$ slice in the $2Q$ phase near the transition. In
this region, we find a vortex-like texture. To identify vorticies and
antivorticies, we compute the vorticity defined by the sum of the four
relative angles ($\in(-\pi,\pi]$) between the neighboring spins surrounding
each square plaquette after projecting onto the $(001)$ plane: It takes
$(-)2\pi$ for the (anti)vortex. As a result, we reveal that the $2Q$ state for
$0.305\lesssim h\lesssim 0.365$ consists of two vortex–antivortex pairs, whose
cores are indicated by the orange and blue plaquettes in Figs. 7(e) and 7(f).
We show that the vortices and antivortices disappear at $h\simeq 0.365$,
suggesting another topological transition. Again $C$ and $dm/dh$ do not show
any anomalies, while both shows a broad hump around the topological
transition, as shown by the blue arrow in Fig. 7(b).

####  IV.2.2 $p=0.5$

Figure 8:  Magnetic field–temperature phase diagrams (left) and $h$–$T$ maps
of the spin scalar chirality $\chi_{{\rm sc}}$ (right) for (a) ${\bf
h}\parallel[100]$, (b) ${\bf h}\parallel[110]$, and (c) ${\bf
h}\parallel[111]$ at $p=0.5$. The magenta dashed line in the left panel of (b)
indicates the parameter range for Fig. 10. The other notations are common to
those in Fig. 5.  Figure 9:  Temperature dependences of (a) $m_{\eta_{\ell}}$
and (b) $C$ and $N_{\rm m}$ at $h=0$ and $p=0.5$. The black solid lines in (a)
indicate functions proportional to $|T-T_{c}|^{1/2}$.

Next, we show the results for $p=0.5$. Figure 8 summarizes the phase diagrams
and $\chi_{{\rm sc}}$ on the $h$–$T$ plane. At $h=0$, the ground state is the
$3Q$ HL with two pairs of monopoles and antimonopoles ($N_{\rm m}=4$); see
Fig. 3(b). With increasing $T$, the system undergoes successive transitions
before entering into the PM state above $T\simeq 0.433$. The temperature
dependences of $m_{\eta}$ and $C$ are shown in Fig. 9. At $T\simeq 0.175$, the
system shows a first-order phase transition to another $3Q$ HL with $N_{\rm
m}=8$, where $m_{\eta}$ show discontinuous changes: The primary $3Q$
components change from $m_{1}>m_{2}>m_{3}$ to $m_{1}>m_{2}=m_{3}$, as shown in
Fig. 9(a). At the same time, $C$ shows a delta function like anomaly, as shown
in Fig. 9(b). Thus, this is a discontinuous phase transition between the $3Q$
HLs with different magnetic and topological properties. In contrast, the next
transition at $T\simeq 0.275$, where $N_{\rm m}$ changes from $8$ to $0$ by
annihilation of four pairs of monopoles and antimonopoles, is a hidden
topological transition: $C$ as well as $m_{\eta}$ does not show any anomalies
as indicated by the green arrow in Fig. 9(b), similar to those with $N_{\rm
m}=8\to 6\to 4$ at $p=0.4$ in the previous section.

When $T$ is further increased, a second-order phase transition occurs from
$3Q$ to $5Q$ at $T\simeq 0.285$. At this transition, two of $m_{1\text{--}3}$
vanish as $m_{\eta}\propto|T-T_{c}|^{1/2}$, leaving one nonzero component of
$3Q$ in addition to the four $4Q$ ones. While the nonzero magnetic components
are common, the $5Q$ state is topologically different from that found in the
ground state in Fig. 2, as the present one have no monopoles ($N_{\rm m}=0$).
Thus, this continuous transition is purely magnetic, taking place between the
topologically-trivial phases. We note that all $m_{4\text{--}7}$ show the
criticality of $|T-T_{c}|^{1/2}$ at the transition to the high-$T$ PM state at
$T\simeq 0.433$, indicating that the $4Q$ components are not secondary but
primary order parameters, unlike those in the $3Q$ phase in Fig. 6.

In an applied magnetic field, the system undergoes complex successive
transitions, as shown in Fig. 8. In particular, the $3Q$ phase experiences
multiple changes of $N_{\rm m}$ at low and intermediate temperatures. Similar
to the case of $p=0.4$, $\chi_{\rm sc}$ has nonzero values in all the $3Q$
phases in the magnetic field, as shown in Fig. 8(b). While increasing $h$ or
$T$, we find another $3Q$ phase dubbed $3Q^{\prime}$ in all the field
directions before going to the $1Q$ or $1Q^{\prime}$ phase. Interestingly, in
the $3Q^{\prime}$ phase, $\chi_{\rm sc}$ is zero for ${\bf h}\parallel[100]$
and $[111]$, but nonzero for ${\bf h}\parallel[110]$. This peculiar behavior
is understood as follows. The spin configuration is similar to the one in the
$3Q^{\prime}$ phase found in the ground state in Sec. IV.1.2, which is two-
dimensional due to the coplanar arrangement of the three ordering wave
numbers. In this situation, $\chi_{{\rm sc}}=0$ when the magnetic field is
parallel to the two-dimensional plane, because $\sum_{{\bf
r}_{0}}{\bm{\chi}}_{{\bf r}_{0}}$ in Eq. (19) becomes perpendicular to the
plane. This condition holds for ${\bf h}\parallel[100]$ and ${\bf
h}\parallel[111]$, but not for ${\bf h}\parallel[110]$. For example,
$\\{1,4,7\\}$ is a set of $\eta$ for nonzero $m_{\eta}$ commonly seen in all
the field directions, and its corresponding plane is $(01\bar{1})$ that is
perpendicular to $[100]$ and $[111]$ but not to $[110]$. In addition, we note
that $\chi_{\rm sc}$ is nonzero in the $5Q$ phase under the magnetic field
despite being topologically trivial, but is zero in the $2Q^{\prime}$ phase
appearing at finite $h$ and $T$ as the $2Q$ phase in the case of $p=0.4$.

Figure 10:  Magnetic field dependences of (a) $m_{\eta_{\ell}}$, (b) $C$ and
$dm/dh$, and (c) $N_{\rm m}$ and $\chi_{\rm sc}$. The calculations are done
for ${\bf h}\parallel[110]$ at $T=0.1$ and $p=0.5$; see the magenta dashed
line in the left panel of Fig. 8(b). The insets in (b) are magnified views of
the blue- and orange-boxed areas in the main panel. The data for $\chi_{\rm
sc}$ in (c) are multiplied by a factor of $10^{2}$ for better visibility. The
green and purple arrows in (b) indicate the hidden topological transitions and
the continuous magnetic phase transitions, respectively.

Let us discuss the magnetic and topological transitions in the $3Q$ phase, by
taking the case of $T=0.1$ in ${\bf h}||[110]$ as an example [magenta dashed
line in Fig. 8(b)]. Figure 10 shows the magnetic field dependences of
$m_{\eta}$, $C$, $dm/dh$, $N_{\rm m}$, and $\chi_{\rm sc}$. When $h$ is
applied, $m_{\eta}$ show a complex field dependence; from the anomalies in
$m_{\eta}$, we identify two magnetic phase transitions within the $3Q$ phase
at $h\simeq 0.309$ and $h\simeq 0.495$. These transitions are of second order
and purely magnetic without a change of $N_{\rm m}$; $C$ and $dm/dh$ show a
jump, as shown by the purple arrows in Fig. 10(b). In addition to these
magnetic phase transitions, we find four topological transitions associated
with the changes of $N_{\rm m}$ as $N_{\rm m}=4\to 6\to 4\to 2\to 0$. Similar
to the case of $p=0.4$ in Fig. 7(b), all these transitions are hidden with no
anomalies in $C$ and $dm/dh$, as shown by the green arrows in Fig. 10(b). It
is, however, worth noting that they exhibit clear humps in both $C$ and
$dm/dh$ except for the one at $h\simeq 0.479$, in contrast to the $p=0.4$
case. As plotted in Fig. 10(c), $\chi_{\rm sc}$ shows a sharp rise at the
transition with $N_{\rm m}=4\to 6$. After the increase through the topological
transitions with $N_{\rm m}=6\to 4\to 2$ and showing a peak in the $N_{\rm
m}=2$ phase, $\chi_{\rm sc}$ rapidly decreases at the transition with $N_{\rm
m}=2\to 0$, and goes to zero discontinuously at the magnetic phase transition
to the $1Q^{\prime}$ phase.

####  IV.2.3 $p=0.6$

Figure 11:  Magnetic field–temperature phase diagrams (left) and $h$–$T$ maps
of spin scalar chirality $\chi_{{\rm sc}}$ (right) for (a) ${\bf
h}\parallel[100]$, (b) ${\bf h}\parallel[110]$, and (c) ${\bf
h}\parallel[111]$ at $p=0.6$. The magenta dashed line in the left panel of (b)
indicates the parameter range for Fig. 12. The other notations are common to
those in Fig. 5.  Figure 12:  Magnetic field dependences of (a) $m_{\eta}$,
modulo $\pi/\Lambda$ of the argument of $S^{z}_{{\bf Q}_{\eta}}$ (${\rm
mod}[\arg(S^{z}_{{\bf Q}_{\eta}}),\pi/\Lambda]$), (b) $C$ and $dm/dh$, and (c)
$N_{\rm m}$ and $\chi_{\rm sc}$ for ${\bf h}\parallel[110]$ at $T=0.12$ and
$p=0.6$. The green arrow in (b) indicates the hidden topological transition.
Spin configurations and positions of monopoles and antimonopoles at (d)
$h=0.45$, (e) $h=0.50$, and (f) $h=0.55$. The insets show distributions of
$m_{\eta}$.

Finally, we show the results for $p=0.6$, which is on the $4Q$ side in the
competing region; see Figs. 2 and 4. Figure 11 shows the $h$–$T$ phase
diagrams and $\chi_{{\rm sc}}$. At $h=0$, the ground state is the $4Q$ HL with
$N_{\rm m}=16$, which remains stable until the phase transition to the PM
phase at $T\simeq 0.52$. In an applied magnetic field, the system undergoes a
direct transition from the $4Q$ phase to the PM phase for ${\bf
h}\parallel[100]$, but it shows an additional phase transition to the
$1Q^{\prime}$ phase for ${\bf h}\parallel[110]$ and $[111]$. In addition, the
$2Q^{\prime}$ phase appears at finite $T$ under ${\bf h}\parallel[110]$,
similar to the case of $p=0.5$ in Fig. 8(b). In the $4Q$ phase, $N_{\rm m}$
varies with $T$ and $h$. $\chi_{\rm sc}$ is nonzero in the entire region of
the $4Q$ phase, and becomes large in the $N_{\rm m}=8$ phase at low
temperature. The largest value of $N_{\rm m}=24$ is found at finite
temperature under ${\bf h}\parallel[110]$.

Let us discuss the transitions including the $4Q$ HL with the largest $N_{\rm
m}=24$, by taking the case of $T=0.12$ in ${\bf h}||[110]$ [magenta dashed
line in Fig. 11(b)]. Figure 12 shows the magnetic field dependences of
$m_{\eta}$, $C$, $dm/dh$, $N_{\rm m}$, and $\chi_{\rm sc}$. In this range of
$h$, $N_{\rm m}$ changes from $16$ to $24$ at $h\simeq 0.479$ and from $24$ to
$8$ at $h\simeq 0.517$. Let us first discuss the transition at $h\simeq
0.479$. We find that this is a hidden topological transition with no anomalies
in $m_{\eta}$, $C$, and $dm/dh$, except for a hump in $C$ as indicated by the
green arrow in Fig. 12(b). Figures 12(d) and 12(e) show the configurations of
spins and monopoles in the $N_{\rm m}=16$ and $N_{\rm m}=24$ phases,
respectively. The intriguing aspect of this topological transition is that the
monopoles and antimonopoles are pair created for the increase of $h$ without a
magnetic phase transition. Although there are several examples of pair
annihilation for the increase of the magnetic field [22, 23, 24, 25], this is
the first example of pair creation in the $4Q$ HLs to the best of our
knowledge.

Next let us discuss the transition at $h\simeq 0.517$. This is a first-order
magnetic phase transition with small jumps of $m_{\eta}$. Through the
discontinuous change of the spin configuration, $N_{\rm m}$ also changes from
$24$ to $8$; see Figs. 12(e) and 12(f). Interestingly, we find that the phase
transition is accompanied by changes in the complex phases of the Fourier
components $S^{\mu}_{{\bf Q}_{\eta}}$, as plotted in Fig. 12(a); we here plot
$\mod[\arg(S^{z}_{{\bf Q}_{4\text{--}7}}),\pi/\Lambda]$, where ${\rm
mod}[x,y]\equiv x-y\lfloor x/y\rfloor$, as the complex phase has arbitrariness
of multiples of $\pi/\Lambda$ corresponding to spatial translation. Note that
the importance of such phase degree of freedom in topological spin textures
has also been pointed out in Refs. [33, 48].

##  V Summary

In summary, we have theoretically investigated topological transitions driven
by external magnetic fields and temperature in emergent magnetic monopole
lattices. Motivated by the recent experimental discovery of the monopole
lattices in magnets, we have proposed a spin model that can stabilize both
cubic $3Q$ and tetrahedral $4Q$ HLs and studied the topological nature of the
magnetic and thermodynamic properties. The ground-state phase diagrams (Figs.
2 and 4) and the magnetic field–temperature phase diagrams (Figs. 5, 8, and
11) have been obtained precisely using the recently developed exact steepest
descent method that is crucial to identify the topological transitions in the
thermodynamic limit. Through the comprehensive analyses, we have found a
variety of hidden topological transitions which do not show anomalies in the
macroscopic physical quantities such as the specific heat, the magnetization,
and the net spin scalar chirality, whereas the other magnetic transitions with
topological changes exhibit critical behaviors like in the conventional phase
transitions. Some of the hidden topological transitions show humps in the
magnetic field and temperature dependences of the macroscopic quantities like
crossovers, but the others do not — the latter are hardly visible in
macroscopic measurements. These findings indicate that one needs to be
extremely careful to identify such topological transitions associated with the
emergent magnetic monopoles.

While our model is constructed by simply integrating the models for the $3Q$
and $4Q$ HLs, it reproduces well some aspects of the experimental results for
MnSi_1-$x$Ge_x by regarding $x$ as the mixing ratio $p$ of the magnetic
interactions; the relative stability of the two HLs is sensitive to $p$ but
almost insensitive to the magnetic field $h$ (Figs. 2 and 4). Furthermore, our
model predicts unprecedented $5Q$ states both with and without magnetic
monopoles in the competing region between the $3Q$ and $4Q$ HLs, and a
different type of topological transition associated with pair annihilation of
two-dimensional magnetic vortices and antivortices in the $2Q$ state appearing
only at finite temperature under a magnetic field. For further quantitative
comparison including these new findings, however, experimental studies of the
single crystals are indispensable. Furthermore, it would be important to
elaborate a more sophisticated model by taking into account, e.g., the single-
ion magnetic anisotropy, short-range magnetic interactions, magnetic field
dependences of the coupling constants as well as the lattice structures, and
randomness by chemical substitutions. We hope the present results stimulate
such future experimental and theoretical studies for developing the emergent
electromagnetism.

###### Acknowledgements.

The authors would like to thank Y. Fujishiro, S. Hayami, N. Kanazawa, Yusuke
Kato, A. Miyake, S. Okumura, K. Shimizu, and M. Tokunaga for fruitful
discussions. Y. K. would like to thank K. Inui for his assistance in using the
JAX-based library, Optax. This work was supported by Japan Society for the
Promotion of Science (JSPS) KAKENHI Grant Nos. JP19H05825 and JP22K03509, and
JST CREST Grant No. JPMJCR18T2.

## References

  * Bogdanov and Yablonskii [1989] A. N. Bogdanov and D. A. Yablonskii, Thermodynamically stable “vortices” in magnetically ordered crystals. the mixed state of magnets, Sov. Phys. JETP 68, 101 (1989).
  * Braun [2012] H.-B. Braun, Topological effects in nanomagnetism: from superparamagnetism to chiral quantum solitons, Adv. Phys. 61, 1 (2012).
  * Seidel [2016] J. Seidel, ed.,  _Topological Structures in Ferroic Materials_ (Springer, Cham, 2016).
  * Bogdanov and Panagopoulos [2020] A. N. Bogdanov and C. Panagopoulos, Physical foundations and basic properties of magnetic skyrmions, Nat. Rev. Phys. 2, 492 (2020).
  * Milde  _et al._ [2013] P. Milde, D. Köhler, J. Seidel, L. M. Eng, A. Bauer, A. Chacon, J. Kindervater, S. Mühlbauer, C. Pfleiderer, S. Buhrandt, C. Schütte, and A. Rosch, Unwinding of a Skyrmion Lattice by Magnetic Monopoles, Science 340, 1076 (2013), https://www.science.org/doi/pdf/10.1126/science.1234657 .
  * Volovik [1987] G. E. Volovik, Linear momentum in ferromagnets, J. Phys. Condens. Matter 20, L83 (1987).
  * Kanazawa  _et al._ [2011] N. Kanazawa, Y. Onose, T. Arima, D. Okuyama, K. Ohoyama, S. Wakimoto, K. Kakurai, S. Ishiwata, and Y. Tokura, Large Topological Hall Effect in a Short-Period Helimagnet MnGe, Phys. Rev. Lett. 106, 156603 (2011).
  * Kanazawa  _et al._ [2012] N. Kanazawa, J.-H. Kim, D. S. Inosov, J. S. White, N. Egetenmeyer, J. L. Gavilano, S. Ishiwata, Y. Onose, T. Arima, B. Keimer, and Y. Tokura, Possible skyrmion-lattice ground state in the $B20$ chiral-lattice magnet MnGe as seen via small-angle neutron scattering, Phys. Rev. B 86, 134425 (2012).
  * Tanigaki  _et al._ [2015] T. Tanigaki, K. Shibata, N. Kanazawa, X. Yu, Y. Onose, H. S. Park, D. Shindo, and Y. Tokura, Real-Space Observation of Short-Period Cubic Lattice of Skyrmions in MnGe, Nano Lett. 15, 5438 (2015).
  * Kanazawa  _et al._ [2016] N. Kanazawa, Y. Nii, X.-X. Zhang, A. S. Mishchenko, G. De Filippis, F. Kagawa, Y. Iwasa, N. Nagaosa, and Y. Tokura, Critical phenomena of emergent magnetic monopoles in a chiral magnet, Nat. Commun. 7, 11622 (2016).
  * Kanazawa  _et al._ [2017] N. Kanazawa, S. Seki, and Y. Tokura, Noncentrosymmetric Magnets Hosting Magnetic Skyrmions, Adv. Mater. 29, 1603227 (2017).
  * Tokura  _et al._ [2017] Y. Tokura, M. Kawasaki, and N. Nagaosa, Emergent functions of quantum materials, Nat. Phys. 13, 1056 (2017).
  * Fujishiro  _et al._ [2019] Y. Fujishiro, N. Kanazawa, T. Nakajima, X. Z. Yu, K. Ohishi, Y. Kawamura, K. Kakurai, T. Arima, H. Mitamura, A. Miyake, K. Akiba, M. Tokunaga, A. Matsuo, K. Kindo, T. Koretsune, R. Arita, and Y. Tokura, Topological transitions among skyrmion- and hedgehog-lattice states in cubic chiral magnets, Nat. Commun. 10, 1059 (2019).
  * Kanazawa  _et al._ [2020] N. Kanazawa, A. Kitaori, J. S. White, V. Ukleev, H. M. Rønnow, A. Tsukazaki, M. Ichikawa, M. Kawasaki, and Y. Tokura, Direct Observation of the Statics and Dynamics of Emergent Magnetic Monopoles in a Chiral Magnet, Phys. Rev. Lett. 125, 137202 (2020).
  * Fujishiro  _et al._ [2020] Y. Fujishiro, N. Kanazawa, and Y. Tokura, Engineering skyrmions and emergent monopoles in topological spin crystals, Appl. Phys. Lett. 116, 090501 (2020).
  * Ishiwata  _et al._ [2020] S. Ishiwata, T. Nakajima, J.-H. Kim, D. S. Inosov, N. Kanazawa, J. S. White, J. L. Gavilano, R. Georgii, K. M. Seemann, G. Brandl, P. Manuel, D. D. Khalyavin, S. Seki, Y. Tokunaga, M. Kinoshita, Y. W. Long, Y. Kaneko, Y. Taguchi, T. Arima, B. Keimer, and Y. Tokura, Emergent topological spin structures in the centrosymmetric cubic perovskite ${\mathrm{SrFeO}}_{3}$, Phys. Rev. B 101, 134406 (2020).
  * Hayashi  _et al._ [2021] Y. Hayashi, Y. Okamura, N. Kanazawa, T. Yu, T. Koretsune, R. Arita, A. Tsukazaki, M. Ichikawa, M. Kawasaki, Y. Tokura, and Y. Takahashi, Magneto-optical spectroscopy on Weyl nodes for anomalous and topological Hall effects in chiral MnGe, Nature Communications 12, 5974 (2021).
  * Shiomi  _et al._ [2013] Y. Shiomi, N. Kanazawa, K. Shibata, Y. Onose, and Y. Tokura, Topological nernst effect in a three-dimensional skyrmion-lattice phase, Phys. Rev. B 88, 064409 (2013).
  * Fujishiro  _et al._ [2018] Y. Fujishiro, N. Kanazawa, T. Shimojima, A. Nakamura, K. Ishizaka, T. Koretsune, R. Arita, A. Miyake, H. Mitamura, K. Akiba, M. Tokunaga, J. Shiogai, S. Kimura, S. Awaji, A. Tsukazaki, A. Kikkawa, Y. Taguchi, and Y. Tokura, Large magneto-thermopower in mnge with topological spin texture, Nat. Commun. 9, 408 (2018).
  * Park and Han [2011] J.-H. Park and J. H. Han, Zero-temperature phases for chiral magnets in three dimensions, Phys. Rev. B 83, 184406 (2011).
  * Yang  _et al._ [2016] S.-G. Yang, Y.-H. Liu, and J. H. Han, Formation of a topological monopole lattice and its dynamics in three-dimensional chiral magnets, Phys. Rev. B 94, 054420 (2016).
  * Okumura  _et al._ [2020a] S. Okumura, S. Hayami, Y. Kato, and Y. Motome, Magnetic hedgehog lattices in noncentrosymmetric metals, Phys. Rev. B 101, 144416 (2020a).
  * Okumura  _et al._ [2020b] S. Okumura, S. Hayami, Y. Kato, and Y. Motome, Tracing Monopoles and Anti-monopoles in a Magnetic Hedgehog Lattice, JPS Conf. Proc. 30, 011010 (2020b).
  * Shimizu  _et al._ [2021a] K. Shimizu, S. Okumura, Y. Kato, and Y. Motome, Phase transitions between helices, vortices, and hedgehogs driven by spatial anisotropy in chiral magnets, Phys. Rev. B 103, 054427 (2021a).
  * Okumura  _et al._ [2022] S. Okumura, S. Hayami, Y. Kato, and Y. Motome, Magnetic Hedgehog Lattice in a Centrosymmetric Cubic Metal, J. Phys. Soc. Jpn. 91, 093702 (2022).
  * Kato  _et al._ [2021] Y. Kato, S. Hayami, and Y. Motome, Spin excitation spectra in helimagnetic states: Proper-screw, cycloid, vortex-crystal, and hedgehog lattices, Phys. Rev. B 104, 224405 (2021).
  * Kato and Motome [2022] Y. Kato and Y. Motome, Magnetic field–temperature phase diagrams for multiple-$Q$ magnetic ordering: Exact steepest descent approach to long-range interacting spin systems, Phys. Rev. B 105, 174413 (2022).
  * Hayami  _et al._ [2017] S. Hayami, R. Ozawa, and Y. Motome, Effective bilinear-biquadratic model for noncoplanar ordering in itinerant magnets, Phys. Rev. B 95, 224424 (2017).
  * Hayami and Motome [2018] S. Hayami and Y. Motome, Néel- and Bloch-Type Magnetic Vortices in Rashba Metals, Phys. Rev. Lett. 121, 137202 (2018).
  * Yasui  _et al._ [2020] Y. Yasui, C. J. Butler, N. D. Khanh, S. Hayami, T. Nomoto, T. Hanaguri, Y. Motome, R. Arita, T.-h. Arima, Y. Tokura, and S. Seki, Imaging the coupling between itinerant electrons and localised moments in the centrosymmetric skyrmion magnet GdRu2Si2, Nat. Commun. 11, 5925 (2020).
  * Yambe and Hayami [2021] R. Yambe and S. Hayami, Skyrmion crystals in centrosymmetric itinerant magnets without horizontal mirror plane, Sci. Rep. 11, 11184 (2021).
  * Hirschberger  _et al._ [2021] M. Hirschberger, S. Hayami, and Y. Tokura, Nanometric skyrmion lattice from anisotropic exchange interactions in a centrosymmetric host, New J. Phys. 23, 023039 (2021).
  * Hayami  _et al._ [2021] S. Hayami, T. Okubo, and Y. Motome, Phase shift in skyrmion crystals, Nat. Commun. 12, 6927 (2021).
  * Hayami and Motome [2021a] S. Hayami and Y. Motome, Square skyrmion crystal in centrosymmetric itinerant magnets, Phys. Rev. B 103, 024439 (2021a).
  * Hayami and Motome [2021b] S. Hayami and Y. Motome, Topological spin crystals by itinerant frustration, J. Phys.: Condens. Matter 33, 443001 (2021b).
  * Hayami and Yambe [2021] S. Hayami and R. Yambe, Field-Direction Sensitive Skyrmion Crystals in Cubic Chiral Systems: Implication to 4$f$-Electron Compound EuPtSi, J. Phys. Soc. Jpn. 90, 073705 (2021).
  * Shimizu  _et al._ [2021b] K. Shimizu, S. Okumura, Y. Kato, and Y. Motome, Spin moiré engineering of topological magnetism and emergent electromagnetic fields, Phys. Rev. B 103, 184421 (2021b).
  * Khanh  _et al._ [2022] N. D. Khanh, T. Nakajima, S. Hayami, S. Gao, Y. Yamasaki, H. Sagayama, H. Nakao, R. Takagi, Y. Motome, Y. Tokura, T.-h. Arima, and S. Seki, Zoology of Multiple-$Q$ Spin Textures in a Centrosymmetric Tetragonal Magnet with Itinerant Electrons, Adv. Sci. 2022, 2105452 (2022).
  * Ruderman and Kittel [1954] M. A. Ruderman and C. Kittel, Indirect exchange coupling of nuclear magnetic moments by conduction electrons, Phys. Rev. 96, 99 (1954).
  * Kasuya [1956] T. Kasuya, A theory of metallic ferro- and antiferromagnetism on Zener’s model, Prog. Theor. Phys. 16, 45 (1956).
  * Yosida [1957] K. Yosida, Magnetic Properties of Cu-Mn Alloys, Phys. Rev. 106, 893 (1957).
  * Dzyaloshinsky [1958] I. Dzyaloshinsky, A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics, J. Phys. Chem. Solids 4, 241 (1958).
  * Moriya [1960] T. Moriya, Anisotropic Superexchange Interaction and Weak Ferromagnetism, Phys. Rev. 120, 91 (1960).
  * Peason [1905] K. Peason, The Problem of the Random Walk, Nature 72, 294 (1905).
  * Kiefer and Weiss [1984] J. E. Kiefer and G. H. Weiss, The Pearson random walk, AIP Conf. Proc. 109, 11 (1984).
  * Bradbury  _et al._ [2018] J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX: composable transformations of Python+NumPy programs (2018).
  * Hessel  _et al._ [2020] M. Hessel, D. Budden, F. Viola, M. Rosca, E. Sezener, and T. Hennigan, Optax: composable gradient transformation and optimisation, in JAX! (2020).
  * Shimizu  _et al._ [2022] K. Shimizu, S. Okumura, Y. Kato, and Y. Motome, Phase degree of freedom and topology in multiple-$Q$ spin textures, Phys. Rev. B 105, 224405 (2022).