# Monoaxial Dzyaloshinskii–Moriya interaction-induced topological Hall effect
in a new chiral-lattice magnet GdPt2B

Yoshiki J. Sato1<EMAIL_ADDRESS>Hikari Manako1 Ryuji Okazaki1 Yukio
Yasui2 Ai Nakamura3 Dai Aoki3 1Department of Physics and Astronomy, Faculty of
Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510,
Japan
2School of Science and Technology, Meiji University, Kawasaki, Kanagawa
214-8571, Japan
3Institute for Materials Research, Tohoku University, Oarai, Ibaraki 311-1313,
Japan

###### Abstract

We investigate the topological Hall effect (THE) in the monoaxial chiral
crystal GdPt2B, a recently discovered compound that exhibits putative
helimagnetism below 87 K. The distinct THE was observed in GdPt2B in the
magnetically ordered state. The scaling relations for anomalous and
topological Hall conductivities differed from those of conventional models
based on the scattering process. We further demonstrate the clear scaling
behavior of the THE in a wide temperature range, which we attribute to the
monoaxial Dzyaloshinskii–Moriya (DM) interaction under external magnetic
fields perpendicular to the screw axis. The THE induced by the monoaxial DM
interaction as well as the THE in a monoaxial chiral crystal of $f$-electron
system are demonstrated in this study.

††preprint: APS/123-QED

##  I Introduction

Chirality is an important concept across many natural sciences. In physics,
the absence of mirror symmetry in matter plays a key role in chirality-induced
phenomena Rikken and Raupach (1997); Göhler  _et al._ (2011); Shiota  _et al._
(2021). In particular, an asymmetric exchange interaction, namely the
Dzyaloshinskii–Moriya (DM) interaction Dzyaloshinsky (1958); Moriya (1960),
arising from relativistic spin–orbit coupling (SOC), stabilizes chiral
helimagnetism in magnetic materials with chiral crystal structures. The
competition between an external magnetic field and magnetic interactions,
including the DM interaction, induces characteristic spin textures, resulting
in exotic spin-charge coupled phenomena such as the topological Hall effect
(THE) Neubauer  _et al._ (2009); Kanazawa  _et al._ (2011); Nagaosa and Tokura
(2013) and electrical magnetochiral effect Yokouchi  _et al._ (2017); Aoki
_et al._ (2019).

In many studies on the magnetic and transport properties of chiral magnetic
materials, B20-type cubic chiral compounds have been intensively investigated
as archetypal chiral magnets Mühlbauer  _et al._ (2009); Neubauer  _et al._
(2009); Yu  _et al._ (2010); Kanazawa  _et al._ (2011); Nagaosa and Tokura
(2013); Yokouchi  _et al._ (2017). They crystallize into the cubic space group
$P2_{1}3$ with four three-fold rotational axes. This peculiarity of multiple
helical axes causes a complex interplay between the external magnetic field
and the DM interaction, resulting in characteristic spin textures Mühlbauer
_et al._ (2009); Yu  _et al._ (2010); Nagaosa and Tokura (2013); Yokouchi  _et
al._ (2017) and complex responses to the direction of the magnetic field
Münzer  _et al._ (2010); Nagaosa and Tokura (2013); Takeuchi  _et al._ (2020).
In contrast to the cubic system, hexagonal, tetragonal, and trigonal chiral
crystals have one principal helical axis. They were classified as monoaxial
chiral crystals.

Although a relatively large number of cubic chiral helimagnets have been
reported Tokura and Kanazawa (2021), there are few known systems of monoaxial
helimagnets with chiral crystal structures, such as intercalated transition
metal dichalcogenides (TMDs) Miyadai  _et al._ (1983); Karna  _et al._ (2019);
Zhang  _et al._ (2021), CsCuCl3 Kousaka  _et al._ (2017), and YbNi3Al9
Miyazaki  _et al._ (2012). In the monoaxial chiral crystals, the SOC causes
“monoaxial” DM interaction along the screw axis Kishine and Ovchinnikov
(2015). The TMD magnet CrNb3S6 (space group $P6_{3}22$) is an intensively
investigated monoaxial helimagnet that exhibits a chiral soliton lattice (CSL)
phase under a magnetic field perpendicular to the monoaxial DM interaction
Togawa  _et al._ (2012). The CSL exhibits strong coupling with conduction
electrons and induces nontrivial magnetotransport Togawa  _et al._ (2013,
2016). In addition, recent studies have shown a topological and planar Hall
effect in the tilted CSL of a TMD Mayoh  _et al._ (2022) and transformations
from a CSL to magnetic skyrmions in monoaxial TMD crystals with confined
geometries Li  _et al._ (2023). However, the question of whether the monoaxial
DM interaction directly causes spin-charge coupled phenomena is an important
issue. We address this issue by exploring a novel monoaxial chiral crystal
system.

In this paper, we report the emergence and robust scaling behavior of the THE
in a monoaxial chiral crystal GdPt2B, under a magnetic field perpendicular to
the screw axis. In a series of CePt2B type chiral materials Sologub  _et al._
(2000, 2003); Lackner  _et al._ (2005); Dias  _et al._ (2007); Khan  _et al._
(2010, 2015); Sato  _et al._ (2021), the $f$-electron based magnet GdPt2B is a
recently discovered compound, which crystallizes in a hexagonal chiral space
group $P6_{4}22$ Sato  _et al._ (2022). The hexagonal crystal structure and
schematic magnetic ($H$-$T$) phase diagram of GdPt2B are shown in Figs. 1(a)
and 1(b), respectively. GdPt2B is a metallic compound and exhibits putative
chiral helimagnetism below $T_{\rm O}$ = 87 K. The electrical resistivity
$\rho_{xx}$, specific heat $C_{\rm p}$, and magnetization $M$ exhibit a
distinct phase transition at $T_{\rm O}$, as shown in Fig. 1(c). In addition,
$M(T)$ shows a weak anomaly associated with spin reorientation at
$T_{\chi_{\rm max}}\sim$ 50 K. The Curie–Weiss analysis gives a positive Weiss
temperature $\theta_{\rm W}\sim 100$ K, indicating ferromagnetic exchange
interaction between the Gd-ions.

Figure 1:  (a) Crystal structure of GdPt2B. (b) Schematic magnetic phase
diagram consisting of ordered state, paramagnetic (PM), and field-polarized
ferromagnetic (FM) phases. (c) Temperature dependence of magnetization $M(T)$,
specific heat ($C_{\rm p}$), and electrical resistivity ($\rho$). $\rho(T)$
and $C_{\rm p}(T)$ were measured in zero field, and $M(T)$ was measured in a
magnetic field of $\mu_{0}H$ = 0.05 T. $C_{\rm p}(T)$ and $M(T)$ data were
taken from Ref.Sato  _et al._ (2022).

A distinct THE was observed in bulk single crystals of GdPt2B subjected to an
external magnetic field perpendicular to the screw axis (i.e., $H$ was
perpendicular to the direction of the monoaxial DM interaction). This
phenomenon, arising from the competition between external magnetic fields and
monoaxial DM interaction, has rarely been reported. We found the scaling
behavior of the THE based on a 1-D Hamiltonian for monoaxial crystals. To the
best of our knowledge, this is the first study to demonstrate the emergence of
the THE induced by the monoaxial DM interaction as well as the THE in
$f$-electron based monoaxial chiral crystals.

##  II Experimental details

Single crystals of GdPt2B were grown using the Czochralski method in a tetra
arc furnace. The single crystals of GdPt2B were oriented using a Laue camera
(Photonic Science Laue X-ray CCD camera). Hall measurements were performed
using bulk single crystals cut into thin plates with dimensions of 1.5 mm
$\times$ 1.5 mm $\times$ 0.1 mm. The longitudinal and Hall resistances were
simultaneously measured using a lock-in amplifier (SR-830) in a physical
property measurement system (PPMS). Magnetization measurements were performed
using a superconducting quantum interference device magnetometer (MPMS).

##  III Results and Discussion

The observed Hall resistivity $\rho_{yx}^{\rm obs}$ of GdPt2B for the magnetic
field $H$ $||$ $[10\bar{1}0]$ and electrical current $J$ $||$ ${[0001]}$ at 50
K is shown in Fig. 2(a). Usually, the Hall resistivity is expressed as
follows:

$\rho_{yx}=\rho_{yx}^{\rm O}+\rho_{yx}^{\rm A}+\rho_{yx}^{\rm T}.$ (1)

The first term $\rho_{yx}^{\rm O}=R_{0}B$ is the ordinary Hall effect (OHE)
due to the Lorentz force, and $R_{0}$ is the ordinary Hall coefficient. The
second term $\rho_{yx}^{\rm A}$ is the anomalous Hall effect (AHE) derived
from the magnetization of the sample. The third term $\rho_{yx}^{\rm T}$ is
the topological Hall resistivity induced by topological spin textures. In a
high field region, the linear field dependence of $\rho_{yx}^{\rm obs}$ of
GdPt2B can be interpreted as the ordinary Hall effect $\rho_{yx}^{\rm O}$.

Figure 2:  (a) Field dependence of anomalous Hall resistivity $\rho_{yx}^{\rm
A}$ after subtraction of ordinary Hall resistivity $\rho_{yx}^{\rm O}$ from
observed Hall resistivity $\rho_{yx}^{\rm obs}$ at 50 K. (b) $\rho_{yx}^{\rm
obs}-\rho_{yx}^{\rm O}$ at several constant temperatures. (c) Topological Hall
resistivity $\rho_{yx}^{\rm T}(H)$ after subtraction of anomalous Hall
component derived from the magnetization ($\rho_{yx}^{\rm A}$ = $S_{\rm
H}\rho_{xx}^{2}M$). (d) Extracted $-\rho_{yx}^{\rm T}(H)$ at several constant
temperatures.

The field dependence of the Hall component after removing the ordinary Hall
effect, namely $\rho_{yx}^{\rm obs}-\rho_{yx}^{\rm O}$, is shown in Figs. 2(a)
and 2(b). A clear hump-like anomaly was observed. To extract the topological
Hall component, we estimated the anomalous Hall effect corresponding to the
magnetization of the sample. Generally, the anomalous Hall resistivity is
described as Nagaosa  _et al._ (2010); Porter  _et al._ (2014):

$\rho_{yx}^{\rm A}=\alpha\rho_{xx}(H)M(H)+S_{\rm H}\rho^{2}_{xx}(H)M(H),$ (2)

where $\alpha$ and $S_{\rm H}$ are material specific constants. The first and
second terms correspond to skew scattering and intrinsic (Berry phase)
contributions, respectively. As shown in Fig. 2(c), the field dependence of
$\rho_{yx}^{\rm A}$ can be reasonably explained by the intrinsic contribution
$S_{\rm H}\rho^{2}_{xx}(H)M(H)$ in the field-polarized FM state. Here, the
$M(H)$ data were corrected using the demagnetizing factor $D_{z}$ $\sim$ 0.8
for precise analysis Aharoni (1998) (see Supplemental Material (SM) for
details SM ). The fact that $\rho_{yx}^{\rm A}$ can be well explained by
assuming an intrinsic contribution above $T_{\rm O}$ indicates the validity of
the current analysis (see SM SM ).

The extracted topological Hall resistivity, namely $\rho_{yx}^{\rm
T}(H)=\rho_{yx}^{\rm obs}(H)-\rho_{yx}^{\rm O}(H)-\rho_{yx}^{\rm A}(H)$, is
shown in Figs. 2(c) and 2(d). The hump-like field dependence of
$\rho_{yx}^{\rm T}$ resembles the Hall effect induced by topological spin
textures such as skyrmion lattices in B20 chiral helimagnets Neubauer  _et
al._ (2009); Kanazawa  _et al._ (2011); Kakihana  _et al._ (2018). The
topological Hall resistivity of GdPt2B is negative and reaches a maximum value
of $|\rho_{yx}^{\rm T}|=0.14$ ${\rm\mu\Omega\,cm}$ at $T$ = 45 K. The order of
magnitude of $\rho_{yx}^{\rm T}$ is comparable to the reported values of
topological Hall resistivity of MnGe ($\sim$ 0.16 ${\rm\mu\Omega\,cm}$
Kanazawa  _et al._ (2011)) and EuPtSi ($\sim$ 0.12 ${\rm\mu\Omega\,cm}$
Kakihana  _et al._ (2018)). As shown in Fig. 2(d), the hump-like anomaly in
$\rho_{yx}^{\rm T}$ vanishes above the transition temperature $T_{\rm O}$,
suggesting that the magnetically ordered state and the spin chirality are
closely related to the emergence of the THE in GdPt2B.

Here, we discuss the scaling relation for AHE between $\sigma_{yx}^{\rm A}$
and $\sigma_{xx}$ arising from the scattering mechanism. Figure 3(a) shows a
log-log plot of $|\sigma_{yx}^{\rm A,s}|$ versus $\sigma_{xx}$.
$\sigma_{yx}^{\rm A,s}$ is defined as ($-\rho_{yx}^{\rm
A,s}$)/[$(\rho_{yx}(H_{\rm s})^{2}+\rho_{xx}(H_{\rm s})^{2}$], where
$\rho_{yx}^{\rm A,s}$ is the saturation value of $\rho_{yx}^{\rm
obs}-\rho_{yx}^{\rm O}$ in a field of $H_{\rm s}$ [see Fig. 2(b)].
$\sigma_{xx}$ is calculated as [$\rho_{xx}(H_{\rm s})$]/[$\rho_{yx}(H_{\rm
s})^{2}+\rho_{xx}(H_{\rm s})^{2}$]. In a "bad metal" regime with a small
longitudinal conductivity, the scaling law $\sigma_{yx}$ $\propto$
$\sigma_{xx}^{1.6}$ has been demonstrated in several materials Kee  _et al._
(2004); Miyasato  _et al._ (2007); Onoda  _et al._ (2008); Takahashi  _et al._
(2018). In contrast, $\sigma_{yx}$ becomes nearly constant in the intermediate
regime ($\sigma_{xx}$ $\sim$ 104 to 106 ${\rm\Omega^{-1}cm^{-1}}$ Miyasato
_et al._ (2007); Onoda  _et al._ (2008)). The data for GdPt2B and several
other compounds are plotted in Fig. 3(a). The scaling plot suggests that
GdPt2B is in the intermediate regime, but the scaling behavior of the AHE in
GdPt2B exhibits a peculiar temperature dependence.

Figure 3: (a) Scaling relations between the Hall conductivity
$|\sigma_{yx}^{\rm A,s}|$ and the longitudinal conductivity $\sigma_{xx}$ for
GdPt2B and several compounds Suzuki  _et al._ (2016); Takahashi  _et al._
(2018); Liu  _et al._ (2018). (b) Temperature dependence of coefficients
$R_{0}$, $-S_{\rm H}$, and maximum value of $|\rho_{yx}^{\rm T}|$.

At the lowest temperature (2 K), $|\sigma_{yx}^{\rm A,s}|$ is approximately 12
${\rm\Omega^{-1}cm^{-1}}$ with a saturation magnetic moment $M_{\rm sat}$ of
7.2 ${\rm\mu_{\rm B}}$/Gd. As the temperature increases, $|\sigma_{yx}^{\rm
A,s}|$ shows a minimum value of $|\sigma_{yx}^{\rm A,s}|$ = 0.2
${\rm\Omega^{-1}cm^{-1}}$ ($M_{\rm sat}$ = 6.5 ${\rm\mu_{\rm B}}$/Gd) at 30 K
and reaches a nearly constant value of $|\sigma_{yx}^{\rm A,s}|\sim$25
${\rm\Omega^{-1}cm^{-1}}$ ($M_{\rm sat}$ = 4.3 ${\rm\mu_{\rm B}}$/Gd) above 60
K. $M_{\rm sat}$ and magnetoresistance vary systematically in the high field
region. Therefore, the temperature dependence of the AHE in GdPt2B cannot be
explained by conventional AHE scenarios. The breakdown of the scaling relation
of AHE is an interesting phenomenon. The mechanisms such as electron-phonon
scatteringXiao  _et al._ (2019) and Kondo coherence in an $f$-electron
systemSiddiquee  _et al._ (2023) have recently been discussed as the origin of
the breakdown of the scaling relation. The mechanism underlying the peculiar
scaling behavior of AHE in GdPt2B remains unclear.

Figure 3(b) shows the temperature dependence of the coefficients $R_{0}$ and
$-S_{\rm H}$, together with the maximum value of topological Hall resistivity
$|\rho_{yx}^{\rm T}|_{\rm max}$. Whereas, the THE of GdPt2B is rapidly
suppressed in the PM and field-polarized FM states, and the OHE and AHE
exhibit a smooth temperature dependence near $T_{\rm O}$. $R_{0}$ was negative
at all measured temperatures, indicating that the electron is a dominant
carrier. The charge carrier density $n$ is estimated to be $n$ =
5.74$\times$1027 m-3 and $n$ = 4.20$\times$1027 m-3 at 80 and 2 K,
respectively. $R_{0}$ takes its maximum value near $T_{\rm O}$. The
coefficient $-S_{\rm H}$ exhibits a weak temperature dependence near $T_{\rm
O}$ and reaches its maximum value at $T$ $=$ 110 K. $|\rho_{yx}^{\rm T}|_{\rm
max}$ peaks near $T_{{\chi}_{\rm max}}$ $\sim$ 50 K.

In order to elucidate the origin of THE in GdPt2B, we examined the field
dependence of topological Hall conductivity $\sigma_{yx}^{\rm T}$ for $H$ ||
$[10\bar{1}0]$ at several constant temperatures, as shown in Fig. 4(a).
$\sigma_{yx}^{\rm T}$ is calculated as [$-\rho_{yx}^{\rm
T}(H)$]/[$\rho_{yx}(H)^{2}+\rho_{xx}(H)^{2}$]. The maximum value of
$\sigma_{yx}^{\rm T}$ reaches 21 ${\rm\Omega^{-1}cm^{-1}}$ at 2 K and
decreases with increasing temperature.

The THE exhibits different scattering time $\tau$ dependence depending on its
mechanism, in conventional scenariosOnoda  _et al._ (2004); Kanazawa  _et al._
(2011); Shiomi  _et al._ (2012); Verma  _et al._ (2022). In the case of the
intrinsic mechanism due to the momentum-space Berry curvature, the anomalous
Hall component is independent of $\tau$ Miyasato  _et al._ (2007); Onoda  _et
al._ (2008). In contrast, $\sigma^{\rm T}$ is expected to show $\tau^{2}$
dependence in the case of the THE induced by topological spin textures such as
magnetic skyrmions because a fictitious magnetic flux in real space is
proportional to the number of topological spin textures in a given area Onoda
_et al._ (2004); Bruno  _et al._ (2004); Verma  _et al._ (2022). We plot
extreme values of $\sigma_{\rm yx}^{\rm T}$ as a function of $\sigma_{xx}$ in
Fig. 4(b), where $\sigma_{xx}$ is proportional to $\tau$ under the assumption
that $n$ and the effective mass of the carrier are constant. The $\tau$
dependence of the topological Hall conductivity of GdPt2B differs from those
of conventional models based on momentum-space and real-space Berry
curvatures. From the power-law fitting, we found that the extreme value of
$\sigma_{\rm yx}^{\rm T}$ is proportional to $1/\sqrt{\tau}$ in GdPt2B. A
previous theoretical study explored the $\tau$ dependence of the length of the
topological spin texture $L_{\rm s}^{\ast}$, which correspond to the length
where $\sigma_{yx}^{\rm T}$ reaches its extreme value Matsui  _et al._ (2021).
For short-pitch skyrmion lattices, it was verified that $L_{\rm s}^{\ast}$ is
proportional to $\sqrt{\tau}$ instead of the conventional $\tau$ dependence
because of the band separation in the magnetic Brillouin zone Matsui  _et al._
(2021). The nontrivial $1/\sqrt{\tau}$ dependence of the THE in GdPt2B may be
attributed to the specific period and configuration of the spin texture.

The most salient feature of the THE in GdPt2B is that its mechanism is closely
related to the monoaxial DM interaction. To discuss the connection between the
THE and monoaxial DM interaction, we first define $H^{\rm T}_{\rm max}$ as the
magnetic field corresponding to the maximum value of $\sigma_{yx}^{\rm T}$.
Figure 4(c) shows the critical field of the ordered state $H_{\rm c}$ as a
function of $H^{\rm T}_{\rm max}$, exhibiting a linear relationship. Thus, the
ratio of $H_{\rm c}$ to $H^{\rm T}_{\rm max}$ remained constant, as shown in
Fig. 4(d). Remarkably, the relationship between $H_{\rm c}$ and $H^{\rm
T}_{\rm max}$ holds for the entire temperature range below $T_{\rm O}$.

To discuss the origin of the constant ratio $H_{\rm c}$/$H^{\rm T}_{\rm max}$,
we consider a simple 1-D Hamiltonian Kishine and Ovchinnikov (2015) for
monoaxial crystal systems as follows:

$\mathcal{H}=-J\sum_{i}\textbf{{S}}_{i}\cdot\textbf{{S}}_{i+1}-\textbf{{D}}\cdot\sum_{i}\textbf{{S}}_{i}\times\textbf{{S}}_{i+1}+{\textbf{{H}}}\cdot\sum_{i}\textbf{{S}}_{i},$
(3)

where $\textbf{{S}}_{i}$ ($|\textbf{{S}}_{i}|=S$) is the local moment at site
$i$, $J$ is the ferromagnetic exchange interaction between two adjacent spins,
and D ($|\textbf{{D}}|=D$) is the monoaxial DM interaction along the screw
axis. The third term represents the effect of the external magnetic field
perpendicular to the screw axis. Based on the above 1-D Hamiltonian, Kishine
and Ovchinnikov derived $H_{\rm c}$ as follows Kishine and Ovchinnikov (2015):

$H_{\rm c}=\left(\frac{\pi}{4}\right)^{2}S\frac{D^{2}}{J}.$ (4)

Equation (4) indicates that the $H_{\rm c}$ is a measure of the ratio of
monoaxial DM interaction $D$ and the ferromagnetic exchange interaction $J$.
Therefore, the constant ratio of $H_{\rm c}$/$H^{\rm T}_{\rm max}$ suggests
that the THE of GdPt2B attains its maximum values at a specific ratio of
$D^{2}/J$ and the external magnetic field perpendicular to the screw axis. The
observed THE could be scaled by the critical field $H_{\rm c}$. Figure 4(e)
shows the normalized topological Hall conductivity $\sigma_{yx}^{\rm
T}$/$\sigma_{yx,{\rm extrm}}^{\rm T}$ as a function of the scaled external
magnetic field $H$/$H_{\rm c}$. The observed robust scaling behavior indicates
that the monoaxial DM interaction plays a significant role in the emergence of
THE in the monoaxial chiral crystal GdPt2B.

To summarize the results of the THE in GdPt2B, we show a contour plot of the
topological Hall conductivity in the $H$-$T$ phase diagram, as shown in Fig.
4(f). An important feature is that the THE of GdPt2B is stable at the lowest
temperature (2 K), without the influence of thermal agitations as observed in
the DM-mediated skyrmions in chiral magnets. A distinct THE is observed only
within the ordered state, and the extreme value of $\sigma_{\rm yx}^{\rm T}$
depends on the scattering time. The results indicate that the spin texture in
the ordered state is related to the emergence of the THE. In addition, $H_{\rm
c}$ and $H^{\rm T}_{\rm max}$ show a constant ratio below $T_{\rm O}$,
suggesting the importance of the monoaxial DM interaction in the THE. It is
noteworthy that the scalar spin chirality
[$\textbf{{S}}_{i}\cdot(\textbf{{S}}_{j}\times\textbf{{S}}_{k})$] vanishes in
coplanar spin configurations such as the simple helimagnetism and CSL. Recent
studies have discussed novel mechanisms such as the noncolinear Hall effect in
coplanar spin configurations Bouaziz  _et al._ (2021). Magnetic structure
analysis is a desirable next step in determining the microscopic origin of the
monoaxial DM interaction-induced THE in GdPt2B.

Figure 4: (a) Topological Hall conductivity $\sigma_{yx}^{\rm T}$ as a
function of magnetic field. (b) Scaling plot of extreme value of
$\sigma_{yx}^{T}(H,T)$ as a function of $\sigma_{xx}(H,T)$. (c) The critical
field $H_{\rm c}$ as a function of the magnetic field $H^{\rm T}_{\rm max}$,
which corresponds to $\sigma_{yx,{\rm extrm}}^{\rm T}$. (d) The ratio of
$H_{\rm c}$ to $H^{\rm T}_{\rm max}$. (e) Scaled topological Hall conductivity
$\sigma_{yx}^{\rm T}$/$\sigma_{yx,{\rm extrm}}^{\rm T}$ as a function of the
scaled external magnetic field $H$/$H_{\rm c}$. (f) Contour plot of
$\sigma_{yx}^{\rm T}$ in the magnetic phase diagram of GdPt2B for $H$ ||
$[10\bar{1}0]$.

Finally, we compare the THE in GdPt2B with those in other material systems. In
Gd-based centrosymmetric magnets such as Gd2PdSi3 Saha  _et al._ (1999);
Kurumaji  _et al._ (2019) and Gd3Ru4Al12 Hirschberger  _et al._ (2019), a
large THE induced by the magnetic frustration in a hexagonal crystal has been
reported. The Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction produces
modulated spin textures including the skyrmion lattice with three spiral spin
modulations, that is, the triple-q state. Owing to the complex magnetic
interactions, the THE in Gd-based frustrated magnets is observed only in the
intermediate magnetic phase, which is different from the present case for
GdPt2B. The dominant exchange interaction is considered to be of the RKKY type
in GdPt2B. The comparison of the THE in Gd-based monoaxial chiral crystal with
those in centrosymmetric Gd-based magnets is important for a better
understanding of the role of the DM interaction and mechanisms of THE. The
present result is the first observation of the THE in a monoaxial chiral
crystal of $f$-electron system, to the best of our knowledge. Another
interesting example is large THE Mayoh  _et al._ (2022) and AHE Ghimire  _et
al._ (2018) in the intercalated TMDs under the external magnetic fields along
the helical axis. In the monoaxial chiral helimagnet CrNb3S6, a large THE
induced by a tilted CSL state has been reported for $H$ $||$ $c$ (helical
axis). In general, the magnetic field along the helical axis leads to a simple
conical structure; however, the monoaxial DM interaction may be effective for
$H$ $||$ helical axis. The anisotropy of the THE in monoaxial chiral crystals
needs to be clarified in the future.

##  IV Conclusions

In conclusion, the large and distinct THE was observed in the monoaxial chiral
crystal GdPt2B. The characteristic $\tau$ dependence of the extreme value of
$\sigma_{\rm yx}^{\rm T}$ and the constant ratio of $H_{\rm c}$/$H^{\rm
T}_{\rm max}$ lead to the clear scaling behavior of the topological Hall
conductivity, indicating that the THE in GdPt2B was induced by the monoaxial
DM interaction under external magnetic fields. The THE induced by the
monoaxial DM interaction and THE in a monoaxial chiral crystal of $f$-electron
system were demonstrated in this study. Our findings extend the range of
material system in which the THE can be observed, and demonstrate the
universal nature of the spin-charge coupled phenomena in topological
materials.

###### Acknowledgements.

We would like to thank S. Ohara, S. Nakamura, and Y. Yamane for discussions.
This work was supported by JSPS KAKENHI (JP22K20360). We acknowledge all
support from the International Research Center for Nuclear Materials Science
at Oarai (IMR, Tohoku University).

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# Turbulence in a magneto-optical trap

Adam Griffin  Institut de Physique de Nice, Université Côte d’Azur, CNRS,
Nice, France  Marius Gaudesius  Institut de Physique de Nice, Université Côte
d’Azur, CNRS, Nice, France  Robin Kaiser  Institut de Physique de Nice,
Université Côte d’Azur, CNRS, Nice, France  Sergey Nazarenko
<EMAIL_ADDRESS>Institut de Physique de Nice, Université
Côte d’Azur, CNRS, Nice, France  Guillaume Labeyrie  Institut de Physique de
Nice, Université Côte d’Azur, CNRS, Nice, France

# Turbulence in a magneto-optical trap

Adam Griffin  Institut de Physique de Nice, Université Côte d’Azur, CNRS,
Nice, France  Marius Gaudesius  Institut de Physique de Nice, Université Côte
d’Azur, CNRS, Nice, France  Robin Kaiser  Institut de Physique de Nice,
Université Côte d’Azur, CNRS, Nice, France  Sergey Nazarenko
<EMAIL_ADDRESS>Institut de Physique de Nice, Université
Côte d’Azur, CNRS, Nice, France  Guillaume Labeyrie  Institut de Physique de
Nice, Université Côte d’Azur, CNRS, Nice, France

###### Abstract

A large number of 87Rb atoms (up to $1.5\times 10^{11}$) is confined and
cooled to $\sim 200~{}\mu$K in a magneto-optical trap. The resulting cloud of
atoms exhibits spatio-temporal instabilities leading to chaotic behaviour
resembling a turbulent flow of fluid. We apply the methods of the turbulence
theory based on the structure functions analysis to classify and quantify the
different degrees of excitation of turbulence, including its scaling and
morphological properties in the moving cloud images.

Fluid properties of the nonlinear light systems and the Bose-Einstein
condensates (BEC) have been noticed and scrutinised over the past two or three
decades. These studies have been particularly well advanced for systems
described by the Gross-Pitaevskii (nonlinear Schödinger) equation for which
the Madelung transformation (the amplitude-phase decomposition of the complex
field) allows to bring this equation to the fluid mass and momentum
conservation form. Among these studies, works devoted to turbulence in optical
and BEC systems Bagnato ; Anderson  are most interesting and important.

However, there are also optical systems, which clearly exhibit turbulent
behaviour, but their dynamical equations are not fully known or too
complicated to be analysed directly mathematically or even numerically. Yet,
as we will show in this Letter using the example of a magneto-optical trap
(MOT) experiment, a lot can be said about such systems using the standard
characterizations from the turbulence theory.

In this Letter, we will analyze the data obtained in an unstable MOT
experiment described further below in the text. Unstable MOTs have been
studied in various groups Wilkowski2000 ; diStefano2003 ; Labeyrie2006 ;
Kim2006 ; Romain2016 ; Gaudesius2020 ; Gaudesius2021 ; Giampaoli2021  and with
different approaches, ranging from atomic physics Pohl2006 , non linear
dynamics Stefano2004  to plasma physics Mendonca2008  and astrophysics
Mendonca2012 . The goal is to study the experimental images using methods from
the field of turbulence. These images at hand visually differ from each other
in being more or less turbulent, as well as approximately statistically
homogeneous and/or isotropic. We stress that images that are visually non-
turbulent may, in reality, correspond to small-scale turbulent fluctuations
masked by presence of a strong mean large-scale background. Thus, a
quantitative tool is needed to be able to distinguish between various
turbulent, laminar and transitional states.

In turbulence one distinguishes the scale-invariant “mean field” behaviour,
usually associated with the Kolmogorov-Obukhov-Corrsin (KOC) theory Kolmogorov
; Obukhov ; Corrsin , and intermittency that is seen in deviations from the
KOC scaling in high-order structure functions (see below) which are sensitive
to presence of coherent quasi-singular structures, such as e.g. shocks, sharp
peaks, solitons etc. Presence of a large-scale mean motion is also a factor
that alters the behaviour of small-scale turbulence in a substantial way. We
will compare the MOT data with the scalings expected for smooth/laminar
fields, KOC turbulence, shocks and features arising when projecting 2D
surfaces in 3D space onto a 2D plane.

Note that our analysis in purely formal, because we do not use the underlying
dynamical equations, and we are not exploring the underlying physics leading
to the MOT dynamics under consideration. However, even tough similarities
between the obtained scalings with the considered reference theories might be
coincidental, we will see that the results of our analysis are strongly
correlated with visual differences observed in the MOT images.

To collect the data analyzed in this Letter, we use a MOT setup able to trap
and cool a large number $N$ of 87Rb atoms ($N$ up to $1.5\times 10^{11}$, see
Camara2014 ). The resulting cloud of atoms is centimeter-sized, with a
temperature around $200~{}\mu$K. For such large $N$ values, the MOT is known
to exhibit spatio-temporal instabilities when the trapping laser frequency is
brought sufficiently close to the atomic transition frequency Labeyrie2006 ;
Pohl2006 , namely when $\left|\delta\right|~{}<~{}\left|\delta_{cr}\right|$.
Here, the negative-valued laser detuning is
$\delta~{}=~{}\omega_{L}~{}-~{}\omega_{at}$ where $\omega_{L}$ and
$\omega_{at}$ are the laser’s and atomic frequencies respectively, and the
threshold detuning $\delta_{cr}$ depends on MOT parameters (such as magnetic
field gradient $\nabla B$ and laser intensity) and on atom number
Gaudesius2020 .

Our analysis of MOT turbulence is based on a set of 50 fluorescence images
recorded by a fast camera (one image every ms) to resolve fast dynamics of the
shape of the MOT. The nine different sets of data presented in this Letter
correspond to different values of the pair of MOT parameters ($\delta$,$\nabla
B$): $(1)\leftrightarrow(\delta=-\Gamma,\nabla B=2.4$ G/cm ;
$(2)\leftrightarrow(\delta=-\Gamma,\nabla B=1.2$ G/cm);
$(3)\leftrightarrow(\delta=-1.4\Gamma,\nabla B=1.2$ G/cm);
$(4)\leftrightarrow(\delta=-\Gamma,\nabla B=4.8$ G/cm);
$(5)\leftrightarrow(\delta=-\Gamma,\nabla B=12$ G/cm);
$(6)\leftrightarrow(\delta=-\Gamma,\nabla B=9.6$ G/cm );
$(7)\leftrightarrow(\delta=-\Gamma,\nabla B=7.2$ G/cm);
$(8)\leftrightarrow(\delta=-2\Gamma,\nabla B=7.2$G/cm);
$(9)\leftrightarrow(\delta=-4\Gamma,\nabla B=2.4$ G/cm). Indeed, we have shown
that qualitatively different unstable MOT regimes can be observed when varying
these parameters Gaudesius2021 . The 50 images are randomly picked from a
larger set, to probe the full dynamics of the cloud. The fluorescence images
are recorded at the MOT operation detuning $\delta$, with the camera’s line of
sight corresponding roughly to the strong axis of the magnetic field gradient.
The images that we record thus correspond to 3D light intensity distributions
integrated along this line of sight (see discussion below). Note that because
the employed detuning values are rather small, the distribution of
fluorescence light intensity may differ from the actual atomic density
distribution, due to multiple scattering of light within the cloud.

The structure functions are by far the most frequently used objects in
analysing turbulence frisch-book . The structure function of order $p$ for the
two-point increment of a field $\rho({\bf x})$ is defined as:

$\displaystyle S_{p}^{(\rho)}({\bf x},{\bm{\ell}})=\langle|\rho({\bf
x})-\rho({\bf x}+{\bm{\ell}})|^{p}\rangle.$ (1)

We will be analysing 2D black-and-white images, so $\rho$ will be the light
intensity on the image on the 2D physical coordinate ${\bf x}=(x,y)$.
Correspondingly,
${\bm{\ell}}=(\ell_{x},\ell_{y})=(\ell\cos(\theta),\ell\sin(\theta))$ is a 2D
coordinate increment. The averaging is performed over the ensemble or time
(assuming ergodicity, these two averages give the same result). However, in
statistically homogeneous systems one can use the space averaging (e.g. over
the the position $\bf x$), and assuming statistical isotropy one can average
over $\theta$. In the latter case, the structure function is independent of
$\bf x$ and $\theta$, i.e. $S_{p}^{(\rho)}({\bf x},{\bm{\ell}})\equiv
S_{p}^{(\rho)}(\ell)$.

In turbulence, the structure function exhibits a scaling behaviour, i.e. it
behaves as a power-law $S_{p}^{(\rho)}(\ell)\propto\ell^{\xi_{p}}$ in a wide
range of scales $\ell$. Often this range is referred to as the inertial range.
The quantities ${\xi_{p}}$ are called the structure function exponents. They
contain a significant information about the turbulence statistics, typical
processes (e.g. turbulent cascades) and presence of coherent and/or singular
structures causing intermittency of the turbulent signal (e.g. shocks or sharp
spikes).

Let us summarise several types of the scalings of the structure functions
which we will use as reference in our analysis later.

Differentiable field (often called smooth ramps): ${\xi_{p}}=p$. This result
trivially follows the leading order Taylor expansion of the structure function
in small $\ell$.

Shocks: ${\xi_{p}}=$ const. This result relies on the fact that for
sufficiently small $\ell$ the main contribution to $S_{p}^{(\rho)}(\ell)$
comes from the pairs of points ${\bf x}$ and ${\bf x}+{\bm{\ell}}$ located at
the different sides with respect to the jump.

Burgulence (random field governed by the Burgers equation). Here smooth ramps
coexist with shocks. Hence, a bifractal behaviour is observed: ramp scaling is
seen for $0<p<1$ and shocks for $1<p<\infty$.

Passive scalar advected by Kolmogorov turbulence. In atmospheric turbulence
the temperature field could be seen as approximately passively advected. The
mean-field theory of Kolmogorov-Oboukhov-Corssin (KOC) predicts
${\xi_{p}}=p/3$ for passive scalar fields advected by the classical Kolmogorov
turbulence. This prediction is made for the inertial range of scales
$\ell_{d}<\ell<\ell_{E}$, where $\ell_{d}$ and $\ell_{E}$ are the dissipative
and the energy-containing (integral) scales respectively. The scaling is
derived under the assumption that the energy dissipation rate is the only
quantity defining the statistical properties in this range Obukhov ; Corrsin .

Intermittent turbulence. This is what is most typical for real turbulent
systems: Kolmogorov type scaling at low values of $p$ is replaced by a more
shallow slope for higher $p$’s due to presence of coherent quasi-singular
structures/events Leveque ; Cliff .

To create the structure function we randomly (with a uniform distribution)
choose a pixel $(x,y)$ from within the box $[0.25N,0.75N]^{2}$, then choose
the second point by selecting a random $\ell$ pixels from $[0,N/2]$ and
$\theta$ from $[0,2\pi)$. With these $\ell$ and $\theta$, we read the value of
$\rho$ corresponding to the pixel into which the second point falls, so that
we can calculate calculate the structure function of Eq. (1). Additional
averaging is done in time by repeating the process for the whole dataset which
consists of 50 frames.

Note that choosing random $(x,y)$ and random $\theta$ (while keeping the
$\ell$-bin constant when computing the structure function) amounts to the
space and direction averaging which make sense only for systems which are
approximately statistically uniform and isotropic in the sampling volume. In
principle, the angle averaging could be dropped if one is interested in
studying the anisotropy effects.

We have studied the effect of detector pixelation using an image with fixed
Gaussian profile and established that the pixelation causes an error less than
3% for $\ell>10$. Thus, in our analysis we take $\ell>10$.

---
Figure 1: Top figures correspond to dataset (1) and bottom—to dataset (9). The
left figures are the structure functions with different values of $p$ plotted
in different colors. Each structure function is overlaid with three lines
corresponding to the low-$\ell$, medium-$\ell$ and high-$\ell$ power-law fits.
The slopes of these fits are presented as a function of $p$ in the right
figure with blue corresponding to the low-$\ell$, green—to the middle-$\ell$
and red—to the high-$\ell$ fits. These are further overlaid with the slopes at
low and high $p$.  Figure 2: Derivatives of the exponents of the structure
functions, $\frac{d}{dp}\xi(p)$, for each dataset (1)-(9). Blue, green and red
lines correspond to the low-$\ell$ ($10<\ell<25$), medium-$\ell$
($25<\ell<50$) and high-$\ell$ ($50<\ell<120$) fits. Solid and dashed lines
correspond to low and high $p$’s respectively. Each image inset is aligned
with the dataset it represents.

We have analysed images from the nine experimental datasets described above,
ordered from visually most turbulent dataset (1) to most laminar dataset (9).
The two extreme limits are presented in Fig. 1 where the upper and the lower
rows correspond to experiments (1) and (9) respectively. On the left, we show
structure functions $S_{p}^{(\rho)}(\ell)$ in log-log scale and a power law
fits in the ranges of low, medium and high separations $\ell$. The right
figures plot the exponents of the latter power law fits of
$S_{p}^{(\rho)}(\ell)$. There is a common feature on all the plots of
$S_{p}^{(\rho)}(\ell)$: they decrease at $\ell\gtrsim 200$ which is close to
the radius of the cloud. But we can also see that the results for the most
turbulent and the most laminar cases are drastically different in other
respects. The most laminar case is characterised by a nearly linear behaviour
of the structure function exponents $\xi(p)$ as a function of $p$ which is
almost independent on the fitting range in $\ell$ (low, medium or high). This
behaviour is typical for the smooth fields which, as we know, should exhibit
$\xi(p)=p$. The slope of $\xi(p)$ for our most laminar case is not as high as
1(it is $\sim 0.8$), which indicates that the corresponding field is not fully
laminar/smooth. Indeed, the set of MOT parameters for the dataset (9)
($\delta=-4\Gamma,\nabla B=2.4$ G/cm) corresponds to a stable MOT, with a
density distribution quite stationary and smooth. In contrast, in the most
turbulent experiment the curves $\xi(p)$ are concave up which is typical for
intermittent turbulence. These curves are different for different fitting
ranges: the slopes are steepest for the low-$\ell$ fitting, which is
indicative of presence of a dissipation at small $\ell$’s, making the fields
smoother at these scales. The slopes at the low $p$ side are consistent with
the KOC scaling ($\xi(p)=p/3$), whereas much smaller slopes are seen at the
high $p$ side which indicates presence of quasi-singular structures such as
shocks (which would result in zero slope) or “cliffs” as in the passive scalar
turbulence described in Cliff .

To summarise and classify the data of all nine experiments, we present Fig. 2
which shows the results for the slopes $\frac{d}{dp}\xi(p)$ for three
different fitting ranges. The plots are overlayed with respective single-shot
images for visual comparison. We see that the smoothness of the insets matches
closely the change in the slopes $\frac{d}{dp}\xi(p)$: with the more turbulent
cases corresponding to smaller slopes. It is interesting, however, that the
low-$\ell$, low-$p$ curve is the least sensitive to the variations in the
different experiments, with the value of $\frac{d}{dp}\xi(p)\sim 0.6-0.7$ for
all the images from most turbulent to most laminar. This points at presence of
small-scale fluctuations which are not noticeable to eye on the visibly
laminar flow images.

Finally, we will also mention another standard quantification of turbulence in
terms of the spectra of the light intensity which are shown in Figure 3 for
the most turbulent and the most laminar image. Here we see that the spectrum
for the more turbulent case has a scaling range where the spectrum
approximately follows a power law with exponent -3. There is no such scaling
range in the most laminar case: the corresponding spectrum decays much faster
than in the case of the turbulent case which corresponds to weakness of the
small-scale fluctuations. Thus, the spectra provide another useful diagnostic
tool for distinguishing between the cases with different levels of turbulence.
However, the information provided by the spectrum is less rich than the one
contained in the structure functions. Indeed, such a spectrum is just the
Fourier transform of $S_{2}^{(\rho)}(\ell)$.

Figure 3: Spectra for the most turbulent case (dataset (1), blue line) and
most laminar case (dataset (9), red line). The straight line is the $k^{-3}$
law.

In this Letter we have applied the methods borrowed from turbulence to the
analysis of various stable and unstable regimes of large clouds of cold atoms
contained in a MOT. We have found that the structure functions of the
intensity field provide an effective tool for classification of the states and
the level of turbulence in self-oscillating MOT clouds. In particular, for
most turbulent MOT states, they allow to observe features typical for
intermittent turbulence of passive scalars, with KOC-type scaling for the
lower-order structure functions and cliff-like coherent structures captured by
the high-order structure functions. Most laminar states are exhibiting
scalings corresponding to smooth large-scale flows with superimposed small-
scale fluctuations. For the intensity field spectra, we have observed a power-
law with exponent $-3$ in the most turbulent case and a non-universal steeply
decaying spectrum in the least turbulent case. These observations could be
used to discriminate between different models aiming at explaining the complex
physics of unstable MOTs.

This work was performed in the framework of the European Training Network
ColOpt, which is funded by the European Union (EU) Horizon 2020 program under
the Marie Sklodowska-Curie (MSC) action, grant No. 721465, and in the
framework of the European project ANDLICA, ERC Advanced grant No. 832219. Adam
Grifin and Sergey Nazarenko were supported by the Chaire D’Excellence awarded
by Université de la Côte d’Azur, France, the EU Horizon 2020 program MSC HALT
project, grant No. 823937, the FET Flagships PhoQuS project, grant No. 820392,
and the Simons Foundation project Collaboration on Wave Turbulence, award ID
651471.

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