# Anisotropy driven response of skyrmion lattice in MnSc2S4 to applied
magnetic fields

H. D. Rosales  Instituto de Física de Líquidos y Sistemas Biológicos
(IFLYSIB), UNLP-CONICET, Facultad de Ciencias Exactas, La Plata, Argentina
Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de
La Plata, La Plata, Argentina  Departamento de Cs. Básicas, Facultad de
Ingeniería, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata,
Argentina  F. A. Gómez Albarracín  Instituto de Física de Líquidos y Sistemas
Biológicos (IFLYSIB), UNLP-CONICET, Facultad de Ciencias Exactas, La Plata,
Argentina  Departamento de Física, Facultad de Ciencias Exactas, Universidad
Nacional de La Plata, La Plata, Argentina  Departamento de Cs. Básicas,
Facultad de Ingeniería, Universidad Nacional de La Plata, C.C. 67, 1900 La
Plata, Argentina  K. Guratinder  Laboratory for Neutron Scattering and
Imaging, Paul Scherrer Institute, CH-5232 Villigen, Switzerland  Department of
Quantum Matter Physics, University of Geneva, CH-1211 Geneva, Switzerland  V.
Tsurkan  Experimental Physics V, University of Augsburg, D-86135 Augsburg,
Germany  Institute of Applied Physics, str. Academiei 5, MD 2028, Chisinau,
Republic of Moldova  L. Prodan  Experimental Physics V, University of
Augsburg, D-86135 Augsburg, Germany  Institute of Applied Physics, str.
Academiei 5, MD 2028, Chisinau, Republic of Moldova  E. Ressouche  Univetsité
Grenoble Alpes, CEA, INAC-MEM, Grenoble, France  O. Zaharko
<EMAIL_ADDRESS>Laboratory for Neutron Scattering and Imaging, Paul
Scherrer Institute, CH-5232 Villigen, Switzerland

###### Abstract

We theoretically and experimentally study the stability of the unconventional
fractional antiferromagnetic skyrmion lattice (AF-SkL) in Mn2S4 spinel under
magnetic fields applied along the $[$1-10$]$ crystal direction. By performing
numerical Monte Carlo simulations for the minimal effective spin model that we
proposed in Ref. [S. Gao, et al., Nature 586, 37-41 (2020)], we show that the
skyrmion lattice is aligned within the equivalent and symmetric $[$1-11$]$ or
$[$1-11$]$ planes, which are equally inclined to the applied magnetic field.
We attribute this behavior to the magnetic anisotropy of the host material.
Neutron single crystal diffraction presents a very good agreement with the
predictions of the effective model. It reveals that the topological spin
texture gets destabilized at low temperatures and moderate magnetic fields and
is replaced by a conical phase for B// $[$1-10$]$. The present study
elucidates the central role of the magnetic anisotropy in the stabilization of
antiferromagnetic skyrmionic states.

##  I Introduction

The notion of skyrmions, first introduced in particle physics to account for
stability of quantized topological detectsSkyrme1961 , is used broadly now
also in magnetic systems. The prospect to create, move and annihilate isolated
nanometer-sized swirls of spins is attractive for future memory
devicesFert2013 . The bulk counterparts, so called skyrmion lattices (SkL) or
skyrmion crystalsTokura2021 , are appropriate model systems to explore the
microscopic mechanisms of formation of such magnetic quasiparticles, their
topological properties and responses to external incitesNagaosa2013 .
Several microscopic mechanisms stabilising SkL are presently identified. The
most established one enacts in noncentrosymmetric crystals through the
asymmetric Dzyaloshinskii-Moriya (DM) interactionBogdanov1989 ; Muelbauer2009
. The famous example is the B20-family, which consists of soft chiral
ferromagnets (such as metallic MnSiMuelbauer2009 , isolating Cu2OSeO3Seki2012
; White2014 ) with Bloch-type SkL following the magnetic field. Another
noncentrosymmetric host is the polar lacunar spinel GaV4X8 with X= S,
SeKezsmarki2015 , where the Neel-type SkL are locked to the polar axis.
However, SkL are found also in centrosymmetric crystals. For example, in the
intermetallic compound Gd2PdSi3 the localised 4f-moments on the triangular
lattice are frustrated yielding a topologically nontrivial triple-$\bf{q}$
stateKurumaji2019 ; Okumura2020 .
Materials hosting SkL bear predominantly ferromagnetic interactions and their
temperature-magnetic field (T, B) phase diagrams share a number of common
features (see Fig.2 of Ref.Seki2012,  for a typical example). The zero field
ground state is usually a helical structure, which in applied magnetic field
attains a ferromagnetic component and transforms first into a conical phase
and then into a field-polarized ferromagnet. The skyrmion phase is commonly
located in an intermediate T-B range, adjacent to the conical phase from the
low-T side and bordering to the paramagnetic state at the high-T side. The
region of the SkL existence varies significantly, ranging from a tiny ’pocket’
in MnSiMuelbauer2009  to an extensive area in Co-Zn-Mn alloys.Karube2016  It
becomes increasingly recognised that magnetocrystalline anisotropy plays a
significant role in the formation and stability of SkL’sPreissinger2021 .
Materials with predominantly antiferromagnetic interactions might also host
skyrmion lattices (AF-SkL)Okubo2012 . The realization of this theoretical idea
was experimentally verified in spinel MnSc2S4Gao2016 ; Gao2020 . The phase
diagram of this material was studied for magnetic fields applied along the
$[$100$]$ and $[$111$]$ crystal directions and the AF-SkL phase existed down
to the lowest temperatures.
Our present study reveals that the MnSc2S4 antiferromagnet obeys the universal
features of bulk skyrmion hosting materials and that magnetic anisotropy
controls the stability of AF-SkL. Our neutron single crystal diffraction and
Monte Carlo simulations demonstrate that for magnetic fields applied along the
$[$1-10$]$ direction the triple-$\bf{q}$ phase vanishes and the conical phase
occurs at very low temperatures which we could reach as in the experiments so
in the simulations.

##  II Recap on MnSc2S4

In the cubic MnSc2S4 spinel the magnetic Mn2+ ions (S=5/2) form a bipartite
diamond lattice, with two face-centred cubic ($fcc$) sublattices A and B (Fig.
1). This lattice can be frustrated when the second-neighbour $J_{2}$ coupling
is antiferromagnetic ($J<0$, AF), as the Mn2+ ions connected by the $J_{2}$
exchange build an AF ${fcc}$ sublattice which is highly frustrated. The ground
state of the $J_{1}-J_{2}$ model on the diamond lattice for the ferromagnetic
(F) nearest-neighbour $J_{1}$ and strong AF $J_{2}$ couplings
($J_{2}$/$|J_{1}|>$1/8) is a manifold of massively degenerate spin spirals,
named a spiral spin liquidBergman2007 .

In MnSc2S4 strong frustration arising due to the large $J_{2}/|J_{1}|$ ratio
is released by a small third-neighbour $J_{3}$ coupling. In zero field the
multi-step order sets in below $T_{N}$=2.3 K. The commensurate collinear state
with the propagation vector ${\bf{q}}=(\frac{3}{4},\frac{3}{4},0)$ transforms
into the incommensurate short-living state with
${\bf{q}}_{icm}=(\frac{3}{4}\pm\delta,\frac{3}{4}\pm\delta,0)$ and then back
to the ground state commensurate helical state with the re-entrant ${\bf{q}}$.
The propagation vector ${\bf{q}}=(\frac{3}{4},\frac{3}{4},0)$ could have
twelve arms listed in Table 1.

Figure 1: (Color online) (a) Diamond lattice of the Mn2+ ions consists of two
interpenetrating face centered cubic ($fcc$) Bravais lattices. $J_{1}$,
$J_{2}$ and $J_{3}$ indicate first, second and third neighbors exchange
interactions, respectively. Panels (b)-(d) show the (1-10), (1-11) and (1-1-1)
lattice planes. Panels (e-g) show the distribution for the arms of the
propagation vector ${\bf{q}}=(\frac{3}{4},\frac{3}{4},0)$ highlighting in
purple (e) two arms ${\bf{q}_{2}}$ and ${\bf{q}_{3}}$, parallel to the
magnetic field in [1-10], indicated by an arrow, (f) six arms corresponding to
d4\- and (g) d1\- domains of the triple ${\bf{q}}$ phase.

Analysis of the intensity of the twelve
$\langle\frac{3}{4},\frac{3}{4},0\rangle$ magnetic reflections unravels that
in zero field the multi-domain single-${\bf{q}}$ helical (HL) structure exists
forming twelve domains. In moderate magnetic fields a multi-domain
triple-$\bf{q}$ phase occurs. One domain of this phase is described by three
coplanar ${\bf{q}}_{j}$ vectors satisfying the rule
$\sum_{j=1}^{3}{\bf{q}}_{j}=0$ and four possible domains (di, $i$=1-4) are
disclosed in Table 1 and Fig. 1. Magnetic field along the $[$100$]$ crystal
direction is equally inclined to all four domains, while the field along
$[$111$]$ complies with the symmetry of only one domain. Exactly these domain
distributions were observed in single crystal neutron diffraction
experimentsGao2016 ; Gao2020 .

Table 1: The twelve arms of the propagation vector ${\bf{q}}=(\frac{3}{4},\frac{3}{4},0)$ with twelve domains of the helical or conical phases or four domains d1-4 of the triple-$\bf{q}$ phase.  q1=($\frac{3}{4}$ $\frac{3}{4}$ 0) |  q7=(-$\frac{3}{4}$ 0 -$\frac{3}{4}$) |  q12=(0 -$\frac{3}{4}$ $\frac{3}{4}$) | q4 | q5 | q11 |  | d1
---|---|---|---|---|---|---|---
q2=($\frac{3}{4}$ -$\frac{3}{4}$ 0) |  q8=(-$\frac{3}{4}$ 0 $\frac{3}{4}$) |  q11=(0 $\frac{3}{4}$ -$\frac{3}{4}$) | q3 | q6 | q12 |  | d2
q3=(-$\frac{3}{4}$ $\frac{3}{4}$ 0) |  q5=($\frac{3}{4}$ 0 $\frac{3}{4}$) |  q10=(0 -$\frac{3}{4}$ -$\frac{3}{4}$) | q2 | q7 | q9 |  | d3
q4=(-$\frac{3}{4}$ -$\frac{3}{4}$ 0) |  q6=($\frac{3}{4}$ 0 -$\frac{3}{4}$) |  q9=(0 $\frac{3}{4}$ $\frac{3}{4}$) | q1 | q8 | q10 |  | d4

These experimental observations could be explained by an effective spin
Hamiltonian

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\sum_{ij}J_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+3\,J_{||}\sum_{ij\in
NN}\left(\mathbf{S}_{i}\cdot\hat{r}_{ij}\right)\left(\mathbf{S}_{j}\cdot\hat{r}_{ij}\right)$
(1)
$\displaystyle+A_{4}\sum_{i,\alpha=x,y,z}(S^{\alpha}_{i})^{4}-g\mu_{B}\,{\bf
B}\sum_{i}\mathbf{S}_{i}$

where $J_{ij}$ is the exchange coupling (up to the third neighbours) between
the Heisenberg spins $\mathbf{S}_{i}$ and $\mathbf{S}_{j}$. $J_{||}$ is the
anisotropic coupling between the nearest-neighbours connected by the unitary
vectors $\hat{r}_{ij}$ along the $\langle$110$\rangle$ axes, $A_{4}$ is
fourth-order single-ion anisotropy constant. The last term in Eq. (1) is the
Zeeman coupling between the spins and a magnetic field ${\bf B}$; $g$ and
$\mu_{B}$ are the Landé g-factor and the Bohr magneton.

From combined inelastic neutron scattering and Monte-Carlo simulations
studyGao2020  a set of exchange parameters $J_{1}=-0.31$K, $J_{2}=0.46$K,
$J_{3}=0.087$K, $J_{||}=-0.01$K and $A_{4}=0.0016$K was extracted. It
reproduces the zero field helical ground state, the spin-wave excitations and
the (T, B) phase diagram for magnetic fields applied along the $[$100$]$ and
$[$111$]$ crystal directions.
Moreover, the triple-$\bf{q}$ phase with the finite scalar spin chirality
could be rationalized as a fractional antiferromagnetic skyrmion lattice and
the analytical expression for this AF-SkL was found.

##  III Monte Carlo simulations

The present study aims to explore the stability of the fractional AF-SkL
lattice in MnSc2S4 applying field along the [1$\overline{1}$0] direction. The
classical spin model of Eq. (1) and the set of $J_{1,3}$-$J_{||}$-$A_{4}$
parameters as reported in Ref. [Gao2020, ] were implemented on a $N=8\times
L^{3}$ ($L=8$) lattice with periodic boundary conditions. In this calculation,
the spin lattice has been cooled slowly to low temperatures and the averaging
over $100$ independent attempts has been performed. In order to identify
different phases and to compare with neutron diffraction results, we computed
the magnetic structure factor defined as

$F(\mathbf{q})=\frac{f(\mathbf{q})^{2}}{N}\sum_{ij}\langle\mathbf{S}_{i}\cdot\mathbf{S}_{j}\rangle
e^{i\mathbf{q}\cdot\hat{r}_{ij}}$ (2)

and the measured quantity $F_{\perp{\bf q}}^{2}({\bf q})$ where $\perp{\bf q}$
indicates that the spin-spin correlation only includes spin components
perpendicular to ${\bf q}$. In Eq. (2)
$\noindent\langle\mathbf{S}_{i}\cdot\mathbf{S}_{j}\rangle$ are the
correlations between the magnetic moments at the sites $i$ and $j$, $N$ is the
number of magnetic ions and $f(\mathbf{q})$ is the Mn2+ magnetic form factor.

Fig. 2 presents the theoretical (T, B1-10) phase diagram with three ordered
phases - helical (HL), antiferromagnetic skyrmion lattice (AF-SkL) and conical
(CN). At zero and low fields ($B\leq 1.2$T) only the helical phase exists.
Then, on field growth a strong competition between the HL and CN phases
occurs. As field increases further to $B$=3.5 T the AF-SkL phase emerges. At
lowest temperatures there is an additional non-zero contribution from the
conical phase in the predominantly AF-SkL region. The CN phase is favored at
higher magnetic fields, where the AF-SkL phase vanishes. The boundaries of
these and other phases were determined from the intensity distribution of the
$\langle\frac{3}{4},\frac{3}{4},0\rangle$ magnetic peaks, as we will discuss
in detail in the following subsections.

Figure 2: (Color online) Magnetic phase diagram of the model in Eq. (1) as a
function of the temperature T(K) and external magnetic field B(T) along the
[1$\overline{1}$0] direction. Figure 3: Neutron intensity as a function of
temperature and magnetic field along [1$\overline{1}$0] averaged over 100 MC
runs. (a) Contributions at $B=0.84$T averaged over ten single-${\bf{q}}$
domains of the helical structure (blue) and over two single-${\bf{q}}_{n}$
domains of the conical structure (red). (b) Intensities obtained in two
independent MC runs ($\\#$1, $\\#$2) at $B=1.4$T. (c) Contributions for two
d1,4 domains of triple-${\bf{q}}$ denoted SkL (violete) and two CN domains
(red) at $B=4.9$T. (d) The same intensities for SkL and CN at high fields,
$B=7.84$T.

###  III.0.1 Helical phase

In low magnetic fields ($B<1.2$T) our MC runs result in finite intensity in
the (1-11) and (1-1-1) reciprocal lattice planes (Fig. 1 (f), (g)). These
planes comprise ten arms of the propagation vector
${\bf{q}}=(\frac{3}{4},\frac{3}{4},0)$, which are listed in Table 1 (top and
bottom rows). The intensity distribution for all MC runs and respective
magnetic arrangements imply realisation of the helical phase. One example of
such MC $F_{\perp{\bf q}}({\bf q})$ map is outlined in Fig. 4(a), it
corresponds to the ${\bf{q}}_{6}$ domain. Within a significant (T, B) region
no contributions from the remaining ${\bf{q}_{2}}$, ${\bf{q}_{3}}$ arms emerge
(see Fig. 3(a)).

At higher magnetic fields ($B\gtrsim 1.2$T) the helical and conical phases
compete and the reflections of both phases appear. In our MC runs for $B=1.4$T
either the HL phase or the CN phase with propagation vector
${\bf{q}}_{2}=(\frac{3}{4},-\frac{3}{4},0)$ emerge (see Fig. 3(b)). Fig. 4(b)
presents a $F_{\perp{\bf q}}^{2}({\bf q})$ map for the MC run converging to
the conical phase with ${\bf{q}}_{2}$.

To characterize the single-$\bf{q}$ helical structure we explore the magnetic
arrangement in real space. A typical (1-10) plane obtained from one MC run at
$B=0.56$T and $T=0.92$K is shown in Fig. 5(a). No ordered arrangement is
evident. So we inspect separately the (1-11) and (1-1-1) planes. While no
clear regularity is seen in (1-1-1) (Fig. 5(b)), a distinct ordered structure
is apparent within the (1-11) plane (Fig. 5(c)). Further on, when subdividing
this triangular layer into three sublattices (Fig. 5(d)) (that is twice
depleting the layer as in Ref. [Gao2020, ]) the superposition of three
interpenetrating helices is obvious. Thus the MC run presented in Figs. 5 (c,
d) displays one of the possible single-$\bf{q}$ associated with the helical
structure. The symmetry between the (1-1-1) and (1-11) planes is broken. In
this example, only one domain of the helical structure with spin-correlations
in the (1-11) plane arises, while domains with the HL long-range arrangement
in the (1-1-1) plane are absent. This spontaneous symmetry breaking is
confirmed by other MC runs, where the ordered structure is seen in either
(1-11) or (1-1-1) planes.

The situation is analogous to the observations in the triangularRosales2015 ;
Osorio2017  and kagomeVillalba2019  antiferromagnetic lattices, where the
underlying structures are revealed upon splitting the full arrangement into
simpler sublattices.

Figure 4: (Color online) (a) A typical intensity distribution after one
individual run at $T=0.4$K and $B=0.84$T. The magnetic intensity corresponds
to ${\bf{q}}_{6}=(\frac{3}{4},0,-\frac{3}{4})$ and belongs to the helical
phase. (b) Same at $B=1.4$T, the magnetic intensity appears at
${\bf{q}}_{2}=(\frac{3}{4},-\frac{3}{4},0)$ and this MC run results in the
conical phase. (c) The spin arrangement in the conical phase. Figure 5: (Color
online) Helical texture at $B=0.56$T and $T=0.92$K in different planes. (a):
(1-10), (b): (1-1-1), (c): (1-11); (d) spin texture in one sublattice of the
triangular layer in the (1-11) planes. The color scheme represents the
z-component of the magnetic moment in the local coordinate system of each
panel with S=-1 in blue, S=1 in red. The arrows shown the in-plane spin
component.

###  III.0.2 Conical phase

Further increase of magnetic field along [1$\overline{1}$0] favors the
single-$\bf{q}$ conical phase with the propagation vector along the field
(Fig. 1(e)). The stability of the CN phase in the (T,B) phase diagram could be
monitored following the intensity of the ($\frac{3}{4},-\frac{3}{4},0$)
reflection. Fig. 3(d) shows that $\bf{q}_{2}$ clearly dominates other
$\bf{q}$-arms at field as high as $B=7.84$T. In fact, no other arm is present
in this region of the phase diagram (Fig. 4(b)), thus this is a
single-$\bf{q}$ structure.
The corresponding spin texture in real space is sketched in Fig. 4(c). It is
composed of chains of the Mn moments propagating along the [1,-1,0] directions
with a helical component orthogonal to ${\bf{q}}$ and a ferromagnetic
component along ${\bf{q}}$.

###  III.0.3 Fractional AF-SkL phase

Monte Carlo simulations of the model proposed in Eq. (1) stabilize the
fractional AF-SkL for magnetic fields along [111]Gao2020 . This direction is
perpendicular to the triangular layers of the spinel structure, where the spin
texture develops. Here we show that this AF-SkL phase emerges also when
magnetic field is applied along [1-10]. Remarkably, the skyrmion texture is
not perpendicular to the external field, but spans in the (1-11) and (1-1-1)
planes, i.e. in the triangular layers.

The emergence of the AF-SkL phase could be identified inspecting intensity
distribution of the twelve $\langle\frac{3}{4},\frac{3}{4},0\rangle$
reflections. These reflections are grouped into contributions of four domains
of the triple-$\bf{q}$ listed in Table 1. Fig. 3 (c) presents temperature
dependence of intensity from MC runs at $B=4.9$T. Two separate groups are
apparent - two ($\pm\bf{q}_{2}$) and the rest ten. Such T-dependence implies
that these groups arise from the CN and AF-SkL phases, respectively. The ten
peaks belonging to the d1, d4 AF-SkL domains clearly dominate till TN=1.3K.
However, at low temperatures ($T\lesssim 0.5K$) the $\pm\bf{q}_{2}$ intensity
is non-zero, implying coexistence of AF-SkL with the conical phase. We present
magnetic intensities for the pure AF-SkL phase at $T=0.96K$ and $B=4.9$T in
Fig. 6 (a,b). Here the intensity distribution typical for the d1-domain is
apparent.

Figure 6: (Color online) Magnetic intensities obtained by MC simulations at
$T=0.96$K and $B=4.9$T in two different planes of the reciprocal lattice. This
distribution corresponds to the AF-SkL phase.

Now we look at the corresponding spin texture in real space. In Fig. 7 a
typical arrangement at $T=0.96$K and $B=4.9$T in three different planes:
(1-10), (1-1-1) and (1-11) is shown. Like in the helical phase, the true order
cannot be identified easily in the (1-10) planes (see Fig. 7(a)). However, if
we inspect the (1-11) and (1-1-1) planes, the famous fractional
antiferromagnetic skyrmion lattice Gao2020  is visible (see Figs. 7(c) and
7(d)). Again, like in the helical phase, different numerical simulations show
that the SkL pattern is stabilized either in the (1-11) plane or in the
(1-1-1) plane, never in both planes simultaneously. This means that either the
d1\- or d4\- domain of the AF-SkL phase emerges.

Figure 7: (Color online) Magnetic texture in the triple-$\bf{q}$ phase at
$B=4.9$T and $T=0.96$K in different planes. (a): (1-10), (b): (1-11), (c):
(1-1-1); (d) spin texture in one sublattice of the triangular layer in the
(1-1-1) planes. The color scheme is the same as in Fig. 5.

To further study the AF-SkL phase, we compute the absolute value of the total
scalar spin chirality (discrete topological charge) defined as

$\chi_{\text{tot}}=\langle\frac{1}{8\pi}\sum_{\triangle}\mathbf{S}_{i}\cdot(\mathbf{S}_{j}\times\mathbf{S}_{k})\rangle$
(3)

where $\triangle$ indexes all the elementary triangles of the sites $i$, $j$
and $k$ in the (1-11) and (1-1-1) layers. A finite $\chi_{\text{tot}}$ is
expected for topological textures, whereas it would vanish for the helical and
conical phases. In Fig. 8 $\chi_{\text{tot}}$ as a function of magnetic field
is presented for three representative temperatures. AF-SkL is clearly
identified at 0.55K and 1.06K, but not at 1.70K.

Figure 8: (Color online) Total scalar spin chirality computed for selected MC
final configurations at three representative temperatures.

##  IV Experiment

Neutron single crystal diffraction (ND) was used to generate the experimental
(T,B) phase diagram and to verify the structure of the conical phase. The
single crystals grown with the chemical transport reaction technique and used
in previous diffraction experiments Gao2016 ; Gao2020  were utilized. This
time the experiments were performed on the neutron single-crystal
diffractometer Zebra at the Swiss Spallation Neutron Source, SINQ, Villigen
and on the D23 diffractometer at the Institut Laue-Langevin (ILL), Grenoble
using the normal beam geometry, dilution inserts and vertical magnets. The
crystals were oriented with the [1-10] crystal axis vertical, along the
applied magnetic field. The neutron wavelengths $\lambda$=2.314 Å and 1.383 Å
selected by the PG(200) and Ge(220) monochromators and a 10 T magnet were used
on Zebra. The wavelength 1.273 Å from a Cu(200) monochromator and a 12 T
magnet were used on D23.
The phase diagram constructed from our experiments is presented in Fig. 9 (a).
The boundaries of different phases were determined following the intensity of
the ${\bf{q}_{1}}=(\frac{3}{4},\frac{3}{4},0$) magnetic reflection (Fig. 10
(a)). When the intensity was changing abruptly, increasing, as in the AF-SkL
region, or vanishing, as in the CN phase, we measured intensity of other
${\bf{q}}$ arms to identify which magnetic phase was entered. As the
$(\frac{3}{4},-\frac{3}{4},0$) reflection could not be reached due to the
limits of the experimental setup, the ($\frac{1}{4},-\frac{1}{4},1$)
reflection of the ${\bf{q}_{2}}$ arm was measured instead (Fig. 10 (b)).
In order to determine the antiferomagnetic component of the conical structure
a set of magnetic reflections of the
${\bf{q}_{2}}=(\frac{3}{4},-\frac{3}{4},0)$ arm was measured at $T=0.1$K and
$B=2.5$T. The helical component of the conical structure proposed from the MC
runs gives good fit to the data ($R_{F^{2}}$=7.3 %, $\chi^{2}=2.16$ for 31
measured reflections). The two helices with the origin at the sublattices A
and B are circular, with almost equal components in the $[1,1,0]$\- plane
($M_{[1,1,0]}$=4.65(9)) $\mu_{B}$) and out-of-plane ($M_{z}$=4.7(2)
$\mu_{B}$).

Figure 9: (Color online) Magnetic phase diagram obtained by single crystal
neutron diffraction with magnetic field B[T] along the [1$\overline{1}$0]
direction.

Figure 10: (Color online) (a) Field and temperature evolution of magnetic
intensity at the ($\frac{3}{4},\frac{3}{4},0$) position. Intensity for the
($\frac{1}{4},-\frac{1}{4},1$) peak measured at T=0.3 K is presented in red
and marked by star. (b) Helical part of the cone structure from single crystal
neutron diffraction at T=0.1 K and B=2.5 T.

In the AF-SkL region the magnetic reflections corresponding to two domains of
the AF-SkL phase, d1 and d4, were detected.

##  V Discussion and Summary

Our study brings new insights into the (T, B) phase diagram of the
antiferromagnetic spinel MnSc2S4 and SkL hosting materials in general.
Firstly, for the B//[1-10] field direction the conical phase was found for
MnSc2S4. This phase was not detected for the other principal field
directionsGao2016 ; Gao2020 . In our MC runs the CN phase coexists with the
helical phase at low magnetic fields and with the AF-SkL phase at higher
fields. In the ND experiments the HL-CN transition is more perceptive and a
single-phase region instead of two-phase regions is observed. We suspect that
the mixed regions in the MC simulations is a consequence of freezing of the
adjacent phases at these low temperatures. The conical phase has the helical
component orthogonal to the propagation direction from both, ND experiment and
MC runs, while the ferromagnetic component is along the propagation direction
according to MC.
Secondly, our MC simulations show that the AF-SkL phase for MnSc2S4 with
B//[1-10] is the same as the one found for the [111] field directionGao2020 .
The SkL tubes propagate along the [1-11] or [1-1-1] axes and the spin texture
of the Bloch-type develops in the triangular layers perpendicular to these
directions. The texture is a linear combination of helices. As nearest spins
within the triangular layers are almost anti-parallel, we select next-nearest,
almost parallel spins, which form a depleted triangular sublattice to
visualize the swirls in Fig. 5 (d). The distance between their centers is
$\approx$ 55 Å and they can be viewed as pairs of incipient meron and
antimeron Gao2020 . The scalar spin chirality is finite and non-integer in
this topological texture.

The orientation of skyrmions in MnSc2S4 is locked to the anisotropy axis
defined by the $J_{||}$ anisotropic exchange and fourth-order single-ion
anisotropy $A_{4}$. The relevance of the $J_{||}$ interaction in the formation
of the fractional AF skyrmion lattice was discussed recently in Ref.[Gao2020,
] (see Supp. Material), while the single-ion anisotropy is a key factor
stabilizing skyrmion textures in frustrated magnets Leonov2015 , since it
lowers the energy of SkL with respect to the other phases at moderate magnetic
fields. Skyrmions do not rotate following the applied magnetic field in the
MnSc2S4 host as it happens, for example, in the soft B20 Bloch-type SkL
hostsMuelbauer2009 ; Seki2012 . In our MC runs the domains equally inclined to
the applied magnetic field get stabilised. In this aspect MnSc2S4 is closer to
the family of lacunar spinels GaV4X8 (X=S, Se),Gross2020  where the Neel-type
skyrmions are locked to the polar axis. For this last case rotation of the
applied field modifies domain walls where distortion of the skyrmions and
displacement of their cores takes place.Geirhos2020 .
Furthermore the case of MnSc2S4 allows to generalize that the magnetic
anisotropy is often a decisive factor for stabilization of the SkL phases. The
nature of the dominant coupling: ferromagnetic, antiferromagnetic or RKKY, as
well as, is the type of the texture: Bloch- or Neel-, make each material
hosting skyrmions unique defining the multi-parameter space for the existence
of SkL lattices. Yet, the anisotropy in any feasible form - single-ion
anisotropy, anisotropy of exchange, dipolar interaction - defines the exact
region of stability for the SkL-phase in the (T, B) phase diagram.

Acknowledgments. The neutron diffraction experiments were performed at SINQ,
Paul Scherrer Institute, Villigen, Switzerland and at ILL, Grenoble, France.
This work was supported by the Swiss National Science Foundation (Grant No.
200020-182536). H. D. R. and F. A. G. A. are partially supported by CONICET
(PIP 2021-11220200101480CO), and SECyT UNLP PI+D X792, X788 and X893. F. A. G.
A. acknowledges support from PICT 2018-02968. V. T. and L. D.were partly
supported by the Deutsche Forschungsgemeinschaft (DFG) through Transregional
Research Collaboration TRR 80 (Augsburg, Munich, and Stuttgart), and by the
project ANCD 20.80009.5007.19 (Moldova)

## References

  * (1) T. A. Skyrme Nucl. Phys.31, 556 (1962).
  * (2) A. Fert, V. Cros, J. Sampaio Nat. Nanotechnol. 8, 152 (2013).
  * (3) Y. Tokura, N. Kanazawa, Chem. Rev. 121, 2857 (2021).
  * (4) N. Nagaosa, N. Kanazawa, Nature Nanotechnology 10, 1038 (2013).
  * (5) S. Mülbauer, B. Binz, F. Jopnietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, P. Böni Science 323, 9145 (2009).
  * (6) A. N. Bogdanov, D. A. Yablonskii Sov. Phys. JETP 68, 101 (1989).
  * (7) S. Seki, X. Z. Yu, S. Ishiwata, Y. Tokura Science 336, 198 (2012).
  * (8) J. S. White, K. Prša, P. Huang, et al. Science 113, 107203 (2014).
  * (9) I. Kézsmárki, S. Bordács, P. Milde, et al. Nature Physics 14, 1116 (2015).
  * (10) T. Kurumaji, T. Nakajima, M. Horschberger Science 365, 914 (2019).
  * (11) S. Okumura, S. Hayami, Y. Kato, Y. Motome Phys. Rev. B 101, 144416 (2020).
  * (12) K. Karube, J. S. White, N. Reynolds et al. Nature Mat. 15, 1237 (2016).
  * (13) M. Preissinger, K.Karube, D. Ehlers et al. NPJ Quantum Mater. 6, 65 (2021).
  * (14) T. Okubo, S. Chung, H. Kawamura Phys. Rev. Lett 108, 017206 (2012).
  * (15) S. Gao, O. Zaharko, V. Tsurkan, et al. Nature Physics 10, 1 (2016).
  * (16) S. Gao, H. D. Rosales, F. A. Gómez Albarracín, et al. Nature 586, 37 (2020).
  * (17) D. Bergman, J. Alices, E. Gull, S. Trebst, and L. Balents Nature Physics 3, 487 (2007).
  * (18) H. D. Rosales, D. C. Cabra, and Pierre Pujol, Phys. Rev. B 92, 214439 (2015).
  * (19) S. A. Osorio, H. D. Rosales, M. B. Sturla, and D. C. Cabra, Phys. Rev. B 96, 024404 (2017).
  * (20) M. E. Villalba, F. A. Gómez Albarracín, H. D. Rosales, and D. C. Cabra, Phys. Rev. B 100, 245106 (2019).
  * (21) A. Leonov and M. Mostovoy, Nature Communications 6, 8275 (2015).
  * (22) B. Gross, S. Philipp, K. Geirhos, et al. Phys. Rev. B 102, 104407 (2020).
  * (23) K. Geirhos, B. Gross, B. G. Szigeti et al. NPJ Quantum Mater. 5, 1 (2020).