# Field-induced vortex-like textures as a probe of the critical line in
reentrant spin glasses

N. Martin<EMAIL_ADDRESS>Université Paris-Saclay, CEA, CNRS,
Laboratoire Léon Brillouin, CEA Saclay 91191 Gif-sur-Yvette, France  L.J.
Bannenberg  Faculty of Applied Science, Delft University of Technology, 2629
JB Delft, the Netherlands  M. Deutsch  Université de Lorraine, CNRS, CRM2,
Nancy, France  C. Pappas  Faculty of Applied Science, Delft University of
Technology, 2629 JB Delft, the Netherlands  G. Chaboussant  Université Paris-
Saclay, CEA, CNRS, Laboratoire Léon Brillouin, CEA Saclay 91191 Gif-sur-
Yvette, France  R. Cubitt  Institut Laue Langevin, BP156, F-38042 Grenoble
France  I. Mirebeau  Université Paris-Saclay, CEA, CNRS, Laboratoire Léon
Brillouin, CEA Saclay 91191 Gif-sur-Yvette, France

###### Abstract

We study the evolution of the low-temperature field-induced magnetic defects
observed under an applied magnetic field in a series of frustrated amorphous
ferromagnets (Fe1-xMnx)75P16B3Al3 (a-FeMn). Combining small-angle neutron
scattering and Monte Carlo simulations, we show that the morphology of these
defects resemble that of quasi-bidimensional spin vortices. They are observed
in the reentrant spin-glass (RSG) phase, up to the critical concentration
$x_{\rm C}\approx 0.36$ which separates the RSG and "true" spin glass (SG)
within the low temperature part of the magnetic phase diagram of a-FeMn. These
vortices systematically decrease in size with increasing magnetic field or
decreasing the average exchange interaction, and they finally disappear in the
SG sample ($x=0.41$), being replaced by field-induced correlations over finite
length scales. We argue that the study of these nanoscopic defects could be
used to probe the nature of the critical line between the RSG and SG phases.

## Introduction

The role of disorder is central in condensed matter physics, as it favors the
nucleation of defects which play a crucial role in the evolution and
functionalities of a large variety of systems. Examples are magnetic vortices
in superconductors, skyrmions in helical magnets, Taylor cells in liquid
flows, or twist grain boundary phases in cholesteric liquid crystals. Quite
generally, defects allow new properties to penetrate in the system by forming
intermediate states of matter, precursors of a phase transitions. In this
context, we study here the influence of nm-size magnetic defects on the
evolution from ferromagnetic (FM) and spin glass (SG) ground states. SG are
archetypal disordered magnetic systems that have mobilized a large and
continuous attention for decades. Their physics is mainly driven by atomic
disorder and random sign interactions, i.e. a mixture of FM and
antiferromagnetic (AFM) couplings usually tuned by the concentration of
magnetic ions. In the so-called reentrant spin glasses (RSG), the nature of
the coexistence between SG behavior and ferromagnetism has been much debated.
In this work, we show that the observation of magnetic field-induced vortices,
although not predicted by current theories, could be a key point to
distinguish between SG and RSG ground states at a microscopic level.

Historically, the RSG and SG ground states have been described by two
concurrent theoretical approaches. The infinite range mean field (MF) picture
yields a phase diagram with a tricritical point and a vertical line between
RSG and SG phasesGabay and Toulouse (1981). Below this line, i.e. in the
weakly frustrated case, lowering temperature leads to the occurrence of mixed
RSG phases, where SG and FM order parameters coexist microscopically. At each
magnetic site, "longitudinal" spin component, forming a long-range magnetic
order (LRMO), coexist with a "transverse" one, randomly oriented in the
perpendicular plane. Alternatively, random field (RF) arguments predict the
breakdown of LRMO for an arbitrarily small amount of disorder in dimensions
$d\leq 4$, as formalized by the Imry-Ma (IM) argument Imry and Ma (1975). This
argument was used together with percolation approaches to describe the RSG
phase as randomly oriented clusters spatially separated from the "infinite" FM
one. The latter would break due to RFs at the RSG-SG threshold, in a cross-
over transition Aeppli et al. (1983); Niidera and Matsubara (2007). The IM
argument was recently complemented by a series of Monte-Carlo (MC) simulations
suggesting that, in the case of ferromagnets subject to RF, the IM domains are
protected against a full collapse of the magnetization by the nucleation of
topological defects, such as pinned hedgedhogs in 3 dimensionsProctor et al.
(2014) or a "skyrmion-antiskyrmion glass" in 2 dimensionsChudnovsky and
Garanin (2018).

On the experimental side, magnetic defects -akin to nm-size vortices- have
been observed by small-angle neutron scattering (SANS) in weakly frustrated
RSG under an applied magnetic field. The family of studied compounds
(Ni1-xMnx, AuxFe1-x, Fe1-xAlx, or a-Fe1-xMnx) includes different types of
disorder, magnetic interactions and sample form (single crystal, polycrystal
or amorphous samples)Hennion et al. (1986); Böni et al. (1986); Lequien et al.
(1987); Hennion et al. (1988). In all cases, SANS experiments show that the
transverse spin components rotate over a finite length scale which defines the
average vortex size. These data, supported by MC simulationsKawamura and
Tanemura (1991), also indicate that the vortices shrink with increasing the
applied field, but their behaviour at strong frustration and across the RSG-SG
threshold has not been studied so far.

In order to address this point, we focus here on the series of frustrated
amorphous ferromagnets (Fe1-xMnx)75P16B6Al3 ("a-FeMn"). a-FeMn maps a case of
3d disordered Heisenberg spins, where frustration can be chemically tuned
through the competition of FM and AFM interactions. Using SANS, we follow the
evolution of the field-induced vortices with increasing frustration, as the
magnetic ground state evolves from RSG to SG. We show that non-singular
vortices are characteristic of the RSG ground state. Their average size obeys
scaling laws up to the critical concentration and maximum applied field. They
eventually disappear above the RSG-SG threshold, showing that a non-zero
average exchange is needed for their stabilization. Our results may open a
route to reconcile the MF and RF pictures of the RSG state, opposed for
decades. Indeed, vortices can probe the nature of the frustrated medium and be
the fingerprints of a quantum phase transition at a microscopic level. Our
experimental study is also supported by Monte Carlo simulations, which show
that the occurrence of these defects, as well as their evolution as a function
of magnetic field, can be globally reproduced using a very limited amount of
ingredients. These results suggest that frustration and disorder can be used
to engineer the average size of individual defects in a controlled and
reproducible way in disordered frustrated ferromagnets.

## Samples and their macroscopic magnetic properties

The a-FeMn system is perfectly suitable for our study. Its macroscopic
properties and transition temperatures are well known and almost independent
of the sample synthesis, while the amorphous character guarantees the absence
of structural defects which could otherwise act as pinning centersYeshurun et
al. (1981). Frustration is monitored by the Mn concentration $x$ which
controls the relative amounts of AFM Mn-Mn nearest neighbor (NN) bonds with
respect to the FM Mn-Fe and Fe-Fe ones. Here, we study seven RSG samples of
concentrations ranging from $x=0.22$ to $x=0.35$ and a SG sample with
$x=0.41$, previously studied by neutron depolarization and muon spin
rotationMirebeau et al. (1997). Samples were prepared using the "wheelbarrow"
technique and their amorphous nature was checked using neutron diffraction
(see Supplementary Information with Supplementary Figs. 1 and 2). The
resulting ribbons, of typical thickness $\approx$ 30-70 $\mu$m and $\approx$
8-10 mm width, can be easily cut or piled-up to perform magnetic and neutron
scattering experiments.

The magnetic phase diagram of a-FeMn, inferred from ac-susceptibility (see
Supplementary Information with Supplementary Figs. 4, 5 and 6), is shown in
Fig. 1a. As already found by Yeshurun et al.Yeshurun et al. (1981), transition
lines separating the paramagnetic-FM states ($T_{\rm C}$) and FM-RSG or
paramagnetic-SG states ($T_{\rm F}$) merge at a tricritical point located at
$x_{\rm C}\approx 0.36$ (as also confirmed by magnetic susceptibilitySalamon
et al. (1980), as well as neutron depolarizationMirebeau et al. (1997)
measurements). The Curie ($T_{\rm C}$) and freezing ($T_{\rm F}$) temperatures
are respectively defined by sharp increases and decreases of the real part of
the ac-susceptibility. $T_{\rm F}$ marks the onset of strong irreversibilities
observed in the SG and mixed M2 phases. Below $x_{\rm C}$, another transition
line marks the freezing of transverse spin components and the onset of weak
irreversibilities, observed in the mixed M1 phase. The "canting" temperature
($T_{\rm K}$) associated with this transition is situated between $T_{\rm C}$
and $T_{\rm F}$, as predicted by the MF model of Gabay & ToulouseGabay and
Toulouse (1981). We also performed dc-magnetization measurements in the 0-5 T
field range in order to verify that all samples with $x<x_{\rm C}$ retain a
ferromagnetic character, while the field at which technical saturation takes
place increases with $x$ under the effect of increasing magnetic frustration
(Fig. 1b). From the corresponding Arrott plots (Fig. 1c), we also deduced the
$x$-dependence of the spontaneous moment $M_{\rm 0}=M(\mu_{0}H_{\rm
int}\rightarrow 0)$. As shown in the inset of Fig. 1c, this leads to an
extrapolated zero moment for $x=0.38(3)$, which is consistent with the
literature value of the critical concentration $x_{\rm C}$ where long-range FM
order is lost.

Figure 1: Phase diagram and magnetization of a-FeMn – (a) Magnetic phase
diagram of a-Fe1-xMnx inferred from magnetic ac-susceptibility. For
$0.2\lesssim x\lesssim x_{\rm C}$, cooling from the high-temperature FM state
leads to a sequence of two mixed states: M1, involving the freezing of
transverse spin components Mirebeau et al. (1990) and M2, where replica
symmetry is spontaneously broken (i.e., an analog of the SG state). Above
$x=x_{\rm C}\approx 0.36$, the ferromagnetic (FM) phase is suppressed and
replaced with a "canonical" spin-glass (SG) state at low temperature. Data
from MirebeauMirebeau (1987) and Yeshurun et al.Yeshurun et al. (1981) is
added for the sake of completeness. The green vertical line indicates a
putative critical line between the RSG (M2 phase) and SG regimesGabay and
Toulouse (1981). (b) Low-temperature magnetization curves of a-FeMn. (c) Arrot
plots computed from the data of panel (b). Dashed lines are linear fits to the
high-field data. Inset shows the $x$-dependence of the spontaneous
magnetization $M_{\rm 0}=M(\mu_{0}H_{\rm int}\rightarrow 0)$.

## Small-angle neutron scattering (SANS)

We have carried out a small-angle neutron scattering (SANS) experiment at the
PAXY beamline (Orphée Reactor, Saclay, France). A horizontal magnetic field
$\mu_{0}\mathbf{H}$ up to 4 T was applied transverse to the beam direction
(i.e., in the detector plane). All data were obtained in the zero field cooled
(ZFC) state at $T$ = 3 K. This temperature was chosen because (i) it is well-
below $T_{\rm C}$ and $T_{\rm F}$ for all samples (Fig. 1), and (ii) allows
neglecting the contribution of phonons and magnetic excitations (spin waves)
to the SANS patterns. Data were corrected and calibrated as described in the
Supplementary Information.

Figure 2: SANS data on a-FeMn in the RSG and SG phases – (a-c) Scattering maps
recorded at T = 5 K under an applied magnetic field of 1.5 T. (d-f) Field-
dependence of the transverse magnetic cross section $\sigma_{\rm T}(Q)$. (g-i)
Field-dependence of the longitudinal magnetic cross section $\sigma_{\rm
L}(Q)$. In panels (d-i), we indicate the values of the applied magnetic field.

Typical SANS patterns are shown in Fig. 2, for compositions $x$ respectively
below (Fig. 2a,b) and above (Fig. 2c) $x_{\rm C}$. One can immediately note
differences between these two cases. On the one hand, maxima of intensity
appear parallel to the applied field at a finite value of the momentum
transfer $Q$ when $x<x_{\rm C}$. On the other hand, SANS from the $x=0.41$
sample is typical of field-induced ferromagnetic-like correlations centered at
$Q=0$. In order to separate contributions of magnetic moments transverse ($T$)
and longitudinal ($L$) to the applied field, we make use of the neutron
selection rule which states that only components perpendicular to the
scattering vector $\mathbf{Q}$ contribute to the observed scattering cross
section $\sigma(\mathbf{Q})$. This translates into the following relations

$\sigma_{\rm T}\left(\mathbf{Q}\right)+\sigma_{\rm
bg}\left(\mathbf{Q}\right)/2=\tilde{\sigma}\left(\mathbf{Q}\parallel\mathbf{H}\right)/2\quad\text{and}\quad\sigma_{\rm
L}\left(\mathbf{Q}\right)+\sigma_{\rm
bg}\left(\mathbf{Q}\right)/2=\tilde{\sigma}\left(\mathbf{Q}\perp\mathbf{H}\right)-\tilde{\sigma}\left(\mathbf{Q}\parallel\mathbf{H}\right)/2\quad,$
(1)

where $\sigma_{\rm bg}$ denote the background contributions from the sample
(e.g., nuclear scattering) and $\tilde{\sigma}$ the full observed scattering
within sectors of 60∘ opening angle, parallel or perpendicular to
$\mathbf{H}$. Therefore, a radial integration of the SANS data along the
horizontal and vertical direction allows retrieving the $Q$-dependences of
$\sigma_{\rm T}$ and $\sigma_{\rm L}$ independently, assuming an isotropic
$\sigma_{\rm bg}$.

The result of such procedure is shown in Figs. 2d-i. In the weakly frustrated
$x=0.22$ RSG sample, the intensity is clearly enhanced along the field
direction, i.e. for $\mathbf{Q}\parallel\mathbf{H}$, showing that the
contribution of spin components transverse to the magnetic field are dominant
in the explored $Q$-range, whereas the opposite behavior is observed in the
$x=0.41$ SG sample. As a general feature, we observe field-induced peaks in
$\sigma_{\rm T}(Q)$ at $Q=Q_{\rm max}$ for all compositions $x<x_{\rm C}$.
$Q_{\rm max}$ moves to higher values when the field increases at constant $x$,
and also shows a systematic stiffening as $x$ increases towards $x_{\rm C}$.
$\sigma_{\rm L}(Q)$ shows a broad maximum, but it is more difficult to point
because its intensity is much smaller.

In what follows, we focus on the transverse cross section $\sigma_{\rm T}(Q)$,
which is defined in the most general case as

$\sigma_{\rm T}\left(\mathbf{Q}\right)\propto\langle F_{\rm
T}^{2}(Q)\rangle-\langle F_{\rm T}(Q)\rangle^{2}\left[1-S_{\rm
int}(Q)\right]\quad,$ (2)

where $F_{\rm T}(Q)$ is the form factor of the transverse defects and $S_{\rm
int}(Q)$ is an interference function that expresses the local correlations
between two such defects. Assuming that the latter are organized in a liquid-
like order ($S_{\rm int}(Q)\rightarrow 1$) and noting that the form factor of
a "regular" vortex is null ($\langle F_{\rm T}(Q)\rangle=0$)Mirebeau et al.
(2018), $\sigma_{\rm T}(Q)$ is simply proportional to $\langle F_{\rm
T}^{2}(Q)\rangle$.

In the whole $x<x_{\rm C}$-range, the field-dependence of $Q_{\rm max}$ obeys
a scaling law of the form

$Q_{\rm max}(\mu_{\rm 0}\,H_{\rm int},x)=\kappa(x)\,\left[\mu_{\rm
0}\,\left(H_{\rm int}-H_{\rm 0}(x)\right)\right]^{\gamma}\quad,$ (3)

where $H_{\rm 0}(x)$ is a composition-dependent saturation field, increasing
with $x$. A global fit of Eq. 3 to the data yields a unique exponent
$\gamma=0.39(1)$ and $x$-dependent scaling parameters $\kappa(x)$ (Figs.
3a,b). Results previously obtained on Ni0.81Mn0.19 are also reported in Fig.
3a. In this case, a fit of Eq. 3 to the data yields an exponent
$\gamma=0.34(2)$, quite close to the value derived for the a-FeMn series.

Figure 3: Scaling of vortex properties – (a) Field-dependence of the position
$Q_{\rm max}$ of the maximum in $\sigma_{\rm T}(Q)$ for each studied
compositions. Solid lines are results of a global fit of Eq. 3 to the data.
(b) $x$-dependence of the scaling parameter $\kappa$, extracted from a fit of
Eq. 3 to the data of panel (a). (c) Average canting angle $\langle\varphi_{\rm
C}\rangle$ as a function of the average vortex radius $\langle r_{\rm
d}\rangle$. In panel (a) and (c), data for Ni0.81Mn0.19 is shown for
comparison (pink crosses)Mirebeau et al. (2018). Dashed pink line is the
result of a fit of Eq. 3 to the data.

In order to account for this scaling law, we propose a simple picture and
provide a physical meaning for its parameters. First, we note band structure
calculations of dilute FeMn alloysCampbell and Gomès (1967); Mirzoev et al.
(2006); Schneider et al. (2018) where Mn-Mn NN interactions are AFM whereas
Fe-Fe and and Fe-Mn NN interactions are FM. Based on previous SANS results on
a crystalline Ni0.81Mn0.19 sample, as well as toy models and MC simulations
Mirebeau et al. (2018), we interpret $\sigma_{T}(Q)$ by assuming uncorrelated
defects akin to vortices, nucleated around Mn-Mn first neighbor pairs. In the
simplest picture, the spins components $M_{\rm T}$ are ferromagnetically
correlated and rotate over an average vortex radius $\langle
r_{d}\rangle=\pi/Q_{\rm max}$ to compensate the transverse magnetization
inside the vortexMirebeau et al. (2018).

Using this picture, we can readily interpret the evolution of the SANS
patterns with magnetic field and Mn concentration $x$. At a given $x$,
$\langle r_{d}\rangle$ decreases with increasing the magnetic field (Fig. 3a),
as the vortices progressively align along the field, albeit not necessarily in
a uniform fashion. Their gradual suppression yields a steady increase of the
magnetization (Fig. 1b). At a given field, $\langle r_{d}\rangle$ also
decreases with an increase of $x$, which governs the concentration of AFM NN
pairs within the samples. As we will show below, these features translate the
collapse of the average exchange interaction $\langle J\rangle$.

Altogether, such scaling law suggests that the characteristic size of the
defect is governed by the ratio between the magnetic field and the average
exchange interaction. To check this picture in more details, we have searched
for a common law governing the bulk magnetization curves $M(H)$ at 5 K in the
a-FeMn system. From the experimental curves, we find a
$M\approx\left(\mu_{0}\,\left[H_{\rm int}-H_{\rm
0}\right]\right)^{1/3}$-dependence above a threshold field value $H_{\rm 0}$,
which scales with the saturation field deduced from the magnetization curves
(see Supplementary Information with Supplementary Fig. 3). We can compare this
dependence with that observed in ferromagnets close to saturation, where the
magnetic field suppresses microstructural defects. Here the magnetostatic
exchange length $\Lambda$ which controls the defect size is defined
asWeissmüller et al. (1999)

$\Lambda=\left(\frac{2A}{\mu_{0}M^{2}}\right)^{1/2}\quad,$ (4)

where $A$ is the exchange stiffness and $M$ the bulk magnetization.
Identifying $A$ with the average exchange term $\langle J\rangle$ and
$\Lambda$ with $\langle r_{\rm d}\rangle$ leads to the following dependence
for $Q_{\rm max}$

$Q_{\rm max}\approx\frac{\left(\mu_{0}\,\left[H_{\rm int}-H_{\rm
0}\right]\right)^{1/3}}{\langle J\rangle^{1/2}}\quad.$ (5)

which is quite close to the dependence found experimentally (Eq. 3), noticing
that the experimental value of the exponent $\gamma=0.39(1)$ is slightly above
the value $1/3$ from macroscopic magnetization. This comparison however
confirms that the average exchange interaction and the applied magnetic field
are the main ingredients needed to control the behaviour of the observed
magnetic defects, although additional anisotropic exchange terms (such as the
Dzyaloshinskii-Moriya interaction) could play a minor role. A perfect mapping
of the two cases is in fact not expected, especially for the strongly
frustrated RSGs, where the magnetization does not show any clear saturation
plateau. Taking this analysis into account, one can however tentatively
evaluate the average exchange constant from the scaling parameter $\kappa$ by
$\langle J\rangle=\kappa^{-1/\gamma_{\rm eff}}$, with $1/3\leq\gamma_{\rm
eff}\leq 0.39$.

We can further define the canting angle $\langle\varphi_{\rm C}\rangle$,
averaged over the vortex size, by the expression

$\langle\varphi_{\rm C}\rangle=\frac{\text{arctan}\left(\langle M_{\rm
T}^{2}\rangle/\langle M_{\rm L}^{2}\rangle\right)}{\langle r_{\rm
d}\rangle}\approx\text{arctan}\left(\frac{\sigma_{\rm T}(q_{\rm
max})}{\sigma_{\rm L}(q_{\rm max})}\right)\cdot q_{\rm max}\quad.$ (6)

The result is shown in Fig. 3c, which illustrates the correlation between the
vortex average radius and $\langle\varphi_{\rm C}\rangle$.
$\langle\varphi_{\rm C}\rangle$ is maximum at low fields when the vortex size
is the largest (around 120 Å), and reaches values of 55-60 degs. As for
comparison, the canting angle deduced from the Mössbauer measurements of the
57Fe hyperfine field in the $x$ = 0.235 sample, is around 35(7) degs, with a
large distributionMirebeau et al. (1986). As the field increases, the vortex
size decreases and the canting angles decreases as well, reaching values
around 45 degrees at the smallest vortex size of $\approx$ 20 Å. To summarize
this point, the average canting angle $\langle\varphi_{\rm C}\rangle$
increases as the transverse spin components and magnetic disorder in the
transverse plane increase. Corresponding data, extracted from our previous
SANS experiment on Ni0.81Mn0.19 fall within the same range and follows a very
similar trendMirebeau et al. (2018). This suggests that the observed vortices
are relatively independent on the sample nature (single crystalline or
amorphous) and could represent an immanent feature of the large family of RSG.

## Monte Carlo simulations

In order to get a deeper insight onto the properties of the magnetic defects
evidenced in our SANS experiments, we have carried out a series of Monte Carlo
simulations on 2d square lattices containing 104 spins. The model is described
by the following classical Hamiltonian:

$\mathcal{H}=-\sum_{ij}J_{ij}\,\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\alpha
H\sum_{i}S_{i}^{z}\quad,$ (7)

where $\mathbf{S}_{i,j}$ are Heisenberg spins with $|\mathbf{S}_{i,j}|=1$,
$J_{ij}$ are random independent variables taking the value $\pm 1$,
$\alpha=\mu_{\rm B}/k_{\rm B}\approx 0.672$ is a coupling constant and the
magnetic field $H$ is applied along the $z$ direction. The first sum in Eq. 7
runs over NN pairs. All simulations started by generating random spin
configurations at a temperature $T=2J$, where a concentration $x$ of
"impurities" (i.e., analogs of Mn ions) are scattered within an otherwise
ferromagnetic matrix (i.e., analogs of Fe ions). The following rule is then
applied to calculate the sign of the nearest-neighbors exchange terms: two
nearest-neighbor impurities will be coupled antiferromagnetically
($J_{ij}=-1$) while all other pairs will be coupled ferromagnetically
($J_{ij}=1$). This situation is meant to map quite closely the one expected
from band structure calculationsCampbell and Gomès (1967); Mirzoev et al.
(2006); Schneider et al. (2018), let alone the actual atomic connectivities.
The key quantity describing the MC sample is therefore the concentration
$c_{\rm AFM}$ of AF bonds. In order to stick even more to the experimental
situation, the system is slowly cooled down to $T=0.01J$ at $H=0$ and the
field is further raised in steps $\Delta H=0.01J$. We shall show in the
following that such a simple scheme allows for a "zero-order" simulation of
the properties of the RSG, and a reasonable description of the experimental
observations reported in this paper.

We now focus on the case of a weakly frustrated RSG sample (concentration of
AFM bonds $c_{\rm AFM}\approx 0.05$) to investigate the spin configurations
and the corresponding Fourier maps as the magnetic field increases (additional
cases, displaying essentially similar behaviors, are addressed in the
Supplementary Information with Supplementary Figs. 9 and 10). In the zero-
field or low field region, vortex-like structures are observed around AFM NN
pairs, coexisting with domains walls of large length scale which separate the
magnetic domains. As shown in Fig. 4a, these domain walls can involve
transverse chiral components as well as local defects. However, a Fourier
transform of the spin maps show that they do not yield a maximum of the
scattered intensity as for the vortex-like structures, but rather a huge
increase of the intensity at low $Q$ values. As the field increases, these
walls are rapidly suppressed, leading to a strong increase of the
magnetization, and to the observation of isolated vortices. Such textures are
nucleated randomly in the sample around AFM NN pairs (Fig. 4a-c), so that in
zero (or small) applied field, they could form both in the ferromagnetic
domains and in the domain walls (Fig. 4a). However, they are observed in the
Fourier maps (i.e. by SANS) only when the field is high enough to suppress the
contribution of the domain walls. In other words, vortices emerge from a
ferromagnetic vacuum. In this high field region, magnetization shows a quasi-
plateau (Fig. 4d) while the vortex contributions remains alone, leading to a
clear maximum of the calculated scattered intensity at a finite $Q$-value. The
vortices shrink as the field increases further, and they slowly disappear
together with the AFM pairs which nucleate them. The complete destruction of
all AFM pairs should only occur at very high fields, much larger than the
exchange interaction ($H\gg J$ ). Fourier transforms of the spin maps in the
region of the magnetization plateau shows features very similar to the
experimental ones, both for the transverse and longitudinal contributions to
the cross section (compare Figs. 4e,f and Figs. 2d,e,g,h). Finally, we note
that while being resilient to very large applied fields, the vortex-like
textures obtained in the simulations have a vanishing topological charge, most
likely due to the ill-defined FM vacuum endowing them with irregular shapes.

Figure 4: Spin configurations and Fourier analysis of the MC simulations –
(a-c) Snapshot of a spin configuration ($15\times 15$ spins) obtained on a
$x=0.23$ sample ($c_{\rm AFM}\approx 0.05$) at $T/J=0.01$ for (a) $H/J=0$, (b)
$H/J=0.2$ (slightly above technical saturation) and (c) $H/J=1$. Arrows
represent the in-plane magnetization while color map shows the out-of-plane
(i.e., parallel to the applied magnetic field) component $M_{\rm Z}$. Yellow
lines border the regions where impurity spins are located, anchoring the
field-induced localized vortex-like textures (see text). (d) Field-dependence
of the average longitudinal magnetization $|M_{\rm Z}|$. (e-f) Squared Fourier
transforms of the transverse (e) and longitudinal (f) spin correlations for
different values of $H/J$.

## Discussion

As our key experimental result, we have shown that in a frustrated
ferromagnetic system, vortex-like defects are a characteristic feature of the
RSG ground state. Their average size decreases with decreasing the average
exchange interaction, and the vortices disappear in the true SG. The average
vortex size $\langle r_{\rm d}\rangle$ also decreases with increasing the
applied magnetic field. The whole behaviour is captured by a scaling law
governing $\langle r_{\rm d}\rangle$, where the only ingredient is the ratio
of the internal magnetic field to the average exchange interaction. Similar
laws derived from magnetostatics govern the field behaviour of different
macroscopic quantities. For example, one can quote the quasi saturated
magnetization of ferromagnets with microstructral defects, the magnetization
of type II superconductors, or the thickness of Bloch walls in ferromagnets.

In the RSGs, the presence of vortex-like defects up to the critical
concentration, and their collapse in the SG phase when the average exchange
interaction becomes smaller than the width of its distribution, strongly
supports the existence of a critical line between RSG and SG regions. Our
observations therefore support a MF description of the RSG phase diagram,
rather than the crossover evolution towards FM breakdown predicted by random
field arguments. We however recall that the original MF model of Gabay-
Toulouse Gabay and Toulouse (1981), although being able to correctly describe
the experimental $(x,T)$-phase diagram, cannot predict any defect, since the
transverse spin component is randomly distributed in the transverse plane. The
present observation can therefore help refining the current models for the RSG
problem, by considering the observed magnetic microstructure.

Altogether, our SANS results combined with MC simulations suggest two
complementary phenomena: (i) the vortices emerge from an average ferromagnetic
medium acting as a vacuum field, required for their stabilization and (ii)
they protect the ferromagnetic domains from breaking down under the influence
of magnetic fustration. The MC calculations strongly suggest that the vortices
are nucleated around AFM NN pairs. In real samples, these pairs likely arise
from Mn-Mn first neighbours, as suggested by band structure calculations in
Fe-MnCampbell and Gomès (1967); Mirzoev et al. (2006); Schneider et al.
(2018).

In the weakly frustrated RSG, vortices and domain walls can be clearly
distinguished by combining magnetization, SANS and MC simulations. The domain
walls recall those observed in non-frustrated ferromagnets, but they involve
AFM bonds, which induce magnetic defects where the spin canting is locally
enhanced, and which act as pinning centers at low temperature. Below $T_{\rm
F}$, this process leads to a strong decrease of the susceptibility $\chi(T)$
and to strong irreversibilities of the magnetization $M(H)$. This picture is
supported by electron microscopySenoussi et al. (1988), neutron
depolarisationMirebeau et al. (1990) and recent acoustic absorption
measurementsKustov et al. (2017), which clearly show that LRMO and
$\mu$m-sized domains are preserved in the ground state of the weakly
frustrated RSG. As the field increases, the domain walls are washed out by low
fields, whereas vortices persist up to much higher fields, where their
contribution to the SANS can be clearly identified. In the highly frustrated
RSG, the distinction between the vortices and the domain walls smears; the
average domain size decreases and becomes comparable to the domain wall
thickness, and the magnetization plateau dissappears. The vortex contribution
is still clearly observed in the SANS data, in sharp contrast with the SG
sample.

MC simulations are in turn crucial to refine the above picture, and already
extend our results to a field range (or more precisely $H/J$ range)
inaccessible to experiment. The good agreement between the Fourier transform
of the MC spin maps and our experimental results should be noticed,
considering that the simulated case is over-simplified with respect to the
experimental one. We outline here that the MC spin maps show a huge amount of
disorder around the local defects which nucleate the vortices. Considering the
chemical disorder, many different types of vortices could a priori exist in
the sample and they are indeed observed in the MC spin maps. However, their
average size can be determined without ambiguity, as it leads to a maximum in
$Q$-space, the position $Q_{\rm max}$ of which is tuned by the $H/J$ ratio.

To conclude, we briefly compare the above vortices with the topological
defects observed in ferromagnets submitted to weak random fields. Quite
generally, topological defects are expected when the number of spin components
$n$ is such that $n\leq d+1$ where $d$ is the dimension of spaceToulouse, G.
and Kléman, M. (1976), namely in all experimental cases. For instance, non-
singular skyrmions with a finite topological charge are observed in the
($n=3$, $d=2$) caseProctor et al. (2014). Their existence is a consequence of
crossing points between lines where all RF field components cancel at the same
time. This leads to the very interesting concept of "skyrmion glass", composed
of regions with oscillating positive and negative topological charges, and
sizes scaling that of the IM domainsChudnovsky and Garanin (2018).
Importantly, these defects, which prevent the magnetization from collapsing,
should lead to a measurable topological Hall effect (THE). Conversely, the
vortices stabilized by magnetic frustration (induced by competing interactions
and bare interaction randomness) have a very small topological charge due to
their very irregular shape, but they could also yield a peculiar Hall signal.
In this context, it is worth noting that an anomalous Hall effect was actually
predictedKawamura (2003) and observed in AuFe RSG or SG alloysPureur et al.
(2004); Fabris et al. (2006), as a probe of non coplanar (chiral) spin
configurations. A quantitative study of the field-dependent Hall response of
a-FeMn above and below $x_{\rm C}$ could refine the description of the RSG
ground state, given that the defects involved in the two regimes likely have
different natures.

## Acknowledgements

We thank S. Gautrot (LLB) and M. Bonnaud (ILL) for their assistance during the
small-angle neutron scattering experiments at the PAXY and D33 instruments,
respectively.

## Materials and methods

Materials. The amorphous samples of (Fe1-xMnx)75P16B3Al3 (0.22 $\leq$ x $\leq$
0.41) used in this study were prepared using the "wheelbarrow" technique,
which consists in casting molten alloy with the desired composition on a
spinning wheel. Being a strong neutron absorber, 10B was replaced with
isotopic 11B. Samples were cut in foils of about 1 cm2 surface with
thicknesses varying from 30-70 $\mu$m. These foils were piled up in order to
increase the total sample thickness and yield a large enough sample mass for
the small-angle neutron scattering experiments. Conversely, individual foils
were cut into rectangular pieces, having a height to width ratio close to 2,
for the magnetic measurements.

Magnetic measurements. The ac-susceptibility of the a-FeMn samples have been
obtained using a Quantum Design Physical Properties Measurement System (PPMS,
Dynacool 9 T, Laboratoire Léon Brillouin, France). Magnetization curves were
measured using a Quantum Design Superconducting Quantum Interference Device
magnetometer (SQUID, MPMS-XL 5 T, Technische Universiteit Delft, The
Netherlands).

Small-angle neutron scattering (SANS). SANS experiments were performed on the
PAXY instrument at the Orphée reactor (Laboratoire Léon Brillouin, Gif-sur-
Yvette, France), operated in a standard pinhole geometry. Neutron wavelength
was set to 4 and 6 Å, while keeping the sample-to-detector distance to 2.8 m.
An horizontal magnetic field was applied using a cryomagnet (Oxford SM4000),
allowing to reach fields of 10 T while cooling the sample down to 2 K.
Additionnal SANS measurements were performed on the D33 instrument (Institut
Laue Langevin, Grenoble, France)Mirebeau et al. (2015), as described in the
Supplementary Information and Supplementary Fig. 7.

Monte Carlo simulations. Monte Carlo simulations were carried out using the
"adaptative" algorithm described by Alzate-Cardona et al.Alzate-Cardona et al.
(2019). 40 maps, containing 104 spins sitting on the vertices of a square
lattice were generated at high temperature. A concentration $x$ of "impurity"
spins was scattered across the matrix in order to introduce AFM couplings
within the FM matrix (all couplings had a magnitude $J$). Each sample was
cooled down to $T=0.01J$ in zero-applied field, and the field was then
increased in small steps to study the evolution of the spin configurations
(Figs. 4a-c) and their Fourier transforms (Figs. 4e-f).

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Supplementary Information

In this supplement, we provide information concerning the synthesis and
structural characterization of the (Fe1-xMnx)75P16B6Al3 samples used in this
study (Sec. A). Composition-dependence of the magnetization and AC
susceptibility data is presented in Sec. B. In Sec. C, we describe the
strategy used to scale and analyze the small-angle neutron scattering (SANS)
data. Finally, the details of our Monte Carlo simulations on systems with
varying antiferromagnetic bond concentration are given in Sec. D.

##  Appendix A Samples synthesis and structural characterization

Amorphous samples of (Fe1-xMnx)75P16B6A3 (0.22 $\leq x\leq$ 0.41), herafter
named "a-Fe1-xMnx", were prepared using the "wheelbarrow" technique, which
consists in casting molten alloy with the desired composition on a spinning
wheel, by J. Bigot (Centre d’Études de Chimie—Metallurgie, Vitry sur Seine).
Being a strong neutron absorber, 10B ($\sigma_{\rm abs}=3835$ barn) was
replaced with isotopic 11B ($\sigma_{\rm abs}=0.0055$ barn). Samples were cut
in foils of about 1 cm2 surface with thicknesses varying from 30-70 $\mu$m.
These foils were then piled up in order to increase the total sample thickness
and yield a large enough sample mass for the small-angle neutron scattering
(SANS) experiments (see Sec. C).

Density – The density $d_{\rm a-Fe_{\rm 1-x}Mn_{\rm x}}$ of these materials is
an important value, allowing to calibrate the magnetization and SANS data.
However, it is rather difficult to measure it directly, given the small
thickness of the foils. It can nevertheless be estimated using the _random
close packing_ approximation in a hard-sphere model. The maximum density is 64
% of that of the densest crystalline arrangement, namely fcc, with a compacity
of 0.74. Taking the atomic masses $m_{\rm n}$ (with $n=\,$ {Fe, Mn, P, B, Al})
into account, one gets:

$d_{\rm a-Fe_{\rm 1-x}Mn_{\rm
x}}=\frac{0.75\cdot\left[\left(1-x\right)\,m_{\rm Fe}+x\,m_{\rm
Mn}\right]+0.16\,m_{\rm P}+0.06\,m_{\rm B}+0.03\,m_{\rm Al}}{m_{\rm
Fe}}\cdot\frac{0.64}{0.74}\cdot d_{\rm fcc\,Fe}\quad,$ (8)

where $d_{\rm fcc\,Fe}$ = 8.879 g$\cdot$cm-3 is the density of fcc Fe.
Whenever relevant, Eq. 8 is used to scale the data presented in the main text
and this supplementary material.

Amorphous nature of the samples – The amorphous nature of the samples can be
assessed using neutron diffraction. As shown in Fig. 5a in the case of
a-Fe0.765Mn0.235, the _normalized structure factor_ $S(Q)/S(Q_{\rm max})$
(where $Q_{\rm max}$ is the largest momentum transfer reached in the
experiment) lacks Bragg reflections and is characteristic of an amorphous
(liquid-like) compound. The small-$Q$ region of the pattern (see Fig. 5b) is
marked by an upturn, which is well-described by a power law of the form
$a_{\rm p}/Q^{p}$ with an exponent $p=2.97(12)$ (typical of surface roughness,
yielding $p=3$). The presence of a prepeak, before the main structural one,
indicates a possible clustering of the main chemical specie. Its intensity is
however magnified due to the fact that the scattering lengths of the main
constituents (i.e., Fe and Mn) have opposite signs.

Figure 5:  (a) Diffraction pattern of a-Fe0.765Mn0.235 (taken from
MirebeauMirebeau (1987)). Data is normalized such that $S(Q)\rightarrow 1$ in
the limit of large $Q$s. (b) Zoom into the low $Q$ part of the diffraction
pattern, showing the prepeak at $Q\approx 1.7~{}\text{\AA}^{-1}$. Line is a
fit of a power law to the data (see text).

The _pair distribution function_ (PDF) $g(r)$ is obtained from $S(Q)/S(Q_{\rm
max})$ using

$g(r)=1+\frac{1}{2\pi^{2}\,\rho_{0}\,r}\,\int_{0}^{Q_{\rm
max}}Q\,\left[\frac{S(Q)}{S(Q_{\rm
max})}-1\right]\,\sin\left(Qr\right)\,dQ\quad,$ (9)

where $\rho_{0}$ is the atomic number density. Finally, one obtains the
_radial distribution function_ (RDF) $\rho(r)$ from the PDF via

$\rho(r)=4\pi\,\rho_{0}\,r^{2}\,g(r)$ (10)

This procedure allows determining the coordination numbers $z_{\rm n}$, i.e.
the number of atoms in the nth shell surrounding any central atom, by
integrating $\rho(r)$ within the $r$-range bounded by its first two minima
(shaded region in Fig. 6b). For $n=1$, we find $z_{1}\approx 10.6$ and
$r_{1}\approx 2.6\,$Å, in good agreement with results obtained on amorphous Fe
powder Bellissent et al. (1993) and liquid Fe Waseda and Suzuki (1970)

Figure 6: (a) Pair distribution function $g(r)$ and (b) radial distribution
function $\rho(r)$ calculated from the data of Fig. 5.

##  Appendix B Macroscopic magnetic properties

###  B.1 Magnetization

The field-dependences of the magnetization $M$ of the a-Fe1-xMnx samples were
measured using a MPMS-XL 5T Quantum Design SQUID magnetometer. The samples
were zero-field cooled from $T\gg T_{\rm C},T_{\rm F}$ down to 5 K, and
specific care was taken to avoid the presence of a residual field.
Subsequently, the measurements were performed by stepwise increasing the
magnetic field. Samples masses of the order of several mg were used in order
to be able to accurately scale the magnetic moment in Bohr magneton per
formula unit ($\mu_{\rm B}$/f.u.). As shown in Fig. 7a, the field value at
which magnetization reaches quasi-saturation ($H_{\rm 0}$) increases with
increasing $x$, underscoring the increasing magnetic frustation. In all cases,
$M$ retains a finite slope up to the largest fields, as a result of the
gradual collapse of the vortex-like textures located around the AFM pairs (see
main text). In this regime, $M$ is well-described by a law of the form
$M\approx\left(\mu_{0}H_{\rm int}\right)^{1/3}$ (Fig. 7b). This property is
used in the main text to discuss the possible scaling law governing the field-
evolution of the observed nanoscopic magnetic textures. We can also define the
saturation field $H_{\rm 0}$, at which $M$ acquires the $\left(\mu_{0}H_{\rm
int}\right)^{1/3}$-dependence, and compare it with the $H_{\rm 0}$ extracted
from the SANS scaling laws, i.e. the field above which maxima in the
transverse scattering cross section $\sigma_{\rm T}$ can be defined (Fig. 3a
of main text). The good correlation between these values is illustrated by
Fig. 7c.

The Arrott plots computed from the data of Fig. 7a are shown in Fig. 7d. All
studied samples with $x<x_{\rm C}$ display a non-zero spontaneous
magnetization $M_{\rm 0}$. Theses values are plotted in Fig. 7e. The
extrapolated value of $M_{\rm 0}$ for $x\rightarrow 0$ compares well with
literature values for crystalline Fe (ShullShull and Wilkinson (1955)), and
amorphous Fe75B25 (Cowlam & CarrCowlam and Carr (1985)), Fe (Grinstaff _et
al._ Grinstaff et al. (1993)) and Fe75P12.5B12.5 (Durand & YungDurand and Yung
(1977)), see Fig. 7.

Figure 7: Macroscopic magnetization $M$ of the studied a-FeMn samples as a
function of (a) $\mu_{0}H_{\rm int}$ and (b) $\left(\mu_{0}H_{\rm
int}\right)^{1/2}$. (c) Saturation field value $H_{\rm 0}$ deduced from SANS
data as a function of $H_{\rm 0}$ deduced from magnetization curves. (d) Low
temperature Arrott plots (T = 5 K). (e) $x$-dependence of the saturated
moment, inferred from the Arrott plots. It vanishes for $x\approx 0.38$ and
extrapolates to a value of $\approx 1.86\,\mu_{\rm B}\cdot$f.u.1 for
$x\rightarrow 0$. This value is compared with previous results for crystalline
and amorphous Fe.

###  B.2 ac-susceptibility

The magnetic phase diagram presented in the main text was infered through AC
susceptibility measurements, performed using a Quantum Design Dynacool 9 T
Physical Properties Measurement Systems (PPMS) at the Laboratoire Léon
Brillouin. Unless otherwise stated, data presented in this section were
measured under an AC field of 1 kHz frequency and 10 Oe amplitude, in zero-
applied static field. In order to suppress demagnetizing field effects, we
have cut the individual foils into rectangular pieces, having a height to
width ratio close to 2 in each case. This however lead to very small samples
masses (< 100 $\mu$g) and therefore to relatively weak signals. The AC field
was applied in the sample plane, along its larger dimension. In what follows,
we show how phase boundaries are deduced from maxima in $d\chi^{\prime}(T)/dT$
curves.

Table 1: Dimensions of the a-Fe1-xMnx samples used for the AC susceptibility measurements. $x$ | Height (mm) | Width (mm)
---|---|---
0.22 | 4.7 | 2.3
0.235 | 4.4 | 2.4
0.247 | 4.2 | 2.2
0.3 | 4.6 | 2.2
0.32 | 5.0 | 2.2
0.35 | 4.3 | 1.8
0.41 | 4.0 | 2.1

The temperature-dependence of the real ($\chi^{\prime}$) and imaginary
($\chi^{\prime\prime}$) part of the AC susceptibility of a-Fe1-xMnx samples
with $0.22\leq x\leq 0.41$ is shown in Figs. 8-10. For compositions $x\leq
0.32$, the Curie ($T_{\rm C}$) and spin freezing ($T_{\rm F}$) temperatures
are easily evidenced by well-separated extrema in the first temperature
derivative of $\chi^{\prime}$ (Figs. 8 and 9), (i.e. using the same procedure
as used by YeshurunYeshurun et al. (1981)). The $x=0.35$ case, located very
close to the RSG-SG thershold composition $x_{\rm C}\approx 0.36$, is more
difficult to anlyze. At first glance, its ac-susceptibility is very close to
that of the pure SG with $x=0.41$ (Fig. 10). However, a modest field has a
large impact on the $d\chi^{\prime}/dT$ of the $x=0.35$ sample as opposed to
the $x=0.41$ one. This suggests a remanence of ferromagnetism in the former,
disappearing in the latter case.

In all cases, $\chi^{\prime\prime}$ peaks at temperatures slightly higher than
$T_{\rm F}$, while it falls off to $\approx 0$ around $T_{\rm C}$. Since
$T_{\rm F}$ is known to depend on the ac-field frequency $f_{\rm ac}$, we have
measured the temperature-dependence of the ac-susceptibility of all samples
for 100 Hz $\leq f_{\rm ac}\leq$ 10 kHz (not shown). The various $T_{\rm F}$
shown in Fig. 1a of the main text are the results of extrapolations to $f_{\rm
ac}=1$ Hz.

Figure 8: Zero-field AC suceptibility of a-Fe1-xMnx samples with 0.22
$\mathbf{\leq}$ x $\mathbf{\leq}$ 0.247 in zero applied field – (Left column)
Real part of the AC susceptibility $\chi^{\prime}$ (dots) and its first
temperature derivative $d\chi^{\prime}/dT$ (solid lines). (Right column)
Imaginary part of the AC susceptibility $\chi^{\prime\prime}$. Data is
normalized to samples’ surfaces, such that $\chi^{\prime}$ and
$\chi^{\prime\prime}$ are expressed in emu.g-1.Oe-1.mm-2, and
$d\chi^{\prime}/dT$ in emu.g-1.Oe-1.mm-2.K-1. Figure 9: AC suceptibility of
a-Fe1-xMnx samples with $x$ = 0.3 and 0.32 in zero applied field – (Left
column) Real part of the AC susceptibility $\chi^{\prime}$ (dots) and its
first temperature derivative $d\chi^{\prime}/dT$ (solid lines). (Right column)
Imaginary part of the AC susceptibility $\chi^{\prime\prime}$. Data is
normalized to samples’ surfaces, such that $\chi^{\prime}$ and
$\chi^{\prime\prime}$ are expressed in emu.g-1.Oe-1.mm-2, and
$d\chi^{\prime}/dT$ in emu.g-1.Oe-1.mm-2.K-1. Figure 10: AC suceptibility of
a-Fe1-xMnx samples with $x$ = 0.35 and 0.41 in zero and small applied field –
(a,b) Real ($\chi^{\prime}$) and imaginary ($\chi^{\prime\prime}$) part of the
AC susceptibility of a-Fe0.65Mn0.35. (c,d) Real ($\chi^{\prime}$) and
imaginary ($\chi^{\prime\prime}$) part of the AC susceptibility of
a-Fe0.59Mn0.41. (e,f) First temperature-derivative of $\chi^{\prime}$. While a
weak applied field has a substantial effect of the high-temperature minimum in
$d\chi^{\prime}/dT$ for the $x$ = 0.35 sample, it remains unchanged in the $x$
= 0.41 case. In all panels, data is normalized to samples’ surfaces, such that
$\chi^{\prime}$ and $\chi^{\prime\prime}$ are expressed in emu.g-1.Oe-1.mm-2,
and $d\chi^{\prime}/dT$ in emu.g-1.Oe-1.mm-2.K-1.

##  Appendix C Small-angle neutron scattering

###  C.1 Experimental geometry

The small-angle neutron scattering (SANS) experiment described in the main
text was performed on the PAXY instrument at the Orphée reactor (LLB, Gif-sur-
Yvette, France). We have used a standard pinhole geometry, with parameters
given in Tab. 2.

Table 2: Parameters used for the SANS experiment on the PAXY instrument. Neutron wavelength | $\rightarrow$ | 4 and 6 Å
---|---|---
Source aperture (diameter) | $\rightarrow$ | 16 mm
Collimation length | $\rightarrow$ | 2.25 m
Sample aperture (diameter) | $\rightarrow$ | 5 mm
Sample-to-detector distance | $\rightarrow$ | 2.8 m

A horizontal magnetic field $H=0-4\,$T was supplied using an Oxford 10 T
cryomagnet (Spectromag SM4000). Throughout the experiment, field was applied
perpendicular to the beam direction in order to optimally resolve the
azimuthal asymmetry of the magnetic scattering (see Eq. 1 of main text).
Samples were wrapped into a thin Al foil, sandwiched between two Cd slabs (to
suppress background and get a well-defined sample surface with 5 mm diameter)
and stuck on a Cu frame (to insure a good thermal conduction with the
thermometer).

Figure 11: Experimental sample geometry used during the SANS experiment. The
horizontal magnetic field $\vec{H}$ was applied transverse to the largest
sample dimension.

For practical reasons, the magnetic field could not be applied along the
largest dimensions of the samples (see Fig. 11). In order to obtain scaling
laws depending on the internal magnetic field $\mu_{\rm 0}H_{\rm int}$
experienced by the samples, we have calculated the demagnetizing field using
magnetization data presented in Sec. B (for which demagnetizing field was
negligible) and the demagnetization factor for a very flat ellipsoid given by
OsbornOsborn (1945) (Eq. 2.24). The latter approximation is justified in view
of the effective sample dimensions (Tab. 3).

Table 3: Dimensions of the a-FeMn samples used in the SANS study. $x$ | Thickness (mm) | Width (mm) | Height (mm)
---|---|---|---
0.22 | 3.0 | 9.1 | 30.9
0.235 | 2.1 | 8.0 | 36.6
0.247 | 2.3 | 8.9 | 38.0
0.3 | 1.6 | 8.6 | 37.7
0.32 | 1.5 | 9.0 | 21.6
0.35 | 1.9 | 9.2 | 33.8
0.41 | 4.9 | 8.7 | 33.0

###  C.2 Data correction

In order to get scattering cross sections in absolute units, we follow the
usual data reduction procedure. First, the contribution from the environment
and direct beam are removed using

$I^{\rm sub}(Q)=\frac{I(Q)/t(Q)-\frac{t(Q)}{t_{\rm empty~{}cell}(0)}\cdot
I_{\rm empty~{}cell}(Q)/t_{\rm empty~{}cell}(Q)}{\Omega(Q)}\quad,$ (11)

where $I(Q)$, $t(Q)$ and $\Omega(Q)$ are the $Q$-dependent raw intensities,
sample transmissions and solid angles subtended by the corresponding detector
pixelsBrûlet et al. (2007). In Eq. 11, the subscript "empty cell" denotes a
measurement performed using the same sample holder assembly as for the sample
(including the Al foil).

This subtraction procedure is applied to the SANS from the samples and from a
Ni single crystal. The latter is used to transform the observed intensities in
absolute cross sections, according to

$\sigma(Q)=\frac{I_{\rm a-FeMn}^{\rm sub}(Q)\cdot t_{\rm Ni}(0)\cdot d_{\rm
Ni}\cdot e_{\rm Ni}}{\langle I_{\rm Ni}^{\rm sub}(Q)\rangle\cdot t_{\rm
a-FeMn}(0)\cdot d_{\rm a-FeMn}\cdot e_{\rm a-FeMn}}\cdot\sigma_{\rm Ni}^{\rm
inc}\quad,$ (12)

where $t$, $d$ and $e$ denote transmission, atomic density (see Sec. A) and
sample thickness (see Tab. 3), respectively, while $\sigma_{\rm Ni}^{\rm
inc}=5.2/4\pi$ barn.sr-1 is the incoherent scattering cross section of Ni
($\langle...\rangle$ denotes the average over the detector surface, where
incoherent scattering of Ni is expected to be flat).

###  C.3 Effect of a cooling field on the SANS of the x = 0.22 sample

As noted in seminal experimentalPréjean et al. (1980) and theoreticalFert and
Levy (1980) studies of spin glasses, the anisotropy field maintaining the
remanent magnetization in the direction of an initial applied field strongly
depends of the elements composing the studied material. This comes from
additional terms in the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction for
atoms with large spin-orbit coupling (such as Au, Pt, etc.), leading to
Dzyaloshinskii-Moriya (DM) anisotropy of unidirectional type. In SG and RSG,
this DM anisotropy modifies the torque magnetization, the shape and position
of the hysteresis cycle, and the effect of a cooling field.

When the DM term is small, an applied field causes the magnetization to rotate
"rigidly" against the DM-induced domain anisotropy. On the other hand, when
the DM term is strong, domain configurations may relax, yielding a
distribution in the strength of the resulting frictional torque. In the former
case, this leads to displaced narrow hysteresis loops under field cooling (FC)
conditions while, in the latter case, one observe undisplaced broadened loops,
independent of the cooling conditions.

Rotational magnetization measurements in a-FeMn ($x=0.235$) by Goeckner &
KouvelGoeckner and Kouvel (1991) have revealed that this system is much less
"rigid" than e.g. Ni1-xMnx, where the anisotropy field is much smallerKouvel
et al. (1987).

To test the influence of the DM anisotropy on the vortex size, we have checked
the effect of a cooling field on the scaling law of their average size vs
field $Q_{\rm max}=f(\mu_{\rm 0}H_{\rm int})$ (see Eq. 3 of main text) in both
systemsMirebeau et al. (2015). The result is shown on Fig. 12, clearly
demonstrating that $Q_{\rm max}$ is progressively shifted upwards by a cooling
field in Ni0.81Mn0.19 while it remains unchanged in the case of
a-Fe0.78Mn0.22. In other words, FC induces an extra magnetic field which
reduces the vortex size in NiMn, whereas it has no effect in a-FeMn. This
shows that the DM anisotropy indeed plays a role on the vortex landscape, as
it does on the magnetization, and underscores the strong relation between the
spin vortices and the underlying ferromagnetic vacuum.

Figure 12: Field-dependence of the position $Q_{\rm max}$ of the maximum in
transverse magnetic cross section $\sigma_{\rm T}(Q)$ for (a) Ni0.81Mn0.19
single crystalMirebeau et al. (2018) and (b) a-Fe0.78Mn0.22.

##  Appendix D Monte Carlo simulations

###  D.1 Model and parameters of the simulations

Monte Carlo simulations presented in the main text have been performed using a
standard local update algorithm on a square lattice containing $L\times
L=10^{4}$ spins, allowed to point in all directions of the 3d space (i.e. we
work out a 2d Heisenberg model). "Impurity" spins are randomly spread over the
matrix with the desired concentration $x$ to form a binary alloy with
composition A1-xBx. These impurities emulate the Mn ions and are
antiferromagnetically (AFM) coupled with other first neighbor impurity spins
($J=-1$), while all other couplings are ferromagnetic (FM, $J=+1$). Moments
are taken to be equal for all spins, namely $M=1$.

In order to obtain reasonable convergence times, we have used the "adaptative"
algorithm proposed by Alzate-Cardona et al.Alzate-Cardona et al. (2019). Trial
moves of individual spin orientations are performed within a cone having an
opening angle $\nu=60^{\circ}$ with respect to the initial spin orientation.
After each Monte Carlo step (MCS, corresponding to $10^{4}$ moves), $\nu$ is
modified according to $\nu_{\rm new}\rightarrow\nu_{\rm old}\times f$ with

$f=\frac{0.5}{1-R_{\rm old}}\quad,$ (13)

where $R_{\rm old}$ is the acceptance rate observed during the previous MCS.
This procedure allows keeping the average acceptance rate of the algorithm
close to 50 %, thereby leading to a relatively quick convergence. For each
move, the classical energy

$\mathcal{H}=-\sum_{ij}J_{ij}\,\mathbf{S}_{i}\cdot\mathbf{S}_{j}-0.672\,H\sum_{i}S_{i}^{z}$
(14)

is calculated, where $\mathbf{S}_{i,j}$ are Heisenberg spins, $J_{ij}$ are
random independent variables taking the value $\pm 1$ and the magnetic field
$H$ is applied along the $z$ direction. The first sum in Eq. 14 runs over
nearest-neighbor pairs. The factor 0.672 $(=\mu_{\rm B}/k_{\rm B})$ appearing
in Eq. 14 allows getting energies in K for magnetic fields expressed in T. The
acceptance of each trial move is tested against the Maxwell-Boltzmann
statistics $\exp\left(-\Delta E/T\right)$, where $\Delta E$ is the energy
difference between the "old" and "new" configurations). For each studied
composition, we performed between 24 and 40 realizations of the following
sequence:

  1. 1.

Random generation of the spin matrix at $T/J=2$ and $H=0$ with a chosen
$c_{\rm imp}$,

  2. 2.

Slow cooling down to $T/J=0.01$ at $H=0$ (500 MCS per step),

  3. 3.

Field sweep in the $H/J=0-1$ range, in small steps of $\Delta H/J=0.01$ (500
MCS per step).

###  D.2 Data analysis & x-dependences

In order to compare the result of our MC simulations with that of the SANS
experiment, we compute the square of the Fourier transform of the spin
matrices which is formally equivalent to the scattering cross section in the
absence of correlation between defects (see Eq. 2 of main text). As in the
experimental case, one can separate correlation functions for the transverse
(T) and longitudinal (L) magnetization. Illustrative examples are shown in
Fig. 4 of main text. Note that the explored momentum range is bounded
downwards by the size of the spin maps ($100\times 100$) and upwards by the
nearest neighbor (NN) distance, but remains much larger than the one covered
by SANS (see below).

Figure 13: (a) $x$-dependence of the magnetization curves derived from the MC
simulations. Inset shows the $x$-dependence of the spontaneous magnetization
$M_{\rm 0}$. (b) Field-dependence of the position $Q_{\rm max}$ of the maximum
in $\left|F_{\rm T}(Q)\right|^{2}$ for each studied compositions. Solid lines
are results of a global fit of Eq. 3 of main text to the data. (c)
$x$-dependence of the scaling parameter $\kappa$, extracted from a fit of Eq.
3 of main text to the data of panel (b).

In addition to the $x=0.23$ ($c_{\rm AFM}\approx 0.05$)-case discussed in the
main text, we have also explored different concentrations in order to check
the applicability of our primitive simulations when $x$ changes within the
weakly frustrated side of the phase diagram (i.e., for $\approx 0.05\lesssim
c_{\rm AFM}\lesssim 0.26$). The obtained magnetization curves are plotted in
Fig. 13a, showing the same behavior as a function of increasing frustration as
the experimental ones (Fig. 1b of main text and Fig. 7a of this supplement).
We have also determined the scaling laws of $Q_{\rm max}$ as a function of the
applied magnetic field (Fig. 13b). A global fit of Eq. 3 of main text to the
data yields an exponent $\gamma=0.49(1)$ that is slightly different from the
experimental value ($\approx 0.39$). Of course, our simulations are only
taking NN interactions on a square lattice into account. It is not surprising
that the scaling laws are renormalized by the effect of an increased number of
first neighbors (see Sec. A) and longer-ranged interactions in the real
amorphous metallic samples. However, the simulated scaling laws are very
similar to the experimental ones in the sense that the scaling paramter
$\kappa$, monitoring the "stiffness" of these curves, increases linearly with
$x$ in agreement with the experiment (Fig. 13c). Taking these results
together, we find that such a simplified model already captures many
experimental features. Moreover, the MC simulations show that the scaling laws
are indeed verified up to large concentrations of AF interactions, and to high
values of the $H/J$ ratio.

This simple model could of course be extended to account e.g. for the behavior
at the RSG-SG threshold, taking more realistic values for the moments,
exchange constants and atomic connectivities.

As discussed above, the MC simulations allow extending the explored $Q$-range
to values $Q\approx 1$, sensitive to the smallest interatomic distance. This
is an interesting asset, since one can expect in this Q-range (which naturally
evades the SANS window), a growing AFM contribution to the scattering pattern
as $x$ increases. As shown in Fig. 14, where we have selected field values
such that $Q_{\rm max}$ stays constant for different $x$, this is indeed the
case.

Figure 14: Squared Fourier transforms of the transverse (a) and longitudinal
(b) spin correlations for different $x$ and magnetic field values, chosen to
yield a constant $Q_{\rm max}$ (see text).