# Symmetry-breaking pathway towards the unpinned broken helix

E. Donoway  T. V. Trevisan  A. Liebman - Peláez  R. P. Day  K. Yamakawa  Y.
Sun  Department of Physics, University of California, Berkeley, California
94720, USA  Materials Science Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA  J. R. Soh  Institute of Physics, École
Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland  D.
Prabhakaran  A. T. Boothroyd  Department of Physics, University of Oxford,
Clarendon Laboratory, Oxford, OX1 3PU, UK  R. M. Fernandes  School of Physics
and Astronomy, University of Minnesota, Minneapolis, MN, 55455, USA  J. G.
Analytis  J. E. Moore  J. Orenstein  V. Sunko<EMAIL_ADDRESS>Department
of Physics, University of California, Berkeley, California 94720, USA
Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley,
California 94720, USA

###### Abstract

One of the prime material candidates to host the axion insulator state is
EuIn2As2. Initial first-principles calculations predicted the emergence of
this exotic topological phase based on the assumption of a simple collinear
antiferromagnetic structure. Recently, however, neutron scattering
measurements revealed a much more intricate magnetic ground state,
characterized by two coexisting magnetic wavevectors, reached by successive
thermal phase transitions. The proposed high and low temperature phases were a
spin helix and a state with interpenetrating helical and Néel
antiferromagnetic order, termed a ‘broken helix,’ respectively. Despite its
complexity, the broken helix would still protect the axion state because the
product of time-reversal and a rotational symmetry is preserved. Here we
unambiguously identify the magnetic structure associated with these two phases
of EuIn2As2 using a multimodal approach that combines symmetry-sensitive
optical probes, scattering, and group theoretical analysis. We find that the
higher temperature phase is characterized by a variation of the magnetic
moment amplitude from layer to layer, with the moment vanishing entirely in
every third Eu layer. The lower temperature structure is similar to the
‘broken helix’, with one important difference: due to local strain the
relative orientation of the magnetic structure and the lattice is not fixed,
resulting in an ‘unpinned broken helix’. As a result of the consequent
breaking of rotational symmetry, the axion phase is not generically protected
in EuIn2As2. Nevertheless, we show that it can be restored if the magnetic
structure is tuned with uniaxial strain. Finally, we present a spin
Hamiltonian that identifies the spin interactions that account for the complex
magnetic order in EuIn2As2. Our work highlights the importance of a multimodal
approach in determining the symmetry of complex order-parameters.

##  I Introduction

The search for materials exhibiting novel emergent properties relies on the
identification of their characteristic symmetries. Examples of such properties
include the anomalous and topological Hall effects, arising from Berry
curvature in momentum[1] and real space [2], respectively, momentum-dependent
Zeeman splitting of electronic bands [3], and quantized response functions in
topological systems [4, 5]. These phenomena all depend on the underlying
magnetic order and present the exciting prospect of tuning by manipulating the
symmetry of the magnetic state.

Figure 1:  (a) The side view and (b) the top view of the crystal structure of
EuIn2As2. (c-e) A table showing the three magnetic structures discussed for
EuIn2As2 in previous work: the theoretically assumed A-type antiferromagnet
(AFM), the helix proposed in Phase I, and the broken helix proposed in Phase
II. In all the structures the moments lie in Eu planes, and individual layers
are ferromagnetically aligned. For each structure we show the layer-resolved
spin-orientation, with the hexagon denoting the lattice orientation, as well
as a top view (last row), with spins labeled by the layer number. (f) The
magnetic unit cell, corresponding to three crystallographic unit cells. We
mark odd and even layers with white and colored arrowheads, respectively.

Promising material candidates for observing these phenomena can be identified
by ab initio calculations [6, 7]. Of necessity, such calculations are based on
an assumed magnetic structure, as it is difficult to reliably predict the
magnetic ground state. An effective search strategy is to combine ab initio
calculations with the magnetic order deduced from scattering measurements.
However, in some of the most interesting material systems the interpretation
of scattering data can lead to ambiguities of critical importance. For
example, states that exhibit order characterized by multiple symmetry-related
wavevectors (Q) pose a challenge, since the diffraction pattern does not
readily distinguish phase-sensitive mixed order from an equal population of
domains characterized by a single Q. Similarly, domains complicate determining
the orientation of magnetic moments even in a single-Q structure. The distinct
scenarios consistent with a given diffraction pattern are radically different
from a symmetry perspective, and have distinct consequences for response
functions and topological properties, motivating specialized scattering
experiments [8, 9, 10], and comparisons with complementary experimental
techniques [11, 12].

EuIn2As2, whose structure is shown in Fig. 1(a,b), is a perfect example of the
scenario outlined above. Ab initio calculations, based on an assumed
antiferromagnetic structure (AFM, Fig. 1c), predicted that it hosts the
elusive axion insulator state [13], exhibiting quantized responses to
electromagnetic fields [14, 15]. However, a more complex magnetic behavior was
uncovered by neutron scattering measurements, with two close-by transitions at
$T_{N1}\approx 17.5\,\mathrm{K}$ and $T_{N2}\approx 16\,\mathrm{K}$ [16], in
which two distinct propagation vectors appeared sequentially. Neither of the
two phases appearing at $T_{N1}$ and $T_{N2}$ are consistent with the
previously assumed AFM structure. Nonetheless, the magnetic structures
assigned to the two phases (Fig. 1d,e) were shown to host the axion state
[16].

Here we introduce a multimodal approach that reveals a different picture of
how the magnetic order of EuIn2As2 evolves with temperature. We combine
information from scattering experiments and symmetry-sensitive optical probes
with group theory analysis to identify two magnetic states that would have
remained hidden to the application of any single technique. A higher-
temperature Phase I ($T_{N1}>T>T_{N2}$) is characterized by the onset of a
single wavevector, $\textbf{Q}_{1}$, which we identify as a ‘nodal amplitude-
modulated state’, in which the expectation value of the magnetic moment
vanishes on every third Eu layer. In the lower-temperature Phase II
($T<T_{N2}$) an additional wavevector $\textbf{Q}_{2}$ appears, forming an
‘unpinned broken helix’, in which the orientation of the magnetic moments with
respect to the crystal axes varies continuously with location on the sample.
Only special orientations, with moments aligned to high-symmetry directions of
the lattice, maintain symmetries that protect the topological phase.

Although the experimentally-informed group theory analysis, mentioned above,
is well-suited to identifying the magnetic phases that emerge in EuIn2As2, it
does not reveal their microscopic origin. To address this point, we propose a
minimal spin model that captures the unpinned broken helix as the ground
state. We show that this state requires the exchange interactions,
$J(\textbf{Q})$, to peak sharply at two values of the wavevector Q. This
requires long-ranged interactions, naturally arising from the coupling of
itinerant electrons to the magnetic degrees of freedom. We further show that
such exchange interactions alone cannot account for the higher-temperature
amplitude modulated state. Whether it can be stabilized by thermal
fluctuations close to the two transitions is an interesting question posed by
our work.

This paper is organized as follows. We summarize the current understanding of
EuIn2As2 in Sec. II. In Sec. III we introduce two optical techniques,
sensitive to rotational and time-reversal symmetries, and show the results of
these experiments as a function of temperature, position on the sample, and
cooling protocol. We demonstrate that the symmetries revealed by these
measurements are inconsistent with the previous understanding of magnetic
phases in EuIn2As2, and therefore motivate a new analysis of the order
parameters. In Sec IV we perform a systematic analysis of symmetry-allowed
magnetic structures by combining scattering and optical data with group
theory. We uniquely identify Phases I and II as a nodal amplitude-modulated
state and an unpinned broken helix, respectively. We complement this analysis
with a phenomenological Landau free energy model (Sec V), demonstrating the
symmetry-breaking pathway towards the ground state. Furthermore, we show
experimentally and theoretically how the symmetry of the magnetic ground state
and electronic topology can be tuned by uniaxial strain (Sec. VI). Finally, in
Sec VII we introduce a microscopic spin-Hamiltonian, pointing to the
importance of the coupling between electronic and magnetic degrees of freedom.
We conclude by emphasizing that our measurements and analyses show that
EuIn2As2 hosts a remarkably rich and tunable system of coupled electrons and
localized moments, opening doors for future efforts to control topological
phases.

##  II Current understanding of $\mathrm{EuIn}_{2}\mathrm{As}_{2}$

EuIn2As2 is a rare-earth-based magnetic material with triangular layers of Eu
stacked along the crystallographic $\bm{c}$ direction and separated by blocks
of In2As2. Hereafter, we set $\bm{c}$ parallel to $\hat{z}$ (Fig. 1a,b).
Magnetism arises from the localized moments of the Eu2+ ions ($S=7/2$, $L=0$).
The moments in each Eu layer are ferromagnetically aligned with respect to
each other, and localized in the Eu-Eu planes. This easy-plane anisotropy,
confirmed by magnetization and neutron scattering experiments [16], is
implicitly assumed throughout this paper; when we refer to ‘moment
orientation’, we are referring to the orientation within the Eu planes.

The low-energy electronic bands, dominated by the 5s orbitals of In and 4p
orbitals of As, were predicted to host topological properties. Combined with
the Eu-based magnetism, this makes EuIn2As2 a promising platform to explore
the interplay between topology and magnetism, as has been proposed and debated
in the context of several Eu-based magnetic materials [17, 18, 19, 20, 21, 22,
23, 24]. More specifically, EuIn2As2 was predicted to host the axion insulator
state [13, 16], a phase characterized by half-quantized magneto-electric
coupling in the bulk and half-quantized Hall surface conductivity. The axion
phase requires $\mathcal{T}$ to be broken in order to gap the surface states,
while the quantization is protected by another symmetry that reverses the sign
of the magnetoelectric coupling constant, such as spatial inversion
($\mathcal{P}$), $\mathcal{T}$ combined with half-translations, or the product
of $\mathcal{T}$ and two-fold rotations ($\mathcal{T}C_{2}$).

The original prediction of an axion insulator state in EuIn2As2 was based on
first-principles calculations that showed an insulating bulk and an A-type
antiferromagnetic order illustrated in Fig. 1c. Note that this Néel-like
magnetic order does not change the periodicity of the lattice, since the
paramagnetic unit cell already contains two Eu layers; as a result, the
magnetic ordering vector coincides with the Bragg wave-vector $(0,0,1)$.
Importantly, this magnetic configuration preserves spatial inversion, which
protects the axion phase [13]. However, experiments uncovered a different, but
intriguing, picture: EuIn2As2 shows metallic DC [25, 26] and optical [27]
transport, it has a Fermi surface [28, 25], and the magnetic structure is
considerably more complex. Both neutron [16] and resonant x-ray [29]
scattering experiments found two consecutive magnetic transitions
($T_{N1}\approx 17.5\,\mathrm{K}$, $T_{N2}\approx 16\,\mathrm{K}$),
corresponding to the onsets of two distinct wave vectors. At $T_{N1}$ a single
propagation vector, $\bm{Q}_{1}=(0,0,1/3)$, is observed, while at $T_{N2}$ an
additional propagation vector, $\bm{Q}_{2}=(0,0,1)$, emerges. The higher and
lower temperature phases that we refer to as Phase I and Phase II,
respectively, were interpreted as a 60°-helix (Fig. 1d) and a novel magnetic
state exhibiting interpenetrating Type A-AFM and spin-helical orders,
respectively. The latter phase was termed ‘broken helix’ (Fig. 1e). Despite
this complexity, both phases still host an axion insulator state that, in this
case, is protected by a magnetic symmetry $\mathcal{T}C_{2}$ that the product
of time-reversal and two-fold rotation [16]. In the following, we demonstrate
that the findings of our optical experiments challenge this established
picture.

##  III Optical probes

We measure optical reflectance to extract information on the three-fold
rotational symmetry around $\hat{z}$ ($C_{3z}$) and time-reversal symmetry
($\mathcal{T}$); the latter we probe in two complementary ways. In this
section we briefly describe the physical principles behind these measurements,
and the relevant experimental considerations. We then show the findings of
these measurements performed on EuIn2As2 as a function of temperature,
position on the sample and cooling protocol. We discuss how the unique
capability of these probes reveals information incompatible with the current
understanding of EuIn2As2.

Figure 2: Schematic of the optical setup used for measurements of the
polarization rotation as a function of incident polarization. For thermal
modulation (T-mod), an optically chopped second (pump) beam is spatially
overlapped with the probe beam on the sample surface and used to modulate the
temperature of the sample by heating it. During field modulation (H-mod), an
AC magnetic field is applied along the c-axis of the sample with a coil.
During thermal (field) modulation, the field (pump beam) is turned off.

###  III.1 Symmetry sensitive experiment

The reflectance at normal incidence is described by a $2\times 2$ matrix,
which can be parameterized as:

$\bm{r}=r_{0}\mathbbm{1}+\delta r_{s}\left(\begin{array}[]{cc}\cos
2\theta_{0}&\sin 2\theta_{0}\\\ \sin 2\theta_{0}&-\cos
2\theta_{0}\end{array}\right)+\delta r_{a}\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right),$ (1)

where $\mathbbm{1}$ denotes the unit matrix, and $\theta_{0}$ and
$\theta_{0}+\pi/2$ correspond to the principal optical axes. $r_{0}$ is the
isotropic contribution allowed in all materials, whereas $\delta r_{s}$ and
$\delta r_{a}$, known as birefringence and the Kerr effect, respectively,
contain distinct symmetry information. The crucial difference between them is
their behavior with respect to the exchange of indices: $\delta r_{s}$ is
symmetric and $\delta r_{a}$ anti-symmetric. The general reciprocity relations
[30, 31] enforce $\delta r_{a}=0$ in $\mathcal{T}$-invariant systems; thus
$\delta r_{a}\neq 0$ is an unambiguous probe of time-reversal symmetry
breaking. In a reflectivity experiment the Kerr effect manifests as a change
of polarization orientation upon reflection by an angle of $\delta
r_{a}/r_{0}\leavevmode\nobreak\ \,\mathrm{rad}$.

In contrast, $\delta r_{s}$ in not sensitive to $\mathcal{T}$, but is
forbidden by rotational symmetry $C_{nz}$, if $n\geq 3$. A non-zero $\delta
r_{s}$ means that the reflectance depends on the relative orientation of the
light polarization and the lattice. Like $\delta r_{a}$, $\delta r_{s}$ also
induces a polarization rotation upon reflection, however the magnitude of the
change depends on the incident polarization. In particular, the rotation
vanishes when the incident polarization is aligned to the principal optical
axes. As we show below, this fact is the basis for our detection of rotational
symmetry breaking.

Further symmetry-sensitive information can be gleaned from the change of
reflectance in an applied magnetic field $H_{z}$. To linear order the
symmetric and antisymmetric components of the field-induced change to
reflectance, $r_{H}$, can be parameterized as:

$\bm{r}_{H}=\delta\alpha_{s}H_{z}\left(\begin{array}[]{cc}\cos
2\theta_{H}&\sin 2\theta_{H}\\\ \sin 2\theta_{H}&-\cos
2\theta_{H}\end{array}\right)+\delta\alpha_{a}H_{z}\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right).$ (2)

With the additional $\mathcal{T}$-odd factor, $H_{z}$, the reciprocity
relations now allow the anti-symmetric term ($\delta\alpha_{a}$) in all
materials, while the symmetric term ($\delta\alpha_{s}$) indicates breaking of
time reversal symmetry. A nonzero ($\delta\alpha_{s}$) is usually referred to
as the linear magneto-optic effect [32], or linear magneto-birefringence
(LMB). The principal axes of the LMB response are given by $\theta_{H}$ and
$\theta_{H}+\pi/2$. Although both $\delta r_{a}$ and $\delta\alpha_{s}$
require time-reversal to be broken, the symmetry conditions that enforce them
to be non-zero are different. This follows from the first term of Eq. 2, which
shows that $C_{3z}$ must be broken in addition to $\mathcal{T}$ for
$\delta\alpha_{s}$ to be non-zero.

The three distinct symmetry-sensitive quantities that we obtain through
reflectance measurements are therefore $\delta r_{s}$, $\delta r_{a}$, and
$\delta\alpha_{s}$. All of them induce changes of light polarization upon
reflection, and can be distinguished from each other by our optical techniques
[33, 34, 35]. The fundamental observable is the rotation of the angle of
linear polarization ($d\phi$) about the optical axis at normal incidence, as a
function of sample orientation. Since we cannot physically rotate the sample,
we access the same information by rotating the incoming light polarization
($\phi$). The change of polarization upon reflection at temperature $T$ and
field $H_{z}$ is given by:

$d\phi=A\left(T,H_{z}\right)\sin\left[2\left(\phi-\theta\left(T,H_{z}\right)\right)\right]+B\left(T,H_{z}\right).$
(3)

The amplitude of the sinusoidal variation, $A\left(T,H_{z}\right)$, and the
principal axis orientation, $\theta\left(T,H_{z}\right)$, are determined by
the symmetric part of the reflectance tensors ($\delta r_{s}$,
$\delta\alpha_{s}$), while the constant offset $B\left(T,H\right)$ originates
from the anti-symmetric $\delta r_{a}$ and $\delta\alpha_{a}$. It is now clear
how the quantities of interest can be experimentally distinguished: in a
$H_{z}=0$ measurement the sinusoidally-varying and constant $d\phi$ originate
from $\delta r_{s}$ and $\delta r_{a}$, respectively, while $\delta\alpha_{s}$
is captured by a sinusoidal variation proportional to a magnetic field.

In practice, we perform measurements in two modes (Fig. 2; see Appendix S2 for
details). In the temperature modulated mode ($T$-mod) we modulate temperature
at a frequency $f\approx 2\,\mathrm{kHz}$ using a second laser beam as a
heater, therefore measuring the temperature derivative of Eq. 3. This
experimental procedure reveals the same information about symmetry as the
unmodulated experiment at $H_{z}=0$ ($\delta r_{s}$, $\delta r_{a}$), while
enhancing sensitivity and rejecting contributions from inevitable setup
imperfections.

In the field-modulated mode ($H$-mod) we modulate the magnetic field supplied
by a copper coil ($H_{z}\approx 3\,\mathrm{mT},f\approx 100\,\mathrm{Hz}$),
and measure the field derivative of Eq. 3. This experiment is sensitive to
$\delta\alpha_{s}$. Therefore, the symmetry constraints to observe non-zero
temperature and field derivatives of $A$ ($\partial_{T}A$ and $\partial_{H}A$,
respectively), differ. As we will demonstrate below, the ability to
simultaneously measure these quantities is crucial to determine the symmetry
of the two ordered phases in EuIn2As2.

###  III.2 Onset of the two phases

Figure 3: (a) Polar plots of thermally modulated (T–mod, red circles) and
field modulated (H–mod, blue diamonds) polarization rotation $\delta\varphi$
at different temperatures reveal distinct signatures of broken rotational
symmetry in each of the two magnetic phases. To illustrate the remnant
180$\degree$ rotational symmetry, $\delta\varphi$ measured as a function of
the incident polarization angle from $-90\degree\rightarrow 90\degree$ is
replicated for $90\degree\rightarrow-90\degree$. (b) Temperature dependence of
T–mod and H–mod amplitude. The T–mod signal onsets at the transition
temperature associated with the higher temperature phase, whereas the H–mod
signal onsets at the lower transition temperature. The amplitudes are
normalized at $10\,\mathrm{K}$. (c) Temperature dependence of the principal
axes orientations corresponding to the T–mod and H–mod signals. The principal
axes associated with the two signals are oriented $45\degree$ relative to each
other and remain constant with temperature.

Fig. 3 illustrates the onset with decreasing temperature of the optical
signatures of broken symmetry. The polar plots in Fig. 3a show the $T$-mod and
$H$-mod rotation of polarization as red and blue symbols, respectively,
measured at three temperatures. The orientations of the principal axes
obtained by the two modulation modes differ by
$\theta_{H}-\theta_{0}=45\,\mathrm{\degree}$, a point we return to later. The
temperature dependence of the $T$-mod and $H$-mod amplitude and principal axis
orientation are plotted in Fig. 3b and Fig. 3c, respectively. Both amplitudes
exhibit a sharp onset, separated by $1.4\,\mathrm{K}$. The onset of the
$T$-mod birefringence coincides with the appearance of the $\bm{Q}_{1}$ peak
(Phase I) and the $H$-mod signal to the appearance of antiferromagnetic
$\bm{Q}_{2}$ peak (Phase II), as detected by x-ray scattering measurements
performed on the same crystal [29].

The two optical experiments, which probe two distinct symmetries, revealed the
two transitions, and unambiguously showed that the symmetry-breaking pathway
proposed on the basis of scattering experiments cannot be correct. The
$60\,\mathrm{\degree}$-helix structure previously assumed to describe Phase I
preserves the $C_{3z}$ symmetry of the lattice, and is thus incompatible with
the birefringence onset at $T_{N1}$. In fact, birefringence was expected to
arise only at $T_{N2}$, in contrast to the observation. Furthermore,
spatially-resolved measurements, discussed in the following section, reveal
additional information, which is incompatible with the past understanding of
both phases in EuIn2As2.

###  III.3 Spatial distribution of optical signals

To investigate the spatial dependence of the two optical signals, we raster
scanned the sample under the beam focus and repeated the polarization rotation
measurements across a $240\times 260\mathrm{\mu m^{2}}$ sample region, with
measurements taken every $20\,\mathrm{\mu m}$. These measurements revealed two
surprising facts: (a) the principal optical axes in EuIn2As2 can assume any
orientation with respect to the lattice, and (b) despite that breadth of
orientations, the fundamental $H$–mod signal is the same in every position on
the sample.

####  III.3.1 $T$-mod signal

First we focus on the $H=0$ signal. In Figs. 4(a,b) we show a map of the
spatial distribution of the principal axis orientation $\theta_{0}$, and the
corresponding histogram. We find a single broad peak in the distribution of
principal axis orientations, spanning more than $30\,\mathrm{\degree}$ (Fig.
4b), as emphasized by the polar plot showing the polarization dependence of
the $T$-mod signal taken at two sample positions (Fig. 4c). Since the two
high-symmetry directions in the Eu planes are separated by
$30\,\mathrm{\degree}$, these measurements show that the principal optical
axes can take any orientation with respect to the lattice.

Figure 4: Principal axis orientations associated with T–mod and H–mod signals:
(a), (d), (g) Maps of the spatial distributions of the principal axis
orientations for T–mod ($\theta_{0}$), H–mod ($\theta_{H}$), and their
difference ($\theta_{H}-\theta_{0}$), respectively. (b), (e), (h) Histograms
of the distributions in (a), (d), (g), respectively. The orientations of the
T–mod and H–mod signals are broadly distributed and continuously varying,
whereas the difference between the orientations, $\theta_{H}-\theta_{0}$, is
sharply peaked at $45\degree$. (c), (f) Polar plots of the polarization
rotation at points labeled $P1$ and $P2$ in (a), (d), respectively, overlaid
on hexagons to demonstrate the lack of registration to the crystalline axes.
The principal axes of the two points are oriented $\approx 30\degree$ from
each other in each of the T–mod and H–mod maps. (i) Polar plot of the T–mod
and H–mod polarization rotations at point $P1$, demonstrating the principal
axes of the T–mod and H–mod signals to be oriented $45\degree$ relative to
each other.

The observation of a broad distribution of principal axes is surprising, since
breaking of the discrete $C_{3z}$ symmetry is expected to yield three domains
related by $C_{3z}$, which would manifest as three narrow peaks in the
histogram, separated by $60\,\mathrm{\degree}$ [35, 36, 33, 37]. This
observation could be interpreted in two ways, which are important to
distinguish. Either the mapped region is dominated by a single domain, and the
broadening of the histogram is caused by contributions of the other two
domains which cannot be resolved with our diffraction-limited resolution of 1
$\mu$m (micro-domain scenario), or the magnetic order microscopically assumes
a continuum of orientations (orientation continuum scenario). We note that the
broken helix structure, proposed for Phase II (Fig. 1e), protects the axion
state only for one orientation of the structure with respect to the lattice;
the orientation continuum scenario would therefore drastically change the
topological properties. Below, we show that only the orientation continuum
scenario is consistent with the data, requiring a re-evaluation of symmetry
and topology in EuIn2As2.

The orientation continuum scenario is proven by analyzing the amplitude of
birefringence as a function of position. In the micro-domain scenario the
amplitude is the largest when the signal is dominated by a single domain, and
is reduced by averaging over micro-domains, yielding a well-defined prediction
for the relationship between the birefringence amplitude and orientation (see
Appendix S4 for more details). In contrast, the amplitude is independent of
the angle in the orientation continuum scenario. As evident in Fig. 5a, the
amplitude is uniform across the sample region in which the principal axis
orientation spans an angle larger than $30\,\mathrm{\degree}$, i.e. the full
range between the high-symmetry directions of the lattice. In contrast, in the
microdomain scenario the amplitude would systematically vary across this
region (simulated in Fig. 5b).

The confirmation of the orientation continuum scenario suggests that the local
principal axes are chosen by a built-in strain, and that the distributions in
Figs. 4(a,b) reflect the distribution of built-in strain axes. A somewhat
similar scenario was proposed to explain the rotation of the nematic director
in the superconducting nematic phase of twisted bilayer graphene [38, 39].
However, it is unusual to observe an orientation continuum in a magnetic
system, since magneto-crystalline anisotropy (MCA) tends to pin moments to the
high-symmetry direction of the lattice. Our observation therefore suggests a
very weak MCA, such that an inevitable built-in strain dominates, and selects
an arbitrary orientation at a generic sample position. This hypothesis is
proven below by deliberate application of uniaxial strain (Sec. VI), showing
that the magnetic space group can be manipulated, with important consequences
for topology.

Figure 5: Measured and simulated birefringence amplitude (T–mod): (a)
Thermally modulated birefringence amplitude in the same region as the maps in
Fig. 4. The amplitude is nearly constant over the region. (b) Birefringence
amplitude simulated using the microdomain model prediction for birefringence
amplitude from orientation. The microdomain model predicted amplitude is
expected to vary as the measured birefringence angle is continuously tuned, in
contrast with the observed experimental amplitude.

####  III.3.2 $H$-mod signal

The $H-$mod signal measured over the same sample region reveals an angle
distribution $\theta_{H}$ of principal optical axes of similar width (Figs.
4(d-f)). However, it is quite striking that the distribution of the difference
of the two angles ($\theta_{H}-\theta_{0}$) is remarkably narrow, and centered
at $45\,\mathrm{\degree}$ (Figs. 4(g-i)).

This finding raises two questions, which we address in the remainder of this
section: why the distribution is so narrow, and why time-reversed domains of
$\theta_{H}-\theta_{0}=-45\,\mathrm{\degree}$ are not observed anywhere on the
sample.

The narrow distribution of the difference
$\theta_{H}-\theta_{0}=45\,\mathrm{\degree}$ suggests that this relationship
is enforced by symmetry, as has been observed in some other magnetic systems.
For example, in Ref. [33] it was shown that the magnetic point groups $2/m$
and $2^{\prime}/m^{\prime}$ allow only for $\theta_{H}-\theta_{0}=\pm
45\degree$ and $\theta_{H}-\theta_{0}=0,90\degree$, respectively. However,
further consideration reveals that this cannot be the case in EuIn2As2: since
the principal axis orientation generically does not coincide with the high-
symmetry directions of the crystal, no in-plane rotational axes or vertical
mirror planes remain valid symmetries at a generic sample position, alone or
in combination with time reversal. Therefore, there is no symmetry that can
enforce a relationship between $\theta_{H}$ and $\theta_{0}$, and a
microscopic mechanism must instead underlay the sharp angle distribution in
Fig. 4h. Identifying why $\theta_{H}-\theta_{0}=0,90\degree$ is not observed,
although it is allowed by symmetry, is beyond the scope of our work, and we
hope that our findings motivate further ab initio investigation to identify
the origin of the observation reported in Figs. 4(g-i).

Figure 6: (a) $H$-mod signal as a function of incident polarization angle
measured at $5\,\mathrm{K}$ in a sample cooled in a training field of
$-3\,\mathrm{mT}$ and $3\,\mathrm{mT}$ (blue and green symbols, respectively).
(b) $H$-mod signal measured using incident polarization of
$50\,\mathrm{\degree}$, as a function of a training DC field applied during
cooling. Both panels clearly demonstrate that the signal can be trained by a
magnetic field, proving its TR-odd nature.

The remaining puzzle is the absence of the time-reversed domain
($\theta_{H}-\theta_{0}=-45\,\mathrm{\degree}$). A closely related observation
is that the same $\theta_{H}-\theta_{0}=45\,\mathrm{\degree}$ domain was
observed on each cool-down through the transition, which is not expected of a
$\mathcal{T}$-breaking order parameter that condensed in the absence of a
magnetic field. To explore this, we measured $H$-mod birefringence at
$5\,\mathrm{K}$ following cooling in a DC magnetic field (Fig. 6a). The sign
of the signal can be trained by the field, confirming its $\mathcal{T}$-odd
nature. The two time-reversed states are characterized by values of
$\theta_{H}$ that differ by $90\,\mathrm{\degree}$, and correspond to
$\theta_{H}=\theta_{0}\pm 45\,\mathrm{\degree}$.

We further monitor the maximum signal, at $50\,\mathrm{\degree}$ of incident
polarization, as a function of the training field (Fig. 6b), and find that
even at a vanishing applied training field the signal reaches a value close to
saturation. This indicates that the Earth’s field is sufficient to train the
domains, consistent with the observation that the sign of the signal does not
change with sample position, or between different cooldowns. We conclude that
the $\mathcal{T}$-breaking domains in EuIn2As2 are easily switched, and their
sign can be detected through LMB.

##  IV Determining the magnetic structures

In this section, we use a multimodal approach to determine the magnetic
structures in EuIn2As2. Both optical and scattering experiments show evidence
of two magnetic transitions as a function of temperature, yielding two
magnetic phases. These experiments impose complementary constraints on the
corresponding magnetic structures. Our symmetry-sensitive optical measurements
show that: (i) $C_{3z}$ is broken in Phases I and II; (ii) $\mathcal{PT}$ is
broken in Phase II, but preserved in Phase I; (iii) the in-plane magneto-
crystalline anisotropy is negligible. Additionally, scattering experiments
[16, 29] revealed the propagation vectors: (iv) $\mathbf{Q}_{1}=(0,0,1/3)$ in
Phase I and (v) $\mathbf{Q}_{1}=(0,0,1/3)$ and $\mathbf{Q}_{2}=(0,0,1)$ in
Phase II. While observation (i) is inconsistent with the 60°-helix previously
associated with Phase I [16, 29], we show in this section that it is possible
to reconcile all the experimental results for both phases.

In Phase I we find a collinear state with varying amplitude of the magnetic
moments that we hereafter refer to as nodal amplitude-modulated collinear
order, while in Phase II we find a state similar to the previously proposed
broken-helix [16], but with one key difference: the moments in the
experimentally observed structure are not pinned to the crystalline axes. We
refer to this state as the unpinned broken helix, and show that it can be
manipulated with uniaxial strain (Sec. VI), raising the possibility of on-
demand control of electronic topology in EuIn2As2. This tunability is a direct
consequence of weak magneto-crystalline anisotropy, which can be overpowered
even by modest built-in strain.

In the remainder of this section, we highlight the main steps involved in our
experimentally guided symmetry analysis, while more details are given in Sec.
S5 and Sec. S6 of the Appendix. In Sec. V we complement the symmetry analysis
with a phenomenological free energy model, which captures the sequence of
broken symmetries, and the corresponding evolution of magnetic structure.
Finally, we summarize the main results of the symmetry and free energy
analyses; a reader more interested in those results than in the reasoning that
led to them may immediately proceed to Sec. V.2. Illustrations of the magnetic
structures associated with the two phases are shown in Fig. 7d and Fig.
8(c,e), respectively.

###  IV.1 Multimodal Approach

Scattering experiments reveal the successive onset of two propagation vectors,
$\mathbf{Q}_{1}$ and $\mathbf{Q}_{2}$, resulting in a non-zero expected value
of Eu magnetic moment that can be expressed as the sum over Fourier
components:

$\mathbf{M}_{\alpha}(\mathbf{r}_{i})=\sum_{\mathbf{Q}}e^{i\mathbf{Q}\cdot\mathbf{r}_{i}}\mathbf{M}_{\alpha}(\mathbf{Q}),$
(4)

where $\mathbf{r}_{i}$ denotes the position of a Eu site, and $\alpha=1,2$ is
the Eu sublattice index. $\mathbf{Q}=\pm\mathbf{Q}_{1}$ in Phase I and
$\mathbf{Q}=\pm\mathbf{Q}_{1},\pm\mathbf{Q}_{2}$ in Phase II. However, the
scattering experiments are not sufficient to uniquely determine
$\mathbf{M}_{\alpha}(\mathbf{Q})$; removing this ambiguity is the objective of
our analysis. We note that since both propagation vectors are perpendicular to
the Eu layers ($\mathbf{Q}_{1}||\mathbf{Q}_{2}||\mathbf{\hat{z}}$), it is
sufficient to consider the average magnetic moment in each Eu layer,
$\mathbf{M}_{\alpha}(z_{i})=\left\langle\mathbf{M}_{\alpha}(\mathbf{r}_{i})\right\rangle_{z=z_{i}}$,
rather than the magnetic moment at each site.

The first step is to identify all magnetic structures
$\mathbf{M}_{\alpha}(z_{i})$ consistent with the scattering experiments, and
with the fact that the magnetic moments lie in Eu planes. The next step is to
identify the magnetic space groups associated with each of those magnetic
configurations. The symmetry-sensitive information obtained from the optical
experiments can then be used to rule out a vast majority of structures
obtained in the previous step. In Phase I of EuIn2As2 this symmetry-based
approach has proven sufficient to uniquely identify the magnetic structure,
while in Phase II it needs to be complemented with further considerations, as
explained in Sec.V.

###  IV.2 Phase I: Amplitude modulated collinear order

Each magnetic structure lowers the system symmetry from the parent
paramagnetic group $P6_{3}/mmc1^{\prime}$ (No. 194.264) to a magnetic space
group (MSG) that is a subgroup of the paramagnetic group, and has the symmetry
properties of one or more of its irreducible representations (Irreps). We find
that all magnetic configurations $\mathbf{M}_{\alpha}(z_{i})$ that are
consistent with the observed magnetic Bragg peaks and in-plane magnetic
moments transform according to the $m\Delta_{6}$ Irrep (see Appendices S5 and
S6 for a detailed analysis). One of them is the 60°-helix proposed in earlier
works [16, 29], which is inconsistent with the observed birefringence. We now
show that there is only one magnetic structure for Phase I consistent with all
the experimental results (i)-(iii) outlined above.

It is useful to rewrite the order parameter of Phase I,
$\mathbf{M}_{\alpha}(\mathbf{Q}_{1})$ as a vector in the four-dimensional
representation space of $m\Delta_{6}$, whose basis functions are any four
linearly independent arrangements of the magnetic moments that transform as
the $m\Delta_{6}$ Irrep. One possible choice of the basis functions is the
four 60°-helix states of opposite helicities and orthogonal orientations,
corresponding to the states $H_{1}$, $H_{2}$ and $\phi=0$, $\phi=\pi/2$ in
Fig. 7(a-c). For our purposes, however, it is more convenient to consider the
amplitude-modulated collinear magnetic states of different phase, shown in
Fig. 7(d-f) and labeled by $A_{1}$, $A_{2}$ and $\phi=0$, $\phi=\pi/2$. In
these structures, all magnetic moments point in the same direction, while
their amplitude varies sinusoidally from layer to layer.

There are two classes of amplitude-modulated structures, hereafter called
nodeless (denoted by $A_{1}$) and nodal (denoted by $A_{2}$). In the nodeless
structure, $\mathbf{M}_{\alpha}(z_{i})$ is non-zero in all Eu layers, while in
the nodal structure it vanishes in every third Eu layer (Fig. 7(d,e)). In
principle, the magnetic moments associated with each of the orders $A_{1,2}$
could assume any orientation. Therefore, the four structures obtained by
$A_{1,2}$ with moments pointing along two orthogonal directions form a valid
basis of the $m\Delta_{6}$ representation space (Fig. 7f). Written in this
basis, the order parameter for Phase I takes the form

$\bm{\Psi}_{I}=\left(a_{1}\cos{\phi_{1}},a_{1}\sin{\phi_{1}},a_{2}\cos{\phi_{2}},a_{2}\sin{\phi_{2}}\right),$
(5)

where $\phi_{1,2}$ and $a_{1,2}$ set the orientation and amplitude,
respectively, of the moments associated with $A_{1,2}$. Any magnetic structure
consistent with the scattering pattern observed in Phase I can be obtained by
specific choices of $a_{1,2}$ and $\phi_{1,2}$, and the relationship between
$\bm{\Psi}_{I}$ and $\mathbf{M}_{\alpha}(\mathbf{Q}_{1})$ is derived in
Appendix S6. For instance, while the amplitude-modulated phases $A_{1}$ and
$A_{2}$ in Fig. 7(d-e) are described by $a_{1}\neq 0$, $a_{2}=0$ and
$a_{2}\neq 0$, $a_{1}=0$, respectively, the 60°-helix phases $H_{1}$ and
$H_{2}$ are parametrized by $a_{1}=a_{2}$ and $\phi_{2}=\phi_{1}+\pi/2$, with
$\phi_{1}=\pi$ for the $H_{1}$ phase and $\phi_{2}=\phi_{1}-\pi/2$ with
$\phi_{1}=0$ for the $H_{2}$ phase.

Figure 7: Top row: The layer-resolved moment orientation in (a, b) the two
helical structures of opposite helicities ($H_{1}$ and $H_{2}$); (d,e) the two
symmetry-distinct amplitude-modulated structures ($A_{1}$ and $A_{2}$,
respectively). Middle row: Two equivalent choices for the basis function for
the $m\Delta_{6}$ Irrep: (c) helical and (f) amplitude-modulated states.
Bottom row: symmetries associated with the four different states: helical,
$A_{1}$ and $A_{2}$. We note only the symmetries that are present regardless
of the relative orientation of the lattice and the moments (for more details,
see Table S1 in the Appendix). We mark in red and blue the symmetries
consistent with the $T$-mod and $H$-mod measurements, respectively.

While $\bm{\Psi_{I}}$ is compatible with the scattering experiments for any
choice of $a_{1,2}$ and $\phi_{1,2}$, optical measurements impose further
constraints. Birefringence is non-zero only in structures that break the
three-fold rotation symmetry along $z$ ($C_{3z}$), which can be re-cast as a
non-zero value of the composite three-state Potts nematic order parameter
[35],

$\bm{\eta}=\frac{1}{2}\left(\begin{matrix}a_{1}^{2}\cos(2\phi_{1})+a_{2}^{2}\cos(2\phi_{2})\\\
a_{1}^{2}\sin(2\phi_{1})+a_{2}^{2}\sin(2\phi_{2})\end{matrix}\right).$ (6)

The condition $\bm{\eta}\neq 0$ for Phase I excludes the 60°-helical states,
and allows any of the amplitude-modulated states. This is because, as
explained above, the 60°-helical states have $a_{1}=a_{2}$ and
$\phi_{2}=\phi_{1}\pm\pi/2$, which makes $\bm{\eta}=0$.

The absence of an $H$-mod signal in Phase I enables us to distinguish between
the two amplitude-modulated states, as this signal is forbidden by the product
of inversion and time reversal symmetry ($\mathcal{PT}$). Therefore, the state
$A_{1}$ is ruled out, as the Eu sites remain centers of inversion, as they
were in the parent space group, and $\mathcal{P}\mathcal{T}$ is broken. In
contrast, in $A_{2}$ inversion symmetry is broken, while
$\mathcal{P}\mathcal{T}$ is preserved. In fact, this nodal amplitude-modulated
structure is the only one belonging to the $m\Delta_{6}$ Irrep that allows for
non-zero $\bm{\eta}$ and vanishing $H$-mod signal. Therefore, the only
magnetic structure that is consistent with all experimental findings in Phase
I is $A_{2}$, characterized by $a_{1}=0$ and $a_{2}\neq 0$ in Eq. 5.

One remaining degree of freedom is the direction of the magnetic moments
within the $ab$-plane, set by $\phi_{2}$. Setting $a_{1}=0$ in Eq. (6), we
note that the direction of the magnetic moment coincides with the orientation
of the nematic director. Therefore, the principal axis extracted from our
optical birefringence measurements corresponds to the direction of the
magnetic moments. Although the broad distribution of principal axis
orientations shown in Fig. 4 was measured in Phase II, we have demonstrated
that the orientation does not change as a function of temperature (Fig. 3e).
Therefore, the distribution of orientations is equally broad in Phase I,
indicating that $\phi_{2}$ is not constrained. This observation is surprising
from a symmetry perspective, since there are always symmetry-allowed terms in
the free energy, which can be thought of as magneto-crystalline anisotropy,
that favor $\phi_{2}$ pointing along the high-symmetry directions. The
experimental observation therefore indicates that the magneto-crystalline
anisotropy is weak compared to built-in strain.

Our work is the first to suggest an amplitude modulated phase in EuIn2As2, and
our conclusion is supported by the results of Mössbauer spectroscopy (reported
in the Supplementary Fig. 10 of Ref. [16]). These measurements also confirm
that the valence Eu2+ in EuIn2As2 is temperature independent, fixing spins on
individual sites to $S=7/2$ in all the phases. The amplitude modulation
therefore arises from the sinusoidal variation of the thermal average of the
magnetic moments. Mössbauer measurements are not sensitive to the phase of
amplitude modulation and, therefore, cannot distinguish between $A_{1}$ and
$A_{2}$. We note that a nodal amplitude-modulated phase had previously been
observed in ground state of the Na-doped SrFe2As2 [40], where it originates
from itinerant magnetism, in contrast to the local moment magnetism of
EuIn2As2.

###  IV.3 Phase II: Mixed order

We now turn to the analysis of Phase II, whose onset is characterized by the
appearance of an additional ordering vector $\mathbf{Q}_{2}=(0,0,1)$,
corresponding to the emergence of Néel order (illustrated in Fig. 8a). In-
plane Néel order transforms as the $m\Gamma_{5}^{+}$ Irrep of the paramagnetic
group of EuIn2As2 and its order parameter takes the form of a two-component
vector,

$\bm{\Psi}_{L}=l\left(\cos{\theta},\sin{\theta}\right),$ (7)

in the space of $m\Gamma_{5}^{+}$. Here, $l$ denotes the amplitude of the
moments in the Néel state and $\theta$ determines their orientation with
respect to the in-plane crystal axes (Fig. 8(a,b)).

Phase II is, therefore, parameterized by both $\bm{\Psi}_{I}$ defined in
Eq.(5) and $\bm{\Psi}_{L}$, and all candidate magnetic states for this phase
can be described by different choices of $a_{1,2}$, $\phi_{1,2}$, $l$ and
$\theta$. One example with $\bm{\Psi}_{I}=(0,0,a_{2},0)$ and
$\bm{\Psi}_{L}=(0,l)$ is shown in Fig. 8(c,d). The blue arrows indicate the
direction of the Néel component. A complete list of states compatible with
Phase II can be found in Table S2 in the Appendix, where the relationship
between $\mathbf{M}_{\alpha}(\mathbf{Q}_{2})$ and $\bm{\Psi}_{L}$ is also
derived.

Figure 8: Top row: The layer-resolved spin orientation in (a) the Néel state;
(c) in the mixed structure obtained by combining $A_{2}$ order with the Néel
order; the blue arrows indicate the orientation of $\bm{\Psi}_{L}$; (e) The
mixed structure obeying the equal moment condition (Eq. 9). Bottom row: (b)
the basis functions of the $m\Gamma_{5}^{+}$ Irrep, corresponding to the two
perpendicular orientations of the Néel order. (d,f) The top view of two
orthogonal orientations of the mixed order shown in (c,e), respectively.
$\theta$ denotes the orientation of the Néel order. The structure shown in
panel f with $\theta=\pi/2$ corresponds to the broken helix considered in
previous work [16].

To analyze the optical responses in Phase II, we first note that any non-zero
$\bm{\Psi}_{L}$ also introduces a Potts-nematic order parameter:

$\bm{\xi}=\frac{l^{2}}{2}\left(\begin{matrix}\cos 2\theta\\\ \sin
2\theta\end{matrix}\right)\text{ .}$ (8)

Therefore, in the lowest temperature phase, both $\bm{\xi}$ and $\bm{\eta}$
[Eq.(6)] contribute to the observed birefringence via the combination
$\bm{\eta}+\bm{\xi}$ (see Appendix S7 for details). Furthermore, the Néel
component breaks $\mathcal{PT}$, accounting for the onset of the $H$-mod
signal in Phase II.

From a symmetry perspective only, we cannot completely constrain the
magnitudes of $a_{1,2}$ and $l$, nor their relative orientations. Many
different structures with $\bm{\Psi}_{I}$ and $\bm{\Psi}_{L}$ both non-zero
break $C_{3z}$ symmetry and $\mathcal{PT}$ symmetry, and are therefore
consistent with the optical measurements. Indeed, Table S2 shows various
phases characterized by different combinations of the order parameters that
result in different magnetic space groups.

However, by taking into account the local moment nature of the EuIn2As2
magnetism, we can narrow down the candidates for the magnetic ground state.
The magnetism originates from the Eu2+ localized magnetic moments ($S=7/2$),
with no valence mixing with Eu+3[16]. At the lowest temperatures, where
fluctuations associated with the magnetic phase transitions are completely
suppressed, the moments in each layer should have the same magnitude. Note
that this statement is not in contradiction with our proposed amplitude-
modulated Phase I, where the magnetic moments are small and fluctuate strongly
because of proximity to magnetic phase transitions. Indeed, as we show below,
the amplitude-modulated phase can smoothly evolve from a configuration near
$T_{N2}$ in which the moments in each layer are not the same to a
configuration at lower temperatures in which the moments all have the same
magnitude, consistent with the aforementioned Mössbauer spectroscopy data.

Enforcing the moments to have the same magnitude corresponds to the additional
conditions on the components of the $\bm{\Psi}_{I}$ and $\bm{\Psi}_{L}$ order
parameters:

$a_{1}=\sqrt{a_{2}^{2}+2l^{2}}-\sqrt{2}l,\hskip
5.0pt\phi_{1}=\phi_{2}\pm\frac{\pi}{2},\hskip
5.0pt\theta=\phi_{2}\mp\frac{\pi}{2}.$ (9)

We note that the broken helix state, proposed in Refs. [16, 29], is a special
combination of $\bm{\Psi}_{I}$ and $\bm{\Psi}_{L}$ which fulfills the equal-
moment condition (Fig. 8(e,f)). Although the equal-moment condition cannot be
obeyed throughout Phase II (in contrast to assumptions in the previous work),
the fact that there are no phase transitions between $T_{N2}$ and the lowest
measured temperature (2K) indicates that the symmetry at $T\lesssim T_{N2}$ is
the same as in the equal-moment ground state. The free energy model presented
in the following section will provide a perspective on the evolution of the
magnetic structure from the amplitude modulated state of Phase I to the equal-
moment ground state.

In addition to the moment-length variation, the key difference between the
broken helix considered previously and our findings is that the nematic order
in Phase II is not constrained by the underlying lattice, and we therefore
refer to this structure as the unpinned broken helix. As we discussed in the
context of Phase I, this observation indicates that the magneto-crystalline
terms in the free energy are weak compared even to the inevitable built-in
strain.

##  V Landau free-energy expansion

In the previous section, we identified the symmetries of the two magnetic
phases in EuIn2As2. Here, we introduce a phenomenological description of the
evolution of the magnetic structure from the nodal amplitude-modulated state
to the unpinned broken helix. The key finding is that it is not necessary to
include two separate instabilities to describe the two phase transitions.
Instead, the order $\Psi_{L}$ naturally arises from coupling to the order
$\Psi_{I}$ of Phase I.

###  V.1 Phenomenological description of the two transitions

Our starting point is the Landau free energy

$F\left(\bm{\Psi}_{I},\bm{\Psi}_{L}\right)=F_{A}\left(\bm{\Psi}_{I}\right)+F_{L}\left(\bm{\Psi}_{L}\right)+F_{AL}\left(\bm{\Psi}_{I},\bm{\Psi}_{L}\right).$
(10)

It has three classes of terms: $F_{A}$ involves only the order parameter
$\bm{\Psi}_{I}$ of Phase I (Eq. 5), which transforms as the $m\Delta_{6}$
Irrep; $F_{L}$ depends only on the Néel component $\bm{\Psi}_{L}$ of the order
parameter of Phase II, which transforms as the $m\Gamma_{5}^{+}$ Irrep; and
$F_{AL}$ accounts for the coupling between them. The most general expressions
for these three terms are given in the Appendix S8. For the discussion in this
Section, it suffices to know that only combinations of the components of
$\bm{\Psi}_{I}$ and $\bm{\Psi}_{L}$ that are invariant under all symmetries of
the parent paramagnetic group, including time-reversal symmetry, are allowed.
In particular, $F_{A}$ has a quadratic term,

$F_{A}\left(\bm{\Psi}_{I}\right)=\alpha_{I}\left(T-T_{N1}\right)\left|\bm{\Psi}_{I}\right|^{2}+(...)$
(11)

whose coefficient changes sign to induce the transition into Phase I at
$T_{N1}$. The remaining parameters of $F_{A}$ are temperature independent, and
chosen to favor the experimentally-observed nodal amplitude-modulated state.

Similarly, $F_{L}$ has a quadratic term

$F_{L}\left(\bm{\Psi}_{L}\right)=\alpha_{L}\left|\bm{\Psi}_{L}\right|^{2}+(...),$
(12)

and the second transition, at $T_{N2}$, could in principle be induced by a
change of its sign. However, obtaining two transitions so close in temperature
would require fine-tuning of model parameters. Instead, $\bm{\Psi}_{L}$ can be
induced by the coupling term $F_{AL}$. The form of $F_{AL}$ determines whether
these two transitions take place simultaneously or at different temperatures.
Because $\bm{\Psi}_{I}$ and $\bm{\Psi}_{L}$ transform as different Irreps,
there can be no bilinear coupling between them. This, combined with the fact
that the free energy is time-reversal-even, implies that all couplings
$\bm{\Psi}_{I}$ and $\bm{\Psi}_{L}$ happen at quartic order. As we show in
Appendix S8, there are two types of quartic coupling: a linear-cubic coupling
between $\bm{\Psi}_{L}$ and $\bm{\Psi}_{I}$ and a biquadratic coupling. The
former implies that the onset of $\bm{\Psi}_{I}$ necessarily triggers
$\bm{\Psi}_{L}$, i.e. that the two transitions are simultaneous. Since two
transition temperatures are observed in EuIn2As2, linear-cubic coupling has to
vanish in Phase I. Indeed, as we show in Appendix S8, the linear-cubic term
generically vanishes if the condition $a_{1}=0$ is imposed. Therefore, the
observation of two transitions, combined with the analysis of the Landau free
energy, provides independent evidence that $A_{2}$ is the state realized in
Phase I.

The subsequent transition to Phase II can arise from the biquadratic coupling
between $\bm{\Psi}_{I}$ and $\bm{\Psi}_{L}$, which is allowed even if
$a_{1}=0$. As the temperature is lowered and $\left|\bm{\Psi}_{I}\right|^{2}$
increases, the biquadratic coupling renormalizes the quadratic term of the
Néel order parameter according to:

$F\left(\bm{\Psi}_{I},\bm{\Psi}_{L}\right)=(\alpha_{L}-\beta\left|\bm{\Psi}_{I}\right|^{2})\left|\bm{\Psi}_{L}\right|^{2}+(...),$
(13)

such that it eventually changes sign if $\beta>0$, triggering the second phase
transition at $T_{N2}$ (see Appendix, Sec. S8.3 for a detailed discussion).

We have therefore shown that a symmetry-based Landau approach can reproduce
the observed sequence of phase transitions. However, this approach does not
uniquely predict the evolution of the magnetic structure, as it does not
capture the tendency towards equal moment magnitude. To develop a description
consistent with this tendency, we introduce constraints on $F$ beyond those
that follow from symmetry alone. In Appendix S8.4 we demonstrate that fixing
the ratios between some of the coefficients of $F_{A}$ and $F_{AL}$ results in
a term whose minimization gives precisely the equal-moment condition. We
introduce a temperature dependence of this term to model the increase in
stiffness with respect to variation in moment amplitude with reduced
temperature, and show that it is sufficient to induce the transition into
Phase II.

###  V.2 The symmetry-breaking pathway

In the previous two sections we identified the order parameters in the two
phases of EuIn2As2, derived the most general form of the free energy, and
constrained its parameters based on experimental findings and physical
arguments. We now analyze the magnetic structures obtained by minimization of
the free energy as a function of temperature, and compute quantities that can
be compared with experimental results.

In Fig. 9a we show the evolution of the magnetic structure with temperature.
The transition between the paramagnet and Phase I at $T_{N1}$ is characterized
by breaking of the rotational $C_{3z}$ symmetry. The magnetic structure in
Phase I is identified as the nodal amplitude modulated state, in which the
average magnetic moment vanishes in every third Eu layer ($A_{2}$, Fig. 7e).
This moment length variation is enabled by strong fluctuations in the vicinity
of the transition, but it becomes increasingly unfavorable as the temperature
is lowered. The non-linear coupling between the nodal amplitude modulated
state and the Néel order ($\bm{\Psi}_{L}$; Fig. 8a) triggers the second
transition, and both the Néel order and the nodeless $A_{1}$ component (Fig.
7d) develop at $T_{N2}$. At this point, the moments are neither of the same
magnitude nor point along the same direction, as can be verified in Fig. 9a.
As the temperature is lowered, the minimization of the constrained free energy
($\tilde{F}$, Eq. S8.4), gives the results of Fig. 9a, revealing indeed that
the moments tend to acquire the same magnitude far away from the transition.

Figure 9: Results of the free energy minimization of the constrained free
energy $\tilde{F}$ as a function of temperature: (a) Magnetic structure; (b)
Scattering intensity at $\mathbf{Q}_{1}$ (red) and $\mathbf{Q}_{2}$ (blue),
computed as $\left|\bm{\Psi}_{I}\right|^{2}$ and
$\left|\bm{\Psi}_{L}\right|^{2}$, respectively; (c) The magnitude of the
nematic order parameter, computed as $\left|\bm{\eta}+\bm{\xi}\right|$; (d)
The mean (red) and standard deviation (green) of the magnetic moment length,
extracted from the magnetic structures obtained by energy minimization. The
values for the parameters of the Landau free-energy are given in Appendix S8.
For illustrative purposes, the coefficients of the magneto-crystalline
anisotropy terms in the Landau functional were set to zero (see Appendix,
Sec.S8), so the magnetic moments in the resulting structure do not point along
high-symmetry directions, consistent with the experiment. Of course, in the
experiments, the magnetic moments are ultimately pinned by the local strain.
All the axes are in arbitrary units.

To compare the results of energy minimization with experimental findings, we
compute $\left|\bm{\Psi}_{I}\right|^{2}$ and $\left|\bm{\Psi}_{L}\right|^{2}$,
(Fig. 9b), the nematic order parameter ($\left|\bm{\eta}+\bm{\xi}\right|$;
Fig. 9c), and the mean magnetic moment length, as well as its standard
deviation (Fig. 9d). The results of minimization of the constrained Landau
free energy as a function of temperature capture the main features of the
experimental findings remarkably well: (a) $\left|\bm{\Psi}_{L}\right|^{2}$
onsets at a slightly lower temperature than $\left|\bm{\Psi}_{I}\right|^{2}$,
and shows a steeper temperature dependence, consistent with scattering
experiments [16, 29]; (b) nematic order ($\left|\bm{\eta}+\bm{\xi}\right|$)
onsets at $T_{N1}$, consistent with the observed birefringence reported in
this work; (c) the expected value of the magnetic moment varies substantially
close to the transition, but the variation is suppressed by the onset of Phase
II, consistent with Mössbauer spectroscopy [16]. We conclude that our free
energy model captures the essential aspects of all the optical and
spectroscopic measurements on EuIn2As2 to date.

##  VI Strain-tuning the magnetic symmetry

The sharp onset of birefringence at $T_{N1}$ proves its magnetic origin. On
the other hand, the experimental observation of the broad distribution of the
principal axis orientation (Fig. 4) implies that the orientation of magnetic
moments at each sample location is determined by built-in strain, which
locally breaks the $C_{3z}$ symmetry (Fig. 4). This in turn suggests that
uniaxial strain can be used to control the principal axis orientation. As we
demonstrate below, this is indeed possible, with implications for the tuning
of magnetic symmetry.

To test the hypothesis of axis tunability, we applied compressive and tensile
uniaxial strain to a bar-shaped sample using a commercial strain device (see
Appendix S3 for experimental details). It is immediately clear from the
$T$-mod birefringence measured at several strain values that the principal
axes orientation can indeed be rotated by strain (Fig. 10a). Systematic
strain-dependent measurements (Fig. 10b) reveal a smooth strain dependence of
the orientation, spanning the range of $\sim 0-90\degree$, corresponding to
the exchange of fast and slow optical axes.

To show that this is a consequence of weak magneto-crystalline anisotropy, we
include strain $\bm{e}$ in our free energy model through coupling to the
nematic order parameters $\bm{\eta}$ and $\bm{\xi}$ (Eqs. 6 and 8):

$F_{\epsilon}=-\epsilon\bm{e}\cdot\left(\bm{\eta}+\bm{\xi}\right),$ (14)

where $\epsilon$ is the coupling constant, $\bm{e}$ the total strain assumed
to be the same for both $\bm{\eta}$ and $\bm{\xi}$, since they both originate
from the Eu2+ moments (Appendix S7). The total strain is a sum of built-in
strain $\bm{e_{0}}$ and applied strain $\bm{e_{a}}$. To obtain the curve in
Fig. 10b we vary the applied strain $\bm{e_{a}}$, and minimize the total free
energy (other parameters are the same as in Fig. 9, and $T-T_{N1}=-3.5$,
$\epsilon=0.1$).We find excellent agreement with the measured strain-
dependence of principal axis orientation for $\left|\bm{e_{0}}\right|=0.056\%$
(Fig. 10b). The principal axis rotates smoothly, in agreement with the
expectations of a 3-state Potts order parameter coupled to a conjugate field
in the limit of weak magneto-crystalline anisotropy [41].

Figure 10:  (a) The polarization rotation as a function of incident
polarization measured at four different values of applied strain. (b) Symbols:
Principal axis orientation as a function of applied strain (positive values
denote tensile strain). Agreement between the data taken while increasing and
decreasing strain indicate that the tuning is reproducible, with no plastic
deformations. The line is calculated by minimizing the free energy (see text
for more detail; parameters are the same as in Fig. 9, and $T-T_{N1}=-3.5$,
$\epsilon=0.1$), assuming a built-in strain, whose magnitude and orientation
($0.056\%$, oriented at -65°) were extracted from the fit. The applied strain
is oriented at -82°, indicating a slight misalignment ( 8°) between the
applied strain axis and the principal optical axes of the laboratory. (c) The
magnetic structures found by free energy minimization for different
orientations of total strain. The rotation of the magnetic structure has stark
consequences for symmetry, with the vast majority of orientations belonging to
MSG 4.7; only when $L$, denoted by the blue arrows, is oriented along the Eu-
Eu bond direction, or perpendicular to it, are different MSGs found: 20.31 and
20.33, respectively.

To explore the consequences of this tunability, we calculate the magnetic
structure that minimizes the free energy as a function of the orientation of
total strain, while keeping its magnitude fixed (Fig. 10c). As expected, we
find that the relative orientation of the magnetic structure and the lattice
can be freely tuned. Furthermore, we analyze the symmetry of all of the
structures, and we find that for most orientations the magnetic point group is
P21 (No. 4.7), which contains $C_{2z}$ combined with a translation by one Eu
layer and identity as the only symmetries. However, when the Néel vector is
aligned to one of the high symmetry directions of the lattice, additional
symmetries are restored. In particular, when the Néel vector is perpendicular
to the Eu-Eu bond direction the magnetic space group identified for Phase II
in Ref. [16] is restored, protecting the axion insulator phase. We therefore
find that it is possible to use uniaxial strain to switch on and off the axion
insulator state in EuIn2As2.

##  VII Spin Hamiltonian

So far we have shown that our measurements and symmetry analysis lead to a
unique identification of the two magnetic phases, as well as describing the
two phase transitions. A question that remains is the microscopic origin of
this behavior, which we address by developing an effective spin Hamiltonian.

As we show in the following sections, there are two prerequisites for forming
the broken helix order: (i) a long-range Heisenberg interaction
$\mathcal{J}_{ij}$ and (ii) fourth-order exchange, $\mathcal{J}_{ijkl}$. Our
proposed spin Hamiltonian,

$H=\sum_{ij}\mathcal{J}_{ij}\bm{S}_{i}\cdot\bm{S}_{j}+\sum_{ijkl}\mathcal{J}_{ijkl}\left(\bm{S}_{i}\cdot\bm{S}_{j}\right)\left(\bm{S}_{k}\cdot\bm{S}_{l}\right),$
(15)

includes these terms, and neglects terms that depend on the spin orientation
with respect to the lattice. To capture the effects of the easy-plane
anisotropy, and the fact moment length is the same for every Eu2+ ion in the
ground state, we parameterize each spin by an angle,
$\bm{S}_{i}=(\cos{\theta_{i}},\sin{\theta_{i}})$.

Our two theoretical approaches are complementary: the Landau free energy
expansion captures the sequence of broken symmetries, and is valid in the
vicinity of the transitions. One the other hand, the spin Hamiltonian
describes the microscopic origin of the magnetic ground state, but provides no
information about the pathway between the paramagnetic state and the ground
state. In Sec. VII.3 we will complement the spin Hamiltonian with atomistic
magnetic simulations at non-zero temperatures, hence connecting the two
approaches.

###  VII.1 Heisenberg term

The Heisenberg term in Eq. 15 can induce modulated magnetic structures if
coupling beyond nearest neighbors are included, with the propagation vector
determined by the Fourier transform, $\mathcal{J}_{Q}$, of the spatially
dependent exchange interaction, $\mathcal{J}_{ij}$. The two periodicities in
the ground state of EuIn2As2 imply that $\mathcal{J}_{Q}$ is peaked at both
$\bm{Q}_{1}$ and $\bm{Q}_{2}$, while the sequence of transitions indicates
that $\bm{Q}_{1}$ is energetically more favorable. We capture this physical
intuition with a Ruderman–Kittel–Kasuya–Yosida (RKKY)-inspired expression for
the effective exchange coupling (for more details see the Appedix S9.1):

$\mathcal{J}_{ij}=J_{1}\frac{\cos{\left(k_{1}\left(i-j\right)\right)}}{k_{1}\left(i-j\right)}+J_{2}\frac{\cos{\left(k_{2}\left(i-j\right)\right)}}{k_{2}\left(i-j\right)}.$
(16)

In Fig. 11a we show the energy of a single $\bm{Q}=(0,0,q_{z})$ state as a
function of $q_{z}$, calculated from Eq. 16 assuming a ten-neighbor Heisenberg
Hamiltonian. The parameters, $k_{1,2}$ and $J_{1,2}$ (given in the caption)
are chosen so that the energy has two local minima, at $q_{z}=1/3$ and
$q_{z}=1$, and the minimum at $q_{z}=1/3$ is slightly lower.

An energy landscape with two well-defined minima as shown in Fig. 11a requires
narrow energy wells and consequently long-ranged real-space interactions. Two
possible mechanisms for the long-range coupling are the RKKY interaction
mediated by conduction electrons, and the dipolar interaction, which has been
suggested to induce A-type antiferromagnetism in several Eu- compounds [42]. A
possible scenario is that the dipolar interaction promotes the AFM order,
while the RKKY interaction promotes $\bm{Q}_{1}$. Regardless, the exchange
parameters used here (Fig. 11a) capture the tendency towards ordering at two
$\bm{Q}$-values, and RKKY interaction is required for at least $\bm{Q}_{1}$.

Figure 11: (a) Main plot: Heisenberg energy of a single $\bm{Q}=(0,0,q_{z})$
magnetic structure, as a function of $q_{z}$. The distance-dependence of the
Heisenberg interaction is given by Eq. 16, with parameters ($k_{1}=0.4\pi$,
$k_{2}=\pi$, $J_{1}=-0.1$, $J_{2}=0.125$) chosen to reproduce the minima at
$q_{z}=1/3$ and $q_{z}=1$, and favor the $q_{z}=1/3$ state. We consider
interactions up to tenth neighbor (the numerical values of the exchange
parameters are given in Table S3 of the Appendix S9.2). Top panel: magnetic
structures corresponding to the helix and antiferromagnetic states. (b) Main
panel: The energy of two forms of the fourth-order exchange: biquadratic
($\mathcal{J}_{1212}=0.2$, green curve) and a combination of the biquadratic
and a 4-spin-3-site exchange ($\mathcal{J}_{1212}=\mathcal{J}_{1223}=0.2$,
purple curve), as a function of the magnetic structure interpolated between
the helical ($l=0$) and the antiferromagnetic ($l=1$) state, while maintaining
the spin normalization condition (Eq. 18), and setting the length of each spin
to 1. Biquadratic exchange exhibits degenerate minima at the helical ($l=0$)
and mixed ($0<l<1$) states, and therefore is not sufficient to stabilize the
mixed state. However, biquadratic exchange combined with the 4-spin-3-site
exchange exhibits one minimum in the mixed state, and can stabilize it. The
top row shows the magnetic structures corresponding to $l=0$ (helix), $l=0.5$
(mixed order) and $l=1$ (AFM).

###  VII.2 Fourth-order terms

Crucially, no parameter choice within a purely Heisenberg model will promote
the coexistence of two periodicities - the system simply chooses the lower of
the two energy minima in Fig. 11a, i.e and antiferromagnet or a helix. To
account for the coexistence of the two periodicities, it is necessary to move
beyond the Heisenberg model. It has been shown in several itinerant systems
that various forms of the four-spin exchange ($\mathcal{J}_{ijkl}$; Eq. 15)
can promote multi-$\bm{Q}$ states composed of symmetry-equivalent
single-$\bm{Q}$ states [43, 44, 45, 46]. As we show below, fourth order terms
can stabilize the coexistence of states even if they belong to different
Irreps, as is the case with the $\bm{Q}_{1}$ and $\bm{Q}_{2}$ orders in
EuIn2As2. We note that first principles calculations [47], effective Hubbard
models [45] and perturbative expansions of Kondo models [48] have shown that
coupling described by $\mathcal{J}_{ijkl}$ naturally arise in itinerant
systems, and can be crucial for promoting exotic magnetic order.

First, we find that the biquadratic term, $(\bm{S}_{i}\cdot\bm{S}_{i+1})^{2}$,
is not sufficient to promote the mixed order. However, when combined with the
symmetry-allowed three-site-four-spin term it leads to an an interaction which
can stabilize the mixed ground state:

$H_{4}=\mathcal{J}_{1212}(\bm{S}_{i}\cdot\bm{S}_{i+1})^{2}+\mathcal{J}_{1223}(\bm{S}_{i}\cdot\bm{S}_{i+1})(\bm{S}_{i+1}\cdot\bm{S}_{i+2}).$
(17)

We demonstrate this by showing that the energy of the fourth-order term is
lower in the mixed state than in either of the single-$\bm{Q}$ states
characterized by equal moments (helix and AFM). To that end we construct a
continuum of magnetic structures with varying ratios between the amplitudes of
the two order parameters. In these structures equal moment length in each
layer ($M$) is maintained by applying the constraint expressed in Eq. 9,
leading to:

$a_{1}=\frac{M-l}{\sqrt{2}},\hskip
10.0pta_{2}=\frac{\sqrt{-3l^{2}+2lM+M^{2}}}{\sqrt{2}},$ (18)

with the orientations $\phi_{1}=\phi_{2}\pm\pi/2$ and
$\theta=\phi_{2}\mp\pi/2$. For $l=0$ we find $a_{1}=a_{2}=M/\sqrt{2}$,
corresponding to a helix, while for $l=M$ the structure is antiferromagnetic
($a_{1}=a_{2}=0$). For each value of $0<l<M$ we calculate $a_{1}$ and $a_{2}$,
determine the corresponding layer-dependent spin orientation, and compute the
energy of the different Hamiltonian terms.

In Fig. 11b we plot the energy of the fourth order term (Eq. 17,
$\mathcal{J}_{1212}=\mathcal{J}_{1223}=0.2$, purple curve) as a function of
$l$, for $M=1$. The fourth-order exchange clearly exhibits a minimum for
$0<l<1$, and therefore favors the mixed state over either of the
single-$\bm{Q}$ states. In contrast, the biquadratic term alone exhibits two
degenerate minima, at $l=0$ and $0<l<1$, and therefore is not sufficient to
stabilize the mixed state ($\mathcal{J}_{1212}=0.2,\mathcal{J}_{1223}=0$,
green curve).

###  VII.3 Evolution of the magnetic structure

The above analysis suggests that combining the long-range Heisenberg exchange
with the 4-spin terms can induce the mixed structure. To verify this, we found
the magnetic ground state by solving the Landau-Lifshitz-Gilbert (LLG)
equation, as implemented in the Spirit framework for atomistic spin
simulations [49]. The broken helix is indeed the ground state, and its Fourier
transform (Fig. 12a) shows good agreement with the scattering data [16, 29].
We have therefore identified a Hamiltonian whose ground state is the observed
mixed order.

We extend the calculation to non-zero temperature, implemented within the
Spirit framework by adding a stochastic thermal field, whose distribution is
determined by the fluctuation-dissipation theorem [49]. In Fig. 12b we show
the amplitudes of the two scattering peaks as a function of temperature, and
find two second-order transitions, with $\bm{Q}_{2}$ vanishing at a lower
temperature upon warming, in agreement with the experiments. However, the
simulations reveal a helical, rather than amplitude-modulated (AM) state in
Phase I, in contrast to our experimental findings. It is perhaps not
surprising that the effective spin Hamiltonian cannot capture the AM phase,
since neither of its terms is likely to favor it: the Heisenberg energies of
the helix and AM phases are degenerate, and the four-spin exchange terms are
negligible just below the second order transition at $T_{N1}$, when the AM
phase is stabilized. Magneto-crystalline anisotropy, which has been suggested
to promote the AM state in a few gadolinium compounds [50, 51], is not
included in the calculation because our data suggest it is not experimentally
relevant.

Figure 12: (a-b) Results of atomistic LLG simulations [49], based on
Hamiltonian shown in Eq. 15, with the same parameters used in Fig. 11 (details
are given in Sec. S9.2 of the SM). (a) The Fourier transform of the spin
structure found at $T=0$, revealing two peaks, at $\bm{Q}_{1}$ and
$\bm{Q}_{2}$. (b) The temperature dependence of scattering intensity at
$\bm{Q}_{1}$ and $\bm{Q}_{2}$. revealing two second-order transitions,
consistent with the experiment. (c) The Fourier transform of snapshots of the
magnetic structure in Phase I, as well as a the Fourier transform of the
structure found by averaging $10^{5}$ such snapshots. This demonstrates
antiferromagnetic fluctuations in Phase I, despite vanishing static AFM order.
(d) The temperature dependence of the temperature derivative of reflectivity
measured at $633\,\mathrm{nm}$, showing a sharp change of electronic structure
at $T_{N2}$. This indicates that the AFM fluctuations shown in panel d are
accompanied by fluctuations of the electronic structure.

Although the free energy originating from the spin Hamiltonian alone does not
capture the nodal amplitude modulated phase, the strong coupling of magnetic
and electronic degrees of freedom can account for additional contributions to
entropy. To illustrate why this is the case, we first look at individual
snapshots of the magnetic structure in Phase I, obtained by the atomistic
simulations, which reveal peaks at both $\bm{Q}_{1}$ and $\bm{Q}_{2}$. In
contrast, the average over an ensemble of $10^{4}$ such snapshots, shows only
the helical $\bm{Q}_{1}$ peaks, as is characteristic of Phase I. Therefore,
the transition into Phase II is anticipated by strong AFM fluctuations. Since
magnetic fluctuations are slow on electronic time scales, AFM fluctuations are
accompanied by electronic ones.

Crucially, due to the strong coupling of itinerant and localized degrees of
freedom in EuIn2As2, the fluctuations of the electronic structure that
accompany AFM fluctuations are significant. Infrared studies [27] have shown
that electronic structure dramatically reconstructs with the onset of the AFM
order in Phase II. This is consistent with our own measurement of the
temperature derivative of reflectivity on the same sample used for symmetry-
sensitive measurements (Fig. 12d), which exhibits a sharp kink at $T_{N2}$.
Entropic effects in EuIn2As2 therefore cannot be correctly captured by
localized spin models alone. Our findings suggest that the entropy associated
with the joined system of localized spins and itinerant electrons stabilizes
the nodal amplitude-modulated state, providing a strong motivation for
explicit calculations considering the magnetic and electronic degrees of
freedom in EuIn2As2 on equal footing. It also yields constraints for such
models: a correct calculation needs to reproduce both the nodal amplitude
modulated state in Phase I, and the broken helix ground state.

The key message of our spin model is that the itinerant electrons are crucial
for stabilizing the magnetism of EuIn2As2: the long-range Heisenberg
interaction and the fourth order term coupling three Eu layers are the
simplest terms that can stabilize the magnetic ground state, but neither can
arise just from direct exchange of the localized $4f$ orbitals. Instead, they
are likely to be mediated by conduction electrons, with possible contributions
from coupling to the lattice, as proposed in Ref. [29]. The understanding of
the origin of the magnetic interactions offers a natural way to resolve a
slight difference in $\bm{Q}_{1}$ reported in the two scattering experiments:
only Soh et al. [29] found $\bm{Q}_{1}=(0,0,1/3)$ within experimental
resolution, while Riberolles et al. [16] found $\bm{Q}_{1}\sim(0,0,0.303)$.
Since the magnetism is mediated by conduction electrons, such differences can
naturally be explained by variations in stoichiometry. In addition to
explaining the existing data, this suggests that electron density could be a
powerful tuning knob for magnetism in EuIn2As2.

##  VIII Discussion and Conclusions

To summarize, we demonstrated a multimodal approach that combines scattering
and symmetry sensitive optical measurements with group theory analysis to
uniquely determine the magnetic structures in two phases of EuIn2As2. In
addition to answering the open questions regarding the magnetism of EuIn2As2,
we set the scene for future research in materials with strongly intertwined
magnetic and electronic degrees of freedom. We have further shown that,
although the axion state is not reached simply by cooling a sample through the
two transitions, it can be retrieved by application of uniaxial strain.
Finally, we have identified the coupling of local magnetic and itinerant
electronic degrees of freedom as the origin of the rich magnetic behavior
observed in EuIn2As2.

The multimodal approach has proven invaluable to determine the nature of the
broken symmetry phases, offering information beyond that available to any of
the techniques alone. While scattering is unique in its ability to determine
ordering wavevectors, spatially-resolved probes are crucial to distinguish
systems that possess a given symmetry from those in which apparent symmetry is
restored by averaging over domains. This has proven critical for recognizing
that the magnetic structure in Phase I is an amplitude-modulated state, rather
than a helix. More broadly, the multimodal approach to identifying broken
symmetry phases can be employed regardless of the nature of the order: here it
was demonstrated on a material with complex magnetism, but it is also
applicable to systems exhibiting charge density or orbital ordering.

Each of the magnetic phases that we have identified has unique aspects.
Amplitude-modulated structures in systems of localized moments are rare. They
have been deduced on the basis of thermodynamic measurements in several Gd3+
compounds [52, 50, 51, 53], however the phase of the amplitude modulation was
not determined. Magneto-crystalline anisotropy, thought to be the driver of
amplitude modulation in the Gd compounds, is experimentally irrelevant in
EuIn2As2, raising the intriguing possibility that its amplitude modulated
state is stabilized by entropic effects.

A further consequence of the decoupling of magnetic states from the EuIn2As2
lattice is that the magnetic symmetry can be controlled by uniaxial strain.
Thus EuIn2As2 can be tuned into and out of a topologically non-trivial axion
state. This capability allows the exploration of approximate symmetries in
topological systems, addressing the following question: if a symmetry protects
a topological state for a particular orientation of the magnetic structure and
the lattice, how do the response functions change when the orientation is
infinitesimally changed? In other words, how robust are topological responses
to continuous tuning parameters? Our work identifies Eu2+ \- based compounds
as ideal platforms for such studies.

Our understanding of itinerant electrons as mediators of magnetic interactions
radically changes the picture of EuIn2As2: while the initial interest in this
material was spurred by the notion that a simple magnetic structure will
generate exotic electronic features [13], we have shown that electronic states
instead generate an exotic magnetic ground state. Of course, this does not
mean that the electronic states themselves do not possess interesting
properties. In addition to the topological phase restored for specific
relative orientations of the magnetic structure and the lattice, AFM order was
shown to induce altermagnetic momentum-dependent spin polarization of the
electronic bands in EuIn2As2 [23]; the influence of such band-splitting on
electron-mediated magnetic interactions is an important open question. Our
work therefore challenges ab-initio calculations to study the interplay of
magnetic and itinerant degrees of freedom in EuIn2As2 in a realistic way, in
order to capture the experimental findings on both subsystems.

###### Acknowledgements.

This research was primarily funded by the Quantum Materials program under the
Director, Office of Science, Office of Basic Energy Sciences, Materials
Sciences and Engineering Division, of the U.S. Department of Energy, Contract
No. DE-AC02-05CH11231. E. D. received additional support from the National
Science Foundation Graduate Research Fellowship Program under Grant No.
DGE-2146752 and the Ford Foundation Predoctoral Fellowship. R.D. is supported
by the Canadian Government under a Banting Fellowship. R.M.F.
(phenomenological modeling) was supported by the U.S. Department of Energy,
Office of Science, Basic Energy Sciences, under award no. DE-SC0020045. D. P.
and A.T.B. would like to acknowledge the Engineering and Physical Sciences
Research Council (EPSRC), UK and the Oxford-ShanghaiTech collaboration project
for financial support. J.O received support from the Gordon and Betty Moore
Foundation’s EPiQS Initiative through Grant GBMF4537 to J.O. at UC Berkeley.
V.S. is supported by the Miller Institute for Basic Research in Science, UC
Berkeley.

## References

  * Nagaosa  _et al._ [2010] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous Hall effect, Reviews of Modern Physics 82, 1539 (2010).
  * Nagaosa and Tokura [2013] N. Nagaosa and Y. Tokura, Topological properties and dynamics of magnetic skyrmions, Nature Nanotechnology 8, 899 (2013).
  * Šmejkal  _et al._ [2022] L. Šmejkal, J. Sinova, and T. Jungwirth, Altermagnetism: spin-momentum locked phase protected by non-relativistic symmetries, Physical Review X 12, 031042 (2022), arXiv:2105.05820 [cond-mat].
  * Deng  _et al._ [2020] Y. Deng, Y. Yu, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H. Chen, and Y. Zhang, Quantum anomalous hall effect in intrinsic magnetic topological insulator MnBi2Te4, Science 367, 895 (2020), publisher: American Association for the Advancement of Science.
  * Serlin  _et al._ [2020] M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Intrinsic quantized anomalous hall effect in a moiré heterostructure, Science 367, 900 (2020), publisher: American Association for the Advancement of Science.
  * Frey  _et al._ [2020] N. C. Frey, M. K. Horton, J. M. Munro, S. M. Griffin, K. A. Persson, and V. B. Shenoy, High-throughput search for magnetic and topological order in transition metal oxides, Science Advances 6, eabd1076 (2020), publisher: American Association for the Advancement of Science.
  * Xu  _et al._ [2020] Y. Xu, L. Elcoro, Z.-D. Song, B. J. Wieder, M. G. Vergniory, N. Regnault, Y. Chen, C. Felser, and B. A. Bernevig, High-throughput calculations of magnetic topological materials, Nature 586, 702 (2020).
  * Choi  _et al._ [2022] J. Choi, Q. Wang, S. Jöhr, N. B. Christensen, J. Küspert, D. Bucher, D. Biscette, M. H. Fischer, M. Hücker, T. Kurosawa, N. Momono, M. Oda, O. Ivashko, M. v. Zimmermann, M. Janoschek, and J. Chang, Unveiling unequivocal charge stripe order in a prototypical cuprate superconductor, Phys. Rev. Lett. 128, 207002 (2022).
  * Bluschke  _et al._ [2022] M. Bluschke, R. Basak, A. Barbour, A. N. Warner, K. Fürsich, S. Wilkins, S. Roy, J. Lee, G. Christiani, G. Logvenov, M. Minola, B. Keimer, C. Mazzoli, E. Benckiser, and A. Frano, Imaging mesoscopic antiferromagnetic spin textures in the dilute limit from single-geometry resonant coherent x-ray diffraction, Science Advances 8, eabn6882 (2022).
  * Kim  _et al._ [2005] J. W. Kim, A. Kreyssig, L. Tan, D. Wermeille, S. L. Bud’ko, P. C. Canfield, and A. I. Goldman, Imaging antiferromagnetic domains in GdNi${}_{\textrm{2}}$Ge${}_{\textrm{2}}$ with x-ray resonant magnetic scattering, Applied Physics Letters 87, 202505 (2005).
  * Takagi  _et al._ [2023] H. Takagi, R. Takagi, S. Minami, T. Nomoto, K. Ohishi, M. T. Suzuki, Y. Yanagi, M. Hirayama, N. D. Khanh, K. Karube, H. Saito, D. Hashizume, R. Kiyanagi, Y. Tokura, R. Arita, T. Nakajima, and S. Seki, Spontaneous topological hall effect induced by non-coplanar antiferromagnetic order in intercalated van der waals materials, Nature Physics 19, 961 (2023).
  * par [2023] Tetrahedral triple-Q magnetic ordering and large spontaneous Hall conductivity in the metallic triangular antiferromagnet Co${}_{\textrm{1/3}}$TaS${}_{\textrm{2}}$ (2023), arxiv:2303.03760 .
  * Xu  _et al._ [2019] Y. Xu, Z. Song, Z. Wang, H. Weng, and X. Dai, Higher-Order Topology of the Axion Insulator EuIn${}_{\textrm{2}}$As${}_{\textrm{2}}$, Physical Review Letters 122, 256402 (2019).
  * Essin  _et al._ [2009] A. M. Essin, J. E. Moore, and D. Vanderbilt, Magnetoelectric polarizability and axion electrodynamics in crystalline insulators, Phys. Rev. Lett. 102, 146805 (2009).
  * Turner  _et al._ [2012] A. M. Turner, Y. Zhang, R. S. K. Mong, and A. Vishwanath, Quantized response and topology of magnetic insulators with inversion symmetry, Phys. Rev. B 85, 165120 (2012).
  * Riberolles  _et al._ [2021] S. X. M. Riberolles, T. V. Trevisan, B. Kuthanazhi, T. W. Heitmann, F. Ye, D. C. Johnston, S. L. Bud’ko, D. H. Ryan, P. C. Canfield, A. Kreyssig, A. Vishwanath, R. J. McQueeney, L.-L. Wang, P. P. Orth, and B. G. Ueland, Magnetic crystalline-symmetry-protected axion electrodynamics and field-tunable unpinned Dirac cones in EuIn${}_{\textrm{2}}$As${}_{\textrm{2}}$, Nature Communications 12, 999 (2021).
  * Wang  _et al._ [2019] L.-L. Wang, N. H. Jo, B. Kuthanazhi, Y. Wu, R. J. McQueeney, A. Kaminski, and P. C. Canfield, Single pair of Weyl fermions in the half-metallic semimetal EuCd2As2, Physical Review B 99, 245147 (2019).
  * [18] J. Ma, H. Wang, S. Nie, C. Yi, Y. Xu, H. Li, J. Jandke, W. Wulfhekel, Y. Huang, D. West, P. Richard, A. Chikina, V. N. Strocov, J. Mesot, H. Weng, S. Zhang, Y. Shi, T. Qian, M. Shi, and H. Ding, Emergence of Nontrivial Low-Energy Dirac Fermions in Antiferromagnetic EuCd2As2, Advanced Materials , 1907565.
  * Wang  _et al._ [2021] H. Wang, N. Mao, X. Hu, Y. Dai, B. Huang, and C. Niu, A magnetic topological insulator in two-dimensional EuCd2Bi2: Giant gap with robust topology against magnetic transitions, Materials Horizons 8, 956 (2021).
  * Li  _et al._ [2021] R. Li, H. Wang, N. Mao, H. Ma, B. Huang, Y. Dai, and C. Niu, Engineering antiferromagnetic topological insulator by strain in two-dimensional rare-earth pnictide EuCd2Sb2, Applied Physics Letters 119, 173105 (2021).
  * [21] G. M. Pierantozzi, A. De Vita, C. Bigi, X. Gui, H.-J. Tien, D. Mondal, F. Mazzola, J. Fujii, I. Vobornik, G. Vinai, A. Sala, C. Africh, T.-L. Lee, G. Rossi, T.-R. Chang, W. Xie, R. J. Cava, and G. Panaccione, Evidence of magnetism-induced topological protection in the axion insulator candidate EuSn2P2, Proceedings of the National Academy of Sciences of the United States of America , e2116575119.
  * [22] D. Santos-Cottin, I. Mohelský, J. Wyzula, F. L. Mardelé, I. Kapon, S. Nasrallah, N. BarišIć, I. Živković, J. R. Soh, F. Guo, M. Puppin, J. H. Dil, B. Gudac, Z. Rukelj, M. Novak, A. B. Kuzmenko, C. C. Homes, T. Dietl, M. Orlita, and A. Akrap, EuCd2As2, arxiv:2301.08014 .
  * Cuono  _et al._ [2023] G. Cuono, R. M. Sattigeri, C. Autieri, and T. Dietl, Ab initio overestimation of the topological region in Eu-based compounds, Physical Review B 108, 075150 (2023).
  * Valadkhani  _et al._ [2023] A. Valadkhani, M. Iraola, A. Fünfhaus, Y.-J. Song, L. Smejkal, J. Sinova, and R. Valenti, Influence of magnetism, strain and pressure on the band topology of EuCd2As2 (2023), arXiv:2308.08619 .
  * Regmi  _et al._ [2020] S. Regmi, M. M. Hosen, B. Ghosh, B. Singh, G. Dhakal, C. Sims, B. Wang, F. Kabir, K. Dimitri, Y. Liu, A. Agarwal, H. Lin, D. Kaczorowski, A. Bansil, and M. Neupane, Temperature-dependent electronic structure in a higher-order topological insulator candidate EuIn${}_{\textrm{2}}$As${}_{\textrm{2}}$, Physical Review B 102, 165153 (2020), publisher: American Physical Society.
  * Yan  _et al._ [2022] J. Yan, Z. Z. Jiang, R. C. Xiao, W. J. Lu, W. H. Song, X. B. Zhu, X. Luo, Y. P. Sun, and M. Yamashita, Field-induced topological Hall effect in antiferromagnetic axion insulator candidate EuIn${}_{\textrm{2}}$As${}_{\textrm{2}}$, Physical Review Research 4, 013163 (2022), publisher: American Physical Society.
  * Xu  _et al._ [2021] B. Xu, P. Marsik, S. Sarkar, F. Lyzwa, Y. Zhang, B. Shen, and C. Bernhard, Infrared study of the interplay of charge, spin, and lattice excitations in the magnetic topological insulator EuIn${}_{\textrm{2}}$As${}_{\textrm{2}}$, Physical Review B 103, 245101 (2021), publisher: American Physical Society.
  * Sato  _et al._ [2020] T. Sato, Z. Wang, D. Takane, S. Souma, C. Cui, Y. Li, K. Nakayama, T. Kawakami, Y. Kubota, C. Cacho, T. K. Kim, A. Arab, V. N. Strocov, Y. Yao, and T. Takahashi, Signature of band inversion in the antiferromagnetic phase of axion insulator candidate EuIn${}_{\textrm{2}}$As${}_{\textrm{2}}$, Physical Review Research 2, 033342 (2020), publisher: American Physical Society.
  * Soh  _et al._ [2023] J.-R. Soh, A. Bombardi, F. Mila, M. C. Rahn, D. Prabhakaran, S. Francoual, H. M. Rønnow, and A. T. Boothroyd, Understanding unconventional magnetic order in a candidate axion insulator by resonant elastic x-ray scattering, Nature Communications 14, 3387 (2023), number: 1 Publisher: Nature Publishing Group.
  * Shelankov and Pikus [1992] A. L. Shelankov and G. E. Pikus, Reciprocity in reflection and transmission of light, Physical Review B 46, 3326 (1992).
  * Halperin [1992] B. I. Halperin, The Hunt for Anyon Superconductivity, in  _The Physics and Chemistry of Oxide Superconductors_, Springer Proceedings in Physics, edited by Y. Iye and H. Yasuoka (Springer, Berlin, Heidelberg, 1992) pp. 439–450.
  * Eremenko and Kharchenko [1987] V. V. Eremenko and N. F. Kharchenko, Magneto-optics of antiferromagnets, Physics Reports 155, 379 (1987).
  * Sunko  _et al._ [2023] V. Sunko, Y. Sun, M. Vranas, C. C. Homes, C. Lee, E. Donoway, Z.-C. Wang, S. Balguri, M. B. Mahendru, A. Ruiz, B. Gunn, R. Basak, S. Blanco-Canosa, E. Schierle, E. Weschke, F. Tafti, A. Frano, and J. Orenstein, Spin-carrier coupling induced ferromagnetism and giant resistivity peak in EuCd${}_{\textrm{2}}$P${}_{\textrm{2}}$, Physical Review B 107, 144404 (2023).
  * Lee  _et al._ [2022] C. Lee, P. Vir, K. Manna, C. Shekhar, J. E. Moore, M. A. Kastner, C. Felser, and J. Orenstein, Observation of a phase transition within the domain walls of ferromagnetic Co${}_{\textrm{3}}$Sn${}_{\textrm{2}}$S${}_{\textrm{2}}$, Nature Communications 13, 3000 (2022).
  * Little  _et al._ [2020] A. Little, C. Lee, C. John, S. Doyle, E. Maniv, N. L. Nair, W. Chen, D. Rees, J. W. F. Venderbos, R. M. Fernandes, J. G. Analytis, and J. Orenstein, Three-state nematicity in the triangular lattice antiferromagnet Fe${}_{\textrm{1/3}}$NbS${}_{\textrm{2}}$, Nature Materials 19, 1062 (2020).
  * Xu  _et al._ [2022] Y. Xu, Z. Ni, Y. Liu, B. R. Ortiz, Q. Deng, S. D. Wilson, B. Yan, L. Balents, and L. Wu, Three-state nematicity and magneto-optical Kerr effect in the charge density waves in kagome superconductors, Nature physics 18, 1470 (2022), publisher: Nature Publishing Group UK London.
  * [37] L. Ye, Y. Sun, V. Sunko, J. F. Rodriguez-Nieva, M. S. Ikeda, T. Worasaran, M. E. Sorensen, M. D. Bachmann, J. Orenstein, and I. R. Fisher, Elastocaloric signatures of symmetric and antisymmetric strain-tuning of quadrupolar and magnetic phases in DyB2C2, Proceedings of the National Academy of Sciences , e2302800120Publisher: Proceedings of the National Academy of Sciences.
  * Fernandes and Venderbos [2020] R. M. Fernandes and J. W. F. Venderbos, Nematicity with a twist: Rotational symmetry breaking in a moiré superlattice, Science Advances 6, eaba8834 (2020).
  * Cao  _et al._ [2021] Y. Cao, D. Rodan-Legrain, J. M. Park, N. F. Q. Yuan, K. Watanabe, T. Taniguchi, R. M. Fernandes, L. Fu, and P. Jarillo-Herrero, Nematicity and competing orders in superconducting magic-angle graphene, Science 372, 264 (2021).
  * Allred  _et al._ [2016] J. M. Allred, K. M. Taddei, D. E. Bugaris, M. J. Krogstad, S. H. Lapidus, D. Y. Chung, H. Claus, M. G. Kanatzidis, D. E. Brown, J. Kang, R. M. Fernandes, I. Eremin, S. Rosenkranz, O. Chmaissem, and R. Osborn, Double-q spin-density wave in iron arsenide superconductors, Nature Physics 12, 493 (2016).
  * Chakraborty and Fernandes [2023] A. R. Chakraborty and R. M. Fernandes, Strain-tuned quantum criticality in electronic Potts-nematic systems, Physical Review B 107, 195136 (2023), arxiv:2303.00685 .
  * Berry  _et al._ [2022] T. Berry, V. J. Stewart, B. W. Y. Redemann, C. Lygouras, N. Varnava, D. Vanderbilt, and T. M. McQueen, A-type antiferromagnetic order in the Zintl-phase insulator EuZn${}_{\textrm{2}}$P${}_{\textrm{2}}$, Physical Review B 106, 054420 (2022), publisher: American Physical Society.
  * Hayami and Motome [2021a] S. Hayami and Y. Motome, Noncoplanar multiple-q spin textures by itinerant frustration: Effects of single-ion anisotropy and bond-dependent anisotropy, Physical Review B 103, 054422 (2021a), publisher: American Physical Society.
  * Hayami and Motome [2021b] S. Hayami and Y. Motome, Topological spin crystals by itinerant frustration, Journal of Physics: Condensed Matter 33, 443001 (2021b), publisher: IOP Publishing.
  * Hoffmann and Blügel [2020] M. Hoffmann and S. Blügel, Systematic derivation of realistic spin models for beyond-Heisenberg solids, Physical Review B 101, 024418 (2020), publisher: American Physical Society.
  * Takagi  _et al._ [2018] R. Takagi, J. S. White, S. Hayami, R. Arita, D. Honecker, H. M. Rønnow, Y. Tokura, and S. Seki, Multiple-q noncollinear magnetism in an itinerant hexagonal magnet, Science Advances 4, eaau3402 (2018).
  * Mendive-Tapia and Staunton [2019] E. Mendive-Tapia and J. B. Staunton, Ab initio theory of the Gibbs free energy and a hierarchy of local moment correlation functions in itinerant electron systems: The magnetism of the Mn3A materials class, Physical Review B 99, 144424 (2019), publisher: American Physical Society.
  * Hayami  _et al._ [2017] S. Hayami, R. Ozawa, and Y. Motome, Effective bilinear-biquadratic model for noncoplanar ordering in itinerant magnets, Physical Review B 95, 224424 (2017), publisher: American Physical Society.
  * Müller  _et al._ [2019] G. P. Müller, M. Hoffmann, C. Dißelkamp, D. Schürhoff, S. Mavros, M. Sallermann, N. S. Kiselev, H. Jónsson, and S. Blügel, Spirit: Multifunctional framework for atomistic spin simulations, Physical Review B 99, 224414 (2019), publisher: American Physical Society.
  * Blanco  _et al._ [1991a] J. A. Blanco, D. Gignoux, and D. Schmitt, Specific heat in some gadolinium compounds. II. Theoretical model, Physical Review B 43, 13145 (1991a), publisher: American Physical Society.
  * Bouvier  _et al._ [1991] M. Bouvier, P. Lethuillier, and D. Schmitt, Specific heat in some gadolinium compounds. I. Experimental, Physical Review B 43, 13137 (1991), publisher: American Physical Society.
  * Blanco  _et al._ [1990] J. A. Blanco, D. Gignoux, P. Morin, and D. Schmitt, Incommensurate phase transitions in Gd compounds, Journal of Magnetism and Magnetic Materials 90-91, 166 (1990).
  * Blanco  _et al._ [1991b] J. A. Blanco, D. Gignoux, P. Morin, and D. Schmitt, Thermodynamical Properties of Incommensurate Magnetic Systems, Europhysics Letters 15, 671 (1991b).
  * [54] Application Notes, http://razorbillinstruments.com/download-page/.
  * Ikeda  _et al._ [2018] M. S. Ikeda, T. Worasaran, J. C. Palmstrom, J. A. W. Straquadine, P. Walmsley, and I. R. Fisher, Symmetric and antisymmetric strain as continuous tuning parameters for electronic nematic order, Physical Review B 98, 245133 (2018), publisher: American Physical Society.
  * Straquadine  _et al._ [2022] J. Straquadine, M. Ikeda, and I. Fisher, Evidence for Realignment of the Charge Density Wave State in ErTe3 and TmTe3 under Uniaxial Stress via Elastocaloric and Elastoresistivity Measurements, Physical Review X 12, 021046 (2022), publisher: American Physical Society.
  * Stokes  _et al._ [2017] H. T. Stokes, D. M. Hatch, and B. J. Campbell, ISOTROPY Software Suite, http://iso.byu.edu (2017).
  * Roth  _et al._ [1966] L. M. Roth, H. J. Zeiger, and T. A. Kaplan, Generalization of the Ruderman-Kittel-Kasuya-Yosida Interaction for Nonspherical Fermi Surfaces, Physical Review 149, 519 (1966), publisher: American Physical Society.

## Supplementary Material for ‘Symmetry-breaking pathways to the broken helix
state’

##  Appendix S1 Crystal growth and characterization

The optical data in Figs. 2\- 5 were taken on the same crystal used for the
resonant x-ray scattering in Ref. [29], synthesized at the University of
Oxford by a method described in Ref. [29]. Since scattering and optical
experiments were performed on physically the same crystal, there are no
ambiguities about sample-sample variation that could complicate our multimodal
approach. Samples made at UC Berkeley gave qualitatively the same optical
response, with $T_{N1}$ and $T_{N2}$ revealed by the $T$-mod and $H$-mod
measurements, respectively. The strain experiment (Fig. 10) was performed on
one of those samples, with the corresponding crystal growth method described
below.

Single crystals of EuIn2As2 were prepared by a flux method similar to [16].
Europium (99.9999$\%$ from Ames Lab) was cut up and combined in an alumina
crucible under a dry nitrogen glove box atmosphere with arsenic pieces
(99.99$\%$ from Alfa Aesar) and indium shot (99.9999$\%$ from Alfa Aesar) in a
1:3:12 molar ratio with total mass 4g. The crucible was sealed in an evacuated
quartz ampule without being exposed to air. The ampule was heated in a box
furnace from room temperature to $300\,\mathrm{\degree C}$ over 2 hours, held
for 1 hour, then heated to $580\,\mathrm{\degree C}$ over 3 hours, held for 2
hours, then ramped to $900\,\mathrm{\degree C}$ over 20 hours before soaking
at $900\,\mathrm{\degree C}$ for 4 hours. Finally, the ampule was slowly
cooled to $770\,\mathrm{\degree C}$ at $1.8\,\mathrm{\degree C}$/hour, and
spun in a centrifuge to remove excess indium flux. The method is similar to
the one used in Ref. [29], with a few differences: the highest temperature in
that work was $950\,\mathrm{\degree C}$, the centrifuging temperature at
$750\,\mathrm{\degree C}$, and the Eu purity 99.99$\%$. Both methods used heat
soaks before reaching the peak temperature to avoid any high vapor pressure of
arsenic and increase the homogeneity of the flux solution.

##  Appendix S2 Optical experiments

Figure S1: Schematic of optical setup for modulated birefringence
measurements.

A schematic of the optical setup is shown in Fig. S1. The incident
polarization angle, $\varphi$, of the probe laser (633 nm) is set by a
polarizer followed by a half-wave plate (HWP1). The measured polarization
rotation from the sample is independent of the pump (780 nm) polarization.
After reflecting off the sample, the pump beam is rejected by a color filter
(F), and the probe beam polarization is further rotated by an angle
$\varphi+45\degree$ by a second half-wave plate (HWP2). In the case where the
polarization state is not altered by the setup or the sample, the polarization
of the reflected probe beam upon exiting the second half-wave plate is an
equal superposition of vertical (V) and horizontal (H) linearly polarized
light. The beam is then sent through a Wollaston prism (WP), which spatially
separates vertical and horizontal components of the light, and the two
orthogonal components are each focused onto separate, unbiased photodiodes of
an balanced optical bridge detector. When the V and H components have equal
intensity, which occurs when the polarization state of the light is unchanged
by the sample, the net photocurrent is zero; in contrast, any measured signal
indicates a change of polarization, such that the technique has a high
sensitivity to detecting such changes.

While changes in the final polarization state can also be introduced by
birefringence or ellipticity of the setup—resulting in artifacts—we largely
mitigate this issue by performing thermally-modulated (T–mod) and field-
modulated (H–mod) experiments, as these setup effects are independent of both
temperature and field. For thermal modulation, a second (pump) laser is
focused onto the same spot on the sample as the probe beam and optically
chopped at kHz frequencies to locally modulate the temperature of the sample.
For field modulation, the sample is placed in a coil that is driven with an
alternating current to create an oscillating magnetic field. The experiment is
sensitive only to effects which are proportional to the modulation parameter,
which the artifacts arising from the setup are not. Despite this, small cross-
coupling terms can still occur if there is more than one optical constant
proportional to the modulation parameter, as discussed in detail in the
Supplementary Material of Ref. [33].

Figure S2: Thermally modulated (a) offset, (b) $d\mathrm{R}/d\mathrm{T}$, and
(c) birefringence amplitude.

In Phase I, the sample becomes birefringent, rotating the polarization of the
probe beam, which unbalances the contributions of the orthogonally polarized
light components entering the balanced photodiode and causes a signal to be
measured. The intensity of the light admitted to each of the photodiodes can
be calculated using the Jones calculus formalism, in which the polarization
state of the light is represented by a vector in the (V, H) basis and each
optical component is represented by a 2x2 matrix. The Jones matrix for a half-
wave plate with its fast axis rotated by $\theta$ with respect to the
horizontal axis, thereby rotating the polarization by $\varphi=2\theta$, is
described by:

$J_{hwp}\left(\theta\right)=\left(\begin{array}[]{cc}\cos(2\theta)&\sin(2\theta)\\\
\sin(2\theta)&-\cos(2\theta)\end{array}\right).$ (S1)

The Jones matrix representing the sample is:

$J_{sam}\left(r,b,k,\varphi_{0}\right)=R\left(-\varphi_{0}\right)\left(\begin{array}[]{cc}r+b&k\\\
-k&r-b\\\ \end{array}\right)R\left(\varphi_{0}\right),$ (S2)

where $r$ represents the sample reflectivity, $b$ the birefringence
(difference in reflectivity between V and H polarizations), and $k$ for the
polar Kerr effect (difference in reflectivity between the left- and right-
circularly polarized light caused by an out-of-plane magnetization).
$R\left(\varphi_{0}\right)$ is a rotation matrix, which encodes the different
orientations of the sample and its domains with respect to the lab coordinate
system:

$R\left(\varphi_{0}\right)=\left(\begin{array}[]{cc}\cos(\varphi_{0})&-\sin(\varphi_{0})\\\
\sin(\varphi_{0})&\cos(\varphi_{0})\\\ \end{array}\right)$ (S3)

Modeling the experiment in this way, the final polarization state of the
reflected light is given by,

$\begin{pmatrix}E_{V}\\\
E_{H}\end{pmatrix}=J_{\mathrm{HWP2}}(\varphi/2+22.5\degree)\cdot
R(-\varphi_{0})\cdot J_{\mathrm{sample}}\cdot R(\varphi_{0})\cdot
J_{\mathrm{HWP1}}(\varphi/2)\cdot\begin{pmatrix}1\\\ 0\end{pmatrix}.$ (S4)

The balanced photodiode detector measures the intensity difference between the
two orthogonal polarization components, $I=|E_{V}|^{2}-|E_{H}|^{2}$. In the
experiment, we rotate the two half-wave plates, which rotates the polarization
of the incident probe beam relative to the sample to simulate rotation of the
sample.

The data were fit to the following function:

$I(\varphi)=A\mathrm{sin}(2(\varphi-\varphi_{0}))+B\mathrm{sin}(4(\varphi-\varphi_{1}))+C.$
(S5)

The term $A\mathrm{sin}(2(\varphi-\varphi_{0}))$ describes the birefringence:
$A$ is the amplitude of the birefringence and $\varphi_{0}$ is the orientation
of the sample’s principal axes relative to the lab coordinate frame. The term
$B\mathrm{sin}(4(\varphi-\varphi_{1}))$ is small and arises from second order
couplings of the sample birefringence as well as weak residual birefringence
and ellipticity of the setup that can cross-couple with modulated
birefringence from the sample.

The offset term, $C$, typically describes the derivative of the sample
magnetization with respect to the modulation parameter; this corresponds to
the magneto-optical Kerr effect for the thermally modulated experiment and the
magnetic susceptibility for the field modulated experiment. However, cross-
couplings between the sample and setup can result in a small offset artifact
that is unrelated to these derivatives and which can be resolved when they are
equal to zero. Fig. S2 shows the thermally modulated offset, reflectivity, and
birefringence amplitude. In the thermally modulated measurements, $C$ is small
(two orders of magnitude smaller than the birefringence amplitude $A$ and
thermally-modulated, temperature-dependent reflectivity $dR/dT$).
Contributions to this term arise from cross-coupling between the sample and
setup birefringence, ellipticity, and $dR/dT$, resulting in a small offset
(typically at least an order of magnitude smaller than the birefringence
amplitude) when no magneto-optical Kerr effect is detected in the material, as
is the case for EuIn2As2. See Ref. [33] for an in-depth analysis of the ways
in which these cross-coupling terms arise, the material and optical properties
they couple, the artifacts they can create, their influence on the measured
signal, and methods for mitigating them to isolate the true signal.

##  Appendix S3 Strain experiment and calibration

Figure S3: (a) sample loaded into strain cell. (b) strain dependence of
birefringence vs temperature.

###  S3.1 Experiment

For the measurement of $T$-mode birefringence under applied strain, a EuIn2As2
sample (600$\times$500 $\mu$m) was loaded into a Razorbill Instruments CS130
strain cell. The sample was secured between two titanium plates at each end
with Stycast 2850FT epoxy, which were in turn bolted to the piezo-actuated
jaws of the cell (Fig. S3a). A custom thermal link to the cryostat base was
mounted onto the strain cell, from which two copper wires were used to heat
sink the sample via EPO-TEK H20E silver epoxy. Two additional wires were
routed from the sample to a Cernox thermometer and resistive heater
respectively, both of which were mounted onto the strain cell body and
isolated from the thermal link by an air gap. The CS130 is equipped with a
capacitive sensor to monitor the displacement of the jaws, which we measured
using a Keysight E4980AL LCR meter.

To explore the temperature-strain phase diagram of EuIn2As2, we measured the
birefringence as a function of both temperature and strain across the two
transitions. Fig. S3b shows the temperature dependence of the $T$-mod
birefringence for different strains at a characteristic spot on the sample.
For each curve, the sample was cooled to 16 K in zero applied strain, after
which the voltage applied to the piezos was ramped to a fixed value at $\sim
0.5$ V/s. The birefringence was measured while warming under the applied
voltage, allowing the reading on the strain cell capacitor to stabilize at
each temperature. Because the voltage-strain relationship is temperature
dependent, strictly speaking the strain changes with temperature for each
curve, however we report the average strain and find that the variance is
negligible over this small temperature range. Importantly, the transition
temperature and overall shape of the birefringence with temperature is
unmodified by strain, indicating that the essential character of the phase
transition remains intact with strain.

Furthermore, to demonstrate strain tunability of the magnetic structure, we
swept the strain at a fixed temperature of 16K (Fig. 10). After cooling the
sample in zero applied strain, we swept the voltage on the piezos from 0
$\rightarrow$ 20 V $\rightarrow$ $-20$ V $\rightarrow$ 0, measuring the
capacitance at each point.

###  S3.2 Calibration

The methods for calculating the strain applied to a sample mounted in a CS130
strain cell are described in detail in the Razorbill Instruments application
notes [54]. The strain cell incorporates a capacitive displacement sensor to
quantify the strain in the sample for a given applied voltage to the peizo
stacks. In the ideal scenario, the displacement of the capacitor plates is
equivalent to the displacement of the crystal, from which the strain can be
calculated as $\epsilon=\Delta L/L$. However several caveats must be taken
into account: (1) the capacitance of the sensor exhibits a temperature
dependence independent of the true gap between the jaws of the cell, primarily
due to a contraction of the distance between the capacitor plates at lower
temperatures, (2) the differential thermal expansion between the sample and
the titanium cell leads to an offset to the displacement of the sample, and
(3) the finite stiffness of the epoxy and bolting plates securing the sample
to cell manifests as an additional reduction to the sample displacement.
Obtaining an accurate strain measurement thus requires careful consideration.
Here, we describe the specific procedure used towards these corrections.

The displacement of the piezo stacks corrected for temperature dependent
contractions in the capacitor can be parameterized as:

$\Delta L_{\textnormal{piezo}}(T,C)=\frac{\alpha}{C(T)-C_{p}(T)}-d_{0},$ (S6)

where $T$ and $C$ are the temperature and measured capacitance respectively,
$\alpha$ and $d_{0}$ are provided by Razorbill, and $C_{p}(T)$ represents a
temperature dependent parallel capacitance. $C_{p}(T)$ can be obtained by
loading a stiff titanium dummy sample, which isolates the temperature
dependence of the measured capacitance to changes in the capacitor itself, and
measuring the capacitance with no applied voltage,
$C_{\textnormal{dummy}}(T)$, as a function of temperature:

$C_{p}(T)=C_{\textnormal{dummy}}(T)-\frac{\alpha}{d_{0}}.$ (S7)

The experimental values for $C_{\textnormal{dummy}}(T)$ measured in our system
are shown in Fig. S4.

Figure S4: Measured $C_{\textnormal{dummy}}(T)$ vs. $T$.

As mentioned above, $\Delta L_{\textnormal{piezo}}$ ought to be corrected to
obtain the true sample displacement $\Delta L$. However, as neither the
thermal expansion nor the Young’s modulus are available for EuIn2As2, we have
chosen to disregard these corrections in reporting strain with the
understanding that the values are relative and should be taken as an upper
bound to the true strain. These need not be small corrections; simulations for
the case of the iron-pnictide supercondctors show strain transmission of 70%
due to the pliability of the epoxy, and the thermal expansion between titanium
and a test sample can differ by several multiples [55, 56] . However, the
qualitative observation that the optical axis is tunable under strain is
independent of these corrections.

##  Appendix S4 Microdomain scenario

Figure S5: (a) Birefringence amplitude and (b) principle axis orientation as a
function of $x_{0}$ and $x_{1}$ with $x_{2}$ set by the normalization
condition. The ellipsoid indicates a possible phase-space trajectory that
yields a constant amplitude and changing orientation. (c) Population fraction
phase-space trajectories around the constant amplitude ellipsoid versus
expected principal axis.

In any symmetry-breaking transition, energetically equivalent but
distinguishable configurations of the order parameter, i.e. domains, can be
obtained by the application of the broken symmetry operators. In EuIn2As2, the
transition into Phase I breaks the $C_{3z}$ symmetry of the paramagnetic
state, so three domains related by $C_{3z}$ rotation are generally expected.
However, our experiment does not show the expected three domains, and reveals
a broad orientation continuum instead (Fig. 4).

One possible explanation of this observation is that the size of such domains
is smaller than our resolution ($\sim$ 5 $\mu$m), so that the measured
birefringence results from averaging over many domains. Here we model this
microdomain scenario to illustrate that it in fact cannot naturally account
for the smoothly varying principle axis orientation and constant birefringence
amplitude observed in EuIn2As2 as shown in Figs. 4 and 5.

We consider a collection of small domains within the area of our laser spot.
Each domain is described by a nematic vector with its principle axis oriented
along one of the three $C_{3z}$-equivalent directions of the crystal, which we
label $i=0,1,2$; taking orientation 0 to point along the $x$-axis, these
correspond to the nematic vector pointing along $i\pi/3$. We denote the
population fraction of domains at orientation $i$ as $x_{i}$, such that
$\sum_{i}x_{i}=1$. The reflectivity matrix for a domain at orientation $i$ can
be expressed as

$r_{i}=R(-i\pi/3)\begin{pmatrix}1+\delta r&0\\\ 0&1-\delta
r\end{pmatrix}R(i\pi/3),$ (S8)

where $R(\theta)=\begin{pmatrix}\cos(\theta)&-\sin(\theta)\\\
\sin(\theta)&\cos(\theta)\end{pmatrix}$ is a typical rotation matrix.

As discussed in Sec. S2, each domain at orientation $i$ contributes a
polarization rotation as a function of incident polarization angle of
$d\phi_{i}=\delta r\sin(2(\phi-i\pi/3))$ such that the total signal is
$d\phi=\sum_{i}x_{i}d\phi_{i}.$ To first order in $\delta r$, this evaluates
to

$d\phi=A(\mathbf{x})\sin(2(\phi+\alpha(\mathbf{x}))),$ (S9)

where the effective birefringence amplitude $A(\mathbf{x})$ and principle axis
orientation $\alpha(\mathbf{x})$ are given by

$\displaystyle A(\mathbf{x})$ $\displaystyle=\delta
r\sqrt{3(x_{1}-x_{2})^{2}+(x_{1}+x_{2}-2x_{0})^{2}}$ (S10)
$\displaystyle\alpha(\mathbf{x})$
$\displaystyle=\frac{1}{2}\tan^{-1}\left(\frac{\sqrt{3}(x_{1}-x_{2})}{x_{1}+x_{2}-2x_{0}}\right)$
(S11) $\displaystyle\mathbf{x}$ $\displaystyle=(x_{0},x_{1},x_{2}).$ (S12)

Having developed a model for the microdomain scenario, we want to determine
whether a continuously changing principle axis orientation can be consistent
with a constant amplitude as observed in the experiments. To this end, we
explore both quantities in the $\mathbf{x}$ phase space, which represents the
possible spread of domains at different spots on the sample. The normalization
constraint $\sum_{i}x_{i}=1$ allows us to rewrite Eqns. S10 and S11 in terms
of just two components of $\mathbf{x}$, arbitrarily taken as $x_{0}$ and
$x_{1}$. Fig. S5a-b shows the effective birefringence amplitude and principle
axis within this reduced parameter space. As indicated by the red ellipses, we
find that the only trajectories that are constant in amplitude with changing
principle axis are very fine-tuned, involving contrived changes in populations
of all three domains (Fig. S5c). While this scenario is technically feasible,
the strict restriction on the phase space has seemingly no physical basis. On
these grounds we discard the microdomain scenario as a viable explanation of
the birefringence in EuIn2As2.

The predictions based on this model in the main text show the case of only two
competing domain orientations. Specifically, we define the orientation with
maximum weight in Fig. 4c to be $\theta_{0}=0$ with respect to the crystal
with population $x_{0}$, and consider the broad distribution to be a
consequence of mixing in some of the other two domains; for orientations
clockwise to $\theta_{0}$ we allow a population $x_{2}$ of $\phi_{2}=2\pi/3$
domains, and for orientations counterclockwise to $\theta_{0}$ we allow a
population $x_{1}$ of $\phi_{1}=\pi/3$ domains, taking the population of the
third domain to zero. For example, $x_{2}=0$ corresponds to the bottom edges
of the phase space in Fig. S5. For each point in Fig. 5b, we then use Eqns.
S10 and S11 to uniquely specify the amplitude expected for the measured
orientation.

##  Appendix S5 Symmetry analysis

In this section, we provide details of the symmetry analysis and group theory
methods that we applied to pinpoint the magnetic configuration for Phase I and
Phase II of EuIn2As2.

Two ingredients are important to characterize a magnetically ordered state:
the propagation vector(s) $\mathbf{Q}$, which determines the periodicity of
the magnetic structure, and the symmetries that they preserve. The spatial
dependence of the expectation values of the magnetic moments in the ordered
state is given by

$\mathbf{M}_{\alpha}(\mathbf{r}_{i})=\sum\limits_{\mathbf{Q}}e^{i\mathbf{Q}\cdot\mathbf{r}_{i}}\mathbf{M}_{\alpha}(\mathbf{Q})\text{
.}$ (S13)

Here, $\alpha$ denotes the sublattice index and $\mathbf{r}_{i}$ is the
position of the atom $i$ of the underlying lattice.
$\mathbf{M}_{\alpha}(\mathbf{r}_{i})$ transforms as one or more Irreducible
representations (Irreps) of the parent paramagnetic space group (also called
gray group). The gray group is formed by the crystal space group $G$ combined
with time-reversal ($\mathcal{T}$), $G\oplus\mathcal{T}G$. The development of
a magnetic order breaks $\mathcal{T}$ alone and certain spatial operations
$g\in G$ 111$g$ can be proper or improper rotations, spatial inversions, and
also non-symmorphic operations in which point group operations are combined
with half-lattice translations., but may preserve certain combinations of
spatial operations and time-reversal $\mathcal{T}g$. Therefore, the magnetic
order lowers $G\oplus\mathcal{T}G$ down to one of its subgroups that define
the magnetic space group (MSG) $G_{M}$ of the ordered state.

For EuIn2As2, the gray group is $P6_{3}/mmc1^{\prime}$ (No. 194.264),
$\mathbf{Q}=\mathbf{Q}_{1}=\pm(0,0,1/3)$ for Phase I and
$\mathbf{Q}=\mathbf{Q}_{1},\mathbf{Q}_{2}=\pm(0,0,1)$ for Phase II. Besides,
$\alpha=1,2$ since there are two non-equivalent Eu atoms per crystallographic
unit cell. For the discussion that comes later in this section, we can focus
on the dependence of $\mathbf{M}_{\alpha}(z_{i})$ on the $z$-component of the
position of the Eu layer $i$, since the moments are ferromagnetically aligned
within each plane and the ordering vectors are perpendicular to the direction
$\hat{z}$ of stacking of the Eu planes (see Fig.S6).

Note that throughout the paper, we write $\mathbf{Q}$ in terms of the
reciprocal lattice vectors,
$\mathbf{Q}=(h,k,l)=h\mathbf{b}_{1}+k\mathbf{b}_{2}+l\mathbf{b}_{3}$, where
$\mathbf{b}_{1}=-\frac{4\pi}{3a}\hat{x}$,
$\mathbf{b}_{2}=\frac{2\pi}{3a}\left(-\hat{x}+\sqrt{3}\hat{y}\right)$.

Magnetic structures with the same $\mathbf{Q}$ but distinct
$\mathbf{M}_{\alpha}(\mathbf{Q})$ have different symmetry properties and
transform as distinct Irreps of the parent gray group. In this section, we
explain how to get a list of all magnetic structures, and corresponding
$\mathbf{M}_{\alpha}(\mathbf{Q})$, that are candidates for Phases I and II
from a symmetry perspective only. We further narrowed it down based on
experimental evidence. Our multimodal protocol can be summarized as follows:

  * •

Step 1\- given the paramagnetic space group and the ordering vector(s) for the
phase in case, the list of all magnetic space groups that are subgroups of the
paramegnetic groups and can be reached by a magnetic phase transition can be
readily obtained using the software Isotropy [57]. The corresponding Irreps
and order parameters are also listed.

  * •

Step 2\- For each Irrep, there are multiple magnetic structures belonging to
different magnetic space groups. Coupled Irreps are also possible. We
constrained which Irrep to focus on by requiring that the magnetic structure
has magnetic moments lying on the Eu planes and magnetic Bragg peaks
consistent with scattering experiments[16, 29]. These constraints singled out
Irrep $m\Delta_{6}$ for Phase I and $m\Delta_{6}$ coupled with
$m\Gamma_{5}^{+}$ for Phase II.

  * •

Step 3\- All the magnetic structures that transform as the Irrep(s) selected
in step 2 are listed in Table S1 for Phase I and S2 for Phase II.

  * •

Step 4\- The symmetries of each magnetic structure from step 3 are compared
with the constraints set by the optical measurements, which are used to rule
out a subset of them.

Figure S6: Representative moment configurations of the class of magnetic
structures listed in Tables S1 and S2. The moments in each Eu layer are in-
plane and ferromagnetically aligned, pointing along the direction indicated by
the arrows. The top view of each structure is shown in the top row of the
figure (panels a-e). In panels (e-f), the broken helix structure fulfills the
equal moment condition in Eq. S49

.

For Phase I, our multimodal approach allowed us to identify three states that
are consistent with both scattering and optical experiments. They correspond
to nodal amplitude-modulated states with different orientations of the
magnetic moments. One of the possibilities is that the moments are parallel to
$[100]=\hat{x}$, as illustrated in Fig.S6b and characterized by an order
parameter $\bm{\Psi}_{I}=(a,a,0,0)$. Due to the high-symmetry direction of the
moments, this structure preserves the two-fold rotation around $\hat{y}$
($C_{2[010]}$), as well as a non-symmorphic two-fold rotation around $\hat{z}$
($C_{2z}$) and the product between time-reversal and inversion
($\mathcal{T}\mathcal{P}$), and belongs to the MSG
$Cm^{\prime}c^{\prime}m^{\prime}$ (No. 65.564). Note that any structure
obtained from $\bm{\Psi}_{I}=(a,a,0,0)$ by a three-fold rotation around $z$
($C_{3z}$) is a related domain and belongs to the same MSG. The second allowed
nodal structure is characterized by the order parameter
$\bm{\Psi}_{I}=(0,0,a,-a)$ and has moments parallel to $\hat{y}$. This is also
a high-symmetry direction in the Eu plane. The moments break $C_{2[010]}$, but
preserve a mirror symmetry $M_{[010]}=\mathcal{P}C_{2[010]}$. The third
allowed nodal structure is parametrized by $\bm{\Psi}_{I}=(a,a,b,-b)$ and is a
linear superposition of the two structures mentioned before. As a result, the
magnetic moments point along a direction that is not a high-symmetry one, and
neither $M_{[010]}$ nor $C_{2,[010]}$ are preserved. The only symmetries left
in the generic orientation nodal structure are the non-symmetric $C_{2z}$ and
$\mathcal{P}\mathcal{T}$, lowering the MSG to $P2_{1}/m^{\prime}$ (No. 11.53).

Note that the sixth-order terms in the Landau functional (see Sec.S8) enforce
the moments to point along a high-symmetry direction, reflecting the
underlying crystalline anisotropy. However, in the regime where built-in
strain dominates over such a crystal anisotropy, the nodal structure with
lower symmetry is generically favored by uniaxial strain along an arbitrary
direction. Motivated by the observed broad distribution of crystal axes, we
will consider this regime.

For Phase II, on the other hand, all structures in Table S2 are consistent
with both scattering and optical experiments. Among them, the three broken-
helix structures are more likely to describe Phase II since they are favored
by the equal moment condition, which in turn is expected if the magnetism of
EuIn2As2arises from localized $S=7/2$ Eu2+ moments (see Sec.S8.4). Similarly
to the three nodal amplitude modulated structures possible for Phase I, the
three broken helices in Phase II are distinguished by the orientation of the
Néel component with respect to the crystal axis. Once again, crystal
anisotropy favors moments parallel to the in-plane high-symmetry direction,
but the dominant built-in strain favors a less symmetric orientation of the
moments (MSG $P2_{1}$ (No. 4.7)). Given the broad distribution of built-in
strain direction suggested by our optical experiments, the broken helix in
Phase II is essentially unpinned from the crystal axes, presumably due to
built-in strain.

Representatives of each type of state appearing in Tables S1 and S2 are shown
in Fig. S6. Note that in Phase I there is a class of structures labeled
“distorted helix”, which is obtained by a superposition of an amplitude-
modulated structure and a 60°-helix. The moments get distorted in comparison
with the more symmetric 60-°counterpart, but there is still a well-defined
single-handed circulation of the moments’ direction around $\hat{z}$. In Phase
II, an AFM component is also added to the magnetic structure, in agreement
with the development of the additional $\mathbf{Q}_{2}=(0,0,1)$ ordering
vector. When strong enough, this AFM component leads to the interchange of
four moment directions, leading to a broken helix similar to that originally
introduced in Ref. 16 (compare Fig. S6d and e). However, as mentioned in the
main text, a key difference is that the broken helix proposed in this work is
not pinned to a high-symmetry direction in the Eu plane due to the strong
effect of the built-in strain.

Table S1: Candidate magnetic structures for Phase I that are consistent with
the experimentally observed Bragg peaks $\mathbf{Q}_{1}=(0,0,1/3)$ and
magnetic moments lying within the Eu planes. For each structure, we specify
the form of the order parameter $\bm{\Psi}_{I}$ written in the basis of the
representation space of $m\Delta_{6}$ adopted by Isotropy (helical basis). The
transformation matrix between this basis and the collinear basis that we adopt
in this work is shown in Sec. S6.3. The magnetic space group (MSG) and its
generators are also listed. $1$ denotes the identity operation, $\mathcal{P}$
denotes spatial inversion, and $\mathcal{T}$ represents time reversal.
$C_{n,[uvw]}$ denotes a rotation of $2\pi/n$ around the axis
$[uvw]=u\mathbf{a}_{1}+v\mathbf{a}_{2}+\mathbf{a}_{3}$, where $\mathbf{a}_{j}$
are primitive crystal axes. We set $[001]\parallel\hat{z}$. The mirror
operations are $M_{[uvw]}=\mathcal{P}C_{2,[uvw]}$. When any of these
operations (generically denoted by $g$) is combined with a translation
$\tau=\tau_{1}\mathbf{a}_{1}+\tau_{2}\mathbf{a}_{2}+\tau_{3}\mathbf{a}_{3}$ by
a fraction of the primitive unit cell, we have a non-symmorphic operation
$\\{\left.g\right|\tau_{1}\tau_{2}\tau_{3}\\}$. The last column shows the two
optical responses considered in this work. $\checkmark$ ($\times$) signifies
that the response is symmetry allowed (forbidden).

Magnetic structure | Order parameter | MSG | Generators of the MSG | Optical responses
---|---|---|---|---
T-mod signal | H-mod signal
60°-helix | $(0,0,0,a)$ | P612’2’ (178.159) |  $1$, $\\{\left.C_{2z}\right|00\frac{3c}{2}\\}$, $\\{\left.C_{3z}\right|00c\\}$, $\mathcal{T}C_{2,[010]}$ | $\times$ | $\times$
$(0,0,a,0)$ | P652’2’ (179.165) |  $1$, $\\{\left.C_{2z}\right|00\frac{3c}{2}\\}$, $\\{\left.C_{3z}\right|002c\\}$, $\mathcal{T}C_{2,[010]}$ | $\times$ | $\times$
$(a,0,0,0)$ | P6522 (179.161) |  $1$, $\\{\left.C_{2z}\right|00\frac{3c}{2}\\}$, $\\{\left.C_{3z}\right|002c\\}$, $C_{2,[010]}$ | $\times$ | $\times$
$(0,a,0,0)$ | P6122 (178.155) |  $1$, $\\{\left.C_{2z}\right|00\frac{3c}{2}\\}$, $\\{\left.C_{3z}\right|00c\\}$, $C_{2,[010]}$ | $\times$ | $\times$
$(a,0,b,0)$ | P65 (170.117) |  $1$, $\\{\left.C_{2z}\right|00\frac{3c}{2}\\}$, $\\{\left.C_{3z}\right|002c\\}$ | $\times$ | $\times$
colinear nodal amplitude modulated | $(a,a,0,0)$ | Cm’c’m’ (63.465) | $1$, $\\{\left.C_{2z}\right|00\frac{3c}{2}\\}$, $C_{2,[010]}$, $\mathcal{T}\mathcal{P}$ | $\checkmark$ | $\times$
colinear nodeless amplitude modulated | $(0,0,a,a)$ | Cm’c’m (63.462) | $1$, $\\{\left.C_{2z}\right|00\frac{3c}{2}\\}$, $\mathcal{T}C_{2,[010]}$, $\mathcal{P}$ | $\checkmark$ | $\checkmark$
| colinear nodal
---
amplitude modulated
$(0,0,a,-a)$ | Cmcm’ (63.461) |  $1$, $\\{\left.C_{2z}\right|00\frac{3c}{2}\\}$, $M_{[010]}$, $\mathcal{T}\mathcal{P}$ | $\checkmark$ | $\times$