# Quantum decoherence by magnetic fluctuations in a candidate axion insulator

Ruben Saatjian  Materials Department, University of California, Santa Barbara,
California 93106, USA  Kohtaro Yamakawa  Department of Physics, University of
California, Berkeley, California 94720, USA  Ryan S. Russell  Materials
Department, University of California, Santa Barbara, California 93106, USA
James G. Analytis  Department of Physics, University of California, Berkeley,
California 94720, USA  John W. Harter<EMAIL_ADDRESS>Materials Department,
University of California, Santa Barbara, California 93106, USA

###### Abstract

In magnetic topological insulators, spontaneous time-reversal symmetry
breaking by intrinsic magnetic order can open an energy gap in the topological
surface spectrum. In the resulting state, exotic properties like axion
electrodynamics, the quantum anomalous Hall effect, and other topological
magnetoelectric responses are expected to emerge. A detailed understanding of
the magnetic order and its coupling to the topological surface states is
essential to harness and tune these properties. Here, we leverage near-
resonant electric quadrupole optical second harmonic generation to probe
magnetic fluctuations in the candidate axion insulator EuSn2(As,P)2 across its
antiferromagnetic phase boundary. We observe a pronounced dimensional
crossover in the quantum decoherence induced by magnetic fluctuations, where
two-dimensional in-plane ferromagnetic correlations at high temperatures give
way to three-dimensional long-range order at the Néel temperature. We also
observe the breaking of rotational symmetry within the long-range-ordered
antiferromagnetic state and map out the resulting spatial domain structure.
More generally, we demonstrate the unique capabilities of nonlinear optical
spectroscopy to study quantum coherence and fluctuations in magnetic quantum
materials.

Magnetic topological insulators represent a fascinating class of quantum
materials that combine a topologically nontrivial electronic structure with
magnetic order [1]. Under these conditions, spontaneous time-reversal symmetry
breaking by the magnetism can promote a range of novel phenomena, including
axion electrodynamics [2, 3, 4, 5, 6], the quantum anomalous Hall effect [7,
8, 9, 10], chiral topological superconductivity [11], and the generation of
ideal Weyl nodes [12, 13, 14]. These states have unique functionalities and
tunability, with potential applications in spintronic devices and other
advanced technologies. The first experimental observation of such physics
occurred in conventional topological insulators extrinsically doped with
dilute magnetic ions [8, 9, 10], but this necessarily introduces substantial
disorder into the material. More recently, intrinsic magnetic topological
insulators, which are stoichiometric compounds with native long-range magnetic
order, have garnered significant interest as clean and tunable material
platforms that may facilitate the observation and study of topological
magnetoelectric phenomena at higher temperature scales.

Although still rare, two dominant intrinsic magnetic topological insulator
material families have emerged in recent years: MnBi2Te4 and related
materials, the subject of considerable early work [6, 15, 16, 17, 18, 19], and
a series of Eu-based Zintl materials including EuIn2As2, EuCd2As2, EuSn2As2,
and EuSn2P2 [5, 12, 13, 14, 20, 21, 22, 23]. This latter family has a layered
structure with Eu2+ ions forming a triangular lattice sandwiched between two
covalently-bonded buckled honeycomb layers, such as (SnAs)${}_{2}^{2-}$, from
which the topological bands are derived. Magnetism arises from half-filled
$4f^{7}$ subshells on the Eu2+ ions. These localized high-spin ($S=7/2$)
moments align ferromagnetically in the plane and antiferromagnetically out of
the plane [Fig. 1(a)], resulting in long-range A-type antiferromagnetism below
a Néel temperature $T_{N}$ that ranges from 10 K to 30 K, depending on
material. The distinct spatial separation of magnetic and topological
electronic states in these materials raises the possibility of precise
magnetic control over the electronic topology, for example through the
direction of magnetization [5].

In this work, we study fluctuations and symmetry breaking by the magnetic
degrees of freedom in the Eu-based materials using near-resonant electric
quadrupole optical second harmonic generation (SHG). We focus on the
EuSn2(As,P)2 compounds, which have the highest reported $T_{N}$ (24 K and 30 K
for the Sn and P endmembers, respectively [20, 21]). We measure the quantum
decoherence induced by magnetic fluctuations across $T_{N}$ and find a
dimensional crossover from short-range two-dimensional in-plane ferromagnetic
correlations above $T_{N}$ to three-dimensional long-range antiferromagnetic
order below it. Within the low-temperature antiferromagnetic phase, we detect
broken rotational symmetry due to global alignment of the in-plane magnetic
moments, and map out the corresponding spatial domain patterns. Our results
motivate several routes to tune the magnetism in this series of materials,
paving the way for the potential control of topological magnetoelectric
properties. More broadly, we showcase the capabilities of nonlinear optical
spectroscopy to access the degree of quantum coherence of magnetic
fluctuations in quantum materials.

Figure 1:  Crystal and electronic structure of EuSn2(As,P)2 and RA-SHG
patterns. (a) Crystal structure showing half of the conventional magnetic unit
cell. Red arrows depict Eu2+ local magnetic moments, which lie within the
plane along the $a$ axis. Moments are aligned within each plane and
antialigned with neighboring planes, forming long-range A-type
antiferromagnetism. The inset illustrates the local environment surrounding
the Eu2+ ions, showing $D_{3d}$ site symmetry. (b) Orbital-resolved electronic
band structure of EuSn2As2 from DFT. The calculation was performed for the
primitive unit cell (ferromagnetic moment alignment), but k-points are
referenced to the conventional unit cell. Vertical arrows highlight optical
transitions near the $M$ point contributing to near-resonant SHG. The
irreducible representations of the states participating in the resonances are
labeled. The right inset shows a simplified energy level diagram of SHG
involving both electric dipole (ED) and electric quadrupole (EQ) optical
transitions. (c) Parallel polarization RA-SHG patterns for all three materials
at 50 K. The top left panel defines the oblique incidence scattering geometry.
The top row shows Sin-Sout (SS) polarization patterns and the bottom row shows
Pin-Pout (PP) polarization patterns.

Single crystals of EuSn2As2, EuSn2P2, and the 1:1 alloy EuSn2AsP were prepared
by a flux method similar to Ref. 24. All three materials show nearly-identical
nonlinear optical responses, and magnetic susceptibility measurements show
clear signatures of antiferromagnetic transitions with $T_{N}$ closely
matching previously reported values [20, 21, 22, 24]. Further details of
sample synthesis and characterization can be found in the Supplemental
Materials. Density functional theory (DFT) calculations were performed with
the Vienna Ab initio Simulation Package (vasp) using the supplied projector
augmented wave potentials within the generalized gradient approximation, the
experimental structural parameters [20], a 400 eV energy cutoff, and a
$6\times 6\times 6$ k-point mesh. To match the experimental energy position of
the Eu $4f$ bands, a Hubbard parameter $U=5.0$ eV was employed [22]. Measuring
the rotational anisotropy of optical SHG (RA-SHG) is a precise method to
access the nonlinear optical susceptibility tensor and deduce the point group
symmetries of a material. Using RA-SHG, tensor elements can be tracked across
a wide temperature range to reveal minute changes in order, symmetry,
dimensionality, and the strength of interactions in a sample [25, 26, 27, 28].
With this goal, RA-SHG experiments were performed on freshly-cleaved samples
in an optical cryostat using an ultrafast laser supplying 40-fs pulses at a
50-kHz repetition rate with a center wavelength of 800 nm and a fluence of
$\sim$1 mJ/cm2. The extreme sensitivity of SHG can be further enhanced near
resonances, strengthening the SHG response by orders of magnitude and
amplifying signatures of states involved in the resonant transition. As Fig.
1(b) demonstrates, 800 nm excitation (1.55 eV) is nearly resonant with several
energy transitions involving Eu $4f$ and $5d$ orbitals at the $M$ point in the
first Brillouin zone, gaining us direct access to the magnetic degrees of
freedom.

Figure 1(c) illustrates the RA-SHG experimental scattering geometry and
corresponding data for all three samples. In the Sin-Sout channel, six equal
lobes appear, while in the Pin-Pout channel, the underlying 3-fold rotational
symmetry of the crystallographic point group $D_{3d}$ ($\bar{3}m$) is revealed
through alternating large and small lobes. Oblique incidence is key to the
appearance of this alternation; normal incidence measurements would fail to
capture this lower symmetry and instead yield six equal lobes as in the Sin-
Sout data. The measured SHG intensity is relatively strong, which is unusual
because $D_{3d}$ is centrosymmetric and leading-order bulk electric dipole
SHG—the usual source of strong nonlinear optical responses—is hence excluded
by symmetry. Among the most likely higher-order nonlinear responses are
surface electric dipole, bulk magnetic dipole, and bulk electric quadrupole
SHG (EQ-SHG) [26]. Surface SHG typically arises from slight atomic distortions
at the surface of a material and is therefore generally weakly dependent on
temperature. We exclude this possibility based on the striking temperature
dependence that we observe and discuss below. We also exclude magnetic dipole
SHG due to the presence of time-reversal symmetry (in combination with a
lattice translation below $T_{N}$). This leaves EQ-SHG as the primary
nonlinear response channel in EuSn2(As,P)2. Recently, EQ-SHG has proven to be
a powerful tool for probing the intricate magnetic properties of topological
materials such as magnetic Weyl semimetals [28].

EQ-SHG is described by a fourth-rank nonlinear optical susceptibility tensor
$\chi^{\mathrm{EQ}}_{ijkl}$ that relates the induced nonlinear polarization
density in a material $P_{i}(2\omega)$ to two powers of the incident electric
field $E_{i}(\omega)$ via the equation
$P_{i}(2\omega)=\chi^{\mathrm{EQ}}_{ijkl}E_{j}(\omega)\nabla_{k}E_{l}(\omega)$
[26]. Within point group $D_{3d}$ and with additional permutation symmetries
adapted from Ref. 29, we may calculate the explicit angular dependence of the
RA-SHG patterns. For the two parallel polarization channels shown in Fig.
1(c), we obtain $I_{SS}(\phi)=|\alpha\sin(3\phi)|^{2}$ and
$I_{PP}(\phi)=|\alpha\cos(3\phi)+\beta|^{2}$, where
$\alpha=\chi^{\mathrm{EQ}}_{xxzx}\cos(\theta)$,
$\beta=(\chi^{\mathrm{EQ}}_{xxxx}-\chi^{\mathrm{EQ}}_{xxzz})\sin(\theta)$,
$\phi$ is the azimuthal angle of the scattering plane, and $\theta$ is the
angle of incidence relative to the sample surface normal. Fits of the data to
these functional forms show excellent agreement, as demonstrated in Fig. 1(c).
Further details of the tensor symmetry analysis can be found in the
Supplemental Materials.

A defining characteristic of EQ-SHG is the inclusion of an optical transition
between two states of the same parity, a good quantum number in
centrosymmetric systems. Such excitations are forbidden in conventional
electric dipole transitions. With this symmetry condition in mind, we examine
the irreducible representations of wave functions calculated by DFT for
EuSn2As2 and identify the $M$ point as the likely source of near-resonant EQ-
SHG at 800 nm. At this high-symmetry point, with little group $C_{2h}$
($2/m$), an electron occupying a Eu $f_{xyz}$, $f_{xz^{2}}$, or
$f_{x(x^{2}-3y^{2})}$ orbital (odd-parity $a_{u}$ symmetry) can be excited to
an unoccupied even-parity $a_{g}$ virtual state through resonant electric
dipole absorption at $\sim$1.5 eV, further excited by a parity-preserving
electric quadrupole transition to a higher $a_{g}$ or $b_{g}$ virtual state
(both are allowed), and then return to its initial state through resonant
electric dipole emission at $\sim$3 eV. The overall EQ-SHG process is
illustrated in Fig. 1(b). The participating unoccupied virtual states have
partial ($\sim$20%) Eu $5d$ character but majority Sn and As $p$ character.
Thus, in real space we may interpret the initial optical transition as a
charge-transfer excitation in which an electron is removed from a tightly-
bound Eu $4f$ orbital and added to the extended (SnAs)2 network. This leaves
behind a Eu atom in a $4f^{6}$ configuration with a reduced spin. The reduced
moment on the spin lattice represents a localized magnetic excitation that
will propagate outward as it becomes entangled with neighboring spin. This
process of magnetic decoherence of the virtual states is expected to be
sensitive to the strength and nature of the magnetic order in the system, as
we detail below.

Figure 2:  Temperature dependence of SHG. Upon cooling from room temperature,
the SHG intensity increases dramatically. Below $\sim$100 K, a broad hump
(most apparent in EuSn2AsP) and large fluctuations (most apparent in EuSn2P2)
are observed. We identify these features with the emergence of fluctuating
short-range ferromagnetic correlations within the Eu planes (blue). Within
$\sim$2 K of $T_{N}$, a pronounced peak appears, which we attribute to
critical slowing down at the magnetic phase transition, as discussed in the
main text. Below $T_{N}$, the SHG intensity decreases as long-range three-
dimensional antiferromagnetic order freezes in (green). The inset shows a
close-up view of the low-temperature region.

Now that near-resonant EQ-SHG has been established as the source of the strong
nonlinear optical response, we turn to its temperature dependence. Figure 2
shows the total SHG intensity integrated over $\phi$ as a function of
temperature. Several remarkable features are apparent. First, we observe a
significant growth in SHG as the sample is cooled, more than doubling between
room temperature and 100 K. Second, below $\sim$100 K a broad high-intensity
hump exists with large associated fluctuations. Third, upon traversing the
magnetic transition temperature $T_{N}$, an extremely sharp SHG peak occurs.
Finally, as temperature is further lowered in the magnetically-ordered phase,
the SHG intensity decreases again. Such behavior resembles a linear magnetic
susceptibility measurement, but we emphasize that here we are measuring a
nonlinear optical susceptibility that is not directly sensitive to the
magnetic degrees of freedom. Generally, one may expect a slight change in the
SHG intensity below the magnetic ordering temperature, but that is not what we
observe. Instead, we appear to be sensitive to fluctuations of the magnetic
order. In particular, we observe an increase in SHG associated with the growth
of short-range two-dimensional ferromagnetic correlations within the
individual Eu layers above $T_{N}$. Large fluctuations are expected in this
regime because true long-range order is forbidden by the Mermin-Wagner theorem
[30]. Strong in-plane ferromagnetic fluctuations below $\sim$100 K have also
been detected in the related material EuCd2As2 by muon spin relaxation, and it
has been suggested that these short-range magnetic fluctuations are sufficient
to induce topological properties even in the absence of long-range magnetic
order [12, 31]. At $T_{N}$, we observe a dimensional crossover as the short-
range in-plane ferromagnetic correlations give rise to true long-range three-
dimensional antiferromagnetic order. At the transition, critical fluctuations
are expected to be largest, and this is reflected in the observed sharp SHG
peak. Finally, upon further cooling, the long-range magnetic order grows in
strength and fluctuations accordingly diminish, as does the SHG intensity.
Thus, taken as a whole, our observations are consistent with a pronounced
sensitivity of EQ-SHG to fluctuations of the magnetic degrees of freedom,
rather than the magnetic order itself.

To interpret this result, we examine the expression for the EQ-SHG nonlinear
optical susceptibility tensor derived from perturbation theory [32]:

$\chi^{\mathrm{EQ}}_{ijkl}(2\omega,\omega,\omega)=\frac{N}{2\epsilon_{0}\hbar^{2}}\sum_{mvn}\left(\rho^{\mathrm{(eq)}}_{mm}-\rho^{\mathrm{(eq)}}_{vv}\right)\frac{\left\langle
m\right|\mu_{i}\left|n\right\rangle\left\langle
n\right|Q_{kl}\left|v\right\rangle\left\langle
v\right|\mu_{j}\left|m\right\rangle}{\left[(\omega_{nm}-2\omega)-i\gamma_{nm}\right]\left[(\omega_{vm}-\omega)-i\gamma_{vm}\right]},$

where $N$ is the electron density, $m$, $v$, and $n$ label eigenstates,
$\rho^{\mathrm{(eq)}}_{nm}$ is the equilibrium density matrix, $\mu_{i}$ and
$Q_{ij}$ are components of the electric dipole and quadrupole operators,
respectively, $\omega_{nm}=(E_{n}-E_{m})/\hbar$, and $\gamma_{nm}$ is a
phenomenological parameter introduced to account for environmental degrees of
freedom that contribute to quantum decoherence but are not explicitly included
in the density matrix. More precisely, the density matrix obeys the quantum
master equation

${\dot{\rho}_{nm}=\frac{1}{i\hbar}\left[\mathcal{H},\rho\right]_{nm}-\gamma_{nm}\left(\rho_{nm}-\rho^{\mathrm{(eq)}}_{nm}\right),}$

showing that $\gamma_{nm}$ parameterizes quantum coherence through the rate at
which off-diagonal density matrix elements decay in time: $\rho_{nm}(t)\propto
e^{-i\omega_{nm}t}e^{-\gamma_{nm}t}$. Here, the density matrix captures
orbital (charge) eigenstates but not magnetism, so quantum decoherence
primarily arises from interactions with the background magnetic environment.
As shown above, the SHG signal is dominated by a near resonance at the $M$
point, where both $\omega_{nm}\approx 2\omega$ and $\omega_{vm}\approx\omega$.
This significantly amplifies the effects of the $\gamma$ parameters appearing
in the denominator of the susceptibility expression. In fact, these are the
only parameters that can appreciably influence the temperature dependence of
SHG, as $\rho^{\mathrm{(eq)}}_{mm}-\rho^{\mathrm{(eq)}}_{vv}=1$ because $m$ is
occupied and $v$ is unoccupied, and the electric dipole and quadrupole matrix
elements in the numerator are not expected to change significantly with
temperature. Thus, we see that near-resonant EQ-SHG is acutely sensitive to
the quantum decoherence induced by magnetic fluctuations in the system, as we
have inferred from our temperature dependence measurements.

A hallmark of second-order phase transitions is the phenomenon of “critical
slowing down,” a significant increase in the time it takes for a system to
respond to perturbations or to return to equilibrium. In the critical regime,
the free energy is flat and the dynamics of the system become extremely slow.
Within the context of a magnetic transition, this occurs when locally-
fluctuating moments become entangled with their neighbors at increasingly
large distances. Both the correlation length
$\xi\propto\left|T-T_{c}\right|^{-\nu}$ and the correlation time
$\tau\propto\xi^{z}$ diverge, with critical exponents $\nu$ and $z$,
respectively [33]. In an ideal system, this results in $\gamma\rightarrow 0$
at the critical temperature, which we propose as the likely origin for the
sharp peak in SHG that we observe at $T_{N}$. We emphasize that within this
interpretation, the growth of SHG as temperature approaches $T_{N}$ is the
result of an increase in the lifetime of magnetic fluctuations, not an
increase in their amplitude. We can place a rough upper bound on the coherence
time $\tau$ of the $4f^{7}\rightarrow 4f^{6}$ magnetic excitation caused by
the charge-transfer optical transition. We observe an $O(1)$ change in SHG
over the measured temperature range, which implies that $\gamma$ must be at
least of the same order as $\Delta\omega$, the spectral width of the ultrafast
laser. Thus, we calculate $\tau\sim 1/\gamma<1/\Delta\omega\sim 16$ fs. This
value is comparable to the excitation lifetime of an isolated magnetic atom
bound to a metallic substrate, as measured by inelastic tunneling
spectroscopy, where it was suggested that the short lifetime was due to
hybridization with metallic bands, leading to efficient electron-electron
interactions that relax the magnetic state of the atom [34]. EuSn2(As,P)2 is
also metallic, and indeed it has been suggested that the magnetic exchange
coupling in the material is mediated by conduction electrons via the
Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction [20, 35]. Thus, we may also
expect a similarly short magnetic excitation lifetime due to coupling to
conduction electrons.

According to the celebrated fluctuation-dissipation theorem, there is a close
relationship between the imaginary part of a linear susceptibility and the
power spectral density of fluctuations of that susceptibility’s conjugate
observable [33]. No such theorem exists for nonlinear susceptibilities, where
the imaginary part does not represent dissipation because the time-average of
$\mathbf{E}(\omega)\cdot\mathbf{J}(2\omega)$ is necessarily zero [36]. It has
been theoretically shown, however, that nonlinear responses are much more
sensitive than linear responses to semiclassical chaotic behavior through the
interference of multiple dynamical Liouville-space trajectories [37]. The
second-order susceptibility, for example, is directly related to the stability
matrix of a system and its associated Lyapunov exponents quantifying the
system’s sensitivity to initial conditions. Our measurements largely affirm
this notion, demonstrating that, at least near resonances, the nonlinear
optical response of a material can be significantly influenced by the dynamics
of fluctuations of some order parameter of the material, even when that order
is not the conjugate observable of the susceptibility. We expect that this
principle is generally applicable, and that under sufficiently resonant
conditions, the nonlinear optical response of a material can be harnessed to
study fluctuations, dynamics, and quantum coherence across many types of
second-order phase transitions.

Figure 3:  Spontaneous rotational symmetry breaking in EuSn2As2. (a)
Temperature dependence of the 2-fold rotational anisotropy that emerges at the
antiferromagnetic phase transition. The inset shows Sin-Uout (SU) RA-SHG
patterns for each temperature, and the main panel shows the corresponding
2-fold anisotropy versus temperature. The gray curve is a guide to the eye,
showing an order-parameter-like onset below $T_{N}$. (b) Spatial map of
rotational symmetry breaking. RA-SHG patterns where acquired on a $10\times
10$ grid spanning a $300\times 300$ $\mu$m area. At each grid point, the
amplitude and direction of the 2-fold anisotropy was extracted (represented by
the length and direction, respectively, of the double-headed arrows). The
upper left inset shows a photograph of the sample with the mapped region
identified, and the lower left inset shows a polar plot of the anisotropy
sorted by direction into three symmetry-equivalent antiferromagnetic domains
(red, green, and blue). The map reveals static domains with sizes on the order
of $\sim$100 $\mu$m.

Within the long-range ordered antiferromagnetic phase below $T_{N}$, the Eu2+
moments of EuSn2(As,P)2 become in-plane aligned. This necessarily results in
spontaneous breaking of the $\bar{3}$ symmetry axis of the crystal, as the
moments must choose one of three equivalent crystallographic directions along
which to orient themselves. Rotational symmetry breaking can generally be
revealed through RA-SHG patterns, which are exceptionally sensitive to point
group symmetries [26]. As Fig. 3(a) demonstrates, we indeed observe the onset
of rotational symmetry breaking below $T_{N}$ in RA-SHG patterns, with a clear
order-parameter-like temperature dependence. Here we measure the unpolarized
SHG output (Uout) because, in this case, a fully isotropic (i.e. circular)
response is expected in the high-symmetry phase, degenerating into a 2-fold
“peanut-like” shape when the rotational symmetry axis is lost. Further details
on the quantification of the 2-fold anisotropy can be found in the
Supplemental Materials. We attribute this observation to two possible
mechanisms, both involving spin-orbit coupling. In the first mechanism, strong
coupling of the globally-aligned spins to the initial-state $f$ orbitals
($f_{xyz}$, $f_{xz^{2}}$, and $f_{x(x^{2}-3y^{2})}$) shifts their energies,
and this manifests as slight changes to the optical resonance condition for
different polarization angles. In the second mechanism, the global alignment
of moments influences the quantum coherence of different initial-state $f$
orbitals. Given that the 2-fold anisotropy we observe is almost of the same
magnitude as the SHG intensity itself ($\sim$20%), we believe the second
scenario is more likely. Accompanying spontaneous symmetry breaking is the
formation of magnetic domains transforming into each other by the broken
symmetry element. Optical imaging of antiferromagnetic domains is typically
very challenging because of the absence of a net magnetization, with SHG being
one of the few known techniques for doing so [38, 39]. Figure 3(b) shows a
spatial RA-SHG mapping over a $300\times 300$ $\mu$m area that reveals these
domains. We find a roughly equal distribution of all three possible domain
orientations, with an approximate domain length scale of $\sim$100 $\mu$m.
Thermal cycling across $T_{N}$ generates completely new domain patterns, as
shown in the Supplemental Materials, suggesting domain boundaries are not
pinned by disorder.

In conclusion, we have harnessed near-resonant EQ-SHG to probe the intricate
magnetic dynamics in the candidate axion insulator EuSn2(As,P)2. By examining
the material across its antiferromagnetic phase boundary, we have revealed a
pronounced dimensional crossover in the quantum decoherence induced by
magnetic fluctuations, with two-dimensional in-plane ferromagnetic
correlations at high temperatures transitioning to three-dimensional long-
range order below $T_{N}$. Moreover, we have observed an unusually sharp SHG
peak right at the magnetic phase boundary that we attribute to the phenomenon
of critical slowing down, where the dynamics of the system become extremely
slow and the correlation length and time diverge. The quantum coherence time
of the charge-transfer $4f^{7}\rightarrow 4f^{6}$ magnetic excitation is
estimated to be no more than $\sim 16$ fs. We conjecture that such a short
lifetime is due to efficient coupling between the localized magnetic moments
and the conduction electrons in EuSn2(As,P)2, supporting a scenario whereby
RKKY interactions mediate the magnetic exchange [20, 35]. This conclusion has
important implications for the coupling strength between the magnetic order
and the topological electronic states, where prior work has reported only a
very weak hybridization [22]. Exotic topological boundary modes have been
predicted at magnetic domain boundaries in this class of materials [5]. By
leveraging the extreme sensitivity of RA-SHG, we have uncovered the clear
breaking of rotational symmetry within the antiferromagnetic phase and used it
to map out the spatial domain structure that emerges. Our findings help to
clarify the nature of the magnetic order in this class of materials and its
interactions with topological surface states, which play a critical role in
realizing topological magnetoelectric phenomena. More generally, our work
showcases the unique capabilities of nonlinear optical spectroscopy as a
powerful tool for investigating quantum coherence and fluctuations in magnetic
quantum materials.

## Acknowledgments

This work was supported by the U.S. Air Force Office of Scientific Research
(AFOSR) under Award No. FA9550-22-1-0270. The research reported here made use
of the shared facilities of the Materials Research Science and Engineering
Center (MRSEC) at UC Santa Barbara: NSF DMR–2308708. The UC Santa Barbara
MRSEC is a member of the Materials Research Facilities Network (www.mrfn.org).

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