# Spatial Coherence Properties of Organic Molecules Coupled to Plasmonic
Surface Lattice Resonances in the Weak and Strong Coupling Regimes

L. Shi  COMP Centre of Excellence, Department of Applied Physics, Aalto
University, FI-00076 Aalto, Finland  T. K. Hakala  COMP Centre of Excellence,
Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland  H.
T. Rekola  COMP Centre of Excellence, Department of Applied Physics, Aalto
University, FI-00076 Aalto, Finland  J.-P. Martikainen  COMP Centre of
Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto,
Finland  R. J. Moerland  COMP Centre of Excellence, Department of Applied
Physics, Aalto University, FI-00076 Aalto, Finland  Department of Imaging
Physics, Faculty of Applied Sciences, Delft University of Technology,
Lorentzweg 1, NL-2628 CJ, Delft, The Netherlands  P. Törmä  Electronic
address<EMAIL_ADDRESS>COMP Centre of Excellence, Department of
Applied Physics, Aalto University, FI-00076 Aalto, Finland

###### Abstract

We study spatial coherence properties of a system composed of periodic silver
nanoparticle arrays covered with a fluorescent organic molecule (DiD) film.
The evolution of spatial coherence of this composite structure from the weak
to the strong coupling regime is investigated by systematically varying the
coupling strength between the localized DiD excitons and the collective,
delocalized modes of the nanoparticle array known as surface lattice
resonances. A gradual evolution of coherence from the weak to the strong
coupling regime is observed, with the strong coupling features clearly visible
in interference fringes. A high degree of spatial coherence is demonstrated in
the strong coupling regime, even when the mode is very excitonlike (80 $\%$),
in contrast to the purely localized nature of molecular excitons. We show that
coherence appears in proportion to the weight of the plasmonic component of
the mode throughout the weak-to-strong coupling crossover, providing evidence
for the hybrid nature of the normal modes.

###### pacs:

33.80.-b, 73.20.Mf, 42.50.Nn

Spatial coherence properties of waves can be probed by passing a wave front
through distant slits and observing interference. Inspired by this phenomenon
well known for classical radiation, interference experiments were crucial in
establishing the wave-particle nature of single photons, as well as massive
particles Davisson and Germer (1927); Arndt et al. (1999); Juffmann et al.
(2012), within quantum mechanics. In the experiments Davisson and Germer
(1927); Arndt et al. (1999); Juffmann et al. (2012) the quantum mechanical
wave properties of matter became visible at low temperatures. Here, we
consider a different question: the spatial coherence properties of objects, or
modes, that are hybrids of wavelike and particlelike components. Mixing a
localized matter component with light may possibly give the hybrid object a
nontrivial spatial coherence length.

Examples of light-matter hybrids include coherent superpositions of atoms and
cavity photons Rempe et al. (1987); Thompson et al. (1992), semiconductor
cavity polaritons, which have been brought to quantum degeneracy and
condensation Kasprzak et al. (2006), and cavity photon mediated strong
coupling between spatially separated localized molecular excitons Lidzey et
al. (2000). Recently, delocalized electromagnetic modes supported by metal
surfaces (surface plasmon polaritons) or periodic arrays of metallic
nanoparticles [surface lattice resonances (SLRs) Zou et al. (2004); García de
Abajo (2007); Auguié and Barnes (2008); Zhou et al. (2013)] have been shown to
strongly couple with localized emitters Bellessa et al. (2004); Dintinger et
al. (2005); Hakala et al. (2009); Gomez et al. (2010); Schwartz et al. (2011);
Vasa et al. (2013); Rodriguez and Gomez Rivas (2013); Väkeväinen et al.
(2013). The strong coupling in these plasmonic systems involves a large number
$N$ of emitters. The normal mode splittings observed are consistent both with
classical linear dispersion theory and with the vacuum Rabi splitting obtained
as the low excitation limit of the Dicke model, similarly to the early
experiments on many atoms in cavities Zhu et al. (1990). The collective
behavior of many emitters has been clearly demonstrated in these systems,
manifested as the $\sqrt{N}$ dependence of the splitting. The observed
splittings in dispersions strongly support the interpretation that the new
normal modes are hybrid modes formed by strong coupling of lightlike (the
surface plasmon polariton/SLR) and matterlike (the molecular excitation)
components. Observations of the dispersions alone, however, cannot directly
test whether the new modes carry all the essential properties of the original
modes, as should be the case if the hybrids are linear, coherent combinations
of the original modes. In particular, spatial coherence is the specific
characteristic of an extended light mode: in order to prove that the new modes
carry this property, interference experiments are needed. To be conclusive, it
is necessary to show that the coherence appears in proportion to the weight of
the light mode in the hybrid. This in turn requires a systematic study of
coherence throughout the weak-to-strong coupling crossover. This is the goal
of the present work.

The spatial interference effects of light-matter hybrids have been studied in
a few experiments in the context of exciton-polariton condensates Richard et
al. (2005); Kasprzak et al. (2006); Deng et al. (2007). In plasmonic systems,
only one experiment has been reported Aberra Guebrou et al. (2012): signatures
of coherence were observed in the strong coupling regime in a planar metal
surface—molecular film system. However, that work does not prove the
connection of the spatial coherence with the weight of the light component
since there was no study of the weak-to-strong coupling crossover (a different
system, namely quantum dots, was given as the weak coupling reference). Here,
we study the spatial coherence properties of a system composed of periodic
silver nanoparticle arrays covered with fluorescent organic molecules (DiD) by
employing a double slit experiment. We gradually increase the molecule
concentration to investigate both the strong and the weak coupling coherence
properties within the same system.

Figure 1(a) shows a scanning electron micrograph (SEM) of a typical array (for
fabrication details, see Supplemental Material Supplementarynote ). The
$d_{y}$ = 50 nm, $p_{y}$ = 200 nm, while $d_{x}$ and $p_{x}$ were varied
between 133–400 nm and 380–500 nm, respectively. The DiD concentration in
poly(methyl methacrylate) film was varied between 20–800 mM.

The measurement setup is depicted in Fig. 1(b). $y$-polarized white light was
incident on the sample; see Fig. 1(a). Angle and wavelength-resolved
transmission spectra $T=I_{\textrm{Structure}}/I_{\textrm{Reference}}$ [Fig.
1(b)] were measured and subsequently used for calculating the dispersion for
each array. The entrance slit of the spectrometer and the in-plane wave vector
$k$ is parallel to the $x$ axis of the sample with magnitude
$k=2\pi/\lambda\sin(\theta)$, where $\lambda$ is the wavelength in the medium
and $\theta$ is the angle between the optical axis and the light propagation
direction.

Figure 1: (a) A SEM of a typical sample. The scale bar is 1 $\mu$m (200 nm for
the inset). (b) The measurement setup. Angle resolved transmission spectra for
each array were measured by placing the back focal plane of the sample at the
entrance slit of the spectrometer. For spatial coherence measurements, a
double slit was placed at the first intermediate image plane of the system.
Figure 2: The dispersions of three different nanoparticle arrays with
inreasing DiD concentration. (a)–(e) Array 1, $(d_{x})\times(d_{y})$ = 50 nm
$\times$ 220 nm, $p_{x}$ = 500 nm. (f)–(j) Array 2, $(d_{x})\times(d_{y})$ =
50 nm $\times$ 355 nm, $p_{x}$ = 500 nm. (k)–(o) Array 3,
$(d_{x})\times(d_{y})$ = 50 nm $\times$ 167 nm, $p_{x}$ = 380 nm. The first
column corresponds to a case without DiD molecules, while the second, third,
fourth, and fifth columns have 20, 200, 400, and 800 mM concentrations of DiD,
respectively. White areas correspond to maximum extinction. The blue
horizontal lines depict the absorption maximum of the DiD film. The yellow
lines correspond to peak positions obtained from fitting a Gaussian curve to
the line cuts of dispersions while keeping $k$ constant, and the red lines are
obtained from the coupled oscillator model. (p) The SLR-exciton coupling
strength as a function of square root of concentration. The blue plus signs,
red crosses, and green circles correspond to arrays 1, 2 and 3, respectively.
(q)-(s) The relative SLR-exciton weights of the arrays 1-3, respectively. The
solid (dashed) line corresponds to exciton (SLR) percentage and black, orange
and purple to concentrations of 200, 400, and 800 mM, respectively.

In Figs. 2(a)–2(o) are shown the measured angle resolved extinction ($1-T$)
spectra for different nanoparticle arrays. Several observations can be made
from these figures. First, the energy of the $\Gamma$ point ($k=0$) can be
changed by changing the periodicity [see for example Figs. 2(a) and 2(k)].
Second, upon coupling of the <+1, 0> and <-1, 0> diffractive orders Barnes et
al. (1996), a band gap is formed in Fig. 2(f) and the associated new modes can
be made either dark or bright by changing the filling fraction [$d_{x}/p_{x}$,
see Fig. 1(a)]. For details, see Supplemental Material Supplementarynote .

The dispersions in Fig. 2(b)–2(e) illustrate how the system gradually evolves
from the weak to the strong coupling regime with increasing molecular
concentration. A clear modification of the system energies is observed in
Figs. 2(b)–2(d), which then in Fig. 2(e) develops into a distinctive band
bending and anticrossing at the energy corresponding to the absorption maximum
of the molecule, a behavior that is characteristic for the strong coupling
regime. Similar evolution from weak to strong coupling regime can readily be
identified for arrays 2 [Figs. 2(f)–2(j)] and 3 [Figs. 2(k)–2(o)], but now the
system energies are drastically different due to different filling fraction
(array 2) and periodicity (array 3). These results demonstrate how the choice
of geometry and molecular concentration provides excellent control over the
system properties.

In the strong coupling theory, the new modes are linear combinations of the
uncoupled SLRs and the molecular excitations. To describe such hybrid modes,
we employ a coupled oscillator model satisfying the equation

$\left(\begin{array}[]{cc}E_{SLR}(k)+i\gamma_{SLR}&\Omega\\\
\Omega&E_{DiD}+i\gamma_{DiD}\end{array}\right)\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)=0,$ (1)

where $E$ and $\gamma$ are the energies and the widths of the uncoupled modes,
$\Omega$ is the coupling strength between the SLR and DiD, and $\alpha$ and
$\beta$ are the coefficients of the linear combination of SLR and the DiD
exciton (for details see Supplemental Material Supplementarynote ). The SLR-
exciton coupling strength $\Omega$ and the linewidth $\gamma_{\mathrm{DiD}}$
of the exciton are used as fitting parameters. The resulting mode energies are
plotted in Figs. 2(c)–2(e), 2(h)–2(j), and 2(m)–2(o) for different arrays and
are in good agreement with the experimentally observed mode energies. The SLR-
exciton coupling is expected to scale as $\sqrt{N/V}$, where $N$ is the number
of molecules and $V$ is the mode volume Agranovich et al. (2003); González-
Tudela et al. (2013): this is confirmed in Fig. 2(p). Notably, the size of the
observed splitting is in reasonable agreement with microscopic theory
Agranovich et al. (2003) (see Supplemental Material Supplementarynote ). Note
that spectrally broad emitters coupled to spectrally selective (plasmon) modes
can produce luminescence spectra reminiscent of those observed in strongly
coupled systems (see, e.g., Ref. Gruber et al. (2013)). That we observe strong
coupling instead of this phenomenon is proven by the series of different
concentrations that we studied, showing the $\sqrt{N/V}$ dependence expected
for strong coupling.

Figure 3: (a)–(d) The spatial coherence images for the array 2 with
concentrations 0, 20, 400, and 800 mM, respectively. Here white areas
correspond to transmission maximum. The yellow lines have the same meaning as
in Fig. 2. (e) A sample having a random distribution of nanoparticles with 800
mM DiD concentration. Two transmission minima are seen at 1.85 eV (yellow
line) and 2.25 eV, corresponding to DiD absorption and the single particle
plasmon resonance, respectively.

In Figs. 2(q)–2(s), we plot the relative weights of the hybrid modes as
functions of the in-plane wave vector $k$ for arrays 1–3, respectively, with
molecular concentrations of 200, 400, and 800 mM. For arrays 1 and 2, the SLR-
exciton hybrid is mostly SLR-like for $k\sim 0$, and becomes increasingly
excitonlike for higher $k$ values. The relative exciton contribution at $k\sim
0$ increases with concentration due to stronger hybridization of the SLR with
the exciton. Note, however, that for array 3 [Fig. 2(s)] the mode is
excitonlike at $k\sim 0$, and then gradually evolves to SLR-like mode at
higher $k$. This is due to the SLR $\Gamma$-point energy being above the
molecular excitation energy [compare, for example, Figs. 2(g) and 2(l)). These
results demonstrate how the relative weights of the hybrid mode at a given
energy and wave vector can be tailored by choice of geometry and molecular
concentration.

To investigate coherence, angle resolved transmission spectra are recorded
with a double slit placed on the image plane of the sample; see Fig. 1(b).
This forms the crucial test for the presence of spatial coherence in the new
modes: if the spatial coherence length of the mode is greater than the
interslit distance, a distinctive fringe pattern would be expected in the
Fourier plane of the imaging system. In Fig. 3(a)–3(d) are shown the
wavelength-resolved spatial coherence images obtained from array 2 and with
molecular concentrations ranging from 0 to 800 mM. Intriguingly, bending of
the interference pattern is observed towards the strong coupling regime. In
other words, one of the destructive interference fringes in spatial coherence
images always overlaps with the extinction maxima of the dispersion (yellow
symbols); see Figs. 2(g)-2(j). This allows to make an important connection
with the original modes: If a spatially coherent light source (i.e., the
sample) is radiating through a double slit, the interference fringes can be
interpreted as replicas of the original dispersion (Fig. 2) created by the
diffracted orders from the double slit. At high frequencies the interference
pattern becomes complex due to the close spacing of the crossing points of the
replicas (see also Supplemental Material Supplementarynote ). Thus, the fact
that band bending with increasing concentration is seen both in the
dispersions and the spatial coherence images provides a clear signature that
the interference fringes are directly related to the modes of interest and are
not due to any secondary reason. We have thus conclusively shown that the
system modes have prominent spatial coherence throughout the crossover, also
deep in the strong coupling regime.

We want to point out the important role of the array periodicity, i.e the
existence of the dispersive SLR modes, for the emergence of long-range
coherence. Figure 3(e) shows a spatial coherence image of a sample having a
random interparticle spacing (for a SEM image, see Supplemental Material
Supplementarynote ) while the molecular concentration, nanoparticle size,
orientation and number are the same as in the sample in Fig. 3(d). Evidently,
no interference fringes are present in this case. Also, they are absent in DiD
films without nanoparticles.

Notably, the fringes become less visible with increasing concentration at
energies above 1.8 eV; see Figs. 3(b)–3(d). Higher molecular concentration
induces stronger hybridization between the delocalized SLR and localized
molecular excitons. At higher energies, these hybrid modes become increasingly
excitonlike and localized as the energy gets closer to DiD dye absorption,
reducing the spatial coherence length below the interslit distance. Note,
however, that the fringe pattern persists below 1.8 eV energies, even with 800
mM concentration. We have thus demonstrated that the SLR-exciton hybrid modes
display long-range coherence even when the mode is very excitonlike: from Fig.
2(r) the exciton weight can be deduced to be $80\%$ at high $k$-vector values.

In the rest of this Letter, we consider the crucial question of whether there
is a systematic, quantitative connection between the spatial coherence and the
expected weight of the light component in a hybrid mode. First, we want to
show that detailed structure of the interference fringes can be produced by
assuming hybrid modes, with weights of the light and matter parts as obtained
by fitting the experimental dispersion with the coupled oscillator model Eq.
(1) (the obtained dispersion was then used to provide the energy and wave
vector specific information of the mode radiating through the double slit, see
Supplemental Material Supplementarynote ). In Fig. 4(a) we show a close-up of
the spatial coherence image of Fig. 3(d) (800 mM concentration) and in Fig.
4(b) we show the interference image obtained from calculations based on the
coupled oscillators model. While the intensities in both Figs. 4(a) and 4(b)
are of comparable magnitude, at high energies, the experimental data have less
transmission intensity. This can be due to additional absorption of the
molecules that are not contributing to strong coupling Agranovich et al.
(2003); Bellessa et al. (2004). In general, however, the correspondence of the
model with the most prominent features of the experimental data is excellent.
This is the first step of systematically proving the connection between the
hybrid structure and the coherence: the model with weights of matter and light
parts in the hybrid as given by strong coupling theory indeed reproduces the
interference pattern observed experimentally.

Figure 4:  (a) A close-up of the spatial coherence image (800 mM
concentration). (b) The interference image obtained from the coupled
oscillator model. (c) The $\Delta k$ obtained from the experiments (red empty
circles) and from the coupled oscillator model (blue crosses). Dashed and
solid lines correspond to the SLR and exciton weights of the mode,
respectively. (d) The spatial coherence length obtained from the experiments
(red circles) and from the coupled oscillator model (blue empty circles). The
dashed line is the effective interslit distance at the sample plane.

Second, we consider the important connection between the interference fringes,
mode delocalization, the width of the mode $\Delta k$, and the relative
weights of the strongly coupled modes. In Fig. 4(c) we show the $\Delta k$ of
the mode as a function of the energy obtained from the experiments (Fig. 2)
and from the model. The $\Delta k$ was obtained as FWHM of constant-energy
line cuts from the dispersions. Also shown are the relative SLR and exciton
weights of the hybrid mode. In Fig. 4(d) we show the spatial coherence lengths
of the mode obtained as $L_{x}=2\pi/\Delta k$ Mandel and Wolf (1965). Because
the momentum and position are Fourier related, a small $\Delta k$ at energies
around 1.6 eV [see Fig. 4(c)] suggests a delocalized mode and large spatial
coherence length. The delocalization is also evident from the high SLR
fraction (80 $\%$) of the mode. In the spatial coherence image, the
delocalization manifests itself as a distinct interference pattern [Figs. 4(a)
and 4(b)]. As $\Delta k$ increases at energies $E>1.65$ eV, however, the
hybrid mode becomes more localized and more excitonlike, which gradually
yields a less prominent interference pattern in accordance with the increasing
weight of the matter component. At energies above 1.7 eV, the spatial
coherence length decreases below the interslit distance [Fig. 4(d)], and,
consequently the interference pattern disappears; see Fig. 3(d).

In both classical optics and quantum mechanics, modes are characterized not
only by their energies, observable in dispersions, but also by the coherent
modes or wave functions forming linear superpositions. Both aspects should be
considered in identifying physical phenomena, cf. the observation of Bose-
Einstein condensation by evidence in momentum distribution Anderson et al.
(1995) and in interference patterns Andrews et al. (1997). The strong coupling
regime of various types of surface plasmon modes and emitters has been widely
studied by observing dispersion relations. Splittings in the dispersions have
been attributed to hybridization of plasmonic and excitonlike modes. Here we
provide the first systematic study of the evolution of the spatial coherence
in a plasmonic-molecule system when transiting from the weak to the strong
coupling regime. The evolution of spatial coherence is shown to be directly
connected to the hybrid mode structure. Significant spatial coherence lengths
in the strongly coupled system are observed even when the mode is very
excitonlike. Complementing the energy dispersions and dynamics observed
earlier, our interference results provide conclusive evidence for the hybrid
nature of the normal modes in strongly coupled surface plasmon—emitter
systems. In general, our results demonstrate the potential of hybridization in
creating nanosystems with designed properties, in this case long range
coherence for modes that are largely matterlike.

###### Acknowledgements.

We thank Dr. Shaoyu Yin for useful discussions. This work was supported by the
Academy of Finland through its Centres of Excellence Programme (Projects No.
251748, No. 263347, No. 135000, and No. 141039) and by the European Research
Council (ERC-2013-AdG-340748-CODE). Part of the research was performed at the
Micronova Nanofabrication Centre, supported by Aalto University.

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# Hybrid-order topology with tunable chiral hinge modes and unpinned Dirac
surface states in the altermagnetic insulator Eu3In2As4

Yufei Zhao  Department of Condensed Matter Physics, Weizmann Institute of
Science, Rehovot 7610001, Israel  Hyeonhu Bae  Department of Condensed Matter
Physics, Weizmann Institute of Science, Rehovot 7610001, Israel  Yiyang Jiang
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot
7610001, Israel  Yongkang Li  Department of Condensed Matter Physics, Weizmann
Institute of Science, Rehovot 7610001, Israel  Chao-Xing Liu  Department of
Physics, the Pennsylvania State University, University Park, PA 16802, USA
Binghai Yan  Corresponding author<EMAIL_ADDRESS>Department of
Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001,
Israel

###### Abstract

The exploration of magnetic topological states is instrumental in exploring
axion electrodynamics and intriguing transport phenomena, such as the quantum
anomalous Hall effect. Here, we predict that the recently-synthesized material
Eu3In2As4 exhibits as both an axion insulator and a 3D Stiefel-Whitney
insulator with an altermagnetic order. When spins align in the $ab$ plane, we
find an unpinned surface Dirac cone on the $ab$ plane and chiral hinge states
along the $c$ direction, where hinge states can generate a half-quantized
surface anomalous Hall effect on the $ac$ and $bc$ facets. When spins align
along $c$, we observe a mirror-protected topological crystalline insulator.
Furthermore, the ferromagnetic phase, in which spins are aligned in the same
direction by an external in-plane magnetic field, presents an ideal Weyl
semimetal with a single pair of type-I Weyl points and no extra Fermi pocket.
Our work predicts rich topological states tuned by magnetic structures in
Eu3In2As4, supporting the further study of the topological transport and
Majorana fermions in proximity to a superconductor.

##  I Introduction

Topological quantum states of matter are usually characterized by unique
boundary states following the bulk-boundary correspondence [1, 2, 3]. For
example, a topological insulator (TI) exhibits 2D Dirac-type surface states
[4, 5] and a higher-order TI (HOTI) shows 1D hinge states [6, 7, 8]. Here, we
report a magnetic TI material that hosts both topological surface states and
chiral hinge states, dubbed a hybrid-order magnetic TI, which is protected by
the combined symmetry between rotational and time-reversal symmetry [9, 10,
11, 12].

Unlike nonmagnetic HOTIs [13, 6, 14, 7, 15, 16], hinge modes in a magnetic
topological crystalline insulator (TCI) are intimately related to magnetic
structures, by which the direction or chirality of currents can be manipulated
[8]. For instance, applying a magnetic field to flip or rotate spin moments is
equivalent to reversing the mass signs of the gapped surface, and thus
switching the chirality of hinge states and the mapped 2D Chern number [17,
18, 19]. Identifying the nature of topological invariants under a combined
magnetic crystalline symmetry by the translation ($\tau$) or $n$-fold rotation
($C_{n}$) with time-reversal ($\mathcal{T}$), is evidently a central issue,
exemplified by $\tau\mathcal{T}$ protected $\mathbb{Z}_{2}$ antiferromagnetic
(AFM) TI [20, 21, 22, 23]. Recent theory shows that a $C_{2}\mathcal{T}$
protected TCI holds both the robust gapless surface states and chiral hinge
states [9, 10, 11, 12] while its material realization is still missing.

Furthermore, a magnetic TCI can generate axion electrodynamics, described by
the so-called $\theta$ term [24, 25]

$S_{\theta}=\int dtd^{3}x\frac{\theta e^{2}}{4\pi^{2}\hbar
c}\bm{E}\cdot\bm{B},$ (1)

where $\bm{E}$ and $\bm{B}$ are external electric and magnetic fields,
$\theta$ is the axion field defined in a 3D system. When $\mathcal{T}$ is
broken, but inversion ($\mathcal{I}$) or a combined symmetry (e.g.
$\tau\mathcal{T}$, $C_{n}\mathcal{T}$, $C_{n}\mathcal{I}$) is preserved, the
axion field is odd so that $\theta$ is quantized to be 0 or $\pi$, which
corresponds to trivial insulators and axion insulators [26, 27, 28, 20, 29,
21]. With gapped surface states, axion insulators manifest half-quantized
surface anomalous Hall conductance (AHC) of $\pm e^{2}/2h$. When two adjacent
gapped surfaces hold opposite signs of mass, the Hall effect is carried by the
1D chiral mode at the hinge [3, 30, 31, 24, 7, 32, 33].

Despite the intriguing interplay of topology and magnetism, the material
realization of axion insulators is still challenging. External magnetic
doping, proximity, and a few classes of stoichiometric intrinsic magnetic
candidates have been proposed in the past few years [34, 3, 35, 36, 37, 38,
39, 40, 41, 42]. In EuIn2As2 and EuSn2(As,P)2, the experimental signatures of
band inversion, AFM order, and spin spiral states have been reported [18, 43,
44, 45, 46, 47]. Unlike MnBi2Te4, Eu-based Zintl compounds belong to another
axion insulator family. As discussed later, their altermagnetic order, where
the opposite-spin sublattices are connected by rotation symmetry instead of
$\mathcal{IT}$ or $\tau\mathcal{T}$ [48, 49, 50, 51], require us to further
understand the symmetry-related topology.

In this work, we predict that Eu3In2As4, an altermagnet in the ground state,
is a hybrid-order TCI with tunable chiral hinge states and unpinned Dirac
surface states, classified as a $C_{2}\mathcal{T}$-protected Stiefel-Whitney
(SW) insulator [10, 11]. Rotating the magnetic moments in the $ab$ plane can
manipulate the Dirac cone position on the $ab$ plane. Chiral hinge states
appear along the $c$ direction and lead to half-quantized surface anomalous
Hall effect in the $ac$ and $bc$ facets, manifesting the axion insulator
state. Rotating the magnetic field from in-plane to out-of-plane gaps the
Dirac point. We further reveal an ideal Weyl semimetal phase in the
ferromagnetic (FM) phase and identify the universal topological feature in the
Eu-based Zintl family of topological materials. Additionally, Eu3In2As4 was
recently grown from InAs nanowires in a mutual-exchange method [52] and
inherits the ideal proximity with a superconductor [53]. Thus, this material
will provide a platform to study possible Majorana modes due to the interplay
between chiral hinge states/Dirac surface states and superconductivity.

##  II Crystal, magnetic, and bulk band structures

The Zintl compound Eu3In2As4 has an orthorhombic lattice and the corresponding
space group ($Pnnm$, No. 58) includes inversion $\mathcal{I}$, a twofold
rotation $C_{2z}$ and a mirror symmetry $M_{z}$, and nonsymmorphic symmetries
$\\{C_{2x,2y}|\tau\\}$, $\\{M_{x,y}|\tau\\}$, where $\tau$ is the translation
of one-half of a body diagonal $(\frac{a}{2},\frac{b}{2},\frac{c}{2})$. As
shown in Fig. 1(a), all atoms are located at $z=0$ and $z=0.5$ planes, and one
In atom and four nearest neighboring (NN) As atoms form a tetrahedron. The
tetrahedrons ([In2As4]6-) constitute a 1D chain by sharing their corners along
the $c$ axis, between which the divalent Eu2+ cations are dispersed. Three Eu
atoms (a trimer linked by the dashed line) with two NN tetrahedrons tend to
form a sublattice. Within one unit cell, two sublattices are related by
$\\{C_{2x,2y}|\tau\\}$ or $\\{M_{x,y}|\tau\\}$ operations. In density-
functional theory (DFT) calculations, lattice parameters are optimized within
1% error compared to experimental values [54]. The energy bands near the Fermi
level are mainly from In-$5s$ and As-$4p$ orbitals.

Figure 1: (a) Crystal structure of Eu3In2As4 in the AFM$a$ phase. As an
altermagnet, spin sublattices are related by the $\\{C_{2y}|\tau\\}$ symmetry
rather than inversion or translation. The $a,b,c$ crystallographic axes point
to $x,y,z$ directions. (b) The relative energy difference per Eu for different
magnetic phases and corresponding $\mathbb{Z}_{4}$ topological invariant from
the parity eigenvalues. showing an axion insulator in the AFM phase and a Weyl
semimetal in the FM phase. (c) The band structure of Eu3In2As4 without (grey
dashed) and with (black solid) SOC for the AFMc phase. The inset is the bulk
Brillouin zone (BZ). (d) The band structure for the AFM$a$ phase with the
projection of In-$s$ and As-$p$ orbitals.

We first examine the magnetic ground state of Eu3In2As4. Eu2+ shows $\bm{S}$ =
7 $\mu_{B}$ magnetic moment. Localized Eu-4$f$ orbitals appear between $-2.5$
eV and $-1.5$ eV below the Fermi energy for DFT+$U$ calculations tested with
different $U$ values [Supplemental Material (SM) Fig. S2]. We found that the
AFM phase is the magnetic ground state while the FM phase is energetically
less favorable (see Fig. 1(b)). In the AFM phase, Eu2+ cations exhibit FM
coupling along the $c$-axis but a staggered AFM order inside the trimer. Two
sublattices are linked via $\\{C_{2y}|\tau\\}$ and also AFM coupled. Thus,
Eu3In2As4 is an altermagnet [48, 49, 50, 51], and allows the lifted spin
degeneracy along some $k$-path. A small magnetic anisotropy exists for
magnetic moments along $a,b,c$ directions. When the easy axis is along ${a/b}$
axis (AFM$a/b$ state), a time-reversal $\mathcal{T}$ requires to combine with
$C_{2z}$, $M_{z}$, $\\{C_{2x/y}|\tau\\}$ and $\\{M_{x/y}|\tau\\}$. Complete
symmetry operations can be found in Supplementary Table II for different
magnetic structures. The AFM order is consistent with a recent experiment
[52].

We next investigate the band structure and topology under the altermagnetic
order (dubbed “AFM” hereafter). In Fig. 1(c), without spin-orbit coupling
(SOC), a trivial bandgap of 52 meV is found at $\Gamma$ point, and spin-split
bands caused by the altermagnetic order are found along $\Gamma-S$ direction
(see SM Fig. S3). In the AFMc phase, the presence of SOC induces an inverted
gap of 21 meV at $\Gamma$. The two-fold degenerate bands are found at
$k_{z}=0$ plane and along $\Gamma-Z$, since $\\{M_{x,y}|\tau\\}$ anti-commutes
with $M_{z}$, and $\\{C_{2x,2y}|\tau\\}$ anti-commutes with $C_{2z}$. For the
AFM$a$ phase (Fig. 1(d)), we find a gap of nearly 10 meV at $\Gamma$ with band
inversion and an indirect global bandgap of 5 meV. The double-degeneracy here
is removed along $\Gamma-X$, $\Gamma-Y$, and $\Gamma-Z$, but remains along
$X-S-Y$ because of antiunitary operators $\\{C_{2x}|\tau\\}\mathcal{T}$ and
$\\{M_{x}|\tau\\}\mathcal{T}$. Details of symmetry analysis can be found in
the SM Sec. III.

The presence of inversion symmetry enables us to confirm band topology by
computing the parity-defined $\mathbb{Z}_{4}$ invariant [17, 18, 33],

$\mathbb{Z}_{4}=\sum_{\bm{k}=1}^{8}\frac{n_{\bm{k}}^{+}-n_{\bm{k}}^{-}}{2}\
\rm{mod}\ 4,$ (2)

where $n_{\bm{k}}^{\pm}$ is the number of occupied states with $\plus/\minus$
parity at one of the eight inversion-invariant momenta $\bm{k}$.
$\mathbb{Z}_{4}=0,2$ indicate a trivial insulator and axion insulator,
respectively, while $\mathbb{Z}_{4}=1,3$ leads to a semimetal phase. We find
that the parity change only occurs at $\Gamma$ after including SOC, and
$\mathbb{Z}_{4}$ invariant indicates that Eu3In2As4 is an axion insulator in
AFM phase and a topological semimetal in the FM phase, regardless of the
magnetic moment orientation.

Figure 2: Gapless surface and chiral hinge states of AFM$a,b$ phases. (a-b)
(001) Surface states. (c) Schematic illustration of the massless Dirac point
under $C_{2z}\mathcal{T}$ and $k$-path we apply in (a-b). The BZ here is
zoomed in at $\Gamma$ to depict their relative positions. (d) Numerical
calculations of the hinge state along $c$-axis, with a finite 35-unit-cell
along $b$-axis ($\hat{y}$) and semi-infinite along $a$-axis ($\hat{x}$). (e)
The intensity of the in-gap states projected onto each layer (unit cell),
where #0 and #35 represent two hinges. The purple line denotes the real space
location of the hinge states. (f) Illustration of the hinge and surface states
and their Chern insulating phases projected at 2D limit along three
directions. (g-h) Same as (d-e) but in AFM$b$ phase with an 80-unit-cell along
$b$ direction. (i) Top view: chirality and position of hinge states switched
between AFM$a$ (blue) and AFM$b$ (red) phase.(j) The 1D $k_{x}$-integrated
Wilson bands as a function of $k_{y}$ in $k_{z}=0/\pi$ planes for the AFM$a$
phase. (k) The inversion symmetric structure of AFM$a$ phase implemented in
hinge mode and local Chern marker calculations. (l) Local Chern number and
integrated Hall conductance as a function of layer index in a 30-unit-cell
slab.

##  III Topological Surface and hinge states

###  III.1 Topology of AFM$a,b$ phases

The AFM$a$ and AFM$b$ phases of Eu3In2As4 belong to the orthorhombic magnetic
space group $Pnn^{\prime}m^{\prime}$, the generators of which include
$\mathcal{I}$, $C_{2z}\mathcal{T}$ and $M_{z}\mathcal{T}$. Thus, we discuss
them together in the following. In addition, the $a,b,c$ crystallographic axes
refer to [100], [010], and [001] directions here.

The antiunitary symmetry $C_{2z}\mathcal{T}$ is helpful to understand the
bulk-surface-hinge correspondence. Let us begin with a $\mathcal{T}$-preserved
$\mathbb{Z}_{2}$ TI where all surfaces would present Dirac surface states. For
the orthorhombic crystal, magnetism opens a surface gap on $bc$ and $ac$
facets, $i.e.$, introduces a mass term in the Dirac equation. Because they
transform to each other by $C_{2z}\mathcal{T}$, two $bc$ facets (also two $ac$
facets) exhibit opposite masses. For four hinges between $ac$ and $bc$ facets
along the $c$ direction, we can always find two diagonal hinges at the domain
boundary between surfaces with opposite signs of mass, as illustrated in Fig.
2(f), leading to chiral hinge modes inside the surface gap. Although it is not
respected on a single $bc$ or $ac$ facet, the $C_{2z}\mathcal{T}$ symmetry is
preserved on the $ab$ plane (see Fig. 2(c)) and leads to crystalline symmetry-
protected unpinned Dirac surface states. In other words, the surface Dirac
cone can only be pushed away from time-reversal-invariant momenta (e.g.
$\bar{\Gamma}$) rather than gapped out by magnetism. As illustrated by Fig.
2(f), two chiral hinge states terminate at gapless $ab$ or (001) surfaces
while the equilibrium current persists, different from an ordinary higher-
order TI that shows only hinge states.

Such a TCI is equivalent to a 3D strong SW insulator with vanishing Berry
curvatures [10, 11, 55, 56]. Because of the antiunitary nature of
$C_{2z}\mathcal{T}$, we can define a $\mathbb{Z}_{2}$-type topological
invariant $\Delta_{3D}=w_{2}(\pi)-w_{2}(0)$, where $w_{2}(k_{z})$ is the
second SW number defined on a 2D plane. The first SW number $w_{1}$ is
equivalent to the quantized Berry phase and a nontrivial $w_{2}$ characterizes
a double band inversion. $w_{2}$ is well-defined only when $w_{1}=0$. $w_{1}$
and $w_{2}$ can also be obtained by tracing the Wilson loop spectra $\varphi$,
where $w_{2}$ is the number of crossings at $\varphi=\pi$ mod 2. When the
inversion symmetry is preserved, $\Delta_{3D}$ can be alternatively inferred
from $\mathbb{Z}_{4}$ index. Fig. 2(j) indicates AFM$a/b$ phase as a 3D strong
SW insulator with $w_{2}(0)=1$ and $w_{2}(\pi)=0$. The SW insulator can
exhibit an odd number of Dirac cones [9, 11] on the surface with
$C_{2z}\mathcal{T}$ symmetry, consistent with our analysis above.

To verify these surface and hinge states, we perform the Green’s function
calculations using Wannier functions extracted from DFT calculations. A single
gapless Dirac cone exists on the (001) surface of AFM$a/b$ phases, shown in
Figs. 2(a) and 2(b). Because there is no additional symmetry to constrain the
Dirac point besides $C_{2z}\mathcal{T}$, the Dirac point is unpinned from
$\bar{\Gamma}$ or any high-symmetry line. As depicted in Fig. 2(c), the top
and bottom surface Dirac cones are transformed to each other by $\mathcal{I}$
and thus, they are located at opposite $k$ points. The Dirac point positions
can be continuously shifted by synchronously rotating all the magnetic moments
on the $ab$ plane, for which $C_{2z}\mathcal{T}$ is maintained. In this case,
the in-plane magnetic moments act as a massless perturbation for the surface
Dirac cone [9].

We next investigate the hinge modes of two adjacent side facets by building a
half-infinite slab model(Fig. 2(k)) [33, 8]. The obtained spectrum includes
the states from bulk, one (100) surface, two (010) surfaces, and two hinges,
as sketched in the inset of Fig. 2(e). Such a slab contains 34 unit cells plus
half a unit cell along the $b$ axis, which respects $C_{2z}\mathcal{T}$ and
$\mathcal{I}$ and is thick enough to suppress inter-facet and inter-hinge
interactions. In Fig. 2(d), an up-moving in-gap state clearly emerges in the
AFM$a$ phase as we expect. By projecting it into each layer, we find that the
spectral weight is mainly distributed at hinge  2 (marked as the purple line
in the inset and its wavefunction distribution on the right in Fig. 2(e)),
confirming it is a hinge mode in the real space. With the assistance of the
inversion symmetry, we schematically show how the gapless surface states
(dark) and hinge modes distribute and flow in the real crystal in Fig. 2(f).

Thin films of such a magnetic TCI can realize the quantum anomalous Hall
insulator (QAHI). By reducing this compound to the 2D in the $ac/bc$ plane,
hinge states will be mapped to a closed-loop edge state, which enables QAHI.
The corresponding Chern numbers $\mathbb{C}=\pm 1$ are opposite for the films
stacking along [100] and [010] directions.

In the AFM$b$ phase, we also discover two hinge modes with opposite
chiralities. As shown in Figs. 2(g) and 2(h), a down-moving state is well
localized at hinge  1 (wavefunction distributes on the left). Hence, such a
magnetic phase transition results in a chirality reversal and a location
switch of the hinge states. Illustrated by Fig. 2(i), the hinge modes at the
upper right and lower left (AFM$a$) will move to the lower right and upper
left (AFM$b$), respectively, leading to the same Chern number
($\mathbb{C}=-1$) for the films stacking along [100] and [010]. Therefore, a
pair of hinge states with opposite chirality could always be found in the
sample with an orthorhombic geometry and their location relies on the magnetic
structure, providing a way of distinguishing the Néel order between AFM$a$ and
AFM$b$.

The obsessing hinge states, and the gapped (100) and (010) surface states
discussed above also attract us to consider what kind of transport signals or
physical quantity they carry. By following [31, 57], we compute the layer-
resolved local Chern marker, $i.e.$, the layer-resolved Chern number,

$\mathbb{C}_{z}(l)=\frac{-4\pi}{A}\Im\frac{1}{N_{k}}\sum_{\bm{k}}\sum_{vv^{\prime}c}X_{vc\bm{k}}Y^{\dagger}_{vc\bm{k}}\rho_{v^{\prime}v\bm{k}}(l),$
(3)

where $X(Y)_{vc\bm{k}}=\bra{\psi_{v\bm{k}}}i\hbar
v_{x(y)}\ket{\psi_{c\bm{k}}}/(E_{c\bm{k}}-E_{v\bm{k}})$ is the position
operator and $\rho_{v^{\prime}v\bm{k}}(l)=\sum_{j\in
l}\psi^{*}_{v\bm{k}}(j)\psi_{v^{\prime}\bm{k}}(j)$ is the projector onto layer
$l$, which sums over the basis $j$ belonging to that layer. Here, we construct
an AFM$a$-phase slab of 29 plus half unit cells along the [010] direction to
apply the above formula, and the contribution of the AHC from the specific
layer can be expressed as $G_{zx}(l)=\frac{e^{2}}{h}\mathbb{C}_{y}(l)$. The
numerical results of $\mathbb{C}_{y}(l)$ and
$\mathbb{C}_{int}=\sum_{l}\mathbb{C}_{y}(l)$ are shown in Fig. 2(l).

We find a saturation behavior of the half-quantized surface AHC. Such a local
signal is known to be the direct evidence of the axion insulator. For the
AFM$a$ case, the thickness of more than 10 layers is enough to stabilize
$\mathbb{C}_{int}$ at 0.4976, and thus corresponds to the penetration depth of
the hinge states shown in Fig. 2(e). $\mathbb{C}_{y}(l)$ is an even function
with its origin at #15 since the sample is inversion symmetric. Distinct from
AFM MnBi2Te4 [33], we find a single and smooth oscillation of
$\mathbb{C}_{int}$ due to the inhomogeneity of the spin moment. The internal
layers from #10 to #20 tame the oscillation and they do not contribute to the
total AHC. Finally, since the spin moments of the top and bottom layers hold
the same direction, the half-quantized surface AHC from the first and last 10
layers together gives rise to a $\mathbb{C}=-1$ Chern insulator with in-plane
magnetic moments.

###  III.2 Topology of AFM$c$ phase

When the spin moments are rotated to the $c$ direction and keep the AFM order,
the magnetic space group will restore to $Pnnm$, the generators of which
include $\mathcal{I}$, $C_{2z}$ and $M_{z}$ but breaks $\mathcal{T}$. Since a
mirror symmetry always satisfies $M_{z}^{2}=-1$ in a spinful system, its
representation has eigenvalues $\pm i$. Hence, the surface states could be
foreseen by calculating the mirror Chern number ($n_{M_{z}=\pm i}$) in two
subspaces of the mirror-symmetric plane. In Fig. 3(a), our Wilson loop
calculations suggest that $n_{M_{z}=\pm i}=\pm 1$ at $k_{z}=0$ plane while
$n_{M_{z}=\pm i}=0$ at $k_{z}=\pi$ plane. Note that $\tau$ in the glide
operator $\\{M_{x,y}|\tau\\}$ is not along $a$ or $b$ direction, so the
surface Dirac point is not fixed at the $\bar{\Gamma}$ point, but along the
$k_{z}=0$ high-symmetry line at the side surfaces. In Figs. 3(b-d), the
surface Green’s function spectra further confirm that the magnetic moments
induced a (001) surface gap ($\sim$ 2 meV) while the massless Dirac cone is
fixed along $\bar{\Gamma}-\bar{X}$ of (010) and $\bar{\Gamma}-\bar{Y}$ of
(100) surface. One exciting finding here is that the gapless and gapped
surface states between the AFM$a/b$ and AFM$c$ phases are completely switched,
which provides a unique playground for spintronic devices.

Figure 3: Mirror-protected TCI in AFM$c$ phase. (a) The evolution of Wannier
charge center along the $k_{y}$ direction for the occupied bands at mirror
symmetric $k_{z}=0$ plane in AFM$c$ phase. Red and blue circles represent the
mirror operation eigenvalue $\braket{M_{z}}=\pm i$. (b-d) The surface states
along the high-symmetry lines of (001), (010), and (100) surfaces,
respectively. The Dirac cones at $k_{z}=0$ for (010) and (100) surfaces are
protected by $M_{z}$ symmetry.

###  III.3 Topology of FM phase

Having established the topology of the AFM phase, we next focus on the FM
phases. In the FM$a$/$b$ case, we find two type-I Weyl points with opposite
chiralities along $\Gamma-X/Y$ connected by $\mathcal{I}$ in Fig. 4(a). The
Wilson loop calculation, shown in Fig. S7, further confirms that
$\mathbb{C}=1$ at $k_{x}=0$ plane and $\mathbb{C}=0$ at $k_{x}=\pi$ plane,
consistent with the band structure. We then calculate the surface state on the
(100) surface (Fig. 4(b)) and project the Fermi surface onto the (001) surface
(Fig. 4(c)). We find a chiral surface state connecting bulk valence and
conduction bands, and the Weyl nodes separated 0.074 Å-1 along $k_{x}$ are
connected by a surface Fermi arc. We also show in SM that the Weyl nodes in
the FM$b$ phase are shifted to locate along the $\Gamma-Y$ direction. Thus,
FM$a/b$ phases provide an ideal Weyl semimetal, which exhibits a single pair
of type-I Weyl points without extra trivial Fermi pockets at the Fermi energy.
When the spin moments align along the $c$ (FM$c$), corresponding magnetic
space group $Pn^{\prime}n^{\prime}m$, the restoration of the mirror $M_{z}$
will protect a nodal ring on the $k_{z}=0$ plane, as shown in Fig. 4(d-f). The
Berry phase integrated along a circle surrounding the nodal ring is checked to
be $\pi$. Such a nodal ring leads to drumhead-like topological surface states
on the (001) plane.

Figure 4: Weyl semimetal in FM phase. The upper and lower panels represent the
FM$a$ and FM$c$ phases, respectively. (a,d) Band structure near $\Gamma$
point. (b,e) (100) surface energy spectrum along $k_{y}$. (c,f) Projected
(001) Fermi surface, which shows that the Fermi arc connects two Weyl points
and $M_{z}$ protects a Weyl nodal ring.

##  IV Topological phase diagram

This section aims to establish a unified theory for various topological phases
revealed in Eu3In2As4 by constructing a $k\cdot p$ model using the invariant
theory [58], and study the global topological phase diagram for different
magnetic configurations. Because $\\{C_{2y}|\tau\\}$ is a key symmetry to
relate two sublattices and the AFM order, three sets of Pauli matrices should
be introduced to describe this system:

  * •

spin $\sigma$: $\ket{\frac{1}{2}}$ and $\ket{-\frac{1}{2}}$

  * •

orbital $\tau$: $\ket{\mathrm{In}\ s}$ and $\ket{\mathrm{As}\ p_{z}}$

  * •

sublattice $\zeta$: $\ket{A}$ and $\ket{B}$

Table 1:  The character table of $\sigma_{i}\tau_{j}\zeta_{k}$ matrices and the polynomials of $k$ under time-reversal symmetry. The exchange-coupling terms $\Delta_{x,y,z}$ describe AFM$a,b,c$ and and $\lambda_{x,y,z}$ describe FM$a,b,c$ orders, respectively. IR | $\mathcal{T}$ | Polynomials | $\sigma_{i}\tau_{j}\zeta_{k}$
---|---|---|---
$\Gamma_{1}^{+}$ | + | $x^{2},y^{2},z^{2}$ | (000)(001)(030)(031)(302)(332)
$\Gamma_{2}^{+}$ | + | $xy$ | (003)(033)
$\Gamma_{4}^{+}$ | + | $xz$ | (102)(132)
$\Gamma_{3}^{+}$ | + | $yz$ | (202)(232)
$\Gamma_{1}^{+}$ | $-$ | $\Delta_{z}$ | (303)(333)
$\Gamma_{2}^{+}$ | $-$ | $\lambda_{z}$ | (002)(032)(300)(301)(330)(331)
$\Gamma_{4}^{+}$ | $-$ |  $\Delta_{x}$, $\lambda_{y}$ | (103)(133)(200)(201)(230)(231)
$\Gamma_{3}^{+}$ | $-$ |  $\Delta_{y}$, $\lambda_{x}$ | (100)(101)(130)(131)(203)(233)
$\Gamma_{1}^{-}$ | $-$ | $xyz$ | (012)(023)(310)(311)
$\Gamma_{2}^{-}$ | $-$ | $z,z^{3},y^{2}z,x^{2}z$ | (020)(021)(313)(322)
$\Gamma_{4}^{-}$ | $-$ | $y,yz^{2},y^{3},x^{2}y$ | (110)(111)(213)(222)
$\Gamma_{3}^{-}$ | $-$ | $x,xz^{2},xy^{2},x^{3}$ | (113)(122)(210)(211)

and the basis are labeled by $\ket{\Lambda_{\alpha},\uparrow(\downarrow)}$,
where $\Lambda$ = In, As; $\alpha$ = $A,B$; $\uparrow(\downarrow)$ =
$\frac{1}{2}(-\frac{1}{2})$. In the absence of magnetic order, Eu3In2As4 is a
trivial insulator. We can have the following orbital basis according to the
band representation (parity $\pm$) and wavefunction at $\Gamma$

$\ket{\mathrm{In}^{+}_{A}\
s}=\frac{1}{\sqrt{2}}(\ket{\mathrm{In}_{1}}+\ket{\mathrm{In}_{1^{\prime}}}),$
(4) $\ket{\mathrm{In}^{+}_{B}\
s}=\frac{1}{\sqrt{2}}(\ket{\mathrm{In}_{1*}}+\ket{\mathrm{In}_{1*^{\prime}}}),$
(5) $\ket{\mathrm{As}^{-}_{A}\
p_{z}}=\frac{1}{\sqrt{2}}(\ket{\mathrm{As}_{2}}+\ket{\mathrm{As}_{2^{\prime}}}),$
(6) $\ket{\mathrm{As}^{-}_{B}\
p_{z}}=\frac{1}{\sqrt{2}}(\ket{\mathrm{As}_{2*}}+\ket{\mathrm{As}_{2*^{\prime}}}),$
(7)

where the atom index (1,2*’) can be found in Fig. S1. The symmetry operators
are thus given by $\hat{\mathcal{T}}=i\sigma_{2}\tau_{0}\zeta_{0}\mathcal{K}$,
$\hat{\mathcal{I}}=\sigma_{0}\tau_{3}\zeta_{0}$,
$\hat{C}_{2z}=-i\sigma_{3}\tau_{0}\zeta_{0}$ and
$\\{\hat{C}_{2y}|\tau\\}=-i\sigma_{2}\tau_{3}\zeta_{1}$, where $\mathcal{K}$
is the complex conjugate operator. Then we can list the irreducible
representation (IR) of each matrix in Table I and Supplementary Table III.
Based on that, the effective Hamiltonian is described by

$\mathcal{H}=\mathcal{H}_{NM}+\mathcal{H}_{ex}=\begin{pmatrix}\mathcal{H}_{A}&\mathcal{H}_{AB}\\\
\mathcal{H}_{AB}^{\dagger}&\mathcal{H}_{B}\\\
\end{pmatrix}+\begin{pmatrix}\mathcal{H}_{\mathrm{M},A}&\mathcal{H}_{\mathrm{M},AB}\\\
\mathcal{H}_{\mathrm{M},AB}^{\dagger}&\mathcal{H}_{\mathrm{M},B}\\\
\end{pmatrix}$ (8) $\displaystyle\mathcal{H}_{A/B}=$ $\displaystyle(\pm
A_{1}k_{x}+A_{2}k_{y})\sigma_{1}\tau_{1}+(B_{1}k_{x}\pm
B_{2}k_{y})\sigma_{2}\tau_{1}\pm
C_{1}k_{z}\sigma_{3}\tau_{1}+D_{1}k_{z}\sigma_{0}\tau_{2}$ (9)
$\displaystyle+(E_{01}+E_{1}k_{x}^{2}+E_{2}k_{y}^{2}+E_{3}k_{z}^{2}\pm
E_{4}k_{x}k_{y})\sigma_{0}\tau_{0}+(F_{01}+F_{1}k_{x}^{2}+F_{2}k_{y}^{2}+F_{3}k_{z}^{2}\pm
F_{4}k_{x}k_{y})\sigma_{0}\tau_{3},$ $\displaystyle\mathcal{H}_{AB}=$
$\displaystyle
A_{3}k_{y}\sigma_{1}\tau_{1}+B_{3}k_{x}\sigma_{2}\tau_{1}+D_{2}k_{z}\sigma_{0}\tau_{2}+(E_{02}+E_{5}k_{x}^{2}+E_{6}k_{y}^{2}+E_{7}k_{z}^{2})\sigma_{0}\tau_{0}+(F_{02}+F_{5}k_{x}^{2}+F_{6}k_{y}^{2}+F_{7}k_{z}^{2})\sigma_{0}\tau_{3}$
(10)
$\displaystyle-i(G_{0}+G_{1}k_{x}^{2}+G_{2}k_{y}^{2}+G_{3}k_{z}^{2})\sigma_{3}\tau_{0}-iH_{1}k_{x}k_{z}\sigma_{1}\tau_{0}-iI_{1}k_{y}k_{z}\sigma_{2}\tau_{0}-iJ_{1}k_{x}k_{z}\sigma_{1}\tau_{3}-iK_{1}k_{y}k_{z}\sigma_{2}\tau_{3}$
$\displaystyle-i(L_{0}+L_{1}k_{x}^{2}+L_{2}k_{y}^{2}+L_{3}k_{z}^{2})\sigma_{3}\tau_{3}-iM_{1}k_{x}\sigma_{1}\tau_{2}-iN_{1}k_{y}\sigma_{2}\tau_{2}-iO_{1}k_{z}\sigma_{3}\tau_{2},$

where $F_{01}$ controls the bandgap [see SM Fig. S8(d)] so it can be
approximately treated as the inverse of the “effective” SOC strength.
Experimentally $F_{01}$ is tuned by pnictide doping or pressure [59, 60, 61].
The bandgap closes when $F_{01}=0.2$ eV. $\mathcal{H}_{A}$ and
$\mathcal{H}_{B}$ are connected by $\\{C_{2y}|\tau\\}$ symmetry. All the other
material-dependent parameters in $\mathcal{H}$ are chosen to make dispersion
close to $\Gamma$ be qualitatively or semi-quantitatively consistent with the
DFT bands. Since we only focus on the bands near the Fermi level, not all the
Pauli matrices above are required to be activated.

We next consider the exchange coupling terms $\mathcal{H}_{ex}$, namely
$\mathcal{T}$-odd terms when introducing magnetic orders. Note that the AFM$a$
phase shares the same IR with FM$b$ (also for AFM$b$ and FM$a$), so both the
on-site terms $\sigma_{1}\tau_{0}\zeta_{3}$ and $\sigma_{2}\tau_{0}\zeta_{0}$
will couple with IR $\Gamma_{4}^{+}$. Generally,
$\bm{\mathrm{m}}\cdot\bm{\sigma}$ could be taken to characterize the
magnetization of one sublattice and thus to distinguish them [17]. Take the
AFM$a$ phase as an example, we have

$\displaystyle\mathcal{H}_{\mathrm{AFM}a,A/B}=$
$\displaystyle\pm\Delta_{x1}\sigma_{1}\tau_{0}\pm\Delta_{x2}\sigma_{1}\tau_{3},$
(11)
$\displaystyle\mathcal{H}_{\mathrm{FM}b,A/B}=\lambda_{y1}\sigma_{2}\tau_{0}+\lambda_{y2}\sigma_{2}\tau_{3},$
(12)
$\displaystyle\mathcal{H}_{\mathrm{AFM}a/\mathrm{FM}b,AB}=\Delta_{3}\sigma_{2}\tau_{0}+\Delta_{4}\sigma_{2}\tau_{3}.$
(13)

Similarly, the exchange terms for the FM$a$, AFM$b$, and AFM$c$ phases can be
obtained, respectively. Additionally, the intermediate phases between them,
namely $C_{2z}\mathcal{T}$-preserved ferrimagnetic and tilted AFM orders, can
be expressed as the linear combination

$\displaystyle\mathcal{H}_{\mathrm{Ferri}a}=\alpha\cdot\mathcal{H}_{\mathrm{AFM}a}+(1-\alpha)\cdot\mathcal{H}_{\mathrm{FM}a},$
(14)
$\displaystyle\mathcal{H}_{\mathrm{AFM^{\prime}}}=\cos{\beta}\cdot\mathcal{H}_{\mathrm{AFM}a}+\sin{\beta}\cdot\mathcal{H}_{\mathrm{AFM}b},$
(15)
$\displaystyle\mathcal{H}_{\mathrm{AFM^{\prime\prime}}}=\cos{\gamma}\cdot\mathcal{H}_{\mathrm{AFM}b}+\sin{\gamma}\cdot\mathcal{H}_{\mathrm{AFM}c}.$
(16)

Figure 5: Topological phase diagram in terms of SOC strength $F_{01}$ and
magnetic orders $\alpha,\beta,\gamma$ (see text). Pentagrams along the gray
line represent DFT results for Eu3In2As4.

By numerically solving the model, we construct a $\mathcal{T}$-broken phase
diagram (Fig. 5) as functions of $F_{01}$ and $\alpha,\beta,\gamma$. We find
Eu3In2As4 can be very well mapped onto the grey horizontal line with
$F_{01}=0.25$ eV. Thanks to the large exchange field of Eu2+ cations, the
phase transition boundary is advanced to lie around $0.27$ eV, accompanied by
a gap closing and reopening. As an intermediate region between two insulating
phases [62], the chance of being a Weyl semimetal is intimately related to the
magnetic configuration. Within the FM order, the Zeeman effect results in a
large semimetallic region. Then it shrinks to a fairly narrow range in the
AFM$a/b$ or tilted phase, and eventually vanishes in the AFM$c$ phase for the
double degeneracy of the bands. We can see the axion angle $\theta=\pi$ is
quite robust against the perturbation on orientations of the AFM order in
Eu3In2As4. The model parameters and DFT verification (marked white stars) are
provided in SM Sec. VI.

##  V Discussions and conclusions

Our results so far demonstrate that this highly tunable platform Eu3In2As4
enables a variety of topological phases. One may next wonder what the
universal feature of the bulk-surface-hinge correspondence for the whole
Eu2n+1In2As(Sb)2n+2 family ($n=0,1,2,\cdots$). We first notice that EuIn2As2
is quite special as a hexagonal lattice but Eu3In2As4 and Eu5In2As6 are
orthorhombic [54, 18, 59, 60, 61]. Since surface and hinge states are closely
associated with geometries, their commonalities must be analyzed under a
unified crystalline representation. More importantly, $C_{2}\mathcal{T}$ is
preserved in their DFT calculated magnetic ground states but absent on the
surfaces of hexagonal prisms. With a lattice transformation of EuIn2As2, we
found nontrivial $w_{2}$ number by the Wilson-loop and $C_{2}\mathcal{T}$
protected Dirac surface states in SM Sec. IV. The Dirac point is constrained
at the $\bar{\Gamma}-\bar{Z}$ line rather than completely unpinned due to the
$\\{M_{y}|(0,0,c/2)\\}$ symmetry.

Figure 6: Schmatics of the axion insulator Eu3In2As4 (cyan-blue) in proximity
to an $s$-wave superconductor (SC) (grey). (a) The nanowire configuration. (b)
The SC proximity to the gapped surface, $e.g.$, the $ac/bc$ surfaces in AFM$a$
phase. (c) The SC proximity to the gapless surface, e.g., the $ab$ surface in
AFM$a$ phase. One chiral hinge state (pink lines) splits into two chiral
Majorana edge modes, providing an interferometer configuration. The white
circle indicates a vortex inside the interferometer. Yellow lines (dots)
represent the chiral Majorana modes (Majorana end modes). The red arrows
represent spin moments in the axion insulator.

Eu3In2As4 nanowires were recently grown from InAs nanowires by a topotaxial
mutual-exchange method [52] and can inherit good interface quality to the
ordinary superconductor ($e.g.$, Al) [53]. Because of the intrinsic magnetism,
strong SOC effect, and topological surface states, Eu3In2As4 paves an
appealing avenue to explore Majorana physics, as illustrated in Fig. 6. The
intrinsic magnetism and SOC together lift the spin degeneracy and an odd
number of Fermi surfaces can be realized at an accessible chemical potential,
which is the precondition to design structure with Majorana [63]. ($i$) In a
nanowire configuration [64, 65], the finite size effect gaps the hinge and
surface states, leading to nondegenerate topology-driven nanowire bands. In
proximity to an $s$-wave superconductor, localized zero-energy Majorana modes
will arise at two ends. Compared to earlier proposals [66, 67], we do not
require an external magnetic field to break time-reversal symmetry here.
($ii$) On the gapped surface (e.g., $ac/bc$ facet in AFM$a$), one can deposit
a superconductor island and expect chiral Majorana edge states around the
island, if the superconductor’s gap $\Delta$ overcomes the magnetization-
induced surface gap $h\sim 2$ meV. Using Fu and Kane’s proposal [63, 3], we
expect a stable nontrivial pairing by covering a superconductor on $ac$ or
$bc$ surfaces (criterion $|h|<\Delta$ when Fermi energy lies inside the gap).
In this case, 1D chiral Majorana modes appear at the interface between the
axion insulator’s surface and a superconductor region [68]. If the Fermi
energy crosses an odd number of Dirac bands (metallic regime), similar chiral
Majorana modes also appear in this case. ($iii$) On the gapless surface (e.g.,
$ab$ facet in AFM$a$), covering a superconductor on the whole surface may lead
to co-propagating 1D chiral Majorana edge modes. Two Majorana modes merge into
one chiral electronic state at the hinge [69]. As illustrated by Fig. 6c, two
chiral Majorana modes and two chiral hinge states naturally form a
Mach–Zehnder interferometer, in which the conductance between two chiral hinge
modes depends on the parity of the number of enclosed vortices in the
superconductor region [68, 70].

In summary, we propose that Eu3In2As4 is an AFM axion insulator and exhibits
hybrid-order topological states. By establishing a $C_{2}\mathcal{T}$ governed
framework, the unpinned Dirac surface, chiral hinge states, and half-quantized
surface AHC are unveiled. Since the single crystal and nanowire of Eu3In2As4
are readily synthesized, angle-resolved photoemission spectroscopy (ARPES),
scanning tunneling spectroscopy (STS) and other techniques are called to
detect these topological surface states, hinge states, and Weyl points/Fermi
arcs and explore their coupling to magnetism and superconductivity.

###### Acknowledgements.

We are inspired by a recent experiment work by Haim Beidenkopf and Hadas
Shtrikman. We thank helpful discussions with Xi Dai, Mingqiang Gu and Hengxin
Tan. B.Y. acknowledges the financial support by the European Research Council
(ERC Consolidator Grant “NonlinearTopo”, No. 815869) and the ISF - Personal
Research Grant (No. 2932/21). C.-X. L. also acknowledges the support of NSF
Grant No. (DMR-2241327)

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