# A Latent space solver for PDE generalization

Rishikesh Ranade, Chris Hill, Haiyang He, Amir Maleki & Jay Pathak
Ansys Inc
Canonsburg, PA 15317, USA
{rishikesh.ranade, chris.hill, haiyang.he, amir.maleki &<EMAIL_ADDRESS>

###### Abstract

In this work we propose a hybrid solver to solve partial differential equation
(PDE)s in the latent space. The solver uses an iterative inferencing strategy
combined with solution initialization to improve generalization of PDE
solutions. The solver is tested on an engineering case and the results show
that it can generalize well to several PDE conditions.

##  1 Introduction

Simulations are important in engineering applications to explore the
underlying physics. But, they can be computationally expensive because they
involve numerical methods to solve partial differential equations (PDEs) for
various conditions associated with the PDE, such as geometry of computational
domain, type of boundary conditions (BC), external source terms etc.
Researchers have explored the idea of coupling machine learning with PDEs for
several decades (Crutchfield & McNamara, 1987; Kevrekidis et al., 2003).
Recently, there is a tremendous focus on improving the predictive capability
and generalizability of ML methods by infusing physics, either during training
or prediction. A substantial portion of research has focused on introducing
physics-based constraints in neural networks through the computation of PDE
derivatives (Raissi et al., 2019; Raissi & Karniadakis, 2018; Rao et al.,
2020; Ranade et al., 2020; Gao et al., 2020; Wu et al., 2018; Qian et al.,
2020; Xue et al., 2020). More recently, there is a huge effort on training
neural networks within frameworks of differentiable PDE solvers. These
approaches use differentiable solvers to learn and control PDE solutions as
well as the dynamics of the system (Amos & Kolter, 2017; Um et al., 2020; de
Avila Belbute-Peres et al., 2018; Toussaint et al., 2018; Wang et al., 2020;
Holl et al., 2020; Portwood et al., 2019; Bar-Sinai et al., 2019; Zhuang et
al., 2020; Kochkov et al., 2021).

There is a growing need for improving generalizability of ML techniques to a
wider range of PDE parameters. The existing ML approaches that learn from PDE
conditions, such as geometry, BCs and source terms have to handle several
challenges. Firstly, the PDE conditions can be high dimensional and sparse,
impacting learning and generalizability. Secondly, the inference procedure is
static and there are limited opportunities to alter the trajectory of PDE
solutions. Finally, the space of physical solutions is very large.

In this paper, we propose a hybrid solver to learning PDE solutions and
address the outlined challenges. We use lower dimensional representations to
tackle the issues of high dimensionality and sparsity of PDE conditions and
solutions. Further, the inferencing methodology uses an iterative procedure to
solve the PDE in a lower dimensional latent space with fixed point iterations.
Latent space learning has been explored in several works, (Wiewel et al.,
2020; Maulik et al., 2020; Kim et al., 2019; Murata et al., 2020; Fukami et
al., 2020b; Champion et al., 2019; Fukami et al., 2020a; He & Pathak, 2020),
but our approach to solve PDEs is different and novel. The iterative procedure
at inference enables us to alter the solution trajectories using existing PDE
solvers. This is useful in ML based techniques to ensure solver robustness,
generalizability and accuracy. The hybrid solver combines solution
initialization using existing coarse grid PDE solvers with this iterative
inferencing procedure and is demonstrated on a practical engineering
application.

##  2 Solution Methodology

###  2.1 Latent space representation

Figure 1: Autoencoders for PDE conditions and solution

Fig. 1 shows the neural network architectures used to determine the compressed
latent space vectors of the various PDE conditions, such as geometry of
computational domain, BCs and source term distributions, and PDE solutions
($u_{1}$ and $u_{2}$). The geometry of the computational domain is represented
using a binarized level set representation (Osher & Sethian, 1988). On the
other hand, boundary conditions and source terms already have spatio-temporal
distributions. The PDE conditions, as well as the PDE solutions are compressed
into their lower dimensional latent vectors, $\eta_{g},\eta_{h},\eta_{b},\eta$
, using CNN encoder-decoder type networks. The loss functions used to
constrain the Autoencoder networks of PDE conditions are purely statistical.
Conversely, the PDE solution Autoencoders are augmented by including PDE based
loss constraints, similar to Ranade et al. (2020). The PDE constraints are
computed using the discretization schemes, available in the Ansys suite of
software. Additionally, since the PDE solutions are dependent on other PDE
conditions, the solution Autoencoders are also conditioned upon the latent
vectors of the PDE conditions.

###  2.2 Hybrid latent space solver methodology

Figure 2: Hybrid solver: Iterative inferencing strategy

Fig. 2 shows the hybrid latent space solver methodology proposed in this work
for using the Autoencoder networks to infer at unknown and unseen conditions.
The iterative procedure based on fixed point iterations and is motivated from
our previous work, (Ranade et al., 2020). The different steps involved in the
solution procedure are outlined below.

  1. 1.

Latent vectors, $\eta_{g}$, $\eta_{h}$ and $\eta_{b}$, are computed for a new
computational domain geometry, boundary condition and source term distribution
using the encoder networks.

  2. 2.

The initial solution of PDE is computed using an existing PDE solver on a very
coarse grid. These solutions are interpolated on the fine grid and encoded to
their latent vector form, $\eta$.

  3. 3.

The initial solution latent vector, $\eta$ combined with $\eta_{g}$,
$\eta_{h}$ and $\eta_{b}$ is passed through the decoder to generate solution
fields, ${u_{1}},{u_{2}}$.

  4. 4.

Solution fields are compressed to a new solution latent vector, $\hat{\eta}$,
using the the encoder.

  5. 5.

Steps $3$ and $4$ are repeated with the new solution latent vector,
$\hat{\eta}$ until $||\eta-\hat{\eta}||_{2}<1e^{-6}$.

  6. 6.

At convergence, the PDE solutions are decoded using the most recent
$\hat{\eta}$.

The hybrid latent space solver described above has two main implications.
Firstly, the iterative procedure used for inferencing allows initialization of
solutions using existing PDE solvers and moreover provides an opportunity to
intervene the solution process and alter the solution trajectory using PDE
solvers. Secondly, the solution procedure is conditioned by richer, lower
dimensional representations of PDE conditions in the form of latent vectors,
thus enhancing the generalizability.

##  3 Results and discussions

Figure 3: Natural convection of electronic chip

The hybrid solver is demonstrated for a 3-D, steady-state electronic cooling
case with natural convection. There are $5$ solution variables, $3$ components
of velocity, pressure and temperature. The geometry of the computational
domain may be observed in Fig. 3. The domain consists of a chip-mold assembly
held by a PCB and the entire geometry is placed inside a fluid domain. The
chip is subjected to heat sources with random spatial distributions due to
uncertainty in electrical heating. Fig. 6 in Appendix A shows an example of
the $8$ different distributions of heat sources. From a physics standpoint,
natural convection results in a two-way coupling between temperature and
velocity. The heat source specified results in a temperature increase, which
generates fluid velocity because of buoyancy effects and in turn cools the
chip. At steady state there is sufficient velocity generation to cancel out
the heat generation. The main challenges in this case are to capture the
complicated physics resulting from the two-way coupling and to generalize to
different spatial distributions of heat sources.

###  3.1 Data generation and training

The Autoencoder for heat source distribution is trained with random spatial
distributions generated using a Gaussian mixture model, where the number of
Gaussians are varied from $1$ to $20$ for randomized mean and variance. The
Autoencoder is set up to achieve a compression ratio of around $12$. The PDE
solution Autoencoder is trained with $200$ solution generated using Ansys
Fluent fluid flow simulation software on a computational mesh with $128^{3}$
elements for random heat sources. A compression ratio of $64$ is achieved. The
normalized reconstruction mean squared error for unseen test samples is to the
order of $1e^{-6}$ for all the solution variables as well as the heat source.

###  3.2 Comparisons with Ansys Fluent

The hybrid solver is compared with Ansys Fluent solution for two heat source
distributions, which are generated randomly and thus, never seen by the
networks. The initial solution is computed at a mesh resolution of $16$
elements in each spatial direction in Ansys Fluent for 200 steady-state
iterations.

####  3.2.1 Computation time

Ansys Fluent takes an average of $200$ minutes on a single CPU to obtain a
converged solution on $128^{3}$ mesh. On the other hand, the hybrid solver
converges in $50$ seconds on average, including the time it takes to generate
the coarse grid solution. Thus, the hybrid solver results in a 200x speedup
over Ansys Fluent in generating solutions on fine girds. The averages are
calculated over runs on 100 cases with different heat source distributions.

####  3.2.2 Contour & Line plots

Fig. 4 and Fig. 7 in Appendix A compare the solution contours on the X-Y plane
for the two test cases. The hybrid solution agrees well with respect to the
Ansys Fluent solutions with small mean squared errors. Although not shown
here, the performance is similar for a variety of random distributions of heat
source, including the examples shown in Figure 6 in the Appendix A. The line
plots in Fig. 5 and Fig. 8 in Appendix A are plotted along the Y direction
through chip center. It can be observed that the overall trends as well as the
peak quantities agree well for both velocity and temperature in both test
cases. The contour and line plots show that the hybrid latent space solver
captures the two-way coupled physics accurately and more importantly, the
methodology generalizes well to a variety of heat source distributions.

Figure 4: Comparisons of velocity and temperature between hybrid solver
solution and Ansys Fluent solution for test case 1. The contours are plotted
in X-Y plane perpendicular to the chip. The velocity contour represents the
entire domain (X$\in$(-0.1525m to 0.1525m), Y$\in$(-0.1525m, 0.1525m)), while
the temperature contour zooms on the chip (X$\in$(-0.03m, 0.03m),
Y$\in$(-0.03m, 0.03m). Figure 5: Comparisons of velocity and temperature
between hybrid solver solution and Ansys Fluent solution for test case 1. The
line plots are plotted along Y direction at (X, Z)$\in$(0.0m, 0.0m). The
velocity line plot represents the entire domain (Y$\in$(-0.1525m, 0.1525m)),
while the temperature line plot zooms on the chip (Y$\in$(-0.03m, 0.03m).

##  4 Conclusion and Future work

In this work we have proposed a hybrid solver that combines latent space
learning with solution initialization to improve generalization of PDE
solutions. The results shows that the methodology is computationally fast,
accurate and generalizable. In its current form, the proposed methodology has
several limitations. Most notably, the approach is restricted to structured
domains with fixed resolutions with relatively simpler computational domain
geometries. In future, we would like to extend this to solve on unstructured
meshes. Furthermore, the latent space iterative strategy will be integrated
with an actual PDE solver to alter solution trajectories and improve
convergence.

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##  Appendix A Appendix

Figure 6: Example of randomness in spatial distribution of heat source terms
Figure 7: Comparisons of velocity and temperature between hybrid solver
solution and Ansys Fluent solution for test case 2. The contours are plotted
in X-Y plane perpendicular to the chip. The velocity contour represents the
entire domain (X=(-0.1525m to 0.1525m), Y$\in$(-0.1525m, 0.1525m)), while the
temperature contour zooms on the chip (X$\in$(-0.03m, 0.03m), Y=(-0.03m,
0.03m). Figure 8: Comparisons of velocity and temperature between hybrid
solver solution and Ansys Fluent solution for test case 2. The line plots are
plotted along Y direction at (X, Z)$\in$0.0m, 0.0m). The velocity line plot
represents the entire domain (Y$\in$(-0.1525m, 0.1525m)), while the
temperature line plot zooms on the chip (Y=(-0.03m, 0.03m).


# Anisotropic magnetism and electronic structure of trigonal EuAl2Ge2 single
crystals

Santanu Pakhira  Ames National Laboratory, Iowa State University, Ames, Iowa
50011, USA  Asish K. Kundu  Condensed Matter Physics and Materials Science
Division, Brookhaven National Laboratory, Upton, New York 11973, USA  Farhan
Islam  Ames National Laboratory, Iowa State University, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  M. A. Tanatar  Ames National Laboratory, Iowa State University, Ames,
Iowa 50011, USA  Department of Physics and Astronomy, Iowa State University,
Ames, Iowa 50011, USA  Tufan Roy  Center for Science and Innovation in
Spintronics, Core Research Cluster, Tohoku University, Sendai 980-8577, Japan
Thomas Heitmann  The Missouri Research Reactor and Department of Physics and
Astronomy, University of Missouri, Columbia, Missouri 65211, USA  T. Yilmaz
National Synchrotron Light Source II, Brookhaven National Laboratory, Upton,
New York 11973, USA  E. Vescovo  National Synchrotron Light Source II,
Brookhaven National Laboratory, Upton, New York 11973, USA  Masahito Tsujikawa
Research Institute of Electrical Communication, Tohoku University, Sendai
980-8577, Japan  Masafumi Shirai  Center for Science and Innovation in
Spintronics, Core Research Cluster, Tohoku University, Sendai 980-8577, Japan
Research Institute of Electrical Communication, Tohoku University, Sendai
980-8577, Japan  R. Prozorov  Ames National Laboratory, Iowa State University,
Ames, Iowa 50011, USA  Department of Physics and Astronomy, Iowa State
University, Ames, Iowa 50011, USA  David Vaknin  Ames National Laboratory,
Iowa State University, Ames, Iowa 50011, USA  Department of Physics and
Astronomy, Iowa State University, Ames, Iowa 50011, USA  D. C. Johnston  Ames
National Laboratory, Iowa State University, Ames, Iowa 50011, USA  Department
of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA

###### Abstract

Understanding the interplay between magnetic and electronic degrees of freedom
is of profound recent interest in different Eu-based magnetic topological
materials. In this work, we studied the magnetic and electronic properties of
the layered Zintl-phase compound ${\rm EuAl_{2}Ge_{2}}$ crystallizing in the
trigonal ${\rm CaAl_{2}Si_{2}}$-type structure. We report zero-field neutron
diffraction, temperature $T$\- and magnetic-field $H$-dependent magnetic
susceptibility $\chi(T,H)$, isothermal magnetization $M(T,H)$, heat capacity
$C_{\rm p}(T,H)$, and electrical resistivity $\rho(T,H)$ measurements,
together with $T$-dependent angle-resolved photoemission spectroscopy (ARPES)
measurements complemented with first-principle calculations. ${\rm
EuAl_{2}Ge_{2}}$ undergoes second-order A-type antiferromagnetic (AFM)
ordering below $T_{\rm N}=27.5(5)$ K, with the Eu moments
(Eu${}^{2+},\,S=7/2$) aligned ferromagnetically in the $ab$ plane while these
layers are stacked antiferromagnetically along the $c$ axis. The critical
fields at which all moments become parallel to the field are 37.5(5) and
52.5(5) kOe for $H\parallel ab$ and $H\parallel c$, respectively. The $H=0$
magnetic structure consists of trigonal AFM domains associated with $ab$-plane
magnetic anisotropy and a field-induced reorientation of the Eu spins in the
domains is also evident at $T=2$ K below the critical field $H_{c1}=2.5(1)$
kOe. The $\rho(T)$ measurements reveal metallic behavior transforming into a
slight resistivity increase on cooling towards $T_{\rm N}$. A pronounced loss
of spin-disorder scattering is observed below $T_{\rm N}$. The ARPES results
show that ${\rm EuAl_{2}Ge_{2}}$ is metallic both above and below $T_{\rm N}$,
and the Fermi surface is anisotropic with two hole pockets at the zone center
and one small electron pocket at each M point. In the AFM phase, we directly
observe folded bands in ARPES due to the doubling of the magnetic unit cell
along the $c$ axis with an enhancement of quasiparticle weight due to the
complex change in the coupling between the magnetic moments and itinerant
electrons on cooling below $T_{\rm N}$. The observed electronic structure is
well reproduced by first-principle calculations, which also predict the
presence of nontrivial electronic states near the Fermi level in the AFM phase
with $Z_{2}$ topological numbers 1;(000).

##  I Introduction

It is rewarding to study different classes of novel quantum materials having a
complex interplay of lattice, spin, and electronic degrees of freedom. These
materials can exhibit a plethora of interesting physical properties including
superconductivity, heavy fermion behavior, quantum phase transitions, complex
magnetic order, magnetic frustration, valence fluctuations, and nontrivial
topological phases. One such family of materials is comprised of Zintl-phase
compounds that have gained significant recent interest owing to the complex
interplay of magnetic and electronic degrees of freedom. These materials
exhibit topological states, proximity between metal-semimetal-semiconductor-
insulator phases, anomalous and topological Hall effects, low-field-induced
spin reorientations within antiferromagnetic (AFM) domains, along with large
thermoelectricity as recently reported in various compounds [1, 2, 3, 4, 5, 6,
7, 8, 9, 10].

Many $AM_{2}X_{2}$-type Zintl-phase compounds have been investigated, where
$A$ is an alkaline or lanthanide element, $M$ is a metallic $sp$ element, and
$X$ is an $sp$-element anion where the $A$ atom has either a planar triangular
or square-lattice structure. These materials have recently been reported to
exhibit electronic states having nontrivial band topology. These states
include a topological insulating state, a Dirac/Weyl-type semimetallic state,
or an axion-insulating state, and are attractive candidates for
dissipationless electron transport [9, 11, 12, 13, 14, 15, 16, 17]. It has
been experimentally found that when the $A$ site of these compounds is fully
or partially occupied by a rare-earth element, the materials show enhanced
carrier mobility and carrier concentration compared to those with $A$ as an
alkaline-earth metal [18, 19]; the origin of this behavior is currently
unknown.

For example, such magnetic Eu-based compounds are of significant interest due
to their complex interplay of magnetism and band topology, as reported for
${\rm EuIn_{2}As_{2}}$, ${\rm EuCd_{2}As_{2}}$, ${\rm EuMg_{2}Bi_{2}}$, and
${\rm EuSn_{2}As_{2}}$ [4, 8, 11, 12, 13, 14, 17, 20, 21, 22, 23, 24]. The
magnetic properites associated with different anisotropy energies could thus
also play an important role in tuning the electronic states in these materials
associated with magnetic ordering. Although the Eu2+ ion with spin $S=7/2$ and
orbital angular momentum $L=0$ exhibits negligible single-ion anisotropy, the
magnetic properties in most of these materials are anisotropic [20, 21, 23,
25, 26, 27, 28]. Here the anisotropy arises from magnetic-dipole and/or
anisotropic RKKY interactions.

To further investigate the properties of this class of materials, here we
report the growth of ${\rm EuAl_{2}Ge_{2}}$ single crystals with the trigonal
CaAl2Si2 crystal structure [29] and studies of their magnetic, electronic-
transport, and electronic-structure properties. These include zero-field
neutron-diffraction measurements of the ordered magnetic structure,
temperature $T$\- and magnetic-field $H$-dependent magnetization $M(H,T)$,
heat capacity $C_{\rm p}(T)$, and electrical-resistivity $\rho(H,T)$
measurements, along with $T$-dependent angle-resolved photoemission
spectroscopy (ARPES) studies of the electronic structure. The experimental
electronic structure is mapped by calculating the band structure of ${\rm
EuAl_{2}Ge_{2}}$ using density-functional theory (DFT).

We find that ${\rm EuAl_{2}Ge_{2}}$ is metallic as revealed by the $\rho(T)$
and ARPES measurements complemented with theoretical band-structure
calculations. The neutron-diffraction experiments demonstrate that ${\rm
EuAl_{2}Ge_{2}}$ exhibits A-type AFM order below its Néel temperature $T_{\rm
N}=27.5(5)$ K. In this magnetic structure the Eu2+ moments $\mu=7~{}\mu_{\rm
B}$ with spectroscopic-splitting factor $g=2$ and spin $S=7/2$ are aligned
ferromagnetically in each $ab$-plane layer, where the moments in adjacent
layers along the $c$ axis are aligned antiferromagnetically. The $C_{\rm p}$
data for $H=0$ exhibit a second-order $\lambda$-type peak at $T_{\rm N}$. The
ARPES results further reveal magnetism-induced band folding and enhancement of
the quasiparticle intensity in the magnetic ground state. Splitting of the
energy bands is observed even above $T_{\rm N}$, possibly related to time-
reversal-symmetry breaking associated with short-range ferromagnetic (FM)
correlations above $T_{\rm N}$.

Over the broad field range $0\leq H\leq 55$ kOe, the $M(H)$ data at $T=2$ K
appear to be linear for both $H\parallel ab$ and $H\parallel c$ with
respective critical fields $H^{\rm c}_{ab}=37.5$ and $H^{\rm c}_{c}=52.5$ kOe,
at which all moments become parallel to the respective field. However,
detailed $M(H_{ab},T=2$ K) measurements at low fields $H_{ab}\leq H_{\rm
c1}=2.5$ kOe exhibit anomalous positive curvature, whereas for $H_{ab}>H_{\rm
c1}$ the data are again linear up to $H^{\rm c}_{ab}$. This behavior is
quantitatively described by a model where the A-type AFM structure occurs in
three trigonal domains in which the Eu moments in each domain are aligned at
120∘ to each other in $H=0$. With increasing $H_{ab}$ the moments in each
domain reorient to become perpendicular to ${\bf H}_{ab}$ until $H_{\rm c1}$
is reached, above with all moments progressively cant towards ${\bf H}_{ab}$
until $H^{\rm c}_{ab}$ is attained.

Experimental and theoretical details are given in Sec. II. The results and
discussion of the various measurements and analyses are presented in Sec. III,
and concluding remarks are provided in Sec. IV.

##  II  Experimental and Theoretical Details

Single crystals of ${\rm EuAl_{2}Ge_{2}}$ were grown using the flux method
with starting composition Eu:Al:Ge = 1:20:2. The Eu (Ames Lab), Al (Alfa
Aesar, 99.9995%), and Ge (Alfa Aesar, 99.9999%) were loaded into a 2 mL
alumina crucible and sealed in a silica tube under $1/4$ atm high-purity
argon. The assembly was heated to 1175 ∘C inside a box furnace at a rate of
100 ∘C/h. After holding the temperature for 6 h, the furnace was cooled to 700
∘C at a rate of 10 ∘C/h. The assembly was then centrifuged to separate the
crystals from the molten flux. Hexagonal plate-like crystals with typical
dimensions $3\times 3\times 2$ mm3 were obtained from the growth. The
homogeneity and chemical composition of the crystals were confirmed using a
JEOL scanning-electron microscope (SEM) equipped with an energy-dispersive
x-ray spectroscopy (EDS) analyzer. The magnetic measurements were carried out
using a Magnetic-Properties-Measurement System (MPMS) from Quantum Design,
Inc., in the $T$ range 1.8–300 K and with $H$ up to 5.5 T (1 T $\equiv 10^{4}$
Oe).

A Physical Properties Measurement System (PPMS, Quantum Design, Inc.) was used
to measure $C_{\rm p}(T)$ and $\rho(T)$ in the $T$ range 1.8–300 K and $H$ up
to 9 T. Four-probe $\rho(T)$ measurements were performed. The measurements
were performed on as-grown single crystals. Due to the sensitivity of EuAl2Ge2
to the ambient environment leading to a rapid sample decomposition, the
crystals were not shaped into resistivity bars with precision geometric-factor
control by polishing and cutting. However, the crystals had natural shapes
suitable for in-plane resistivity measurements, having a length at least 3
times larger than the width and thickness. Resistivity measurements were
performed along arbitrary directions in the $ab$ plane. In all resistivity
measurements the magnetic field was oriented transverse to the current
direction. Contacts to the fresh surfaces of the crystals were made by
attaching 50 $\mu$m-diameter silver wires with In solder and mechanically
reinforcing the contact with DuPont 4929N silver paint [30]. The contact
resistance was typically in the $\Omega$ range. After application of the
contacts was complete, the samples were covered with Apiezon N-grease to
provide temporal protection from degradation. For measurements in magnetic
fields oriented along the $c$ axis and $ab$ plane, the samples were attached
with Apiezon N-grease to the sides of a plastic cube. This provides alignment
with about $\pm 5^{\circ}$ accuracy [31].

Single-crystal neutron-diffraction experiments were performed in $H=0$ using
the TRIAX triple-axis spectrometer at the University of Missouri Research
Reactor (MURR). An incident neutron beam of energy 30.5 meV was directed at
the sample using a pyrolytic graphite (PG) monochromator. A PG analyzer was
used to reduce the background. Neutron wavelength harmonics were removed from
the beam using PG filters placed before the monochromator and in between the
sample and analyzer. Beam divergence was limited using collimators before the
monochromator; between the monochromator and sample; sample and analyzer; and
analyzer and detector of $60^{\prime}-60^{\prime}-40^{\prime}-40^{\prime}$,
respectively. A $\approx 20$ mg ${\rm EuAl_{2}Ge_{2}}$ crystal was mounted on
the cold tip of an Advanced Research Systems closed-cycle refrigerator with a
base temperature of 4 K. The crystal was aligned in the $(HHL)$ scattering
planes. The lattcie parrameters at base temperature are $a=4.19(1)$ and
$c=7.27(1)$ Å.

ARPES experiments were performed at the Electron Spectro Microscopy (ESM)
21-ID-1 beamline of the National Synchrotron Light Source II, USA. The
beamline is equipped with a Scienta DA30 electron analyzer, with base pressure
better than $\sim$ 1$\times$10-11 mbar. Prior to the ARPES experiments,
samples were cleaved inside an ultra-high vacuum chamber (UHV) at $\sim$ 9 K.
All the measurements were performed using horizontally polarized light. The
uncertainty in the temperature values for the ARPES measurements is $\pm 2$ K.

The Vienna ab initio simulation package was used for the first-principles
calculations [32, 33]. For the exchange and correlation energy/potential we
used the PBE functional [34]. The projected-augmented-wave [35] method was
used to represent the core electrons. The cut-off energy for the plane waves
was set to 550 eV. A $k$-mesh of 14$\times$14$\times$7 (AFM phase) and
14$\times$14$\times$12 (PM phase) was used for the Brillouin-Zone integration.
Spin-orbit-coupling (SOC) was considered in all calculations. The GGA $+$ U
method [36] was used to treat the correlation effects of 4$f$ states in Eu.
Furthermore, WANNIER90 was used for the construction of the first-principle
tight binding Hamiltonian and constant energy surfaces [37]. The $s$ and $p$
orbitals of Ge and Al and $s$, $p$, $d$, and $f$ orbitals of Eu were used to
construct maximally-locallized Wannier functions. In the case of the PM phase,
we treated the $f$ electrons of Eu as core states. The WANNIERTOOLS package
was used for the calculation of the $Z_{2}$ topological number [38]. For the
visualization of the Fermi surfaces, we used FermiSurfer [39]

Figure 1: (a) Zero-field neutron-diffraction pattern along $(00L)$ of single-
crystal ${\rm EuAl_{2}Ge_{2}}$ at 6 and 30 K, as indicated. The aluminum Bragg
reflections marked on the figure originate from the sample holder. The
magnetic Bragg reflections are obtained by subtracting the diffraction pattern
at 30 K from the one at 6 K for (b) $(00L)$, (c) $(\frac{1}{2}\frac{1}{2}L)$,
and (d) $(11L)$ scans. The difference patterns in (b) show clear magnetic
peaks at half-integer $L$ up to $L=3.5$. No such peak is observed in (c,d)
along the $(\frac{1}{2}\frac{1}{2}L)$ and $(11L)$ directions. These
observations are consistent with an A-type AFM state, i.e, the $H=0$ ground
state is such that the intraplane moments are ferromagnetically aligned in the
$ab$ plane while the moments in adjacent Eu planes along the $c$ axis are
aligned antiferromagnetically. Note that structure-factor calculations for
this model indicate $(11L)$ at half-integer values of $L$; we argue that their
absence in (d) is due to the form factor of ${\rm Eu}^{2+}$ at these
relatively large momentum transfers. (e) Integrated intensity as a function of
temperature $T$ of the (0 0 $\frac{1}{2}$) magnetic Bragg reflection fitted
with a power-law to yield $T_{\rm N}=(27.3\pm 0.8)$ K and $\beta=0.21\pm
0.01$. (f) Chemical and A-type AFM ground-state structure of ${\rm
EuAl_{2}Ge_{2}}$. Neutron-diffraction data are insufficient to determine the
in-plane moment directions. Therefore, we arbitrarily show the in-plane
moments pointing along the next-nearest-neighbor direction.

##  III  Results and Discussion

###  III.1  Zero-field neutron diffraction

Figure 1(a) shows zero-field neutron-diffraction scans along the $(00L)$
direction in reciprocal-lattice units (r.l.u.) at 6 K and 30 K, where
reflections at half-integer $L$ values are apparent at $T=6$ K. For more
clarity, Fig. 1(b) shows the difference between these two scans, where within
experimental uncertainty, there is no evidence for other reflections
associated with a modulated structure along the $c$ axis. We also note that
the intensities of the new peaks become weaker at larger $L$ values, roughly
following the falloff expected from the magnetic form factor of Eu2+. Similar
differences [i.e., $I{\rm(6~{}K)}-I({\rm 30~{}K)}]$ for scans along
$(\frac{1}{2}\frac{1}{2}L)$ and $(11L)$, shown in Figs. 1(c,d), respectively,
do not reveal any magnetic peaks.

Qualitatively, these newly-emerging Bragg reflections indicate a doubling of
the unit cell along the $c$ axis. These qualitative observations unequivocally
establish that these reflections are associated with A-type AFM ordering with
propagation vector $\vec{\tau}=\left(0,0,\frac{1}{2}\right)$, consisting of
layers of moments aligned ferromagnetically in the $ab$ plane, with moments in
adjacent planes along the $c$ axis aligned antiferromagnetically.

The proposed A-type AFM structure is shown in Fig. 1(f), where adjacent
nearest-neighbor FM layers along the $c$ axis are rotated by 180∘ with respect
to each other. The direction of the FM moment within an Eu layer cannot be
determined from neutron diffraction alone. Using published values, we obtain
good agreement with lattice parameters; however, the peak intensities differ
significantly from the calculated values due to strong absorption effects by
Eu, which are not accounted for in our calculations.

Nevertheless, we are able to confirm the A-type magnetic structure and obtain
an estimate for the Eu ordered magnetic moment $\mu=g\langle
S\rangle~{}\mu_{\rm B}=(6.5\pm 1)~{}\mu_{\rm B}$ at $T=6$ K by calculating the
magnetic and chemical structure factors, where $S$ is the spin magnetic
quantum number, $g$ is the spectroscopic-splitting factor, and $\mu_{\rm B}$
is the Bohr magneton. We note that the large uncertainty in the evaluation of
the ordered magnetic moment is mainly due to strong-absorption effects which
were not accounted for. Within the error, the fitted value of $\mu$ agrees
with the expected value $\mu=7\,\mu_{\rm B}$/Eu using $g=2$ and $S=7/2$.

Figure 1(e) shows the integrated intensity of the (0 0 $\frac{1}{2}$) magnetic
peak as a function of temperature where we use a weighted power-law function
by a Gaussian distribution of $T_{\rm N}$

$\displaystyle I_{\rm(0\,0\,0.5)}(T)=C|1-T/T_{\rm N}|^{2\beta}\propto\mu^{2},$
(1)

yielding $T_{\rm N}=(27.3\pm 0.8$) K and $\beta=0.21\pm 0.01$. The $T_{\rm N}$
is in good agreement with the value $T_{\rm N}=(27.5\pm 0.5)$ K obtained from
the $\chi(T)$ and $C_{\rm p}(T)$ measurements below.

###  III.2  Magnetic Susceptibility

Figure 2: Inverse magnetic susceptibility as a function of temperature
$\chi^{-1}(T)$ measured for $H=1$ kOe, when (a) $H\parallel ab$ and (b)
$H\parallel c$.

The inverse magnetic susceptibility $\chi^{-1}(T)$ data measured under an
applied field $H=1$ kOe for both $H\parallel ab$ and $H\parallel c$ are shown
in Figs. 2(a) and 2(b), respectively. The data for $T\geq 50$ K for both field
directions were fitted by the modified Curie-Weiss law

$\displaystyle\chi_{\alpha}(T)=\chi_{0}+\frac{C_{\alpha}}{T-\theta_{\rm
p\alpha}}\quad(\alpha~{}=~{}ab,~{}c),$ (2)

where $\chi_{0}$ is the temperature-independent contribution, $C_{\alpha}$ is
the Curie constant, and $\theta_{\rm p}$ is the paramagnetic Weiss
temperature. The Curie constant $C_{\alpha}$ is given by

$C_{\alpha}=\frac{N_{\rm A}{g_{\alpha}}^{2}S(S+1)\mu^{2}_{\rm B}}{3k_{\rm
B}}=\frac{N_{\rm A}\mu^{2}_{\rm{eff,\alpha}}}{3k_{\rm B}},$ (3a) where $N_{\rm
A}$ is Avogadro’s number and the effective magnetic moment is given by
$\mu_{\rm{eff,\alpha}}=g_{\alpha}\sqrt{S(S+1)}\,\mu_{\rm B}.$ (3b)

The fits of the $\chi_{\alpha}^{-1}(T)$ data by Eq. (2) is depicted in Figs.
2(a) and 2(b) for $H\parallel ab$ and $H\parallel c$, respectively, and the
fitted parameters are listed in Table 1. The effective moments are close to
the value 7.94 $\mu_{\rm B}$/Eu expected for Eu2+ spins with $S=7/2$ and
$g=2$. The positive values of the Weiss temperatures $\theta\rm_{p\alpha}$ are
consistent with the A-type AFM order revealed by the above zero-field neutron-
diffraction measurements, where the in-plane FM interactions between the Eu
spins are dominant over the interplane AFM interactions.

Table 1:  The obtained Parameters from the fits of the data in Figs. 2(a) and 2(b) by Eq. (2). Listed parameters are the $T$–independent contribution to the magnetic susceptibility $\chi_{0}$, Curie constant per mol $C_{\alpha}$ in $\alpha=ab,c$ directions, effective moment per Eu $\mu{\rm{}_{eff}(\mu_{B}/Eu)}\approx\sqrt{8C}$ and Weiss temperature $\theta\rm_{p\alpha}$ obtained from the $\chi^{-1}(T)$ versus $T$ data for $H=1$ kOe. Field | $\chi_{0}$ | $C_{\alpha}$ | $\mu_{\rm eff\alpha}$ | $\theta_{\rm p\alpha}$
---|---|---|---|---
direction | $\rm{\left(10^{-4}~{}\frac{cm^{3}}{mol}\right)}$ | $\rm{\left(\frac{cm^{3}K}{mol}\right)}$ | ($\mu_{\rm B}$/Eu) | (K)
H $\parallel ab$ | $-2.6(3)$ | 7.86(1) | 7.93(1) | 24.26(7)
H $\parallel c$ | $-1.9(3)$ | 7.99(1) | 7.99(1) | 21.86(7)
Figure 3: (a) Temperature dependence of the magnetic susceptibilities measured
for $H=0.1$ kOe with $H\parallel ab$ (black squares) and $H\parallel c$ (red
circles). The upturns in the $\chi_{ab}(T)$ and $\chi_{c}(T)$ data below $\sim
5$ K may be associated with an additional magnetic ordering of unknown type.
(b) Spherically-averaged Heisenberg magnetic susceptibility $\chi_{J}(T)$ in
the PM state with $T\geq T_{\rm N}$ obtained using Eq. (4) (filled blue
triangles). The blue curve connects the data points. The $\chi_{ab}(T)$ and
$\chi_{c}(T)$ data in (a) for $T\leq T_{\rm N}$ are respectively shifted
vertically to match the values at $T_{\rm N}$ to the value $\chi_{J}(T=T_{\rm
N})$ = 0.96 cm3/mol. The $\chi_{J,ab}(T\leq T_{\rm N})$ for A-type AFM order
predicted by Eqs. (5) for $kd=\pi$ rad and $f=\theta_{\rm p\,ave}/T_{\rm
N}=0.853$ is shown as the green curve. For A-type ordering with the moments
aligned in the $ab$ plane, one theoretically expects $\chi_{J,ab}(0~{}{\rm
K})/\chi_{J}(T_{\rm N})=1/2$, close to the observed value.

The $T$ dependences of the magnetic susceptibilities $\chi$ of ${\rm
EuAl_{2}Ge_{2}}$ measured in $H=0.1$ kOe for the in-plane ($H\parallel ab$)
and out-of-plane ($H\parallel c)$ field directions are shown in Fig. 3(a). A
sharp AFM transition is observed at $T_{\rm N}=27.5(5)$ K, which is the same
as reported earlier for polycrystalline ${\rm EuAl_{2}Ge_{2}}$ [40]. The
anisotropy between $\chi_{ab}$ and $\chi_{c}$ above $T_{\rm N}$ likely arises
from a combination of magnetic-dipole and magnetocrystalline interactions. The
$\chi_{J}(T)$ data above $T_{\rm N}$ for Heisenberg interactions in the
absence of anisotropy are obtained as the average

$\displaystyle\chi_{J}(T\geq T_{\rm
N})=\frac{1}{3}[2\chi_{ab}(T)+\chi_{c}(T)]$ (4)

which is plotted in Fig. 3(b). Then the data at $T\leq T_{\rm N}$ are shifted
vertically until they match the $\chi_{J}(T\geq T_{\rm N})$ data at $T_{\rm
N}$ as shown.

The $\chi_{J,ab}$ data strongly decrease on cooling from $T_{\rm N}$ to $T\sim
5$ K, whereas the out-of-plane susceptibility $\chi_{J,ab}$ is less dependent
on the temperature, signifying that the $ab$ plane is the easy plane. This
observation is in good agreement with the neutron-diffraction results
revealing the A-type nature of the magnetic ground state with the moments
aligned in the $ab$ plane. However, below $\sim 5$ K, both $\chi_{J,c}$ and
$\chi_{J,ab}$ increase sharply, indicating the occurrence of an additional
magnetic transition of unknown nature at $T\sim 5$ K. Our neutron-diffraction
measurements could not examine the additional transition as their low-$T$
limit was 6 K.

Here we utilize the molecular field theory (MFT) [41, 42] for $c$-axis helical
antiferromagnets with the moments aligned in the $ab$ plane with $c$-axis
propagation vector $k$ and interlayer spacing $d$ for which $kd$ is the turn
angle between moments in adjacent layers. The in-plane magnetic susceptibility
$\chi_{Jab}(T)$ associated with Heisenberg spins and spin interactions $J$ for
$T\leq T_{\rm N}$ and no anisotropy can be written as

$\frac{\chi_{J,ab}(T\leq T_{\rm N})}{\chi_{J}(T_{\rm
N})}=\frac{(1+\tau^{*}+2f+4B^{*})(1-f)/2}{(\tau^{*}+B^{*})(1+B^{*})-(f+B^{*})^{2}},$
(5a) where $f=\theta_{\rm p\,ave}/T_{\rm N},$ (5b)
$B^{*}=2(1-f)\cos(kd)\,[1+\cos(kd)]-f,$ (5c) $t=\frac{T}{T_{\rm
N}},\quad\tau^{*}(t)=\frac{(S+1)t}{3B^{\prime}_{S}(y_{0})},\quad
y_{0}=\frac{3\bar{\mu}_{0}}{(S+1)t},$ (5d) the ordered moment versus $T$ in
$H=0$ is denoted by $\mu_{0}$, the reduced ordered moment
$\bar{\mu}_{0}=\mu_{0}/\mu_{\rm sat}$ with $\mu_{\rm sat}=gS\mu_{\rm
B}=7\,\mu_{\rm B}$ here is determined by numerically solving the self-
consistency equation $\bar{\mu}_{0}=B_{S}(y_{0}),$ (5e)
$B^{\prime}_{S}(y_{0})=[dB_{S}(y)/dy]|_{y=y_{0}}$, and the Brillouin function
$B_{S}(y)$ is $B_{S}(y)=\frac{1}{2S}\left\\{(2S+1){\rm
coth}\left[(2S+1)\frac{y}{2}\right]-{\rm
coth}\left(\frac{y}{2}\right)\right\\}.$ (5f)

Using the value of $f$ calculated from the values of $\theta_{\rm p,ave}$ and
$T_{\rm N}$ from Table 1, the calculated $\chi_{J,ab}(T)$ for $T\leq T_{\rm
N}$ is shown by the green curve in Fig. 3(b). As seen in the figure, the
calculated curve deviates somewhat from the experimental $\chi_{J,ab}(T)$
data, likely due to the additional higher-$T$ magnetic precursor contributions
of the anticipated low-$T$ order below 5 K. According to the MFT [41, 42], at
$T=0$ we have

$\frac{\chi_{J,ab}(T=0)}{\chi_{J,ab}(T_{\rm N})}=\frac{1}{2[1+2~{}{\rm
cos}(kd)+2~{}{\rm cos}^{2}(kd)]}.$ (6)

Thus, for an A-type AFM, where the turn angle between adjacent $ab$-plane FM
layers is $kd\to 180^{\circ}$, one expects
$\chi_{J,ab}(T=0)/\chi_{J,ab}(T_{\rm N})\to 1/2$, close to the value in Fig.
3(b).

Figure 4: Magnetic susceptibility $\chi_{\alpha}(T)$, $\alpha=ab,c$, at
different applied magnetic fields for (a) $H\parallel ab$ and (b) $H\parallel
c$. Although, the $\chi_{c}(T)$ is weakly dependent on $H$ below $T\leq T_{\rm
N}$, $\chi_{c}(T)$ is strongly $H$-dependent up to $H=10$ kOe.

The $\chi(T)$ measured at several applied magnetic fields $H$ are shown in
Figs. 4(a) and 4(b) for $H\parallel ab$ and $H\parallel c$, respectively.
Interestingly, although the out-of-plane magnetic susceptibility $\chi_{c}$
remain almost independent of $H$ for $H\leq 10$ kOe, the in-plane
susceptibility $\chi_{ab}$ changes significantly with $H$ for $T<T_{\rm N}$
and $H$ up to 5 kOe. Similar behavior was also observed for the trigonal
A-type AFM compounds ${\rm EuMg_{2}Bi_{2}}$, ${\rm EuMg_{2}Sb_{2}}$, ${\rm
EuSn_{2}As_{2}}$, and tetragonal ${\rm EuGa_{4}}$ with the moments aligned in
the $ab$ plane [10, 20, 23, 43]. We have argued that the A-type ground state
spin structure of these materials consist of three-fold (for trigonal) or
four-fold (for tetragonal) AFM domains associated with in-plane magnetic
anisotropy. As Eu2+ moments with $L=0$ provide negligible single-ion
anisotropy, magnetic dipole interaction and other magnetocrystalline
anisotropy energy may play a critical role for the formation of AFM domains in
these materials. The $H$-dependent change in the $\chi_{ab}(T)$ behavior is
due to the reorientation of the spins with in-plane field $H_{\rm ab}$ up to a
critical field $H_{c1}$, where all the spins in different domains become
perpendicular to the in-plane applied field direction. The spins tend to align
along the field direction for $H>H_{c1}$, as expected for a collinear
antiferromagnet.

###  III.3  Isothermal magnetization versus applied magnetic field
measurements

Figure 5: (a) Magnetic field dependence of isothermal magnetization $M(H)$ in
the hysteresis mode for $-5.5~{}{\rm T}\leq H\leq 5.5$ T measured at $T=2$ K
for both $H\parallel ab$ and $H\parallel c$. $M(H)$ behavior measured at
different temperatures for (b) $H\parallel ab$ and (c) $H\parallel c$. (d)
Low-field $M(H)$ data showing nonlinearity in the $M_{ab}(H)$ behavior for
$T<T_{\rm N}$, whereas $M_{c}(H)$ is linear down to the lowest measured
temperature 2 K. This nonlinearity is clearly reflected in the $dM/dH$ data
shown in (e). (f) The experimental magnetization $M_{ab}(H)$ at $T=2$ K along
with the theoretical prediction for $T=0$ K with $H_{\rm c1}\approx 2.5$ kOe.
The dashed line is the guide to the eye of the high-field extrapolated linear
behavior. The $M_{ab}(H)$ data exhibit positive curvature for $H<H_{\rm c1}$
as predicted by our theory, but the origin of the quantitative difference
between experiment and theory is not clear at present.

####  III.3.1 Overview

The evolution of the ground-state spin configuration in ${\rm EuAl_{2}Ge_{2}}$
is further probed by isothermal magnetization versus applied magnetic field
$M(H)$ measurements. The $M(H)$ behavior in the hysteresis mode for
$-5.5~{}{\rm T}\leq H\leq 5.5$ T measured at $T=2$ K is shown in Fig. 5(a). No
magnetic hysteresis is observed for fields applied either in the $ab$ plane or
along the $c$ axis. Figures 5(b) and 5(c) show the $M(H)$ behavior measured at
different temperatures for $H\parallel ab$ ($M_{ab}$) and $H\parallel c$
($M_{c}$), respectively, for our full field range 0–55 kOe. Here both $M_{ab}$
and $M_{c}$ appear to increase linearly with $H$ and saturate above the
respective critical field $H^{\rm c}_{ab}=37.5(5)$ kOe and $H^{\rm
c}_{c}=52.5(5)$ kOe with a saturation moment $\mu_{\rm sat}=7.0(5)\,\mu_{\rm
B}$/Eu at $T=2$ K. The measured $\mu_{\rm sat}$ value agrees with $\mu_{\rm
sat}=gS\mu_{\rm B}=7~{}\mu_{\rm B}$/Eu expected for Eu+2 ions with
spectroscopic-splitting factor $g=2$ and spin $S=7/2$.

The significant difference between the critical-field values for the two field
directions indicates the presence of a considerable magnetic anisotropy in the
system with $ab$-plane ordering preferred over $c$-axis ordering in the A-type
AFM structure, as also observed in the magnetic susceptibility behavior in
Fig. 4. Figures 5(b) and 5(c) show that the $H^{\rm c}$ values decrease with
increase in the temperature for $T<T_{\rm N}$ as expected. The $M(H)$ data
measured at $T=50$ K, greater than $T_{\rm N}=27.5$ K, are also nonlinear for
both the field directions, suggesting the presence of short-range dynamic
magnetic correlations in ${\rm EuAl_{2}Ge_{2}}$ above $T_{\rm N}$.

####  III.3.2 Low-field $M_{ab}(H)$ data

The $M_{ab}(H)$ data at $T=2~{}{\rm K}\ll T_{\rm N}=27.5$ K in Fig. 5(a)
measured over our maximum field range below $T_{\rm N}$ appear to increase
linearly up to $H^{\rm c}_{ab}=37.5(5)$ kOe above which they saturate.
However, a careful study at low fields revealed that $M_{ab}(H)$ at $T=2$ K
exhibits positive curvature below $H\lesssim 2.5$ kOe as shown in Fig. 5(d).
The positive curvature is more clearly reflected in the $dM_{ab}/dH$ versus
$H$ at $T=2$ K plotted in Fig. 5(e) that exhibits a broad peak at $H_{\rm
c1}=2.5(1)$ kOe. On the other hand, no nonlinearity is observed in the
$M_{ab}(H)$ data at $T>T_{\rm N}$ or in the $M_{c}(H)$ data at any
temperature. A similar behavior of $M_{ab}(H)$ was observed by us at $T\approx
2$ K, far below the respective $T_{\rm N}$ for other Eu-based trigonal
compounds ${\rm EuMg_{2}Bi_{2}}$ and ${\rm EuMg_{2}Sb_{2}}$ containing
triangular Eu layers, as well as for the tetragonal compound ${\rm EuGa_{4}}$
containing square-lattice Eu layers [10, 21, 23, 28, 43], where each compound
exhibits A-type AFM order with the moments aligned in the $ab$ plane as in
${\rm EuAl_{2}Ge_{2}}$.

####  III.3.3 Theoretical modeling of the low-field $M_{ab}(H)$ data

#### a. Overview

Figure 6: (a) Reorientation of the Eu magnetic moments in the three trigonal
$ab$-plane antiferromagnetic domains in a small $ab$-plane magnetic field
$H_{x}<H_{\rm c1}$. Here, the two oppositely-directed arrows in each domain
represent the moment orientations in adjacent layers of the A-type AFM
structure in small fields. The arrows indicate the direction and increment
$\Delta\phi$ of rotation of the moments in domains B and C towards the
vertical orientation, perpendicular to the applied field ${\bf H}_{x}$. The
moments in each domain remain antiparallel to each other for $H_{x}<H_{\rm
c1}$ apart from a small canting ($\lesssim 1^{\circ}$) towards the magnetic
field direction that gives rise to the measured magnetization in this field
range. (b) Orientation of the moments at the critical field $H_{x}=H_{\rm c1}$
where all moments are perpendicular to ${\bf H}_{x}$ except for the small
canting towards ${\bf H}_{x}$. At higher fields, all moments cant toward ${\bf
H}_{x}$ for $H_{\rm c1}<H_{x}<H^{\rm c}_{ab}$ until at the critical field
$H^{\rm c}_{ab}$ all moments are aligned ferromagnetically in the direction of
${\bf H}_{x}$.

In order to model the nonlinear low-field $M_{ab}(H)$ data at $T\ll T_{\rm N}$
for ${\rm EuMg_{2}Bi_{2}}$, ${\rm EuMg_{2}Sb_{2}}$, and ${\rm EuGa_{4}}$, we
previously proposed that the A-type AFM ground state of each contains
threefold or fourfold A-type AFM domains of moments for the trigonal and
tetragonal spin systems, respectively. In the trigonal case, the three domains
are associated with a weak $ab$-plane magnetic anisotropy energy

$\displaystyle E_{\rm anis}=K_{3}\sin(3\phi)$ (7)

with minima in the $ab$-plane azimuthal angle $\phi$ at $\pi/2$, $5\pi/6$ and
$-5\pi/6$ rad, where $K_{3}$ is the anisotropy constant. Thus in $H=0$, the
collinear moments in adjacent layers in ${\rm EuAl_{2}Ge_{2}}$ form three
domains with the collinear moments oriented along these three minima as shown
in Fig. 6(a).

Upon application of $ab$-plane magnetic field ${\bf H}_{x}$, the antiparallel
spins in domains B and C initially rotate in a direction to become
perpendicular to H at $H_{\rm c1}$ as shown by the arrows in Fig. 6(a)
attached to an angular deviation $\Delta\phi$ for a particular value of the
field $H_{x}$. This happens because for a collinear antiferromagnet at $T=0$
K, the magnetic susceptibility parallel to the moments is zero, whereas the
susceptibility if the moments are perpendicular to the field the magnetic
susceptibility $\chi_{\perp}=\chi(T){\rm N}$ is nonzero according to
molecular-field theory (MFT) [42] and hence the lowest energy occurs if the
moments are perpendicular to ${\bf H}_{x}$, as discussed further below. With a
sufficiently large $H_{x}\equiv H_{\rm c1}$, all moments are oriented
perpendicular to ${\bf H}_{x}$ apart from a small canting $\lesssim 1^{\circ}$
towards ${\bf H}_{x}$ that is responsible for the measured magnetization at
this field. As discussed quantitatively below, the positive curvature in
$M_{ab}(H)$ for $H_{x}<H_{\rm c1}$ as seen in Fig. 5(f) arises from this
magnetic-field-induced reorientation of the moments in Domains B and C. At
fields larger than $H_{\rm c1}$, according to MFT [42] $M_{ab}(H)$ increases
linearly up to the critical field $H^{\rm c}_{ab}$ at which all moments are
aligned parallel to ${\bf H}_{x}$ and hence the magnetization saturates to the
value $7\mu_{\rm B}$/Eu, in agreement with the experimental data in Fig. 5(a).

#### b. Calculations

Here we summarize the development of the model in Ref. [10] for ${\rm
EuMg_{2}Bi_{2}}$ and ${\rm EuMg_{2}Sb_{2}}$ as applied to ${\rm
EuAl_{2}Ge_{2}}$. In the small fields $0\leq H_{x}\leq H_{\rm c1}$, the angles
of the moments in domains A, B, and C in Fig. 6(a) with respect to the
positive $x$ axis are respectively given by

$\displaystyle\phi_{\rm A}$ $\displaystyle=$ $\displaystyle\frac{\pi}{2},$
$\displaystyle\phi_{\rm B}$ $\displaystyle=$
$\displaystyle-\frac{5\pi}{6}+\Delta\phi\quad(0\leq\Delta\phi\leq\pi/3),$ (8)
$\displaystyle\phi_{\rm C}$ $\displaystyle=$
$\displaystyle-\frac{\pi}{6}-\Delta\phi.\quad(0\leq\Delta\phi\leq\pi/3).$

The anisotropy energy averaged over the moments in the three domains in the
field range $0\leq H_{x}\leq H_{\rm c1}$ using Eqs. (7) and (8) is

$\displaystyle E_{\rm anis\,ave}$ $\displaystyle=$
$\displaystyle-\frac{K_{3}}{3}[1+2\cos(3\Delta\phi)].$ (9)

The magnetic energy in the regime $0\leq H_{x}\leq H_{\rm c1}$ is given by

$\displaystyle E_{\rm mag}$ $\displaystyle=$
$\displaystyle-\chi_{\perp}H_{x}^{2}\sin(\phi),$ (10a) where $\chi_{\perp}$ is
the $ab$-plane magnetic susceptibility at $T=0$ K when all moments are
perpendicular to ${\bf H}_{x}$, i.e., when $\phi=\pi/2$. Summing over the
angles of the moments in the three domains in Eq. (8) and dividing by $3$
gives the average magnetic energy as $\displaystyle E_{\rm mag\
ave}=-\frac{\chi_{\perp}H_{x}^{2}}{3}\left[1+2\sin^{2}\left(\frac{\pi}{6}+\Delta\phi\right)\right].$
(10b)

The total average energy $E_{\rm ave}=E_{\rm anis\,ave}+E_{\rm mag\,ave}$ is
given by the sum of Eqs. (9) and (10b). Then normalizing $E_{\rm mag\ ave}$ by
$K_{3}$ gives

$\displaystyle\frac{E_{\rm ave}}{K_{3}}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\bigg{\\{}1+2\cos(3\Delta\phi)]$
$\displaystyle+\frac{\chi_{\perp}}{K_{3}}H_{x}^{2}\left[1+2\sin^{2}\left(\frac{\pi}{6}+\Delta\phi\right)\right]\bigg{\\}}.$

Minimizing $E_{\rm ave}/K_{3}$ with respect to the quantity
$\chi_{\perp}H_{x}^{2}/K_{3}$ yields the relationship between $\Delta\phi$ and
$H_{x}$ given by

$\displaystyle
3\csc\left(\frac{\pi+6\Delta\phi}{3}\right)\sin(3\Delta\phi)=\frac{\chi_{\perp}H_{x}^{2}}{K_{3}},$
(12a) which yields
$\displaystyle\frac{\chi_{\perp}H_{x}^{2}}{K_{3}}(\Delta\phi=0)$
$\displaystyle=$ $\displaystyle 0,$ (12b)
$\displaystyle\frac{\chi_{\perp}H_{{\rm c1}}^{2}}{K_{3}}(\Delta\phi=\pi/3)$
$\displaystyle=$ $\displaystyle 9/2.$ (12c)

Equation (12c) allows the anisotropy constant $K_{3}$ in ${\rm
EuAl_{2}Ge_{2}}$ to be calculated from the known values of the molar
$\chi_{\perp}=\chi_{J}(T_{\rm N})=0.96$ cm3/mol from Fig. 3(b) and $H_{\rm
c1}=2.5$ kOe according to

$\displaystyle K_{3}=\frac{\chi_{\perp}H_{\rm c1}^{2}}{(9/2)N_{\rm
A}}=1.4\times 10^{-3}~{}{\rm meV/Eu},$ (13)

where $N_{\rm A}$ is Avogadro’s number. For comparison, $K_{3}=6.5\times
10^{-5}$ meV/Eu in trigonal ${\rm EuMg_{2}Bi_{2}}$ [10], $K_{3}=1.8\times
10^{-5}$ meV/Eu in trigonal ${\rm EuMg_{2}Sb_{2}}$ [10], and $K_{4}=1.4\times
10^{-3}$ meV/Eu in tetragonal ${\rm EuGa_{4}}$ [43].

For $0\leq H_{x}\leq H_{\rm c1}$, the magnetization $M_{x}$ of the collinear
moments in a domain at $T=0$ versus $H_{x}$ only arises from the component of
M perpendicular to the ferromagnetically-aligned layers in the A-type AFM
structure, because the parallel component gives no contribution at $T=0$ K.
The normalized magnetization averaged over the three domains using Eqs. (8) is

$\displaystyle\frac{M_{x\,\rm ave}(\Delta\phi)}{M_{x}(H_{\rm c1})}$
$\displaystyle=$
$\displaystyle\frac{1}{3}\left[1+2\sin^{2}\left(\frac{\pi}{6}+\Delta\phi\right)\right].$
(14)

Solving for $\Delta\phi(H_{x})$ using Eq. (12a) and the known values of
$K_{3}$ and $M_{x}(H_{\rm c1})$, a plot of $M_{x\,\rm ave}$ versus $H_{x}$
over the range $0\leq H_{x}\leq H_{\rm c1}$ is shown in Fig. 5(f). At higher
fields $H_{\rm c1}\leq H\leq H^{\rm c}_{ab}$, one has
$M(H_{x})=\chi_{\perp}H_{x}$, above which the magnetization saturates.

###  III.4  Heat capacity

Figure 7: (a) Temperature dependence of the zero-field $C_{\rm p}(T)$ for
${\rm EuAl_{2}Ge_{2}}$ (filled red circles) along with a fit by Eq. (15)
(solid black curve). Inset: $C_{\rm p}$ vs $T$ in magnetic fields $H$ from 0
to 7 T. (b) Plot of $C_{\rm mag}/T$ vs $T$ in $H=0$ below 100 K (filled red
circles, left ordinate) and the corresponding magnetic entropy $S_{\rm mag}$
vs $T$ (right ordinate) calculated from the $C_{\rm mag}(T)/T$ data using Eq.
(17). Also shown as a blue curve is $C_{\rm mag}(T)/T$ calculated for $S=7/2$
and $T_{\rm N}=27.4$ K using the molecular-field theory prediction in Eq.
(16). The magnetic entropy $S_{\rm mag}(T)$ calculated using Eq. (17) is
plotted as the green triangles with the scale on the right ordinate.

The temperature dependence of the zero-field heat capacity $C_{\rm p}(T)$ of
${\rm EuAl_{2}Ge_{2}}$ is shown in Fig. 7(a). A clear $\lambda$-type peak is
observed in the $C_{\rm p}(T)$ data at $T_{\rm N}=27.5$ K, indicating the
second-order nature of the AFM transition. The peak position shifts to lower
temperature with increasing applied field, as shown in the inset of Fig. 7(a).
The $C_{\rm p}(T)$ tends to saturate at a value of $\approx$ 124 J/mol K, at
$T=300$ K, close to the classical Dulong-Petit high-$T$ limit $3nR=124.71$
J/mol K, where $n=5$ is the number of atoms per formula unit and $R$ is the
molar gas constant.

The molar $C_{\rm p}(T)$ data were fitted by an electronic contribution
$\gamma T$ plus the Debye lattice heat-capacity model according to

$\displaystyle C_{\rm p}(T)$ $\displaystyle=$ $\displaystyle\gamma T+nC_{\rm
V\,Debye}(T),$ (15) $\displaystyle C_{\rm V}(T)$ $\displaystyle=$
$\displaystyle 9R\left(\frac{T}{\Theta_{\rm
D}}\right)^{3}\int_{0}^{\Theta_{\rm D}/T}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx,$

where $\gamma$ is the Sommerfeld electronic specific-heat coefficient and
$\Theta_{\rm D}$ is the Debye temperature. An accurate Padé approximant
expression for $C_{\rm V}(T)$ [44] was used for the fit. The fit is shown by
the black curve in Fig. 7(a), where $\gamma=21(2)$ mJ/mol K2 and $\Theta_{\rm
D}=332(2)$ K. The $\gamma$ value is significantly larger than the value of
$6(1)$ mJ/mol K2 estimated from the theoretical density of states at the Fermi
energy $D(E_{\rm F})$ value below. The enhancement may be due to electron-
electron and/or electron-phonon interactions.

Although the AFM ordering temperature of ${\rm EuAl_{2}Ge_{2}}$ is $T_{\rm
N}=27.5$ K, the $C_{\rm p}(T)$ data exhibit a positive deviation from the fit
in Fig. 7(a) for the electronic and lattice contributions up to $~{}80$ K,
indicating the presence of short-range magnetic correlations up to $\sim 80$
K. The magnetic contribution $C_{\rm mag}(T)$ to the heat capacity is obtained
by subtracting the electronic and lattice contributions from the measured
$C_{\rm p}(T)$ data using the above fit and is shown as the red symbols in
Fig. 7(b). The $C_{\rm mag}(T)$ remains finite for $T_{\rm N}<T\lesssim 80$ K
due to the presence of short-range dynamic magnetic correlations, in
accordance with the observed nonlinear $M(H)$ behavior in Fig. 3(a) for
$T>T_{\rm N}$ discussed earlier.

In Fig. 7(b), we have also shown the theoretical $C_{\rm mag}(T)/T$ behavior
based on the MFT [42] for this system with $S=7/2$ as the blue line. According
to MFT, the molar $C_{\rm{mag}}(t)$ is given by

$\displaystyle
C_{\rm{mag}}(t)=R\frac{3S\bar{\mu}_{0}^{2}(t)}{(S+1)t[\frac{(S+1)t}{3B^{\prime}_{S}(t)}-1]},$
(16)

where the symbols are defined in Eqs. (5). The MFT prediction below $T_{\rm
N}$ in Fig. 7(b) does not agree well with the data, although the overall
shapes below $T_{\rm N}$ are similar. In this regard we must keep in mind the
presence of the additional transition below $\sim 5$ K noted above and also
the presence of substantial short-range magnetic correlations above $T_{\rm
N}$.

The temperature dependence of the magnetic entropy $S_{\rm{mag}}(T)$ is
calculated using the experimental data (red symbols) in Fig. 7(b) and the
relation

$\displaystyle S_{\rm mag}(T)=\int_{0}^{T}\frac{C_{\rm{mag}}(T)}{T}dT,$ (17)

as shown by the green symbols with the scale on the right ordinate of Fig.
7(b). The $S_{\rm{mag}}(T)$ saturates at $T>80$ K to a value of $\approx 18$
J/mol K, which is comparable with the theoretical saturation entropy $S_{\rm
mag}=R{\rm ln}(2S+1)=17.29$ J/mol K for Eu2+ ions with $S=7/2$. The release of
the entropy at temperatures higher than $T_{\rm N}$ is due to short-range
magnetic correlations above $T_{\rm N}$ as indicated from the $C_{\rm
mag}(T)/T$ vs $T$ data in Fig. 7(b), as also previously found in other Eu- and
Gd-based $S=7/2$ compounds [20, 23, 28, 45, 46].

###  III.5  Electrical resistivity

Figure 8: Temperature $T$-dependent in-plane electrical resistivity $\rho$ of
four EuAl2Ge2 crystals in $H=0$ T below 100 K. The inset shows the full
temperature dependence of crystal #A up to room temperature.

While the general trend of the electrical resistivity $\rho$ in the
paramagnetic state of EuAl2Ge2 is a metallic decrease on cooling below room
temperature as illustrated in the inset of Fig. 8, anomalous behavior is
observed on approaching $T_{\rm N}$ from above as shown in the main panel. In
particular, the resistivity develops significant positive curvature from $\sim
80$ K down to $T_{\rm N}=27$ K, corresponding to the development of dynamic
short-range magnetic correlations observed in the heat capacity data in Fig.
7. Loss of spin-disorder scattering due to long-range AFM ordering leads to
the rapid decrease in the resistivity on cooling below $T_{\rm N}$.

####  III.5.1 Electrical resistivity in magnetic fields $H\parallel c$ axis

Figure 9: In-plane resistivity $\rho$ of EuAl2Ge2 crystal #A in magnetic
fields in the $H\parallel c$ configuration. Measurements were taken at
temperatures of 60 K in the paramagnetic state during the initial development
of magnetic correlations (cyan line), at 33 K in the more-correlated
paramagnetic state (purple line), and at 20 K (green line) and 5 K (black
line) in the A-type AFM state. For reference we show data taken in the
$H\parallel ab$ configuration at 5 K (red line) for which the critical field
is about 3.5 T from Fig. 5(c).

In Fig. 9 we show the field-dependent resistivity, measured in magnetic fields
parallel to the crystal $c$ axis. Measurements were taken at characteristic
temperatures of 60 K (in the paramagnetic state with weak magnetic
correlations, cyan line), at 33 K in the correlated paramagnet state (purple
line), and at 20 K (green line) and 5 K (black line) in the type-A AFM state.
Magnetization versus field measurements at 5 K and 20 K in this configuration,
Fig. 5(c) above, show a linear increase at low fields and saturation at fields
at about 5 T and 3 T, respectively, in very good agreement with the features
seen in the $\rho_{a}(H_{c})$ curves. At 20 K the resistivity decreases above
3 T, evidencing the suppression of spin-disorder scattering. At $T=5$ K, the
$\rho(H)$ curve shows a slope change at $\sim 5$ T. For comparison we show
resistivity data measured at 5 K in the $H\parallel ab$ configuration,
revealing a much clearer feature at the saturation field of $\approx 3.5$ T
(red curve in Fig. 9).

Note that in the paramagnetic state at 60 K, the resistivity in Fig. 9
increases monotonically with magnetic field, close to the $\rho\sim H^{2}$
dependence expected for weak-field orbital magnetoresistance [47]. The
symmetry of the curve with respect to the sign of the magnetic field suggests
minimal contribution of a spurious Hall effect signal in the resistivity
measurements. In the correlated paramagnet state at 33 K the resistivity
decreases with field up to a field of $\sim$ 6 T, due to field-induced
suppression of spin-disorder scattering. Positive magnetoresistance is
restored in the spin-polarized state above 6 T.

####  III.5.2 Electrical resistivity in magnetic fields $H\parallel ab$ plane

Figure 10: Temperature-dependent resistivity of EuAl2Ge2 in magnetic fields
$H\parallel ab$. The sharp feature accompanying long-range AFM ordering at
$T_{\rm N}=27$ K in zero field moves to somewhat lower temperature in a field
of 1 T (red) and smears and moves to higher temperatures in fields of 2 T
(green) and 3 T (blue). Measurements in positive and negative fields of 9 T
reveal some contamination of the resistivity signal with the Hall voltage,
suggesting a sign change of the Hall effect at around 30 K in the 9 T field.

In Fig. 10 we show the evolution of the temperature-dependent resistivity of
EuAl2Ge2 with magnetic field applied parallel to the conducting $ab$ plane.
This field effectively alters the interplane alignment of the ferromagnetic
planes in the type A antiferromagnet with respect to the field, as discussed
in Sec. III.3. A strong enough magnetic field of 1 T (red curve) suppresses
the pre-transition resistivity increase and brings the sharp feature observed
in zero field at $T_{\rm N}=27$ K to somewhat lower temperatures. With a
further field increase to 2 T (green curve), the sharp feature at $T_{\rm N}$
is smeared and transforms into a broad crossover. It shifts to higher
temperatures at 3 T (blue curve) and becomes hard to distinguish at higher
fields, clearly showing the importance of the spin-polarized state for its
observation.

Figure 11: In-plane resistivity $\rho$ of EuAl2Ge2 in magnetic fields with the
$H\parallel ab$ configuration. Measurements were taken at characteristic
temperatures of 60 K (in the paramagnetic state with weak magnetic
correlations, blue line), at 33 K in the correlated paramagnet state (green
line), and at 20 K (red line) and 5 K (black line) in the A-type AFM state.

Figure 11 shows the field-dependent resistivity measured in magnetic fields
parallel to the sample $ab$ plane. Measurements were taken at characteristic
temperatures of 60 K in the paramagnetic state above magnetic correlations
development (blue line), at 33 K in the correlated paramagnetic state (green
line), and at 20 K (red line) and 5 K (black line) in the A-type AFM state.
Magnetization measurements at 5 K and 20 K in this configuration, Fig. 5(b),
show positive curvature at the lowest fields, zoomed in Fig. 5(d), followed by
a linear increase and saturation at fields at about 3.5 T and 2.5 T,
respectively. This is in very good agreement with the features seen in
$\rho(H)$ curves. At 20 K the resistivity decreases above 2 T, reaches a
minimum at 3 T and increases on further field increase. Note a tiny
resistivity increase for the 5 K and 20 K curves, presumably related to
magnetic-moment rotations as discussed above in Sec. III.3.

###  III.6  Electronic structure from ARPES measurements and DFT calculations

Figure 12: Electronic structure of EuAl2Ge2. (a) ARPES spectrum of EuAl2Ge2
along the ${\rm{\Gamma}-{K}-{M}-{\Gamma}}$ path measured in the AFM phase (9
K) using $h\nu=$ 91 eV (kz $\sim$ 0). The inset shows the zoomed-in spectra of
the electron pocket at the M point. The arrow indicates the crossing point of
two bands. (b) Theoretical band dispersions including spin-orbit coupling
(SOC), Hubbard $U=$ 5 eV, and A-type AFM spin-configuration using DFT. The
arrow indicates the crossing of bands. (c) Theoretical band dispersions in the
AFM and PM phases are plotted together. The inset shows zoomed-in spectra
around ${\rm\Gamma}$. Band inversion/avoided-crossing features are indicated
by the two blue arrows in the inset. Compared to the PM phase, a few extra
bands appear in the AFM phase and some of them are indicated by an arrow,
star, and triangle symbols. (d) Two-dimensional second-derivative of the ARPES
spectra along ${\rm{\Gamma}-M}$ for AFM and PM phases. Bands within the dashed
box are captured by theoretical calculations in (c). Fermi surface and
constant-energy contours for the AFM phase in the experiment [(e)-(f)] and
theory [(g)-(h)] and similarly, for the PM phase (40 K) in the experiment
[(i)-(j)] and theory [(k)-(l)]. Different energy values are used between the
experiment and theory as the position of the Fermi level is slightly different
between them. The ARPES spectra in Fig. 13 were taken along the cut shown by
the white dashed line in (e). Figure 13: Electronic structure of EuAl2Ge2
across the magnetic transition. (a)–(c) ARPES spectrum around ${\rm\Gamma}$
close to $E_{\rm F}$ along the cut shown by the dashed line in Fig. 12(e) for
various temperatures 9 K, 27 K, and 32 K, respectively. (d) Temperature
dependence of the energy distribution curves (EDCs) at the momentum indicated
by a vertical line in (a). (e)–(g) Zoomed view of the ARPES spectra within the
region as indicated by a dashed rectangle in (c) for 9 K, 32 K, and 40 K,
respectively. Arrows indicate the splitting of bands. (h) Momentum
distribution curves along the dashed line in (e).

In order to understand the interplay of magnetism and electronic structure in
${\rm EuAl_{2}Ge_{2}}$, ARPES measurements have been performed at different
temperatures, with a particular emphasis on the temperature range bridging
$T_{\rm N}$. The experimentally-observed electronic structure was also
compared with the theoretical electronic structure by density-functional-
theory (DFT)-based calculations.

Figure 12(a) shows the ARPES spectrum of ${\rm EuAl_{2}Ge_{2}}$ along the
${\rm{\Gamma}-{K}-{M}-{\Gamma}}$ path, measured in the AFM phase at $T=9$ K.
The spectrum shows two hole-like and one electron-like bands crossing the
Fermi level at the ${\rm\Gamma}$ and M points of the Brillouin zone (BZ),
respectively. These hole-like bands appear to cross at $-0.5$ eV along
${\rm{\Gamma}-{M}}$ [indicated by green arrow in (a)], but they are well
separated along ${\rm{\Gamma}-{K}}$. For better visualization of the electron
pocket, a closer view is shown in the inset of Fig. 12(a). Extremely less-
dispersive bands with high intensity are observed around $-1.5$ eV due to the
localized Eu-$4f$ levels. Most of the experimental features are reasonably
well-reproduced by DFT calculations, which considers the effect of spin-orbit-
coupling (SOC) and a Hubbard $U=5$ eV to account for the effect of strong
localization of the half-filled Eu-$4f$ orbitals of ${\rm EuAl_{2}Ge_{2}}$ in
its A-type AFM spin configuration as obtained from our neutron diffraction
measurements [Fig. 12(b)].

In order to identify potential changes in the electronic structure associated
with the magnetic transition, the AFM and paramagnetic (PM) band structures
are plotted together in Fig. 12(c). In the AFM phase, several new bands appear
compared to the PM phase, due to the folding of electronic states originating
from the doubling of the magnetic unit cell. For example, an electron-like
band is observed in the AFM phase, just above $E_{\rm F}$ at the ${\rm\Gamma}$
point, whereas it is absent in the PM phase. This electron-like band crosses
two hole-like bands and causes various band anticrossings, as indicated by the
arrows in the inset of Fig. 12(c). Unfortunately, these states are
inaccessible by photoemission spectroscopy as they appear above $E_{\rm F}$.
However, potential changes in the electronic states between the PM and AFM
phases are also expected below $E_{\rm F}$ as indicated by asterisk and
triangle symbols with the dashed box, and arrow and that should be directly
accessible by ARPES. Indeed we resolve those folded shallow bands in the AFM
phase of ${\rm EuAl_{2}Ge_{2}}$ as indicated in Fig. 12(d), whereas no such
states are observed in the PM phase. Generally, folded electronic states
appear weaker in photoemission, regardless of whether they are due to
magnetism or charge density waves [48, 49, 50, 51].

Further, to map the dispersion of the electronic states in the $k_{x}$-$k_{y}$
plane, Fermi surface (FS) mapping was performed. Figures 12(e) and 12(i) show
the FS of ${\rm EuAl_{2}Ge_{2}}$ for the AFM and PM phases, respectively. In
both cases, three Fermi pockets are observed, two at the center of the BZ, and
one at the M point. The circular and hexagonal Fermi pockets at the center of
the BZ are formed by the inner and outer hole-like bands, respectively [Fig.
12(a)], and the elongated oval-shaped Fermi pocket at the M point is the
electron pocket. This electron pocket is formed by the bottom of the
conduction band that enters inside the Fermi level. The inner Fermi pocket is
isotropic whereas the other two are very anisotropic that could produce the
anisotropic magnetic properties as observed in our experiments. All these FSs
are well reproduced by theoretically-computed contours at $E_{\rm F}+40$ meV
[Figs. 12(g) and 12(k)]. This energy shift was used to better match the shape
and sizes of the experimental FS features, suggesting that the sample is
slightly electron-doped.

The FS features and dispersion of electronic states suggest that ${\rm
EuAl_{2}Ge_{2}}$ is metallic, both in the AFM and PM phases. Further,
according to the band structure, folded bands between two consecutive BZs
should connect the M point at a deeper energy that cuts the folded bands at
the M point. Indeed, we observe this signature both in our ARPES and
theoretical simulated constant-energy contours, as shown in Figs. 12(f) and
12(h), respectively. In the PM case, no such intensity is observed at the M
point due to the absence of band folding [Figs. 12(j) and 12(i)]. Recently,
magnetism-induced band folding and nontrivial band topology were reported in
the Eu-based AFM system EuCd2As2 [48, 49]. As discussed above, our DFT
calculations also predicted inverted band features in the AFM state of ${\rm
EuAl_{2}Ge_{2}}$ near $E_{\rm F}$, which is typically observed in materials
hosting nontrivial band topology. To correctly verify its nontrivial
topological origin, we have calculated the $Z_{2}$ topological numbers using
the Wilson loop (Wannier charge center) method [37] for the six time-reversal-
invariant momentum planes. The obtained $Z_{2}$ topological numbers
$v_{0}$;$(v_{1}v_{2}v_{3})$ $=$ 1;(000) indicate the presence of nontrivial
electronic states in this system. Further theoretical studies are needed to
determine the exact nature of the ‘topology’ of the system.

To obtain more insight into the electronic structure change across the
magnetic transition, we have performed high-resolution ARPES measurements
close to $E_{\rm F}$ at various temperatures [Figs. 13(a)–13(c)]. While they
exhibit very similar spectral features across the transition, the
quasiparticle weight decreases significantly. This can be better visualized in
their energy-distribution curves (EDCs) in Fig. 13(d). The temperatures at
which the quasiparticle weight drops correlate well with magnetic transition
temperatures. This drop in quasiparticle weight in the PM phase is most
possibly related to the complex interplay between the orbital and spin degrees
of freedom, caused by the change of coupling between magnetic moments and
itinerant electrons across magnetic transitions.

Quasiparticle enhancement in magnetically-ordered states has been reported in
other magnetic materials due to the decrease of spin fluctuations and changes
in the scattering mechanism [52, 53]. Further zooming the ARPES spectra in
momentum reveals that the individual hole-like bands actually split in two.
The splitting is better resolved for the outer bands as indicated by vertical
lines in Figs. 13(e) and 13(f). The momentum distribution curves (MDCs) also
show clear two-peak structures of the outer band. It is interesting to note
that the band splitting survives above $T_{\rm N}$.

However, according to the theoretical calculations, all the bands in the AFM
and PM phases are twofold degenerate, so no such band splitting is expected.
Thus only two hole-like bands are expected to cross the $E_{\rm F}$ [Fig.
12(c)]. Generally, band splitting occurs when either time-reversal symmetry
$T$ or parity $P$ symmetry is broken. Even though $T$ is broken in the AFM
phase, the double degeneracy of the bands is protected by the combination of
$P$, $T$, and translation ($L$) symmetries by one unit along the $c$axis [49].
The observation of band splitting in the PM phase is quite surprising as both
the $T$ and $P$ symmetries should be preserved.

On the other hand, based on our magnetic measurements, the persistence of
short-range FM correlations above $T_{\rm N}$ may cause the $T$ symmetry to
break in the PM phase, leading to band splitting. In EuCd2As2, an analogous
band splitting was reported [48]. The band splitting was explained as
resulting from quasi-static and quasi-long-range FM fluctuations experienced
by the itinerant electrons. In the AFM phase of ${\rm EuAl_{2}Ge_{2}}$, the
magnetic moments align ferromagnetically within a basal plane, which results
from dominant in-plane FM exchange interactions. Since ARPES is a very
surface-sensitive technique, these FM interactions may result in the band
splitting in the magnetically-ordered state, as observed in Fig. 13(e).

##  IV  Concluding Remarks

We find that ${\rm EuAl_{2}Ge_{2}}$ is a metallic antiferromagnet with
nontrivial electronic states in the AFM phase near $E_{\rm F}$. The compound
exhibits A-type AFM order below $T_{\rm N}=27.5(5)$ K with the Eu moments
aligned in the $ab$ plane. The anisotropic magnetic properties exhibited by
the system, associated with the Eu2+ spins, indicate the presence of
substantial magnetic dipole and magnetocrystalline anisotropy. The presence of
in-plane magnetic anisotropy results in trigonal threefold AFM domain
formation in $H=0$. The moments in the domains exhibit a field-induced
reorientation at $H_{c1}\sim 2.5(1)$ kOe to become perpendicular to the field
direction for $T<T_{\rm N}$. The $ab$-plane and $c$-axis critical fields at
$T=2$ K are $H^{\rm c}_{ab}=37.5(5)$ kOe and $H^{\rm c}_{c}=52.5(5)$ kOe at
which all moments polarized along the respective applied-field directions.

The presence of dynamic short-range magnetic correlations within the $ab$
planes is evident above $T_{\rm N}$ from the zero-field heat capacity and
resistivity studies. A slight resistivity increase on cooling before loss of
spin disorder scattering below $T_{\rm N}$ suggests magnetic correlations
which are different from long-range AFM ordering. Similarly, ARPES studies
reveal band splitting even above $T_{\rm N}$, suggesting a possible breaking
of the $T$ symmetry associated with the magnetic correlations above $T_{\rm
N}$ which are therefore identified to be ferromagnetic in nature. The ARPES
results further reveal that ${\rm EuAl_{2}Ge_{2}}$ is metallic with a well-
defined Fermi surface. The Fermi surface is formed by the two pockets at the
zone center (${\rm\Gamma}$) and electron pockets at each M point. The outer
hole pocket and the electron pockets at M are very anisotropic. In addition to
the various dispersive bands, extremely less-dispersive bands are observed
around an energy $-1.5$ eV below the Fermi energy due to the localized Eu-$4f$
levels. Various folded bands are also observed in the AFM phase due to the
doubling of the unit cell. All these electronic states are modeled well by
considering spin-orbit-coupling (SOC), $U$= 5 eV and the A-type $ab$-plane AFM
configuration of the Eu magnetic moments.

###### Acknowledgements.

The research at Ames National Laboratory was supported by the U.S. Department
of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and
Engineering. Ames National Laboratory is operated for the U.S. Department of
Energy by Iowa State University under Contract No. DE-AC02-07CH11358. The
research at Brookhaven National Laboratory was supported by the U.S.
Department of Energy, Office of Basic Energy Sciences, Contract No. DE-
SC0012704. This work was also supported in part by the Center for Spintronics
Research Network, Tohoku University.

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