††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Magnetically tunable Dirac and Weyl fermions in the Zintl materials family

Anan Bari Sarkar  Department of Physics, Indian Institute of Technology,
Kanpur 208016, India  Sougata Mardanya  Department of Physics, Indian
Institute of Technology, Kanpur 208016, India  Department of Physics, National
Cheng Kung University, Tainan 701, Taiwan  Shin-Ming Huang  Department of
Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan  Barun Ghosh
Department of Physics, Northeastern University, Boston, Massachusetts 02115,
USA  Cheng-Yi Huang  Department of Physics, Northeastern University, Boston,
Massachusetts 02115, USA  Hsin Lin  Institute of Physics, Academia Sinica,
Taipei 11529, Taiwan  Arun Bansil  Department of Physics, Northeastern
University, Boston, Massachusetts 02115, USA  Tay-Rong Chang  Department of
Physics, National Cheng Kung University, Tainan 701, Taiwan  Center for
Quantum Frontiers of Research and Technology, Tainan 701, Taiwan  Physics
Division, National Center for Theoretical Sciences, National Taiwan
University, Taipei, Taiwan  Amit Agarwal<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology, Kanpur 208016, India  Bahadur Singh
<EMAIL_ADDRESS>Department of Condensed Matter Physics and Materials
Science, Tata Institute of Fundamental Research, Mumbai 400005, India

###### Abstract

Recent classification efforts encompassing crystalline symmetries have
revealed rich possibilities for solid-state systems to support a tapestry of
exotic topological states. However, finding materials that realize such states
remains a daunting challenge. Here we show how the interplay of topology,
symmetry, and magnetism combined with doping and external electric and
magnetic field controls can be used to drive the previously unreported
SrIn2As2 materials family into a variety of topological phases. Our first-
principles calculations and symmetry analysis reveal that SrIn2As2 is a dual
topological insulator with $Z_{2}=(1;000)$ and mirror Chern number $C_{M}=-1$.
Its isostructural and isovalent antiferromagnetic cousin EuIn2As2 is found to
be an axion insulator with $Z_{4}=2$. The broken time-reversal symmetry via Eu
doping in Sr1-xEuxIn2As2 results in a higher-order or topological crystalline
insulator state depending on the orientation of the magnetic easy axis. We
also find that antiferromagnetic EuIn2P2 is a trivial insulator with
$Z_{4}=0$, and that it undergoes a magnetic field-driven transition to an
ideal Weyl fermion or nodal fermion state with $Z_{4}=1$ with applied magnetic
field. Our study identifies Sr1-xEuxIn2(As, P)2 as a new tunable materials
platform for investigating the physics and applications of Weyl and nodal
fermions in the scaffolding of crystalline and axion insulator states.

Introduction– Since the discovery of $Z_{2}$ topological insulators, exploring
the role of crystalline symmetries in protecting the non-trivial topology of
Bloch Hamiltonians more generally has become a topic of intense research Hasan
and Kane (2010); Qi and Zhang (2011); Bansil  _et al._ (2016); Armitage  _et
al._ (2018a). The interplay of symmetry and topology in the crystalline solids
gives birth to many new topological insulators and metals, such as the mirror
(glide-mirror)-protected topological crystalline insulators (TCIs) Fu (2011);
Hsieh  _et al._ (2012), rotational-symmetry protected TCIs Fang and Fu (2019);
Wang  _et al._ (2019); Hsu  _et al._ (2019), higher-order topological
insulators Schindler  _et al._ (2018a, b), Dirac Liu  _et al._ (2014a, b);
Armitage  _et al._ (2018b); Singh  _et al._ (2018a), Weyl Xu  _et al._ (2015);
Lv  _et al._ (2015); Xu  _et al._ (2017), and unconventional fermion
semimetals Wang  _et al._ (2016); Singh  _et al._ (2018b); Bradlyn  _et al._
(2016), among other possibilities Song  _et al._ (2018); Bradlyn  _et al._
(2017). The TCIs are characterized by a unique set of topological invariants
which depend only on the dimension and symmetry of the crystalline group Fang
and Fu (2019); Wang  _et al._ (2019); Hsu  _et al._ (2019); Song  _et al._
(2018); Bradlyn  _et al._ (2017). The associated topological properties remain
robust under symmetry-protecting perturbations and provide materials platforms
for developing next-generation low-power-consuming high-speed electronic,
optoelectronic, and spintronic devices Jungwirth  _et al._ (2016); Moore
(2018). The need for finding new classes of tunable topological materials in
order to fill the critical gap in the available materials families is thus
clear.

Manipulation of quantum interactions and topology in the crystalline matrix
can provide a promising route for tuning topological properties of materials.
The recent discovery of topological magnets in which the nontrivial topology
is intertwined with magnetism suggests that certain topological responses
could be tuned through external electric and magnetic fields Mong  _et al._
(2010); Otrokov  _et al._ (2019); Li  _et al._ (2019); Hu  _et al._ (2020);
Belopolski  _et al._ (2019); Yin  _et al._ (2020). The topological state in
the currently available topological magnets, however, originates from the
underlying paramagnetic state and for this reason these materials allow only
limited tunability. It is highly desirable, therefore, to find new strategies
for identifying tunable topological materials.

A natural paradigm for discovering new tunable topological materials is to
exploit the strong interplay between magnetism and topology in families of
isovalent and isostructural materials. Such materials families can offer great
flexibility with respect to magnetism, topology, chemical composition, lattice
parameters, and transport properties. In particular, spatial and non-spatial
symmetries of the parent materials can be relaxed by introducing magnetism
with ionic substitutions or crystal structure engineering in a topological
non-magnet to drive various different types of topological magnetic orders.
Wang  _et al._ (2021). Using this approach as a guiding principle, we identify
here the SrIn2As2 class of Zintl materials with the general formula AM2X2 (A =
alkali, alkaline earth, or rare-earth, M = post-transition metal; and X =
Group IV or V element) in $P6_{3}/mmc$ space group as a materials family that
can support magnetic and nonmagnetic topological phases with substantial
flexibility of manipulation through external controls. The Zintl has attracted
great interest since its discovery owing to its fascinating electronic,
topological, and magnetic properties Jiang and Kauzlarich (2006); Mathieu  _et
al._ (2008); Rauscher  _et al._ (2009); Goforth  _et al._ (2008); Guechi  _et
al._ (2013); Xu  _et al._ (2019); Zhang  _et al._ (2020); Regmi  _et al._
(2020); Riberolles  _et al._ (2021); Zhang  _et al._ (2010); Shinozaki  _et
al._ (2021).

In Fig. 1(a)-(f), we demonstrate surface states evolution of three distinct
topological phases that could emerge by breaking symmetries with the magnetic
orders considering SrIn2As2 and EuIn2As2 as the representative materials.
SrIn2As2 is nonmagnetic whereas EuIn2As2 is an A-type antiferromagnetic (AFM)
system with Neel temperature of $16K$ Goforth  _et al._ (2008). SrIn2As2
preserves the time-reversal and spatial symmetries of space group
$P6_{3}/mmc$. It is found to be a dual topological insulator with $Z_{2}=1$
and mirror Chern number $C_{M}=-1$ similar to Bi2Te3 class of topological
insulators Rauch  _et al._ (2014). The associated Dirac cone states are thus
pinned to the $\Gamma$ point (Figs. 1(a) and (d)). Introducing the AFM order
with magnetic moment $m||a$ (aAFM) breaks the time-reversal symmetry but
preserves the spatial mirror symmetries (see details below). Thus, aAFM state
realizes a topological crystalline insulator in which the Dirac cone states
are unpinned from the $\Gamma$ to lie on the mirror invariant line (Figs. 1(b)
and (e)). Considering the out-of-the-plane AFM with magnetic moment $m||c$
(cAFM), the Dirac cone states are fully gapped, realizing an axion or high-
order topological insulator state (Figs. 1(c) and (f)). In this way, a variety
of topological states can emanate from the parent materials with relaxing
symmetries owing to the magnetic order.

Methods– We performed the first-principle calculations within the framework of
the density functional theory (DFT) using the Vienna ab-initio simulation
package (VASP) Kresse and Furthmüller (1996); Kresse and Joubert (1999). The
generalized gradient approximation (GGA) Perdew  _et al._ (1996) with Perdew-
Burke-Ernzerhof (PBE) parametrization was used for considering the exchange-
correlation effects. We considered Eu $f$ electrons as valence electrons and
adopted the GGA+$U$ Anisimov  _et al._ (1991, 1997) scheme with $U_{\rm eff}$
= 5.0 eV for Eu $f$ states to include strong electron-correlation effects. A
kinetic energy cut-off of 350 eV for the plane-wave basis set and a
$\Gamma$-centered $9\times 9\times 9$ $k$-mesh Monkhorst and Pack (1976) for
bulk Brillouin zone (BZ) integration were used. We consider fully relaxed
structural parameters to compute the electronic structure (see SMs for
details) Monkhorst and Pack (1976). The bulk topological properties and
surface spectral functions were calculated using the WannierTools software
package Wu  _et al._ (2018).

Crystal structure and symmetries– The crystal lattice of AM2X2 Zintl compounds
is described by the hexagonal space group $P6_{3}/mmc$ (No. 194) with the
atomic arrangement illustrated in Fig. 1(g)-(h) for SrIn2As2 as an example
Jiang and Kauzlarich (2006); Rauscher  _et al._ (2009); Guechi  _et al._
(2013). The Sr, In, and As atoms occupy Wyckoff $2a$, 4$f$, and 4$f$
positions, respectively. This crystal contains covalently bonded two-
dimensional (2D) networks of SrIn2As2 along the hexagonal $c-$axis. Each of
SrIn2As2 2D network contains [In2As2]2- anion layers which are balanced by
Sr2+ cation layers. The number of valence electrons thus obey the octet rule
with filled valence shells ($d^{10}s^{2}p^{6}$), leading to a semiconducting
or semimetallic ground state. The Sr atoms can be replaced with other divalent
rare-earth magnetic atoms such as Eu to realize magnetic compounds as seen in
experiments Zhang  _et al._ (2010). The crystal lattice contains various
symmetries which are generated by an inversion $\mathcal{I}$, two-fold
rotation axes $\mathcal{C}_{2y}$ and
$\mathcal{G}_{2x}=\\{\mathcal{C}_{2x}|00\frac{1}{2}\\}$, two-fold screw
rotation axis $\mathcal{G}_{2z}=\\{\mathcal{C}_{2z}|00\frac{1}{2}\\}$, and a
three-fold rotation axis $\mathcal{C}_{3z}$ symmetries. These symmetry
generators construct twenty-four symmetry operators including a six-fold screw
rotation $\\{\mathcal{C}_{6z}|00\frac{1}{2}\\}$
($C_{3z}\otimes\\{\mathcal{C}_{2z}|00\frac{1}{2}\\}$), symmetry equivalent
vertical mirror $\mathcal{M}_{100}$ and glide-mirror
$\\{\mathcal{M}_{120}|00\frac{1}{2}\\}$ planes, and a horizontal mirror plane
$\\{\mathcal{M}_{001}|00\frac{1}{2}\\}$. Figure 1(i) shows the bulk BZ and
(001) and (100) surface projected BZs where various high-symmetry points
marked.

Figure 1: Methodology to design magnetically tunable topological states and
crystal structure. (a)-(f) Schematics of the evolution of the topological
surface states as we break the mirror and time-reversal symmetries. (a),(d) In
the presence of both time-reversal and mirror-symmetry in the nonmagnetic
SrIn2As2, the gapless Dirac point appears at the $\overline{\Gamma}$ point.
(b),(e) With time-reversal-symmetry broken but mirror-symmetry intact in aAFM
EuIn2As2, the gapless Dirac point is pinned to the mirror-symmetric line but
shifts away from the $\overline{\Gamma}$ point. (c),(f) On breaking the time-
reversal-symmetry and the mirror-symmetry in cAFM EuIn2As2, the surface states
become gapped at the $\overline{\Gamma}$ point. (g),(h) The crystal structure
of Zintl Eu/SrIn2As2 family showing stacking of the layers of individual atoms
along the $c$ axis. (i) Bulk and projected (001) and (100) surface Brillouin
zone along with high-symmetry points. The three symmetry equivalent mirror
planes $\mathcal{M}_{100}$, $\mathcal{M}_{010}$ and $\mathcal{M}_{110}$ are
highlighted with green shading.

Nonmagnetic topological states– Topological states can be identified by an
inverted band structure owing to the high spin-orbit coupling (SOC) or crystal
field effects. The bulk band structure and band symmetries of SrIn2As2 in the
absence of SOC is shown in Fig. 2(a). It constitutes a band inversion between
$B_{1g}$ and $B_{2u}$ states at the $\Gamma$ point (Fig. 2(b)). The orbital
analysis illustrates that the $B_{1g}$ states are derived from the In $s$
orbitals and lie below the As $p$ orbitals constituted $B_{2u}$ states. This
band order remains normal at other high-symmetry points in BZ. Away from the
$\Gamma$ point, $B_{1g}$ and $B_{2u}$ states are seen to cross at the
$\Gamma-M$ and $\Gamma-K$ lines at $k=0.0035$ $\mathrm{\AA}^{-1}$ and $0.0049$
$\mathrm{\AA}^{-1}$, respectively. A full BZ exploration shows that these band
crossings trace a topological Dirac nodal line on the $k_{z}=0$ plane as shown
in Fig. 2(c). SrIn2As2 thus realizes a band inversion topological nodal line
semimetal without SOC. The robustness of this topological state is further
verified via parallel computations using the hybrid density functional given
the possible underestimation of the bandgap in GGA functional (see
supplementary materials (SMs) for details).

Figure 2: Topological band structure of nonmagnetic SrIn2As2. (a) Calculated
bulk band structure without spin-orbit coupling (SOC). The parity eigenvalues
($\pm$) and the In $s$ (red) and As $p$ (green) orbitals are marked. (b)
Closeup of the bands along the $\Gamma-K$ and $\Gamma-M$ directions in the
$k_{z}=0$ plane. Band crossings between $B_{1g}$ and $B_{2u}$ states with
opposite $M_{z}=0$ mirror-eigenvalue are evident. (c) $E-k_{x}-k_{y}$
rendition of the valence and conduction bands on the $k_{z}=0$ plane. The
nodal line is shown with the red line. (d),(e) Same as (a),(b) but with the
inclusion of SOC. A hybridization gap between the valence and conduction bands
is seen. (f) Evolution of the Wannier charge center on the $k_{z}=0$ and
$k_{x}=0$ planes. The non-zero mirror Chern number confirms the mirror TCI
phase in SrIn2As2. (g)-(h) Band structure of (001) surface without and with
SOC. The inset shows the fermi band contours associated with the calculated
surface states. (i) Surface band structure of (100) surface with SOC.
Symmetry-enforced anisotropic Dirac cone is highlighted in the inset.

The band structure in the presence of SOC (Fig. 2(d)) shows a continuous gap
between the $\Gamma_{8}^{+}{}$ and $\Gamma_{8}^{-}{}$ states at the Fermi
level. Figure 2(e) shows the closeup of the bands along the $\Gamma-M$ and
$\Gamma-K$ directions and highlights band anti-crossings, showing that the SOC
drives the system into an inverted insulator state. Since the system preserves
inversion symmetry $\mathcal{I}$, the parity analysis at the eight time-
reversal invariant momentum (TRIM) points gives $Z_{2}=(1;000)$. On the other
hand, the $\mathcal{M}_{001}$ and $\mathcal{M}_{110}$ mirror Chern numbers on
$k_{z}=0$ and $k_{y}=0$ planes are calculated as -1 and -1, respectively, from
the Wannier charge centers evolution as shown in Fig. 2(f). SrIn2As2 is thus a
dual topological insulator.

The nontrivial topological properties of bulk manifest on the surface. From
the (001) surface energy dispersion in Fig. 2(h), a topological Dirac cone
state with the Dirac point at the $\overline{\Gamma}$ point is resolved. The
associated constant energy contours are circular with isotropic energy
dispersion near the Dirac node and develop a hexagonal shape as one moves away
from the Dirac point. The spin-texture is helical with a left-handed chirality
for the upper Dirac cone as shown in the inset of Fig. 2(h). This surface
state is protected by both the time-reversal symmetry and $\mathcal{M}_{001}$
mirror symmetry. We also get a similar protected topological state on the low-
symmetry (100) side surface. However, the Dirac cone on the (100) surface is
located well within the bulk energy gap [see Fig. 2(i)]. It has an anisotropic
energy dispersion, indicating that the surface Dirac fermions have different
Fermi velocities along different directions.

Magnetic topological states– We now consider the topological state of EuIn2As2
which is a magnetic counterpart of SrIn2As2. Notably, EuIn2(As, P)2 is
isostructural and isovalent to SrIn2As2 but realizes an AFM state below
$~{}16$K Rauscher  _et al._ (2009). EuIn2As2 shows immense potential for
realizing Axion insulator state and for this reason it has attracted attention
recently Xu  _et al._ (2019); Zhang  _et al._ (2020); Regmi  _et al._ (2020);
Riberolles  _et al._ (2021). The neutron diffraction experiments unfold a low-
symmetry helical antiferromagnetic order for EuIn2As2 which is further
amenable for realizing various topological states under external magnetic
field Riberolles  _et al._ (2021). The experimentally saturated magnetic
moment for Eu ions is $\sim 7.0~{}\mu_{\text{B}}$ that is close to the
magnetic moment of $f^{7}$ state and our calculated value of
$6.90~{}\mu_{\text{B}}$. This indicates that Eu oxidation state is +2 similar
to the oxidation state of Sr. A tunable Sr1-xEuxIn2(As, P)2 system thus can be
achievable for realizing distinct topological magnetic states Zhang  _et al._
(2010); Shinozaki  _et al._ (2021). The dependence of the bandgap on the
lattice constant (Fig. 3(a)) shows that EuIn2As2 has an intrinsically inverted
band structure whereas EuIn2P2 has a trivial band structure.

Figure 3 shows the topological states of antiferromagnetic EuIn2(As, P)2. We
consider topological states originating from the formation of commensurate AFM
order in EuIn2As2 with the intra-layer ferromagnetic coupling and inter-layer
anti-ferromagnetic coupling between the Eu spins. The calculated energy for
the aAFM is nearly the same as cAFM, making both these states energetically
favorable. This nearly degenerate energy between different magnetic orders
suggests that external magnetic tuning of the topological state in EuIn2As2
would be possible Zhang  _et al._ (2020); Riberolles  _et al._ (2021).

Figure 3: Topological states and magnetic tunability of antiferromagnetic
EuIn2(As, P)2. (a) Variation of the bandgap as a function of relative volume
at the $\Gamma$ point for AFM EuIn2(As, P)2 and non-magnetic SrIn2As2. Open
markers at $V/V_{0}=1$ show the bandgap of their pristine states. A
topological phase transition from the nontrivial to trivial phase can be
achieved by decreasing the volume. Calculated band structure of AFM EuIn2As2
(b) without and (c) with SOC. The crossings in (b) form a nodal line unhooked
from $k_{z}=0$ plane due to breaking of $\mathcal{M}_{001}$ mirror plane
symmetry. A hybridization gap opens up at the crossing points and an Axion
insulator state with $Z_{4}=2$ develops with SOC. The calculated surface band
structure of aAFM EuIn2As2 for (d) (001) and (e) (100) surfaces. The mirror
symmetry-protected Dirac cones states are seen inside the bulk gap. (f)-(g)
same as (d)-(e) but for cAFM order. The topological surface states are now
gapped with a magnetic gap at the $\overline{\Gamma}$ point. (h)-(i)
Calculated (h) bulk and (i) (001) surface band structure of aAFM EuIn2P2. A
trivial insulating state with $Z_{4}=0$ is resolved.

We emphasize that considering the AFM order reduces the nonmagnetic $D_{6h}$
point-group symmetry to either $D_{2h}$ or $D_{3d}$ depending on the magnetic
configuration of the AFM state. Figure 3(b) shows the band structure of AFM
EuIn2As2 without SOC. It retains the characteristic inverted band features
seen in SrIn2As2 along with the Dirac nodal line in the BZ. Specifically, the
states close to the Fermi level are comprised of In $s$ and As $p$ orbitals,
whereas Eu $f$ states provide the necessary exchange coupling for inducing the
magnetism in the system. Often strong electron correlations in Eu $f$ states
affect band ordering and thus the topological state of materials Bendavid and
Carter (2014); Prokofiev  _et al._ (1996); Yu  _et al._ (2020). Our analysis
by varying Hubbard $U_{\text{eff}}$ in GGA+U in parallel with hybrid
functional calculations reveals that states close to Fermi level and the
topological phase of EuIn2As2 do not depend on the choice of U parameter (see
SMs).  However, magnetism breaks the horizontal $\mathcal{M}_{001}$ mirror
plane, and thus unlike non-magnetic SrIn2As2, the nodal-line in EuIn2As2 for a
particular spin is unhooked from the $k_{z}=0$ plane and wriggles in the
momentum space around the $\Gamma$ point with small $k_{z}$ dispersion (Fig.
3(b)). Inclusion of SOC in computations develops a gap at the band crossing
points, separating valence and conductions bands in both the aAFM and cAFM
states (Fig. 3(c)). Since both the AFM states preserve inversion $\mathcal{I}$
symmetry, we access their topological character by calculating the parity-
based $Z_{4}$ topological invariant. The calculated $Z_{4}$ is 2, indicating
that EuIn2As2 realizes an Axion insulator state. Notably, despite a nonzero
$Z_{4}$ invariant, the topological state of the system can further be
classified based on the magnetic group symmetries as discussed below.

The aAFM state of EuIn2As2 preserves $\mathcal{M}_{001}$ and
$\mathcal{M}_{100}$ mirror planes. The calculated mirror Chern number on the
$k_{z}=0$ and $k_{y}=0$ planes is -1, similar to SrIn2As2. EuIn2As2 is thus a
mirror topological crystalline insulator with aAFM order. This is further
manifested in the surface spectrum on the (001) and (100) planes where Dirac
cones are seen along the mirror invariant lines. Specifically, the mirror-
protected Dirac cone is located on the $\overline{\Gamma}-\overline{M}$ line
of the (001) plane and $\overline{\Gamma}-\overline{Y}$ line of the (100)
plane as depicted in Figs. 3(d) and 3(e). On considering cAFM state, both the
$\mathcal{M}_{001}$ and $\mathcal{M}_{100}$ mirror planes are broken. There is
no symmetry-protected Dirac cone state found on either surface. Instead, a
magnetic gap develops at the surface Dirac node crossings (see Fig. 3(f) and
(g)). The system thus realizes a higher-order topological insulator with
conducting hinge states Ezawa (2018). Figures 3(h) and 3(i) show the bulk and
surface band structure of AFM EuIn2P2. In contrast to EuIn2As2, EuIn2P2
exhibits a trivial state with $Z_{4}=0$. This trivial state may arise due to
strong carrier confinement. EuIn2P2 has a relatively small unit cell volume
(267.03 Å3) than EuIn2As2 (290.77 Å3) which could induce strong carrier
confinement effect. Such effects reduce the bandwidth for both the valence and
conduction bands, thereby avoiding the band inversion in the vicinity of the
Fermi level.

To showcase the novel physics that emerges in the presence of an external
magnetic field, we now discuss the topological states in ferromagnetic (FM)
EuIn2(As, P)2. Our total energy calculations show that the FM state is $\sim$2
(3) meV/unit cell higher than the AFM state of EuIn2As2 (EuIn2P2). Notably, we
have considered the commensurate FM orders with the magnetic moment $m||a$
(aFM) and magnetic moment $m||c$ (cFM). Both the aFM and cFM possess nearly
degenerate energy similar to the AFM states. It is thus possible that FM order
in these materials becomes energetically favorable under a small external
magnetic field. For FM EuIn2As2, we find that the bulk band inversion stays
robust for both the FM orders and the system realizes an Axion insulator state
with $Z_{4}=2$ (see SMs).

Figure 4: Ferromagnetic tunability of topological states of EuIn2P2. (a)
Calculated spin-resolved bulk band structure of A-type ferromagnetic (aFM)
configuration. A single spin-band-inversion drives aFM phase into an ideal
ferromagnetic Weyl semimetal with a pair of Weyl nodes on the $k_{x}-k_{y}$
plane. (b) The surface band structure calculated along the circular path shown
in (c) on the (001) surface enclosing a Weyl point. The chiral surface states
are seen connecting the projected bulk valance and conduction bands. (c) The
Fermi-arc surface states, connecting projected Weyl nodes on the top and
bottom surfaces. (d) Spin-resolved bulk (f) (001) surface band structure of
C-type ferromagnetic (cFM) configuration. The presence of $\mathcal{M}_{001}$
mirror plane in cFM configuration protects the crossing bands to hybridize,
realizing a nodal line with drumhead surface state. (g) The associated Fermi
band contours showing the drumhead surface states. (h) Schematic
representation of the trivial insulator ($Z_{4}=0$) to a Weyl semimetal or
nodal line semimetal ($Z_{4}=1$) transition in EuIn2P2 with applied external
magnetic field.

Figure 4 illustrates the bulk and surface band structure of FM EuIn2P2. Due to
the large exchange coupling of Eu $f$ states, it shows a single spin-band
inversion in both aFM and cFM orders. Our parity-based topological indicator
is obtained to be $Z_{4}=1$, indicating that the system is a nontrivial half-
semimetal. Since this type of topological semimetal state of a magnetic system
depends on crystalline symmetries, one can realize both the time-reversal
broken Weyl semimetal or nodal line semimetal state. On considering aFM state,
we find that aFM state breaks $\mathcal{M}_{100}$ mirror plane symmetry, and
the system realizes an ideal Weyl semimetal with two Weyl nodes in the bulk
BZ, as shown in Fig. 4(a). The Weyl nodes are located on the $k_{z}=0$ plane
at (0.000, 0.0355, 0.0) $\mathrm{\AA}^{-1}$ with chirality $+1$ and at (0.000,
-0.0355, 0.0) $\mathrm{\AA}^{-1}$ with chirality $-1$ (see SMs). The surface
band structure and the surface Fermi arcs connecting the projected Weyl nodes
on the (001) surface are shown in Figs. 4(b)-(c). Figures 4(d)-(f) show the
bulk and surface electronic spectrum of the cFM state. Since the cFM state
preserves the $\mathcal{M}_{001}$ mirror symmetry, the two crossing bands in
Fig. 4(d) stay robust against band hybridization, thereby realizing a single
Weyl nodal line on the $k_{z}=0$ plane. The associated nontrivial drumhead
surface states and Fermi band contours are shown in Figs. 4(e)-(f). We
emphasize an FM state can be considered as a fully polarized final state of an
initial AFM state. It is thus possible to realize the aforementioned ideal
topological half-semimetal states in EuIn2P2. This is shown schematically in
Fig. 4(g). A magnetic field applied parallel to the $y$ axis will drive it
into a nearly ideal Weyl semimetal whereas a magnetic field applied parallel
to the $z$ axis will drive it into a Weyl nodal-line semimetal state.

$k\cdot p$ model analysis– The preceding analysis reveals that the band
inversion happens between In $s$ and As $p$ states at $\Gamma$ in both the
non-magnetic and magnetic compounds. The role of magnetism is to reduce the
symmetries and drive the system into various TCI states. We can thus consider
the Hamiltonian of the nonmagnetic state and regard magnetism as perturbation
allowed by the magnetic symmetries so that $H=H_{0}+\Delta H$, where $H_{0}$
is non-magnetic state Hamiltonian whereas $\Delta H$ originates from the
magnetic order. The irreducible representations of two bands at $\Gamma$ close
to the Fermi level are $B_{1g}$ and $B_{2u}$ of symmetry group $D_{6h}$ as
shown in Fig. 1(b). Based on the theory of invariants, the low-energy
effective Hamiltonian can be described as

$H_{0}(\vec{k})=\epsilon(\vec{k})\tau_{0}\sigma_{0}+A_{1}\tau_{x}\left(k_{x}\sigma_{y}-k_{y}\sigma_{x}\right)+A_{2}k_{z}\tau_{y}\sigma_{0}+M(\vec{k})\tau_{z}\sigma_{0},$
(1)

where
$\epsilon(\vec{k})=\epsilon_{0}+\epsilon_{1}(k_{x}^{2}+k_{y}^{2})+\epsilon_{2}k_{z}^{2}$
and $M(\vec{k})=M_{0}-\beta_{1}(k_{x}^{2}+k_{y}^{2})-\beta_{2}k_{z}^{2}$.
$\tau$ and $\sigma$ are the Pauli spin matrices in the orbital and spin space,
respectively, and $M_{0}$, $\beta_{1}$ and, $\beta_{2}>0$ (see SM for
details). When SOC is off, $A_{1}=0$ which leads to a nodal ring in the
$k_{z}=0$ plane described with $k_{x}^{2}+k_{y}^{2}=M_{0}/\beta_{1}$. A cAFM
order that breaks $\mathcal{T}$, $\mathcal{C}_{2y}$, and $\mathcal{G}_{2z}$
while preserving $\mathcal{TC}_{2y}=\tau_{z}\sigma_{0}\mathcal{K}$ and
$\mathcal{C}_{2y}\mathcal{G}_{2z}=\mathcal{G}_{2x}$ symmetry operations gives

$\Delta
H_{\text{cAFM}}(\vec{k})=\left(B_{1}\tau_{0}+B_{2}\tau_{z}\right)\left[\left(k_{x}^{2}-k_{y}^{2}\right)\sigma_{x}-2k_{x}k_{y}\sigma_{y}\right].$
(2)

A aAFM order that breaks $\mathcal{T}$ and $\mathcal{C}_{3z}$ gives

$\Delta
H_{\text{aAFM}}(\vec{k})=D_{0}\tau_{y}\left(k_{x}\sigma_{y}-\delta_{\varepsilon}k_{y}\sigma_{x}\right).$
(3)

The ferromagnetic cFM and aFM phases can come with $\Delta
H_{\text{cFM}}=\lambda_{c}\tau_{z}\sigma_{z}$ and $\Delta
H_{\text{aFM}}=\lambda_{a}\tau_{z}\sigma_{y}$, respectively.

Summary–

Table 1: Summary of topological states in SrIn2As2, EuIn2As2, and EuIn2P2 Zintl materials for different commensurate magnetic configurations. NM stands for nonmagnetic, aAFM (cAFM) denotes antiferromagnetic state with magnetic moment $m||a$ ($m||c$), and aFM (cFM) represents ferromagnetic state with magnetic moment $m||a$ ($m||c$). TI, TCI, AI, and NI stand for topological insulator, topological crystalline insulator, Axion insulator, and normal insulator, respectively. WSM and NLSM denote Weyl semimetal and nodal line semimetal, respectively. See text for more details.  Compound | Magnetic state | Space group | Topological invariant | Topological state
---|---|---|---|---
SrIn2As2 | NM | $P6_{3}/mmc$ | $Z_{2}=1$ | TI $\&$ TCI
EuIn2As2 | aAFM | $Cmcm$ | $Z_{4}=2$ | TCI
cAFM | $P6_{3^{\prime}}/m^{\prime}m^{\prime}c$ | $Z_{4}=2$ | AI
aFM | $Cmc^{\prime}m^{\prime}$ | $Z_{4}=2$ | TCI
cFM | $P6_{3}/mm^{\prime}c^{\prime}$ | $Z_{4}=2$ | AI
EuIn2P2 | aAFM | $Cmcm$ | $Z_{4}=0$ | NI
cAFM | $P6_{3^{\prime}}/m^{\prime}m^{\prime}c$ | $Z_{4}=0$ | NI
aFM | $Cmc^{\prime}m^{\prime}$ | $Z_{4}=1$ | WSM
cFM | $P6_{3}/mm^{\prime}c^{\prime}$ | $Z_{4}=1$ | NLSM

We have shown that the interplay of topology, symmetry, and magnetism in the
experimentally realized SrIn2As2 materials class can enable access to a
multitude of topological states through the lowering of symmetry via various
magnetic configurations. SrIn2As2, a hexagonal Zintl compound, is found to be
a dual topological insulator with nontrivial $Z_{2}=1$ and mirror Chern number
$C_{M}=-1$. Its isostructural and isovalent magnetic cousin EuIn2 (As, P)2
supports a variety of topological states. Remarkably, we find that EuIn2As2 is
an axion insulator with $Z_{4}=2$, which can also realize a mirror topological
magnetic insulator for aAFM (magnetic moment $m||a$) and a higher-order
topological insulator for cAFM ($m||c$). In contrast, EuIn2P2 is a robust AFM
trivial insulator in its pristine form and undergoes a magnetic field-driven
transition to an ideal ferromagnetic Weyl point ($m||a$) and Weyl nodal line
($m||c$) half-semimetal with $Z_{4}=1$. In conjunction with the bulk
topological states, we also obtain a the variety of surface states viz.
gapless or gapful, pinned or unpinned at the $\Gamma$ point, see Table 1 for a
summary of topological states in (Sr, Eu)In2(As, P)2. Since Sr and Eu are
isovalent, Sr1-xEuxIn2(As, P)2 provides a material that can be tuned
systematically between various topological orders through its inherent
magnetism and external stimuli. A topological phase transition from an Axion
insulator ($Z_{4}=2$) to a trivial insulator ($Z_{4}=0$) can be realized in
EuIn2As2-xPx by varying the doping concentration $x$ as discussed in SMs. Our
results thus highlight that the SrIn2As2 class of topological insulators in
the Zintl family are a promising materials platform for investigating the
interplay of topology, symmetry, and magnetism and for device applications.

Acknowledgements– A. B. S. acknowledges IIT Kanpur for providing a Senior
Research Fellowship. A.B.S. and A.A. thank the CC-IITK for providing the HPC
facility. A.A. acknowledge funding from Science Education and Research Board
(SERB) and Department of Science and Technology (DST), Government of India.
T.-R.C. was supported by the Young Scholar Fellowship Program from the
Ministry of Science and Technology (MOST) in Taiwan, under a MOST grant for
the Columbus Program MOST110-2636-M-006-016, NCKU, Taiwan, and National Center
for Theoretical Sciences, Taiwan. Work at NCKU was supported by the MOST,
Taiwan, under grant MOST107-2627-E-006-001 and Higher Education Sprout
Project, Ministry of Education to the Headquarters of University Advancement
at NCKU. S.M.H. is supported by the MOST-AFOSR program on Topological and
Nanostructured Materials, Grant No. 110-2124-M-110-002-MY3. H.L. acknowledges
the support by the Ministry of Science and Technology (MOST) in Taiwan under
grant number MOST 109-2112-M-001-014-MY3. The work at Northeastern University
was supported by the Air Force Office of Scientific Research under award
number FA955-20-1-0322 and benefited from the computational resources of
Northeastern University’s Advanced Scientific Computation Center (ASCC) and
the Discovery Cluster. The work at TIFR Mumbai was supported by the Department
of Atomic Energy of the Government of India under Project No. 12-R$\&$D-TFR-
5.10-0100.

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