Magnetic wallpaper Dirac fermions and topological magnetic Dirac insulators Yoonseok Hwang1 2 3Yuting Qian1 2Junha Kang1 2 3Jehyun Lee1 2 3 Dongchoon Ryu1 2 3Hong Chul Choi1 2yand Bohm-Jung Yang1 2 3z

2025-05-02 0 0 9.31MB 46 页 10玖币

侵权投诉

Magnetic wallpaper Dirac fermions and topological magnetic Dirac insulators

Yoonseok Hwang,1, 2, 3, ∗Yuting Qian,1, 2, ∗Junha Kang,1, 2, 3 Jehyun Lee,1, 2, 3

Dongchoon Ryu,1, 2, 3 Hong Chul Choi,1, 2, †and Bohm-Jung Yang1, 2, 3, ‡

1Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, Korea

2Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea

3Center for Theoretical Physics (CTP), Seoul National University, Seoul 08826, Korea

Topological crystalline insulators (TCIs) can host anomalous surface states which inherits the

characteristics of crystalline symmetry that protects the bulk topology. Especially, the diversity

of magnetic crystalline symmetries indicates the potential for novel magnetic TCIs with distinct

surface characteristics. Here, we propose a topological magnetic Dirac insulator (TMDI), whose

two-dimensional surface hosts fourfold-degenerate Dirac fermions protected by either the p0

c4mm or

p40g0mmagnetic wallpaper group. The bulk topology of TMDIs is protected by diagonal mirror

symmetries, which give chiral dispersion of surface Dirac fermions and mirror-protected hinge modes.

We propose candidate materials for TMDIs including Nd4Te8Cl4O20 and DyB4based on ﬁrst-

principles calculations, and construct a general scheme for searching TMDIs using the space group

of paramagnetic parent states. Our theoretical discovery of TMDIs will facilitate future research

on magnetic TCIs and illustrate a distinct way to achieve anomalous surface states in magnetic

crystals.

I. INTRODUCTION

The surface states of topological insulators (TIs) have

anomalous characteristics that are unachievable in or-

dinary periodic systems [1]. A representative example

is the twofold-degenerate gapless fermion on the surface

of three-dimensional (3D) TIs protected by time-reversal

symmetry (TRS) [2–5]. Contrary to the case of ordinary

two-dimensional (2D) crystals with TRS in which gapless

fermions appear in pairs, a single gapless fermion can ex-

ist on the surface of TIs through its coupling to the bulk

bands. Such a violation of fermion number doubling [6–8]

is a representative way in which the anomalous charac-

teristics of surface states are manifested at the boundary

of TIs.

In topological crystalline insulators (TCIs) [9,10],

crystalline symmetries enrich the ways in which anoma-

lous surface states are realized. For example, in sys-

tems with rotation symmetry and TRS, variants of the

fermion doubling theorem enabled by symmetries can be

anomalously violated on the surface of TCIs [11]. Ad-

ditionally, in the case of mirror-protected TCIs [12], al-

though the number of surface gapless fermions can be

even, the surface band structure exhibits a chiral dis-

persion along mirror-invariant lines such that anoma-

lous chiral fermions appear in the one-dimensional (1D)

mirror-resolved subspace of the 2D surface Brillouin zone

(BZ). More recently, studies showed that in crystals with

glide mirrors, the anomalous surface states can have an

hourglass-type band connection [13]. Moreover, when the

surface preserves two orthogonal glide mirrors, a single

fourfold-degenerate Dirac fermion [14] was shown to be

achievable as an anomalous surface state [15].

∗These authors contributed equally to this work.

†chhchl@snu.ac.kr

‡bjyang@snu.ac.kr

In magnetic crystals, there is great potential to achieve

a new type of magnetic TCI with distinct anomalous sur-

face states [16–22] because there are abundant magnetic

crystalline symmetries described by 63 magnetic wall-

paper groups (MWGs) and 1421 magnetic space groups

(MSGs) [23,24], which are overwhelmingly larger than

the 17 wallpaper groups and 230 space groups of nonmag-

netic crystals [25–29]. Very recently, exhaustive stud-

ies of magnetic topological phases and their classiﬁcation

have been performed [16–18], and various novel magnetic

topological phases have been systematically categorized.

However, as far as we can tell, all the surface states of

magnetic TCIs reported up to now appear in the form

of twofold-degenerate gapless fermions, whose detailed

band connection depends on the surface symmetry.

Here, we propose a magnetic TCI with fourfold-

degenerate gapless fermions on the surface, coined the

topological magnetic Dirac insulator (TMDI). A fourfold-

degenerate gapless fermion, a Dirac fermion for short

hereafter, can appear on the surface of a magnetic in-

sulator when the MWG of the surface is one of the three

MWGs p40g0m,p0

cmm, and p0

c4mm, among 63 possible

MWGs. Contrary to the surface Dirac fermion in non-

magnetic crystals protected by two orthogonal glides, our

surface Dirac fermion is protected by symmorphic sym-

metries combined with either an antiunitary translation

symmetry or an antiunitary glide mirror.

In particular, in magnetic crystals whose (001)-surface

MWG is either p40g0mor p0

c4mm, the bulk topology is

characterized by the mirror Chern number (MCN) Cxy

about the diagonal mirror planes normal to either the

[110] or [1¯

10] direction. Because of this, in TMDIs, the

way in which the surface anomaly is realized is diﬀerent

from the case of the nonmagnetic Dirac insulator [15] and

more similar to the case of mirror-protected nonmagnetic

TCIs [12]. Namely, along the mirror-invariant line on the

surface BZ, the Dirac fermion develops a chiral dispersion

relevant to the MCN. Moreover, the MCN of TMDIs also

arXiv:2210.10740v2 [cond-mat.mes-hall] 26 Apr 2023

a b c d e

p′cmm p4′g′m

p′c4mm Dirac fermion in 2D

MWG p4′g′m (nonchiral)

Energy

-2

-4

Γ M

Energy

Γ′M

XΓM

Dirac fermion in 2D

surface of 3D TMDI (chiral)

k

3D TMDI

Γ′=2M

Γ′

Conduction band

Valence band

FIG. 1. MWGs and fourfold-degenerate Dirac fermions. (a)-(c) MWGs that protect fourfold degeneracy. Type-IV

MWGs (a) p0

cmm and (b) p0

c4mm, and (c) Type-III MWG p40g0m. Black arrows represent spin conﬁgurations located at generic

positions. The styles of lines indicate the types of symmetry elements: glides (blue dashed lines), antiunitary glides (red dashed

lines), mirrors (blue solid lines), and antiunitary mirrors (red solid lines). (d) Typical band structure of 2D crystals belonging

to MWG p40g0m. Γ = (0,0), X= (π, 0), M= (π, π), and Γ0= (2π, 2π) denote high-symmetry points. A fourfold-degenerate

Dirac fermion appears at Mnear E= 0 (green arrow). Red (blue) lines denote states with eigenvalue +i(−i) of the diagonal

mirror. Along Γ-M-Γ0, each mirror eigenvalue sector of the Dirac fermion has a nonchiral dispersion. (e) Left: a Dirac fermion

appearing as an anomalous surface state of a topological magnetic Dirac insulator (TMDI) with a chiral dispersion in each

mirror sector. Right: schematic depiction of a TMDI, which hosts a fourfold-degenerate Dirac fermion on the (001) surface,

and mirror-protected hinge modes on the sides invariant under diagonal mirrors.

induces hinge modes at open boundaries along the xand

ydirections, which respect diagonal mirrors.

Using ﬁrst-principles calculations, we propose candi-

date materials for TMDIs, including Nd4Te8Cl4O20 and

DyB4. Since the database for magnetic materials only

has a limited number of materials, we construct a sys-

tematic way to ﬁnd candidate magnetic materials for

TMDIs using the space group of paramagnetic parent

compounds.

II. DIRAC FERMIONS AND MAGNETIC

WALLPAPER GROUPS

In 2D magnetic crystals, Dirac fermions with fourfold-

degeneracy can be symmetry-protected at the BZ corner,

M= (π, π), by three MWGs, i.e., Type-III MWG p40g0m

and Type-IV MWGs p0

cmm and p0

c4mm [see Figs. 1(a)-

(c)]. Here, we use the notation of Belov and Tarkhova

(BT) [23] for denoting MWGs and the notation of Belov,

Neronova, and Smirnova (BNS) [30] for denoting MSGs.

Note that Type-III MWGs have antiunitary spatial sym-

metries combining TRS Twith spatial symmetries, while

Type-IV MWGs have antiunitary translation symmetries

combining Tand fractional lattice translations. All three

MWGs have mirror-invariant lines, whose normal direc-

tions are ˆx, ˆy, or ˆx±ˆy.

In 2D systems belonging to the Type-III MWG p40g0m

described in Fig. 1(c), a Dirac fermion is protected at

Mby twofold rotation about the z-axis C2z, antiunitary

glide mirror T Gy≡T{my|1

2,1

2}, and oﬀ-centered diago-

nal mirror f

Mxy ={mxy|1

2,−1

2}. Here, the notation {g|t}

denotes the point group symmetry gfollowed by a partial

lattice translation t.mxy,y are mirror symmetries that

act on real-space coordinates as mxy : (x, y, z)→(y, x, z)

and my: (x, y, z)→(x, −y, z). [See the conventions in

Supplementary Note (SN) 1.] As detailed in SN 3, the

fourfold degeneracy is formed by four states ψ±,T Gyψ±,

Mxyψ±, and T Gyf

Mxyψ±, where ψ±is an energy eigen-

state with C2zeigenvalue ±i.

In contrast, 2D systems belonging to the Type-IV

MWGs p0

cmm and p0

c4mm, described in Figs. 1(a) and

(b), respectively, have common symmetry elements, i.e.,

antiunitary translation TG={T|1

2,1

2}and two mirrors

Mx={mx|0}and My={my|0}, where mx: (x, y, z)→

(−x, y, z). At M, these symmetry elements anticommute

with each other, and T2

G=−1. These relations protect

the fourfold degeneracy formed by ψ±,TGψ±,Myψ±,

and TGMyψ±, where ψ±has Mxeigenvalue ±i(see SN3).

Note that the same symmetry representation was also

studied in Ref. [31].

A typical band structure supported by MWG p40g0m

is shown in Fig. 1(d). Since MWG p40g0mhas diagonal

mirror f

Mxy, the energy bands can be divided into two

diﬀerent mirror eigensectors along the ΓMΓ0direction.

Here, Γ = (0,0), and Γ0= (2π, 2π). Focusing on the band

structures in each mirror sector, we ﬁnd that the num-

bers of upward (chiral) and downward (antichiral) bands

crossing the Fermi level [E= 0 in Fig. 1(d)] at Mare

the same. Otherwise, the mirror-resolved band structure

cannot be periodic along ΓMΓ0. Hence, a Dirac fermion

in 2D crystals belonging to MWG p40g0mis nonchiral

in each mirror sector. Similar phenomena also occur

in MWGs p0

cmm and p0

c4mm. In general, the mirror-

resolved dispersion of Dirac fermion in 2D crystals pro-

tected by MWGs is nonchiral. Although the local dis-

persion near the Dirac point may exhibit either chiral or

nonchiral dispersion, the full band dispersion along the

mirror invariant line is always nonchiral in 2D crystals.

III. CHIRAL SURFACE DIRAC FERMIONS

A 2D Dirac fermion, which is nonchiral in 2D systems,

can be chiral on the surface of 3D magnetic TCIs, as il-

lustrated in Fig. 1(e). Here, we systematically search for

p4′g′m

p′c4mm

Cxy

( )=(0,0)

Cxy

m=0Cxy

m=1Cxy

m=4

-π

π( )=(2,0)

-π

π( )=(0,2)

-π

b c d e

f g h i j

Mxy

ΓX

Mxy

M =2MΓ′

Cxy

m=2

-π

( )=(1,–1)

-π

mCxy

m,Cy

mCxy

m,Cy

mCxy

m,Cy

M Γ′ XM

M Γ′

XMΓM Γ′ XMΓM Γ′ XMΓM Γ′ XMΓM Γ′

FIG. 2. Wilson loop spectra of 3D magnetic TCIs with (001)-surface MWGs p40g0mand p0

c4mm.Classiﬁcation

of Wilson loop spectra, whose connectivity is equivalent to the (001)-surface band structure, based on the MCNs Cxy

mand

m. (a)-(e) Type-III TMDIs with MWG p40g0mon the (001) surface. (a) (001)-surface BZ.

Mxy is the diagonal mirror used

to deﬁne Cxy

m. (b)-(e) Wilson loop spectra corresponding to (b) Cxy

m= 0, (c) Cxy

m= 1, (d) Cxy

m= 2, and (e) Cxy

m= 4. For

convenience, the position of the Dirac fermion at Mis adjusted to be located at θ(k) = 0. The red (blue) lines correspond

to Wilson bands with mirror eigenvalue +i(−i). The green dashed lines are the reference lines used to count the MCN Cxy

In (c)-(e), in which Cxy

m6= 0, the dispersion of the Dirac fermion can be chiral in each mirror sector along Γ-M-Γ0. Note that

Γ0= (2π, 2π). In (b), where Cxy

m= 0, the Dirac fermion is nonchiral. In (d), where Cxy

m= 2, the reference line is crossed by two

chiral (upward) modes with mirror eigenvalue +i. In (c), where Cxy

m= 1, the dispersion is chiral along the entire Γ-M-Γ0line

but locally looks nonchiral near the Dirac point. The Wilson loop structure in (c) can be deformed into that in (h) by pushing

the twofold crossing at Γ upward, which gives a locally chiral dispersion at M. When |Cxy

m|>2, a Dirac fermion must appear

with other gapless surface states along the Γ-M-Γ0line [black arrows in (e)]. (f)-(j) Type-IV TMDIs with MWG p0

c4mm on the

(001) surface. (f) (001)-surface BZ. (g)-(j) Wilson loop spectra corresponding to (g) (Cxy

m,Cy

m) = (0,0), (h) (Cxy

m,Cy

m) = (1,−1),

(i) (Cxy

m,Cy

m) = (2,0), and (j) (Cxy

m,Cy

m) = (0,2). Note that Cxy

m=Cy

m(mod 2) holds for insulators.

3D magnetic insulators that can host a Dirac fermion on

the (001) surface. As a 2D fourfold-degenerate Dirac

fermion can be protected by one of the three MWGs

p40g0m,p0

cmm, and p0

c4mm, we focus on the MSGs whose

(001) surface has one of these three MWGs. By studying

the MSG symbols and the detailed surface symmetries,

we ﬁnd that there are at least 31 MSGs that can be gen-

erated from such MWGs and additional generators com-

patible with the MWGs. (See Supplementary Table 1.)

All 31 MSGs have mirror planes whose normal vectors

are orthogonal to the (001) direction. Thus, the cor-

responding MCNs can give chiral dispersions along the

mirror-invariant lines on the (001) surface. First, MWG

p40g0mhas oﬀ-centered diagonal mirrors [32]f

Mxy =

{mxy|1

2,−1

2}and f

Mxy ={mxy|1

2,1

2}[see Fig. 1(c)]. For

the 11 MSGs relevant to MWG p40g0m, we deﬁne four

MCNs Ckx=ky

±and Ckx=−ky

±, which are deﬁned in the

kx=kyand kx=−kyplanes, respectively. Here, the

±sign denotes the mirror eigenvalues of occupied bands.

All the MCNs are equivalent up to sign because of the

symmetry relations among f

Mxy,f

Mxy, and T C4z. Hence,

the bulk topology can be classiﬁed by Cxy

m≡ Ckx=ky

−Ckx=ky

−. Similarly, we can deﬁne MCNs for the 5 MSGs

related to MWG p0

c4mm, which have four mirrors, Mx,

My,Mxy, and Mxy . Among them, only two MCNs,

Cxy

mand Cy

m≡ Cky=0

+=−Cky=0

−, are independent, and

serve as bulk topological invariants. Finally, for the 15

MSGs related to MWG p0

cmm, the relevant MCNs are

m≡ Ckx=0

+=−Ckx=0

−and Cy

m≡ Cky=0

+=−Cky=0

−. For

more detailed discussions on the MCNs, see SN4.

Now, we classify the Wilson loop spectra [33–37] ac-

cording to the MCNs. We consider the kz-directed Wil-

son loop Wz(k⊥),

Wz(k⊥)nm =hun(k⊥, π)|

π←−π

Pocc(k⊥, kz)|um(k⊥,−π)i,

(1)

where Pocc(k)≡Pnocc

n=1 |un(k)ihun(k)|is a projection op-

erator for occupied bands |un(k)iand k⊥= (kx, ky).

Since the Wilson loop is unitary, its eigenvalue can be

collectively denoted as {eiθ(k⊥)}={eiθa(k⊥)|θa(k⊥)∈

(−π, π], a = 1, . . . , nocc}. Then, {θ(k⊥)}deﬁnes the Wil-

son loop spectrum, or equivalently, the Wilson bands.

Wilson loop spectra and surface band structures have the

same spectral features [35,38]. Thus, the band structure

on the (001) surface can be systematically classiﬁed based

on Wilson loop analysis. (The details on tight-binding

notation and Wilson loop is provided in SN2 and SN5.)

First, let us consider the MSGs related to MWG

p40g0m[see Figs. 2(a)-(e)]. At M= (π, π), four Wilson

bands form a fourfold degeneracy, which can be identiﬁed

as a Dirac fermion on the (001) surface. The connectiv-

ity of Wilson bands is classiﬁed by the MCN Cxy

m, which

is encoded in the slope of Wilson bands in each mirror-

sector crossing a horizontal reference line [a green dashed

line in Fig. 2(b)] along the Γ-Mdirection. In Fig. 2(d),

for example, as two Wilson bands with mirror eigenvalue

−iintersect the reference line with a negative slope, we

obtain Cxy

m= 2. See SN5 for the details on the counting

rules for Cxy

Now, we compare the Wilson loop spectra of topo-

logical phases with nonzero Cxy

mand the trivial phase

with zero Cxy

mby focusing on the region near the four-

fold degeneracy at M. Along the Γ-M-Γ0line, the four

bands are divided into two diﬀerent mirror sectors. When

Cxy

m= 0, as in Fig. 2(b), the dispersion in each mirror sec-

tor is nonchiral, similar to that of Dirac fermions in 2D

crystals in Fig. 1(d). In Fig. 2(b), chiral and antichi-

ral modes in the same mirror sector (e.g., the +isector)

cross the green dashed reference line with opposite signs

of the group velocity. In contrast, their numbers are not

equal in Fig. 2(d), where Cxy

m= 2.

The Dirac fermions in Figs. 2(c) and (e), which cor-

respond to Cxy

m= 1 and 4, respectively, appear with ad-

ditional surface states (black arrows). When Cxy

m= 1,

the dispersion is chiral along the entire Γ-M-Γ0line but

locally looks nonchiral near the Dirac point. However,

if the dispersion along Γ-M-Γ0is deformed such that the

additional surface states near Γ are pushed away from the

Fermi level (which corresponds to θ= 0 in Wilson loop

spectra), the Dirac fermion in Fig. 2(c) becomes chiral,

as in Fig. 2(h). In contrast, such a deformation is im-

possible in Fig. 2(e), where Cxy

m= 4. In general, one can

show that a nonzero MCN |Cxy

m| ≤ 2 manifests as a chi-

ral dispersion of the surface Dirac fermion along Γ-M-Γ0,

provided that there is no additional surface state other

than the Dirac fermion at the Fermi level. In contrast,

when |Cxy

m|>2, additional surface states always appear

along Γ-M-Γ0. Hence, the chiral dispersion of the Dirac

fermion when |Cxy

m| ≤ 2 and the coexistence of additional

surface states when |Cxy

m|>2 are signatures of the non-

trivial bulk topology of 3D TMDIs with nonzero Cxy

An exact formulation of the relation among the chiral

dispersion of the Dirac fermion, MCN Cxy

m, and number

of additional surface states is given in SN6.

Similarly, one can analyze the Wilson loop spectra of

the MSGs related to MWG p0

c4mm [see Figs. 2(f)-(j)].

The MCN Cxy

m(Cy

m) can be determined by examining the

Mxy (My) eigenvalues and the slopes of Wilson bands

crossing a reference line along Γ-M(Γ-X). The rela-

tion between Cxy

mand the chiral dispersion of the Dirac

fermion is identical to the case of the MSGs with MWG

p40g0m. The only additional feature is that Cxy

mand Cy

must be equivalent up to modulo 2, i.e., Cxy

m=Cy

m(mod

2), in insulating phases.

Finally, in the 15 MSGs related to MWG p0

cmm, the

Wilson loop spectra can be classiﬁed by the MCNs on

the kx,y = 0 planes, Cx

m(0) and Cy

m(0), related to Mx,y

mirrors. The MCNs on the kx,y =πplanes, Cx

m(π) and

m(π), are always trivial, while surface Dirac fermions

can be chiral only for nonzero Cx,y

m(π). Thus, the surface

Dirac fermion in the MSGs with MWG p0

cmm is nonchiral

and trivial.

a b c

ΓΓ XM

Energy (eV)

-0.5

0.0

Top surface

e f

127 P4/mbm Γ2: 127.395 P4/m′b′m′ Γ4: 127.392 P4′/m′b′m

2π

ΓM

Energy (eV)

-0.5

0.0

Top surface

Esurf=–0.2eV

Bottom surface

j l

Γ′

Energy (eV)

0.0

1.0

Γ2

0.5

U (eV)

Γ4

3 87654 9

U>6.5

–

U=6.5 eV

Energy (eV)

0.5

0.0

-0.5

Γ M

XΓ R

ZAZ

U=6.5 eV

Energy (eV)

0.15

0.0

-0.35Γ R

2π

ΓM

=2M

Γ′

–

U=6.0 eV

-0.25

i k

Energy (eV)

-0.5

0.0

–

U=6.5 eV

Bottom surface

ΓXM

Energy (eV)

-0.5

0.0

-0.25

FIG. 3. DyB4, a Type-III TMDI candidate. (a)

Tetragonal crystal structure of DyB4. (b)-(c) Magnetic spin

structures in the (b) Γ2and (c) Γ4states. The Γ4state cor-

responds to Type-III MSG 127.392 P40/m0b0mwith (001)-

surface MWG p40g0m. (d) Density functional theory (DFT)

total energy diﬀerences between the Γ4and Γ2states. The

absolute values are adjusted to be in the range between 0

eV and 1 eV. The magnetic phase transition is indicated by

the brown solid line. The Γ4state has lower energy than

the Γ2state when U≤6.5 eV. (e)-(f) Bulk band structure

from DFT+Ucalculations (U=6.5 eV and J=1 eV) for the Γ4

state. The band gap near the Fermi level is indicated by the

red dashed lines. (g)-(h) Wilson loop spectra below the red

dashed line along Γ-M-Γ0. The winding structure exhibits (g)

Cxy

m=−2 at U=6.5 eV and (h) Cxy

m=−1 at U=6.0 eV. (i)-(j)

(001)-surface spectra for the (i) top (B-terminated) and (j)

bottom (Dy-terminated) surfaces obtained using the surface

Green’s function method. The white arrows guide where sur-

face states appear. (k)-(l) (001)-surface band structures from

a 60-layer slab calculation, which are drawn to show surface

states indicated by the white arrows in (i) and (j). Bulk and

surface bands are represented by gray and black, respectively.

Surface Dirac fermions are moved inside the gap by applying

a surface potential of -0.2 eV on the top. (i-l) are calculated

using the Wannierized tight-binding model from the DFT+U

calculation (U=6.5 eV).

IV. TOPOLOGICAL MAGNETIC DIRAC

INSULATORS

According to the Wilson loop analysis, the bulk band

topology of the 11 MSGs with Type-III MWG p40g0mand

the 5 MSGs with Type-IV MWG p0

c4mm can be charac-

terized by Cxy

mand (Cxy

m,Cy

m), respectively. In these 16

MSGs, when Cxy

m6= 0, a Dirac fermion whose mirror-

resolved dispersion is chiral can appear on the (001)

surface. Based on this, we deﬁne TMDIs as 3D mag-

netic TCIs with nonzero Cxy

mhosting a 2D chiral Dirac

fermion on the (001) surface. Additionally, according

to the (001)-surface MWG, TMDIs can be divided into

Type-III and Type-IV TMDIs such that Type-III (Type-

IV) TMDIs have (001)-surface MWG p40g0m(p0

c4mm).

Interestingly, TMDIs also exhibit higher-order topol-

ogy [11,29,39–46] with hinge modes at open boundaries

along the xand ydirections when the entire ﬁnite-size

systems respect the diagonal mirror symmetries. The

MCN for the diagonal mirror Cxy

mfollows the higher-

order bulk-boundary correspondence [41,42,45]. The

dispersion of hinge modes can be both chiral and heli-

cal depending on the details of the systems. Note that

the number of chiral and antichiral hinge modes at each

hinge can be changed by a mirror-symmetric and bulk-

gap-preserving perturbation because such a perturbation

can close and reopen a surface gap [42,44,47]. However,

the MCN Cxy

mprotects at least |Cxy

m|hinge modes at each

mirror-invariant hinge. (See SN7 for more details.) We

provide a tight-binding model for a Type-III TMDI in

SN10, which conﬁrms the bulk-boundary correspondence

described above.

V. CANDIDATE MATERIALS

Using ﬁrst-principles calculations, we propose DyB4

and Nd4Te8Cl4O20 as candidate materials for a Type-

III TMDI and a Type-IV TMDI, respectively, whose

electronic and topological properties are summarized in

Figs. 3and 4, respectively. Although their band structure

are metallic, as the systems have nonzero direct gap at all

momenta, their mirror Chern numbers are well deﬁned.

We note that the two candidate materials, DyB4and

Nd4Te8Cl4O20, are not available in the existing material

databases [20,48–55]. In general, searching for candidate

magnetic materials is more challenging than searching for

nonmagnetic materials because the number of available

materials in magnetic material databases [20,54] is lim-

ited to approximately 1600 [54], which is much smaller

than the number of nonmagnetic materials. To overcome

this limitation, we will also propose a general scheme to

systematically search for candidate materials for TMDIs,

whose magnetic structures and MSGs are derived from

their parent paramagnetic states.

Let us ﬁrst consider DyB4, a member of the rare-earth

tetraborides family, whose crystal structure is shown in

Fig. 3(a). The paramagnetic parent phase of DyB4has

Energy (eV)

Γ MX Γ RZ A Z

a b c

ΓΓ XM

2π

ΓM

Energy (eV)

0.0

2.0

Energy (eV)

e f

Energy (eV)

0.0

1.5

h i

123 P4/mmm

129.421 PC4/nmm

1.0

-1.0

Top surface Bottom surface

1.0

0.5

Energy (eV)

0.0

1.5

1.0

0.5

Esurf=–0.544eV

Bottom surface

Top surface

0.0

2.0

1.0

-1.0

0.0

2.0

1.0

-1.0

–

ΓΓ XM

=2M

Γ′ Γ′ Γ′

FIG. 4. Nd4Te8Cl4O20 , a Type-IV TMDI candidate.

(a)-(b) Crystal structures of NdTe2ClO5and Nd4Te8Cl4O20

(2 ×2×1 supercell). (c) Magnetic structure compatible with

Type-IV MSG 129.421 PC4/nmm and (001)-surface MWG

c4mm. (d) Bulk band structure from DFT+Ucalculations

(U=6 eV). The band gap near the Fermi level is indicated

by the red dashed line. (e)-(f) (001)-surface spectra for the

(e) top (O-terminated) and (f) bottom (Nd-terminated) sur-

faces obtained using the surface Green’s function method.

The surface Dirac fermion is indicated by the white arrow

in (e). (g) kz-directed Wilson loop spectrum along the diago-

nal path Γ-M-Γ0. The winding structure exhibits Cxy

m=−1.

(h)-(i) (001)-surface for the (h) top and (i) bottom surfaces

from a 30-layer slab calculation. Bulk and surface bands are

represented by gray and black, respectively. The fourfold-

degenerate Dirac fermions are located approximately 1 eV

from the Fermi level on the top surface. A surface potential

of -0.544 eV is introduced to push the Dirac point on the

bottom surface into the gap.

space group 127 P4/mbm. In experiments [56], this ma-

terial was reported to have two competing spin conﬁgu-

rations that correspond to the Γ2(with MSG 127.395

P4/m0b0m0) and Γ4(with MSG 127.392 P40/m0b0m)

magnetic states, as shown in Figs. 3(b) and (c). Between

them, the Γ2state was reported to be more favored [56].

Note that the Γ4phase can support a Type-III TMDI

with (001)-surface MWG P40g0m. Here, we investigate

the conditions in which the Γ4state becomes the mag-

netic ground state in DyB4based on DFT+U(where Uis

the onsite Coulomb interaction) calculations. We exam-

ine the total energies of the Γ2and Γ4states as a function

of Uwith ﬁxed Hund’s coupling J= 1 eV. As shown in

Fig. 3(d), DyB4undergoes a magnetic phase transition

from the Γ4to Γ2state when U > 6.5 eV.

An indirect gap near the Fermi level EF=0.0 eV ex-

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MagneticwallpaperDiracfermionsandtopologicalmagneticDiracinsulatorsYoonseokHwang,1,2,3,YutingQian,1,2,JunhaKang,1,2,3JehyunLee,1,2,3DongchoonRyu,1,2,3HongChulChoi,1,2,yandBohm-JungYang1,2,3,z1CenterforCorrelatedElectronSystems,InstituteforBasicScience(IBS),Seoul08826,Korea2DepartmentofPhysicsandAs...

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Magnetic wallpaper Dirac fermions and topological magnetic Dirac insulators Yoonseok Hwang1 2 3Yuting Qian1 2Junha Kang1 2 3Jehyun Lee1 2 3 Dongchoon Ryu1 2 3Hong Chul Choi1 2yand Bohm-Jung Yang1 2 3z

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