All hourglass bosonic excitations in the 1651 magnetic space groups and 528 magnetic layer groups

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Dongze Fan,1, 2 Xiangang Wan,1, 2, 3 and Feng Tang1, 2, ∗

1National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China

2Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

3International Quantum Academy, Shenzhen 518048, China.

The band connectivity as imposed by the compatibility relations between the irreducible representations of

little groups can give rise to the exotic hourglass-like shape composed of four branches of bands and ﬁve band

crossings (BCs). Such an hourglass band connectivity could enforce the emergence of nontrivial excitations like

Weyl fermion, Dirac fermion or even beyond them. On the other hand, the bosons, like phonons, magnons, and

photons, were also shown to possess nontrivial topology and a comprehensive symmetry classiﬁcation of the

hourglass bosonic excitations would be of great signiﬁcance to both materials design and device applications.

Here we ﬁrstly list all concrete positions and representations of little groups in the Brillouin zone (BZ) related

with the hourglass bosonic excitations in all the 1651 magnetic space groups and 528 magnetic layer groups,

applicable to three dimensional (3D) and two dimensional (2D) systems, respectively. 255 (42) MSGs (MLGs)

are found to essentially host such hourglass BCs: Here “essentially” means that the bosonic hourglass BC exists

deﬁnitely as long as the studied system is crystallized in the corresponding MSG/MLG. We also perform ﬁrst-

principles calculations on hundreds of 3D nonmagnetic materials essentially hosting hourglass phonons and

propose that the 2D material AlI can host hourglass phonons. We choose AuX (X=Br and I) as illustrative

examples to demonstrate that two essential hourglass band structures can coexist in the phonon spectra for

both materials while for AuBr, an accidental band crossing sticking two hourglasses is found interestingly.

Our results of symmetry conditions for hourglass bosonic excitations can provide a useful guide of designing

artiﬁcial structures with hourglass bosonic excitations.

I. INTRODUCTION

Various topological band crossings (BCs) in bulk and

boundary electronic systems have attracted extensive interest

in the past nearly two decades [1–9] on which crystallographic

symmetries [10–12] could impose diverse constraints. On the

other hand, bosons, such as phonons, photons, and magnons,

emergent in crystalline materials or artiﬁcial structures, have

also been proposed to carry nontrivial band topology [13–

15]. Topological bosons are expected to be endowed with

novel consequences like novel quantum information storage

and processing [13], low dissipation phonon transport [14],

new type spintronics device application [15] and so on.

Very recently, comprehensive classiﬁcation of BCs for all

the 230 space groups (SGs) [16] or even 1651 magnetic

space groups (MSGs) [17–20] have been obtained. Espe-

cially, by Ref. [18], one can know both the associated low-

energy k·pmodel and band nodal structures given a BC at

some special high-symmetry kpoint in the Brillouin zone

(BZ), though whether the BC exists in a concrete material

usually needs practical ﬁrst-principles calculations [21–26].

Symmetry-based classiﬁcations of band topology for all SGs

or MSGs [27–34] indicated that diverse topological phases

can be formed in the spinful setting as well as the spinless

setting [35–49]. Note that we usually require the BCs to be

close to the Fermi level in electronic systems for observable

phenomena, while there is no such requirement for bosonic

systems. And some BCs exist deﬁnitely (or essentially), for

example, when the BC is located at a high-symmetry point,

the irreducible representation (irrep) carried by the BC is the

only possible irrep of the little group of the high-symmetry

∗fengtang@nju.edu.cn

point. Another type of essential BC is due to nonsymmorphic

symmetry enforcing special band connectivity such as hour-

glass one [50–53] (see Fig. 1for all types of hourglass band

structures). The essential BC is beneﬁcial to searches for ma-

terials realizations of a target BC or complex nodal structures:

For example, in the design of two nested nodal loops, one of

which can already exist due to the essential BC (the essential

BC lies in the nodal loop), we only need to make the other

one. This is obvious much more easier than the method of

making two nodal loops by tuning structure parameters [54].

Such strategy of designing nodal structures might be easily

implemented in artiﬁcial structures [55–73].

In this work, we focus on all possible hourglass bosonic

excitations in all the 1651 MSGs and 528 magnetic layer

groups (MLGs), which can be obtained using compatibil-

ity relations (CRs) to enforce hourglass band connectivity.

As applications, we perform ﬁrst-principles calculations for

hourglass phonons with time-reversal symmetry in realis-

tic three-dimensional (3D) materials [74] and theoretically-

proposed two-dimensional (2D) materials [75], to which re-

sults of type II MSGs and MLGs are applied, respectively. It

is worth pointing out that our tabulation of MSGs or MLGs

realizing hourglass bosonic excitations could be applied to

designing artiﬁcial structures and tuning structure parame-

ters as needed conveniently. Besides, results for type-I, III

and IV MSGs/MLGs are applicable to systems without time-

reversal symmetry, which can be of technological impor-

tance, for example, in realizing phonons with ﬁnite angular

momentum[76–78]. Besides, recently, the spin-space groups

[79–81] have received much attention which might be the

symmetry of magnon Hamiltonian [80]. Ref. [80] shows

that some spin-space groups could be isomorphic with some

MSGs which are of type I, III or IV, thus the results for these

MSGs could be applied to the corresponding cases to identify

arXiv:2210.02954v1 [cond-mat.mtrl-sci] 6 Oct 2022

new-type topological properties of magnons [82–86].

II. STRATEGY

We begin with brieﬂy describing the strategy of exhaus-

tively classifying and listing all hourglass BCs based on CRs.

In this work, single-valued representations for the little groups

of the MSGs are used, applicable for bosonic bands or elec-

tronic bands with spin-orbit coupling negligible. For elec-

tronic bands with spin-orbital coupling, double-valued repre-

sentations should be adopted [87]. Interestingly, we ﬁnd that,

different from the hourglass BCs in the spin-orbital coupled

electronic band structures, whose degeneracy can only take

2 and 4 [87], the bosonic hourglass band crossings can be 2,

3 and 4-fold degenerate. In most cases, the degeneracy of

the hourglass BC is 2. We list all the results of MSGs host-

ing 3-fold and 4-fold bosonic hourglass BCs in Table II. We

should also point out that the so-called hourglass BC through-

out this paper is the one located at the neck of the hourglass

band structure.

As schematically shown in Fig. 1, we consider a trio de-

noted by R-X-B, which means that R and B are connected by

X and deﬁnitely own higher symmetry than X. Then from R

(B) to X, bands should split and the splitting pattern can be

known based on CRs. As shown in Fig. 1, we use two hor-

izontal bars to denote each degenerate energy level at R(B).

The splitting pattern from the energy level at R(B) to X is also

encoded in the colors/styles of the two horizontal bars repre-

senting the energy level. In Sec. IV of the Supplementary

Material [89], we list all such trios allowing hourglass BCs.

In Fig. 1, we use different colors/styles of lines, which repre-

sent energy bands in X, to denote different irreps or co-irreps

of little group of X, carried by the bands in X. To form an

hourglass shape of bands, two energy levels at R and B need

to be considered, all of which split into two branches of bands

along X. We can formally describe such splitting pattern by:

X1⊕X2,X3⊕X4;X5⊕X6,X7⊕X8, namely, the higher (lower)

energy level at R splits into two bands whose (co-)irreps are

X1 and X2 (X3 and X4) while the higher (lower) energy level

at B splits into two bands whose (co-)irreps are X5 and X6 (X7

and X8). Due to the continuity of Bloch wave functions, we

have {X1,X2,X3,X4}={X5,X6,X7,X8}. Besides, to form an

hourglass BC, {X1,X2}6={X5,X6}and {X3,X4}6={X7,X8}.

Next, it should be required that X should allow at least two

different (co-)irreps, thus X can be a high-symmetry line or

high-symmetry plane. When X is a high-symmetry line, R

and B should both be high-symmetry points and the hour-

glass BC can be a nodal point or lie in a nodal line within

a high-symmetry plane [18,54]. When X is a high-symmetry

plane, R and B can be high-symmetry point or line and the

hourglass BC deﬁnitely lies in a nodal line within X, in other

word, each point in the nodal line is the BC point due to

an hourglass band connectivity. For the splitting pattern:

X1⊕X2,X3⊕X4;X5⊕X6,X7⊕X8 which enforces an hour-

glass structure, we have X1=X5, X4=X8, {X2,X3}={X6,X7}

(but X26=X6). Hence, X2 and X3 are interchanged from R

to B and thus constitute the (co-)irreps of the hourglass BC

whose topological character can then be identiﬁed [18]. Based

on the above requirements, we thus obtain ﬁve types of hour-

glass BCs: type A-E as shown in Fig. 1. Based on all (co-

)irreps listed in Ref. [17], we calculate all CRs of all possible

trios R-X-B and then list all possible hourglass BCs. They

are all provided in Sec. IV of the Supplementary Material:

For each MSG or MLG, the coordinates of R, X and B are

given in the convention adopted in Ref. [10] alongside which

the relevant band splitting patterns are also provided. The di-

mensions of relevant (co-)irreps of X are shown simultane-

ously, thus one can quickly know the degeneracy of the hour-

glass BC. The exhaustive list indicates that type-B hourglass

bosonic excitations are very rare while type-A ones are the

commonest cases.

Then we discuss more constraints for hourglass BCs to be

essential, as printed in red in Sec. IV of the Supplementary

Material. As shown in Fig. 1, type D and E hourglass BCs

cannot be essential, since exchange of energy levels could gap

the hourglass BC. In Ref. [90], we focused only on the 230

SGs and obtained an exhaustive list of hourglass BCs and we

also imposed a very strict condition for the hourglass BC to

be essential there: it is required that the hourglass BC is of

type A, B or C and the splitting pattern in the hourglass BC is

the only possible splitting pattern. However, here we ﬁnd that

for some MSGs, the hourglass BC can still exist essentially

in some kpath though there can be different splitting patterns

in this path. All MSGs and MLGs with essential hourglass

bosonic BCs are listed in Table Iand as shown by statistics in

Table III, many of MSGs/MLGs allowing hourglass BC can

host essential hourglass BC. Here we say the splitting pat-

terns are different once the symmetry contents (namely, the

(co-)irreps in the hourglass band structure) are different. Such

interesting case is illustrated in Fig. 1(f) where we also show

that an exchange of energy levels at R would give rise to an-

other accidental BC connecting two hourglasses. A thorough

check of our results reveals that only MSGs 134.481, 138.520

and 138.522 can host coexisting essential hourglass BCs of

different types in some kpaths X, further found to include

types-A, B and C. For other kpaths in these MSGs and for

the rest MSGs and the MLGs in Table I, the essential BCs

in a kpath belong to only one type of hourglass band struc-

ture, but the splitting patterns could be different, listed below:

MSGs 135.483, 135.484, 135.487, 135.493 could host type-

C essential hourglass BCs (of this property); MSGs 26.72,

26.76, 27.86, 31.128, 31.133, 33.149, 34.161, 48.262, 55.360,

55.362, 56.372, 56.376, 58.400, 58.402, 62.450, 62.452,

62.453, 62.455, 106.221, 106.223, 131.445, 133.464, 133.465

and 135.488 could host type-A essential hourglass BCs; For

MLGs based on MSGs 26.72 and 55.360 with the translation

symmetry along aand cbroken, respectively, type-A hour-

glass BCs could essentially exist in some kpath but their

splitting patterns can be different. Among these essential

hourglass BCs with different splitting patterns, we ﬁnd that

MSGs 131.45, 134.481, 135.483, 135.484, 135.487, 135.493,

138.520 and 138.522 can host multiple essential hourglass

band structures which can be tuned to be connected with each

other, as shown in Fig. 1(f).

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Allhourglassbosonicexcitationsinthe1651magneticspacegroupsand528magneticlayergroupsDongzeFan,1,2XiangangWan,1,2,3andFengTang1,2,1NationalLaboratoryofSolidStateMicrostructuresandSchoolofPhysics,NanjingUniversity,Nanjing210093,China2CollaborativeInnovationCenterofAdvancedMicrostructures,NanjingUniver...

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All hourglass bosonic excitations in the 1651 magnetic space groups and 528 magnetic layer groups

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