Encyclopedia of emergent particles in 528 magnetic layer groups and 394 magnetic rod groups Zeying Zhang1 2Weikang Wu3Gui-Bin Liu4 5Zhi-Ming Yu4 5Shengyuan A. Yang2yand Yugui Yao4 5z

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Encyclopedia of emergent particles in 528 magnetic layer groups and 394 magnetic

rod groups

Zeying Zhang,1, 2 Weikang Wu,3, ∗Gui-Bin Liu,4, 5 Zhi-Ming Yu,4, 5 Shengyuan A. Yang,2, †and Yugui Yao4, 5, ‡

1College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China

2Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore

3Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials,

Ministry of Education, Shandong University, Jinan 250061, China

4Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE),

Beijing Institute of Technology, Beijing 100081, China

5Beijing Key Lab of Nanophotonics & Ultraﬁne Optoelectronic Systems,

School of Physics, Beijing Institute of Technology, Beijing 100081, China

We present a systematic classiﬁcation of emergent particles in all 528 magnetic layer groups and

394 magnetic rod groups, which describe two-dimensional and one-dimensional crystals respectively.

Our approach is via constructing a correspondence between a given magnetic layer/rod group and

one of the magnetic space group, such that all irreducible representations of the layer/rod group

can be derived from those of the corresponding space group. Based on these group representations,

we explicitly construct the eﬀective models for possible band degeneracies and identify all emergent

particles, including both spinless and spinful cases. We ﬁnd that there are six kinds of particles

protected by magnetic layer groups and three kinds by magnetic rod groups. Our work provides a

useful reference for the search and design of emergent particles in lower dimensional crystals.

I. INTRODUCTION

The research on topological semimetals in the past

decade has driven extensive eﬀorts in understanding vari-

ous emergent particles enabled by the crystalline symme-

try [1–3]. In these crystals, novel kinds of quasi-particle

states emerge around band degeneracies in the momen-

tum space, and their physical properties are determined

by the character of the band degeneracies. For example,

Weyl and Dirac particles can be realized around twofold

and fourfold band nodal points in crystals, known as

Weyl and Dirac points, respectively [3–6]. These notions

are not limited to electronic systems of real materials, but

also extended to many artiﬁcial crystals such as acous-

tic/photonic crystals [7–9], electric circuit arrays [10–12],

and mechanical networks [13,14].

A main task in this research is to classify all possible

types of emergent particles. The classiﬁcation is typi-

cally based on the dimension of the degeneracy mani-

fold, the degree of degeneracy, the band dispersion, and

the topological charge. For instance, in three dimensions

(3D), besides nodal points, the band degeneracies may

also form nodal lines [15–18] or nodal surfaces [19–21];

the degeneracy for a nodal point could be 2, 3, 4, 6,

and 8 [22–24]; the emergent particle may have linear,

quadratic or cubic dispersion [25–28]; and they could

have a maximal chiral charge of four [24,29]. All these

properties are eventually determined by the symmetry

of the bands that form the degeneracy, or more specif-

ically, how these bands represent the symmetry group

∗weikang wu@sdu.edu.cn

†shengyuan yang@sutd.edu.sg

‡ygyao@bit.edu.cn

of the system. For 3D crystals, the pertinent symme-

try groups are the (magnetic) space groups. Their (co-

)representations have been extensively studied and well

documented in the past [30] (in the remainder of this let-

ter the “representation” means representation for unitary

group and co-representation for magnetic group). Based

on the knowledge of space group representations, an en-

cyclopedia of emergent particles in 3D crystals has been

established in recent works [24,31–34].

Recent years also witnessed a rapid development in

the realization of low-dimensional crystals. Many 2D

layered materials and 1D (or quasi-1D) crystals have

been synthesized in experiment [35–37]. Because lower

dimensions permit a high controllability and easier de-

tection, emergent particles may have even stronger im-

pact in these systems. As a prominent example, many

intriguing properties of graphene can be attributed to its

emergent Dirac fermions [38]. So far, there have been

works on studying speciﬁc kind of emergent particles in

2D [39] and on systematic construction of k·pmodels

for 2D systems [34,40,41]. However, a comprehensive

classiﬁcation for all emergent particles in magnetic layer

groups (MLGs) and magnetic rod groups (MRGs), which

apply for 2D and 1D crystals respectively, has not been

achieved.

One obstacle here is that the irreducible representa-

tions (IRRs) have not been completely derived for these

groups, but only for certain subgroups (such as the type-

I and type-II MLGs) [42,43]. In this work, we develop

an approach to compute IRRs for all 528 MLGs and

394 MRGs. The approach is based on making a cor-

respondence between a given MLG/MRG and one of the

magnetic space groups. We show that all IRRs of the

MLG/MRG can be derived by restricting the IRRs of

the corresponding space group under a constraint. After

obtaining IRRs for these groups, we identify all possible

arXiv:2210.11080v2 [cond-mat.mtrl-sci] 25 Oct 2022

Subperiodic

group

𝒮

Construct

magnetic space

group

𝒢

from

𝒮

𝒢 = 𝒩 ⋉ 𝒮

Find rotation

matrix

𝑄

and shift

of origin

ℓ

align coordinate

system

Obtain the IRR of

𝒮

𝜌 ↓ 𝒮

from the IRR

𝜌

𝒢

Construct

𝒌 ⋅ 𝒑

effective model

around

degeneracy

Classification of

emergent particles

FIG. 1. Flow chart for classiﬁcation of emergent particles in subperiodic groups.

protected band degeneracies and classify the associated

emergent particles, for both spinless and spinful systems.

We ﬁnd six kinds of emergent particles in MLGs and

three kinds in MRGs, as listed in Table I. Our work of-

fers a comprehensive reference for the investigation of

emergent particles in low dimensional crystals.

II. DERIVE IRRS OF SUBPERIODIC GROUPS

MLGs and MRGs are subperiodic groups in 3D, mean-

ing that their translational parts form a 2D or 1D sub-

space of 3D. Each of their point groups remains one of the

3D crystallographic point groups. To derive their IRRs,

one can of course pursue the standard way as in Ref. [30],

e.g., by studying little co-groups and using the method

of projective representations. Here, we shall adopt an

alternative approach, in which we obtain the IRRs of a

magnetic subperiodic group from a constructed magnetic

space group.

A. General approach

Our approach is a uniﬁed treatment for both MLGs

and MRGs. Consider a given subperiodic group S[44].

We ﬁrst construct a magnetic space group Gfrom S.

This is done by taking a lattice translation group N,

which consists of translations normal to the subspace for

S. For a MLG, Nis the 1D lattice translations normal to

the plane. For a MRG, Ncontains the 2D translations

normal to the line. Then the space group Gis constructed

as the semidirect product:

G=NoS.(1)

In the Supplemental Material (SM), we present the cho-

sen Gfor each MLG and MRG in Table S1 [45].

By deﬁnition, Nis a normal subgroup of G, and Sis

isomorphic to G/N, with the isomorphism

φ:s∈ S 7→ N s∈ G/N.(2)

Thus, to get IRRs of Sis equivalent to obtain the IRRs

of the quotient group G/N.

Since Gis one of the magnetic space groups, its IRRs

are already known (here obtained by using the MSG-

Corep package) [46,47]. With this information, all IRRs

of Scan be obtained from restricting IRRs of Gto S.

However, not all IRRs of Glead to IRRs of Sunder

restriction. For some IRRs of G, the restriction to its

subgroup would lead to reducible representations. Then,

what are the suitable IRRs ρof Gthat we need to con-

sider? From group representation theory, these are the

IRRs which satisfy the condition that ker ρcontains N

as a subgroup [48], i.e.,

N ⊆ ker ρ={g∈ G|ρ(g) is identity matrix}.(3)

In other words, the normal translations must be repre-

sented as identity matrix in the IRR ρof G. The restric-

tion of such ρto S, i.e., ρ↓ S, is indeed an IRR of S. This

can be easily veriﬁed from the corresponding restricted

character χ↓ S, where χis the character of ρ. Recall

that a representation ρof Gis an IRR if and only if its

character satisﬁes hχ, χiG= 1, hθ, φiGdenotes the inner

product of characters for group G. Now, the restricted

character χ↓ S satisﬁes

hχ↓ S, χ ↓ SiS=1

|S| X

s∈S

χ(s)χ∗(s)

|S||N | X

n∈N X

s∈S

χ(ns)χ∗(ns)

|G| X

g∈G

χ∗(g)χ(g) = hχ, χiG,

(4)

where in the second step, we used the fact that χ(ns) =

χ(s) since N ⊆ ker ρ. Thus, the restricted representation

ρ↓ S for Sshares the same irreducibility as ρfor G.

In a similar way, one can show that the indicator func-

tion for such ρis also preserved in the process of restric-

tion to S, i.e., Ps∈SA(χ↓ S)(s2) = Pg∈GAχ(g2), where

SAand GAare anti-unitary part of Sand Grespectively.

Thus, the restricted IRR for Sshares the same type of

corepresentation as the original IRR for G.

B. Algorithm for aligning coordinate systems

It is not diﬃcult to identify the IRRs of Gthat satisfy

condition (3). For example, for Sdescribing a 2D system

in the x-yplane, one can easily see that the required

IRR for Gshould correspond to kz= 0 plane of the 3D

Brillouin zone (BZ). Similarly, for a MRG describing a

1D system along z, the IRR for Gshould correspond to

kx=ky= 0 path of the BZ.

There is another technical issue arising in practical cal-

culations. Usually, the coordinate system for the stan-

dard setting of a magnetic space group, as in well-known

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Encyclopediaofemergentparticlesin528magneticlayergroupsand394magneticrodgroupsZeyingZhang,1,2WeikangWu,3,Gui-BinLiu,4,5Zhi-MingYu,4,5ShengyuanA.Yang,2,yandYuguiYao4,5,z1CollegeofMathematicsandPhysics,BeijingUniversityofChemicalTechnology,Beijing100029,China2ResearchLaboratoryforQuantumMaterials,Sin...

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Encyclopedia of emergent particles in 528 magnetic layer groups and 394 magnetic rod groups Zeying Zhang1 2Weikang Wu3Gui-Bin Liu4 5Zhi-Ming Yu4 5Shengyuan A. Yang2yand Yugui Yao4 5z

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