MagneticTB: A package for tight-binding model of magnetic and non-magnetic materials

Abstract

We present a Mathematica program package MagneticTB, which can generate the tight-binding model for arbitrary magnetic space group. The only input parameters in MagneticTB are the (magnetic) space group number and the orbital information in each Wyckoff positions. Some useful functions including getting the matrix expression for symmetry operators, manipulating the energy band structure by parameters and interfacing with other software are also developed. MagneticTB can help to investigate the physical properties in both magnetic and non-magnetic system, especially for topological properties.

Full text

MagneticTB: A package for tight-binding model of magnetic and
non-magnetic materials
Zeying Zhanga, Zhi-Ming Yub,c, Gui-Bin Liub,c,∗, Yugui Yaob,c,∗
aCollege of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China
bCentre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE),
School of Physics, Beijing Institute of Technology, Beijing, 100081, China
cBeijing Key Lab of Nanophotonics & Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology,
Beijing, 100081, China
Abstract
We present a Mathematica program package MagneticTB, which can generate the tight-binding model for
arbitrary magnetic space group. The only input parameters in MagneticTB are the (magnetic) space group
number and the orbital information in each Wyckoff positions. Some useful functions including getting
the matrix expression for symmetry operators, manipulating the energy band structure by parameters
and interfacing with other software are also developed. MagneticTB can help to investigate the physical
properties in both magnetic and non-magnetic system, especially for topological properties.
Program summary
Program title: MagneticTB
Licensing provisions: GNU General Public Licence 3.0
Programming language: Mathematica
External routines/libraries used: ISOTROPY (iso.byu.edu)
Developer’s repository link: https://github.com/zhangzeyingvv/MagneticTB
Nature of problem: Construct the symmetry adopted tight-binding model for the system with arbitrary
magnetic space group.
Keywords: Tight-binding method, Representation theory, Magnetic space group, Mathematica
1. Introduction
Tight-binding method is a powerful tool to investigate the novel properties in condensed mater physics
[1–5]. Compared with first-principles method, tight-binding method can greatly simplify calculations. More-
over, after considering (magnetic) space group symmetry, the tight-binding model can give more reliable
results. For example, in topological materials, symmetry plays an important role to protect the topological
properties, such as Z2topological insulator protected by time reversal symmetry [6], topological crystalline
insulators and topological nodal semimetals protected by space group symmetries [7,8] and magnetic topo-
logical crystalline insulator protected by magnetic space group symmetries, i.e. the combination of space
group operations and time reversal [9,10].
At present, a lot of researchers using symmetry adopted tight-binding model to investigate physical
properties of electronic system [11–21]. However, most of the software packages are mainly focused on the
tight-binding model for non-magnetic materials [22–26] and only a few of them can be used to construct
the symmetry adopted tight-binding model automatically. For the first-principles level, Wannier90 can
generate the Wannier-tight-binding model by interfacing with first-principles software, but the symmetry
∗Corresponding author
Email addresses: gbliu@bit.edu.cn (Gui-Bin Liu), ygyao@bit.edu.cn (Yugui Yao)
Preprint submitted to Elsevier May 21, 2021
arXiv:2105.09504v1 [cond-mat.mtrl-sci] 20 May 2021

adopted Wannier function can not be applied to other first-principles software (such as VASP, ABINIT)
except Quantum-Espresso [27–30]. FPLO can generate symmetry adopted tight-binding model with proper
parameters for given structures [31]. Meanwhile, both Wannier90 and FPLO do not support magnetic
symmetry. For the model level, GTPack, Qsymm and MathemaTB can generate the tight-binding model
with space group symmetry but does not support magnetic symmetry directly [32–34]. WannierTools can
do the symmetrization of non-magnetic tight-binding model but cannot generate the tight-binding model
by itself [35].
Therefore, it is necessary to develop a package which can construct the tight-binding model with magnetic
space group symmetry automatically. Here we introduce a software package: MagneticTB, a tool for
generating the tight-binding model for the system with arbitrary magnetic space group. The required
input information of this package is only the magnetic space group number and the orbital information in
each Wyckoff positions. It can help to investigate the physical properties of the given symmetry. We also
present some useful functions including get the matrix expression for symmetry operators, manipulate the
band structure by parameters and interface with other software.
This paper is organized as follows. In Sec. 2, we give an introduction of symmetry adopted tight-binding
methods. In Sec. 3, The usage of MagneticTB are given, including how to install and run MagneticTB. In
Sec. 4, we give three concrete examples, such examples show the specific capabilities of the MagneticTB.
Finally, conclusions are given.
2. Symmetry adopted tight-binding method
In periodic system, the bases of tight-binding model can be written as Bloch sum [11]
ψn
lmk(r) = 1
√NX
Rj
eik·(Rj+dn
l)ϕn
m(r−Rj−dn
l)(1)
where Nis the number of unit cells in the crystal, Rjis the translation vector of the Bravais lattice, dn
l
is the position of n-th Wyckoff position’s l-th atom in the unit cell (for each Qand dn
lthere exists one
and only one pair of dn
l0and R0which satisfies Qdn
l=dn
l0+R0where Qis arbitrary group element in the
(magnetic) space group and R0is a lattice vector), ϕn
m(r)is the m-th atomic orbital basis for position dn
l
but located at coordinate origin. Then Eq. (1) satisfies the Bloch theorem ψn
lmk(r+Rj) = eik·Rjψn
lmk(r).
The tight-binding Hamiltonian can be written as:
Hnn0
lml0m0(k) = X
Rj
eik·(Rj+dn0
l0−dn
l)Emm0(dn
l,Rj+dn0
l0)
Emm0(dn
j,Rj+dn0
l0) = hϕn
m(r−dn
l)|ˆ
H|ϕn0
m0(r−dn0
l0−Rj)i
(2)
for simplicity, we rewrite the atomic orbitals in vector form: Φn(r−Rj−dn
l) = {ϕn
m(r−Rj−dn
l)},(m=
1, ..., Mn). Then Emm0(dj,Rj+dl0) (m= 1, ..., Mn;m0= 1, . . . , Mn0)form an Mn×Mn0matrix:
E(dn
l,Rj+dn0
l0) = hΦn(r−dn
l)|ˆ
H|Φn0(r−Rj−dn0
l0)i(3)
Then the Hamiltonian can be rewritten as:
Hnn0
ll0(k) = X
Rj
eik·(Rj+dn0
l0−dn
l)E(dn
l,Rj+dn0
l0)(4)
E(dn
l,Rj+dn0
l0)is the hopping matrix between n-th Wyckoff position’s l-th atom to n0-th Wyckoff position’s
l0-th atom. When the lattice is in invariant under some symmetry E(dn
l,Rj+dn0
l0)may ne not independent
for arbitrary dn
land dn0
l0. Fortunately, for symmetry operation Q, group representation theory gives us
explicit expression for the relationship between E(Qdn
j, Q(Rj+dn0
l0)) and E(dn
j,Rj+dn0
l0).
2

(a) (b) Structure
Symmetry
operators
Basis Functions
Bond Hamiltonian
w/o symmetry
D(R) and D(RT) P(Q) Symmetry adopted
Hamiltonian
Qdl’
x
y
dl’
C4
2
2
Figure 1: (a) Sketch of relationship between E(dn
j,Rj+dn0
l0)and E(Qdn
j, Q(Rj+dn0
l0)), in this example we set Q=C4T,
Φ1(r) = {s}locate at d1
l= (0,0),Φ2(r) = {px, py}locate at d2
l0= (λ, 0)(λ6= 0). Then we have D1(C4T)=1, D2(C4T) =
−iσy, and E(Qd1
l, Qd2
l0) = E∗(d1
l,d2
l0)×(−iσy). (b) Workflow of MagneticTB.
For the case that symmetry operation does not contain the time reversal T, i.e. Q={R|t}, where R
and tare the rotation and translation part of Qrespectively, we have
E(Qdn
l, Q(Rj+dn
l0)) = Dn(R)E(dn
l,Rl+dn0
l0)Dn0†(R)(5)
For the case that operation Qcontains time reversal symmetry T, i.e. Q={R|t}T , we have
E(Qdn
l, Q(Rj+dn
l0)) = Dn(RT)E∗(dn
l,Rj+dn0
l0)Dn0†(RT)(6)
Where Dn(R)(Dn(RT)) are the Mn×Mnrepresentation matrices of R(RT)under atomic orbital bases
Φn(r)(not necessarily irreducible representations), E∗(dn
l,Rj+dn0
l0)is complex conjugate of E(dn
l,Rj+dn0
l0)
(see Fig. 1for example). It is clear that for spinless cases with time reversal symmetry T, when the basis
functions are real, D(T)is equal to identity matrix, indicating that E(dn
l,Rj+dn0
l0)are real matrices.
The next step is to get the analytical expressions of Dn(R)(Dn(RA)). For a fixed n, we don’t have to
worry about mixing superscripts of nin Dn(R)because transformations can only occur under the same n.
So we temporarily use D(R)rather than Dn(R)in this step. Consider the following four cases:
i. Spinless system and Qdoes not contain T.
ii. Spinless system and Qcontain T.
iii. Spinful system and Qdoes not contain T.
iv. Spinful system and Qcontain T.
In case. i, D(R)can be obtained simply by solve the linear equation [36]
ˆ
RΦ(r) = Φ(R−1r) = Φ(r)D(R)(7)
in which ˆ
Ris the function operator for the rotation R. In case. ii, we define Φ(r) = ˆ
TΦ(r), for spinless
system T=K, hence, Φ(r) = Φ∗(r)and then solve the linear equation
ˆ
Rˆ
TΦ(r) = Φ(R−1r) = Φ(r)D(RT)(8)
In case. iii, since the spin matrix is orbital independent we define the basis function as
Φs(r) = {Φ(r)↑, Φ(r)↓} (9)
3

The two spinors ↑= (1,0)T, and ↓= (0,1)T,and under the rotation ˆ
Rthey are transformed according to
ˆ
R(↑,↓)=(↑,↓)D1
2(R)(10)
For proper rotation R,D1
2(R) = exp(−1
2iαn·ˆ
σ), where αis the rotation angle of R,nis the unit vector
along rotation axis, for improper rotation S,R=IS,Iis the inversion symmetry, D1
2(S) = D1
2(R)[37,38].
Then D(R)can be obtained by solving the linear equation
ˆ
RΦs(r) = Φs(r)D(R)(11)
Case. (iv) is similar to case. (ii), the only difference is replace the time reversal operator T=Kby T=iˆσyK
and consider the spin rotation matrices. The above four cases cover all the possibilities of D(R)and D(RT).
Then the operator (or representation matrix) for Qcan be defined as
Pnn0
ll0(Q) = 


δnn0˜
δdn
l,Qdn0
l0Dn(R)Q does not contain T
δnn0˜
δdn
l,Qdn0
l0Dn(RT)Q contains T(12)
where˜
δdn
l,Qdn0
l0is equal to 1 only when dn
land Qdn0
l0differ by a lattice vector and to 0 otherwise (it can also
be written as ˜
δdn
l,Qdn0
l0=δdn
l,Qdn0
l0+Rsif a suitable lattice vector Rsis choosen). The Hamiltonian under
constraint of Qcan be written as
P(Q)−1H(k)P(Q) = H(R−1k)(13)
for Q={R|t}, and
P(Q)−1H(k)P(Q) = H∗(−R−1k)(14)
for Q={R|t}T [See Appendix A for proof of Eqs.(5–6) and Eqs.(13–14)). Eqs.(13–14) are key point to
generate the symmetry adopted tight-binding model. In MagneticTB we first generate the Hamiltonian with
only translation symmetry and then use Eqs.(13–14) to the simplify the Hamiltonian (see Fig. 1(b)) for the
workflow of MagneticTB]. The database for tight-binding model of 1651 magnetic space group can be found
in our later work [39].
3. Capabilities of MagneticTB
3.1. Installation
To install the MagneticTB, unzip the "MagneticTB.zip" file and copy the MagneticTB directory to any
of the following four paths:
•FileNameJoin[{$UserBaseDirectory, "Applications"}]
•FileNameJoin[{$BaseDirectory, "Applications "}]
•FileNameJoin[{ $InstallationDirectory , "AddOns", "Packages"}]
•FileNameJoin[{ $InstallationDirectory , "AddOns", "Applications"}]
Then one can use the package after running Needs["MagneticTB‘"]. The version of Mathematica should higher
or equal to 11.0.
4

3.2. Running
3.2.1. Core module
To initialize the program one should identify the magnetic space group and the orbital information in
each Wyckoff positions. Here we provide a function msgop to show the symmetry information of an arbitrary
magnetic space group. The only input of the msgop is the magnetic space group number, one can get the
symmetry information by the following code:
msgop[gray[191]]
msgop[bnsdict[{191, 236}]]
msgop[ogdict[{191, 8, 1470}]]
Magnetic space group (BNS): {191.236,P6’/mm’m}
Primitive Lattice vactor : {{a,0,0},{−a/2,(Sqrt[3] a) /2,0},{0,0, c}}
Conventional Lattice vactor: {{a,0,0},{−a/2,(Sqrt[3] a) /2,0},{0,0, c}}
{{"1" ,{{1,0,0},{0,1,0},{0,0,1}},{0,0,0}, F},
{"3z" ,{{0,−1,0},{1,−1,0},{0,0,1}},{0,0,0}, F},
{"3z−1",{{−1,1,0},{−1,0,0},{0,0,1}},{0,0,0},F},
{"2x",{{1,−1,0},{0,−1,0},{0,0,−1}},{0,0,0},F},
...
here gray[191] return the magnetic space group code of gray space group 191, bnsdict [{191,236}] return the
magnetic space group code of BNS No. 191.236, ogdict [{191,8,1470}] return the magnetic space group code
of OG No. 191.8.1470. Then the msgop will print the standard lattice vector and the symmetry operations
(for primitive cell) of the corresponding magnetic space group, which can be the input of the init function.
Notice the magnetic space group code is build-in constant in MagneticTB, users should use gray , bnsdict ,
ogdict functions rather than inputting the magnetic space group code directly.
Then one can feed the above information to init function then the basic results of input structure
can be generated. The init function have five mandatory options neamly, lattice ,lattpar ,wyckoffposition ,
symminformation and basisFunctions (in ordinary Wolfram language the options for functions are optional, but
such five options must be specified in MagneticTB in order to make the input clear). The lattice is the
lattice vector of magnetic system which can contain parameters, lattpar is the parameters in lattice vector to
determine the bond length of magnetic system,and wyckoffposition is a list to designate atomic position and
magnetization-direction for each Wyckoff positions in the magnetic system. The format of wyckoffposition is:
{{a1,m1},{a2,m2}, ...}
where aiand mirepresent one of the atomic positions and its magnetization-directions of the i-th Wyckoff
position, respectively. The symminformation contain the elements of the coset of magnetic space group with
respect to translation group, which can direct use the output of msgop (notice that the output of msgop is
standard symmetry operation from ISOTROPY [40,41]. However, users can also use the non-standard
structure as input, not limit to the output of msgop). The format of symminformation is:
{{n1, R1, t1, A1},{n2, R2, t2, A2}, ...}
where niis the name of symmetry operation, Riand tiare the rotation and translation part of symmetry
operation, and Airepresents whether the symmetry operation is combined with time reversal symmetry
("T" for true and "F" for false). Finally, the basisFunctions is the basis function for each Wyckoff position, The
format of basisFunctions is:
{b1, b2, ...}
where biis the list of basis functions of the i-th Wyckoff position. The build-in basis functions for spinless
case in MagneticTB is shown in Table 1.
When spin is considered, for build-in basis functions, add "up" or "dn" after basis functions string, e.g.
for |px↑i, the basis function for spin-up case is "pxup". However, users may use other basis functions such
as f,|3/2,1/2iorbitals. In such cases, users can directly input the analytical expression of basis functions.
For example, if only consider fxyz orbital, one should input basisFunctions −> {{x∗y∗z}}]. The analytical
expressions of basis functions can be simply obtained from quantum mechanics or group theory books
[32,42,43].
5

Table 1: String codes representing basis functions and available values for basisFunctions
Basis function String Basis function String
s"s" px"px"
py"py" pz"pz"
px+ipy"px+ipy" px−ipy"px−ipy"
dz2"dz2" dxy "dxy"
dyz "dyz" dxz "dxz"
dx2−y2"dx2−y2"
sgop=msgop[gray[191]];
init [
lattice −> {{a,0,0},{−(a/2),(Sqrt[3] a) /2,0},{0,0, c}},
lattpar −> {a −> 1, c −> 3},
wyckoffposition −> {{{1/3, 2/3, 0}, {0, 0, 0}}},
symminformation −> sgop,
basisFunctions −> {{"pz"}}];
Table 2: Basic results of init
properties illustrate of properties
atompos atomic position and magnetization-direction for each atom
wcc dlfor each basis function
reclatt reciprocal lattice vector for given structure
symmetryops PQfor each symmetry operation
unsymham generate the Hamiltonian with only translation symmetry
symmcompile summary of P(Q), see main text for detail
bondclassify summary of bonds information, see main text for detail
After inputting the above five options appropriately, one can run init and obtain the basic results. Here
we introduce two important basic results: symmcompile and bondclassify , the other properties are given in
Table. 2. The format of symmcompile is
{{N1,{n1, R1, t1, A1}, P1, Rk
1},{N2,{n2, R2, t2, A2}, P2, Rk
2}, ...}
where Ni,Pi,Rk
iare the label, the symmetry operator (Eq.(12)) and the rotation acting on kspace of the
i-th symmetry operation, respectively. For example
symmcompile
{{1,{"1" ,{{1,0,0},{0,1,0},{0,0,1}},{0,0,0}, F},{{1,0},{0,1}},
{{1,0,0},{0,1,0},{0,0,1}}},
{2,{"6z" ,{{1,−1,0},{1,0,0},{0,0,1}},{0,0,0}, F},{{0,1},{1,0}},
{{0,−1,0},{1,1,0},{0,0,1}}},
...}
The format of bondclassify is
{{li, ni,{{pi,j , pi,k}, ...}}, ...}
where liis the bond length of the (i−1)-th neighbour hopping (i= 1 for on-site hopping), niis the number
of the (i−1)-th neighbour’s bonds, {pi,j , pi,k }is the atomic position of nicorresponding bonds.
Till this moment, symmetry adopted tight-binding model for magnetic system is ready to be generated.
By using the symham[n] function, one can obtain the symmetry adopted tight-binding model. When n= 1,
symham[1] return the Hamiltonian with only on-site hopping, n= 2 return the Hamiltonian with only nearest-
neighbour hopping and so on. By default, MagneticTB will check all the input symmetry operations to ensure
the Hamiltonian is correct. However, it may last long time when the structure is complex. In principle, only
6

the generators of the (magnetic) space group are enough to get the Hamiltonian. Therefore, one can specify
the symmetry operations by symmetryset−>list in symham where list is the list of indexes of the symmetry
operations. For example, symham[2,symmetryset−>{2}] will generate the Hamiltonian with only C6zsymmetry
for nearest-neighbour hopping. symmetryset can not only save computing resource but can also investigate
the Hamiltonian for symmetry breaking cases. The parameters for each neighbour in MagneticTB are given
in Table. 3.
Table 3: String codes representing n-th neighbour hoppings for symham
On-site energy Nearest Second-nearest Third-nearest (k−1)-th nearest
e1, e2 ,.. t1,t2 ,... r1 ,r2 ,... s1,s2 ,... pkn1,pkn2 ,...
3.2.2. Plot module
After tight-binding model being generated, there may exist many parameters, one can use bandManipulate
function to manipulate the band structure to investigate the relationship between band structure and
parameters. The format of bandManipulate is
bandManipulate[{{{k1,k2},{name of k1, name of k2 }},...}, np,Hamiltonian]
where np is the number of kpoints per line. Then one can easily check the band structure of different
parameters. When the proper parameters are obtained, one can use bandplot to plot the band structure
bandplot[{{{k1,k2},{name of k1, name of k2 }},...}, np,Hamiltonian,parameters]
see section. 4for concrete example.
3.2.3. IO module
In MagnetTB, one can get the tight-binding model for magnetic system. However, MagneticTB do not
calculate the other properties (such as surface states, finding the gap-less point and so on) directly, since
it will generally cost too much computing resources. It is better to do such heavy calculations by Fortran,
Python or C. Therefore we develop hopp function to convert the symmetry adopted tight-binding model to
"wannier90_hr.dat" format, which is convenient to interface with WannierTools [35], Z2Pack [44], PythTB
[45] and our home-made package Wannflow [46,47]. The "wannier90_hr.dat" in Wannier90 use the following
convention [27], namely conventions II:
˜
ψn
lmk(r) = 1
√NX
Rj
eik·Rjϕn
lm(r−Rj−dn
l)
˜
Hnn0
lml0m0(k) = X
Rj
eik·RjEmm0(dn
l,Rj+dn0
l0)
(15)
which is different form MagneticTB in Eq. (2), the relationship between two conventions is
˜
H(k) =V(k)H(k)V†(k)
Vnn0
ll0(k) =eik·dn
lδll0δnn0
(16)
In MagneticTB (convention I) the operation matrix defined in Eq. (12) is kindependent while the Hamil-
tonian is non-periodic by shifting the reciprocal vector G
H(k+G) =V†(G)H(k)V(G)(17)
By contrast, in conventions II the Hamiltonian is periodic. i.e ˜
H(k+G) = ˜
H(k). The format of hopp
function is
7

hopp[Hamiltonian,parameters]
See section. 4for concrete example. One can also use symmhamII[ham] to generate the Wolfram expression for
Hamiltonian in convention II. Notice that hopp function (but not symmhamII) is only applied to the output of
symham function, and that expressions explicitly including Sin or Cos may not work well. Be careful to use it.
4. Examples
4.1. Three-band tight-binding model for MoS2
MoS2monolayer has direct bandgap in the visible range, strong spin-orbit coupling, and rich valley
related physics, which make it an candidate for nanoelectronic, optoelectronic, and valleytronic applications
[48,49]. The space group of MoS2is P6m2(space group No. 187). Considering the Mo atom at 1aWyckoff
position and using the dz2,dxy , and dx2−y2orbitals, the model can be obtained by
sgop = msgop[gray[187]];
tran = {{1, −1, 0}, {0, 1, 0}, {0, 0, 1}};
sgoptr = MapAt[FullSimplify[tran.#.Inverse@tran] &, sgop, {;; , 2}];
init [
lattice −> {{1, 0, 0}, {1/2, Sqrt [3]/2, 0}, {0, 0, 10}},
lattpar −> {},
wyckoffposition −> {{{0, 0, 0}, {0, 0, 0}}},
symminformation −> sgoptr,
basisFunctions −> {{"dz2", "dxy", "dx2−y2"}}];
mos2 = Sum[symham[i, symmetryset −> {9, 11, 13}], {i, {1, 2}}];
mos2Liu = mos2 /. Thread[{kx, ky, kz} −> ({kx, ky, kz} (2 Pi)) . Inverse@reclatt ];
This gives exactly the same results as in Ref. [14]. The relationship of parameters between Ref. [14] and
MagneticTB are
Ref. [14]12t0t1t2t11 t12 t22
MagneticTB e1 e2 t1 t2 t4 t3 t5 t6
4.2. Graphene
Graphene with linear dispersion around Fermi level is one of the most important materials in spintronics
[50]. The magnetic space group of graphene is P6/mmm10(BNS No. 191.234). There are two C atoms at
2cWyckoff position, and the bands near Fermi energy are mainly from pzorbital. The above information is
enough to establish the tight-binding model near Fermi energy of graphene. The model can be obtained as
follow
Needs["MagneticTB‘"]
sgop = msgop[gray[191]];
init [
lattice −> {{a,0,0},{−(a/2),(Sqrt [3] a) /2,0},{0,0, c }},
lattpar −> {a −> 1, c −> 3},
wyckoffposition −> {{{1/3, 2/3, 0}, {0, 0, 0}}},
symminformation −> sgop,
basisFunctions −> {{"pz"}}];
ham = Sum[symham[i], {i, 3}]; MatrixForm[ham]
output:
"e1+ 2r1(cos(kx+ky) + cos kx+ cos ky)t1ei−2kx
3−ky
3+t1eikx
3−ky
3+t1eikx
3+2ky
3
†e1+ 2r1(cos(kx+ky) + cos kx+ cos ky)#
For spin-orbital coupling case, the only thing which needs to change is the basis functions
basisFunctions −> {{"pzup", "pzdn"}}
8

and the corresponding Hamiltonian reads




e1+h−0h00
0e1+h+0h0
†0e1+h+0
0†0e1+h−




where h±=±2r1(sin kx+ sin ky−sin kx+ky)+2r2(cos(kx+ky) + cos kx+ cos ky),h0=t1ei−2kx
3−ky
3+
t1eikx
3−ky
3+t1eikx
3+2ky
3. Such model is corresponding to the first Z2topological insulator [6]. After get
the Hamiltonian, we can use bandManipulate and bandplot to plot the band structure, and click the "ExportData"
button to print the value of parameters.
path={
{{{0,0,0},{0,1/2,0}},{ "\[ CapitalGamma]","M"}},
{{{0,1/2,0},{1/3,1/3,0}},{"M","K"}},
{{{1/3,1/3,0},{0,0,0}},{"K","\[CapitalGamma]"}}
};
bandManipulate[path, 20, ham]
bandManipulate[path, 20, hamsoc]
bandplot[path, 200, ham, {e1 −> 0, t1 −> 0.5, r1−> 0}]
bandplot[path, 200, hamsoc, {e1 −> 0, t1 −> 0.5, r1−> 0.02, r2−>0}]
ΓM K Γ
-0.8
0.
1.7
0.8
(a) (b)
(d)
(c)
ΓM K Γ
-1.5
-0.8
0.
1.5
0.8
Figure 2: Output of bandManipulate and bandplot for graphene without considering spin (a-b) and with spin (c-d).
Moreover we can get the wannier90_hr.dat by hop function for this model.
hop[hamsoc, {e1 −> 0, r1 −> 0.02, r2 −> 0, t1 −> 0.5}]
Generated by MagneticTB
4
7
9

1111111
−1−1 0 1 1 0.00000000 0.02000000
−1−1 0 2 1 0.00000000 0.00000000
−1−1 0 3 1 0.00000000 0.00000000
−1−1 0 4 1 0.00000000 0.00000000
−1−1 0 1 2 0.00000000 0.00000000
−1−1 0 2 2 0.00000000 −0.02000000
...
4.3. Magnetic C-3 Weyl point
The charge-3 (C-3) Weyl point is a 0D two-fold band degeneracy with Chern number |C|= 3. Encyclo-
pedia of emergent particles tell us that the C-3 Weyl point always appear at least in a pair or coexist with
nodal surface in nonmagnetic systems [51]. Here we confirm that in magnetic system, due to the breaking of
time reversal symmetry T, the C-3 Weyl point can uniquely coexist with conventional Weyl points. Consider
the type IV magnetic space group Pc3(BNS No. 143.3). The generator of the group is C3zand {E|00 1
2}T ,
Put |px+ipy↑i,|px−ipy↓i basis functions at Wyckoff position 2a, and then the symmetry operator for
C3zand E{001
2}T are
C3z=−σz
{E|001
2}T =iσy
Under this bases the effective Hamiltonian at Γpoint can be written as
HC-3 WP =+αkzσx+ck2
k+β(kx+e−iπ
3ky)3σz+h.c. (18)
where , c are real parameters and α, β are complex parameters. Besides, there are another three essential
Weyl points locate at (π, 0,0),(0, π , 0),(π, π, 0). Since the C3zsymmetry does not change the Chern number
of Weyl points, the Chern number at Mhas to be ±1. According to no-go theorem, the Chern number of
Γis ∓3. One can easily check that the Chern number of Eq. (18) is ±3. The degeneracies of Γand Mare
because ({E|00 1
2}T )2=−1at (0/π, 0/π, 0). The model can be obtained as follow
sgop = msgop[bnsdict[{143, 3}]];
init [ lattice −> {{Sqrt[3]/2, −( 1/2), 0}, {0, 1, 0}, {0, 0, 2}},
lattpar −> {},
wyckoffposition −> {{{0, 0, 0}, {0, 0, 1}}},
symminformation −> sgop,
basisFunctions −> {{{x + Iy, 0}, {0, x −Iy}}}];
c3w = Sum[symham[i], {i, {2, 4}}];
c3w2band = Table[c3w[[i, j ]], { i , {1, 4}}, { j , {1, 4}}];
The band structure of c3w2band is shown in Fig. 3(a).
4.4. Magnetic cubic nodal-line
Topological high order nodal line is that the energy difference between the bands are non-linear,and
the order of energy dispersion around the degeneracy points plays an important role in different physical
properties, such as density of states, Berry phase and Landau-level [52]. Recently, Zhang et. al. proposed
high order nodal-line in magnetic system [53]. In this example, we use MagneticTB to generate magnetic
cubic nodal-line. Generally, the magnetic cubic nodal-line is protected by C6zand Mxsymmetries, there
are many magnetic space groups which contains the above two symmetries. Consider the type IV magnetic
space group Pc6cc (BNS No. 184.196), and put |px+ipy↑i,|px−ipy↓i basis functions at Wyckoff position
2a. Then the tight-binding model can be generated by
sgop = msgop[bnsdict[{184, 196}]];
init [ lattice −> {{Sqrt[3]/2, −1/2, 0}, {0, 1, 0}, {0, 0, 2}},
lattpar −> {},
10

(a) (b)
ΓM K ΓA L H A|L M|K H
-1.
-0.5
0.
1.
0.5
Figure 3: (a) Output of bandplot for magnetic C-3 Weyl point, (b) Output of bandManipulate for magnetic cubic nodal-line.
wyckoffposition −> {{{0, 0, 0}, {0, 0, 1}}},
symminformation −> sgop,
basisFunctions −> {{{x + Iy, 0}, {0, x −Iy}}}];
cnl = Sum[symham[i, symmetryset −> {2, 7, 13}], {i, 1, 5}];
cnl2band = Table[cnl[[ i , j ]], { i , {1, 4}}, { j , {1, 4}}];
MatrixForm[cnl2band]
path = {{
{{0, 1/10, 1/4}, {0, 0, 1/4}}, {"Q", "P"}},
{{{0, 0, 1/4}, {−1/10, 1/10, 1/4}}, {"P", "Q"}}};
bandManipulate[path, 20, cnl2band]
One can check that no mater how the parameters change, the dispersion of arbitrary point along Γ-Aon
kx-kyplane are non-linear, see Fig. 3(b) which is consistent with Ref. [53].
5. Conclusion
In conclusion, we have developed a software package to generate the symmetry-adopted tight-binding
model for arbitrary magnetic space group. The input parameters for MagneticTB are clear and easy to set
and both spinless and spinful Hamiltonian can be generated automatically. Besides, some useful functions
such as manipulating the band structure, interfacing with other software are implemented, which can be
used for further study on the magnetic systems. Moreover, MagneticTB can not only be used to investigate
physical properties of electronic systems, but also be used to study photonics, ultracold, acoustic and
mechanical systems [54–56]. Finally, an exciting direction for future is to apply the magnetic field for the
tight-binding model in MagneticTB [57].
Acknowledgments
ZZ acknowledges the support by the NSF of China (Grant No. 12004028), the China Postdoctoral
Science Foundation (Grant No. 2020M670106), the Fundamental Research Funds for the Central Univer-
sities (ZY2018). GBL acknowledges the support by the National Key R&D Program of China (Grant No.
2017YFB0701600). YY acknowledges the support by the National Key R&D Program of China (Grant No.
2020YFA0308800), the NSF of China (Grants Nos. 11734003, 12061131002), the Strategic Priority Research
Program of Chinese Academy of Sciences (Grant No. XDB30000000).
11

Appendix A.
In this appendix, we use the translation operation T(d). For Q={R|v}, we have QT (d) = {R|v}{E|d}=
{R|Rd+v}=T(Qd)R. Thus
E(dn
j,(Rj+dn0
l0)) = hˆ
T(dn
l)Φn(r)|ˆ
Q†ˆ
Hˆ
Q|ˆ
T(dn0
l0+Rj)Φn0(r)i
=hˆ
Qˆ
T(dn
l)Φn(r)|ˆ
H|ˆ
Qˆ
T(dn0
l0+Rj)Φn0(r)i
=hˆ
T(Qdn
l)RΦn(r)|ˆ
H|ˆ
T(Q(dn0
l0+Rj))RΦn0(r)i
=Dn†(R)hˆ
T(Qdn
l)Φn(r)|ˆ
H|ˆ
T(Q(dn0
l0+Rj))Φn0(r)iDn0(R)
=Dn†(R)E(Qdn
j, Q(Rj+dn0
l0))Dn0(R)
(A.1)
which completes the proof of Eq.(5).
For Q={R|v}T , notice time reversal symmetry does not change the real space coordinates, i.e. Td=d,
we have QT (d) = {R|v}T {E|d}={R|Rd+v}T =T(Qd)RT. Use the fact that for anti-unitary operator
ˆ
A,hˆ
Aψ|ˆ
Aφi=hψ|φi∗, then
E∗(dn
j,(Rj+dn0
l0)) = hˆ
Qˆ
T(dn
l)Φn(r)|ˆ
Qˆ
H|ˆ
T(dn0
l0+Rj)Φn0(r)i
=hˆ
Qˆ
T(dn
l)Φn(r)|ˆ
H|ˆ
Qˆ
T(dn0
l0+Rj)Φn0(r)i
=hˆ
T(Qdn
l)RTΦn(r)|ˆ
H|ˆ
T(Q(dn0
l0+Rj))RTΦn0(r)i
=Dn†(RT)hˆ
T(Qdn
l)Φn(r)|ˆ
H|ˆ
T(Q(dn0
l0+Rj))Φn0(r)iDn0(RT)
=Dn†(RT)E(Qdn
j, Q(Rj+dn0
l0))Dn0(RT)
(A.2)
which completes the proof of Eq.(6).
Proof of Eq.(13),
[H(k)P(Q)]nn0
ll0=X
µν
H(k)nν
lµ Pνn0
µl0(Q)
=X
µν X
Rj
eik·(Rj+dν
µ−dn
l)E(dn
l,Rj+dν
µ)δνn0δdν
µ,Qdn0
l0+RsDn0(R)
=X
Rj
eik·(Rj+Qdn0
l0+Rs−dn
l)E(dn
l,Rj+Qdn0
l0+Rs)Dn0(R)
Rj+Rs→RRj
=========== X
Rj
eik·R(Rj+dn0
l0−Q−1dn
l)E(dn
l, Q(Rj+dn0
l0))Dn0(R)
use Eq. (5)
========= X
Rj
eiR−1k·(Rj+dn0
l0−Q−1dn
l)Dn(R)E(Q−1dn
l,Rj+dn0
l0)
(A.3)
12

[P(Q)H(R−1k)]nn0
ll0=X
µν
Pnν
lµ (Q)H(R−1k)νn0
µl0
=X
µν X
Rj
δnν δdn
l,Qdν
µ+RsDn(R)eiR−1k·(Rj+dn0
l0−dν
µ)E(dν
µ,Rj+dn0
l0)
=X
Rj
Dn(R)eiR−1k·(Rj+dn0
l0−Q−1(dn
l−Rs))E(Q−1(dn
l−Rs),Rj+dn0
l0)
=X
Rj
eiR−1k·(Rj+dn0
l0−Q−1dn
l+R−1Rs)Dn(R)E(Q−1dn
l−R−1Rs,Rj+dn0
l0)
=X
Rj
eiR−1k·(Rj+dn0
l0−Q−1dn
l+R−1Rs)Dn(R)E(Q−1dn
l,Rj+dn0
l0+R−1Rs)
Rj+R−1Rs→Rj
============= X
Rj
eiR−1k·(Rj+dn0
l0−Q−1dn
l)Dn(R)E(Q−1dn
l,Rj+dn0
l0)
(A.4)
In the above derivation we use the relation Q−1(dn
l−Rs) = Q−1dn
l−R−1Rs(Q−1is not linear). Compare
the last lines of the above two equations and we can find they are equal to each other, i.e.
[H(k)P(Q)]nn0
ll0= [P(Q)H(R−1k)]nn0
ll0⇒H(k)P(Q) = P(Q)H(R−1k)(A.5)
Proof of Eq.(14): similar to Eq.(A.3) and Eq.(A.4) we have
[H(k)P(Q)]nn0
ll0=X
µν
H(k)nν
lµ Pνn0
µl0(Q)
=X
µν X
Rj
eik·(Rj+dν
µ−dn
l)E(dn
l,Rj+dν
µ)δνn0δdν
µ,Qdn0
l0+RsDn0(RT)
=X
Rj
eik·(Rj+Qdn0
l0+Rs−dn
l)E(dn
l,Rj+Qdn0
l0+Rs)Dn0(RT)
Rj+Rs→RRj
=========== X
Rj
eik·R(Rj+dn0
l0−Q−1dn
l)E(dn
l, Q(Rj+dn0
l0))Dn0(RT)
use Eq. (6)
========= X
Rj
eiR−1k·(Rj+dn0
l0−Q−1dn
l)Dn(RT)E∗(Q−1dn
l,Rj+dn0
l0)
(A.6)
[P(Q)H∗(−R−1k)]nn0
ll0=X
µν
Pnν
lµ (Q)H∗(−R−1k)νn0
µl0
=X
µν X
Rj
δnν δdn
l,Qdν
µ+RsDn(RT)eiR−1k·(Rj+dn0
l0−dν
µ)E∗(dν
µ,Rj+dn0
l0)
=X
Rj
Dn(RT)eiR−1k·(Rj+dn0
l0−Q−1(dn
l−Rs))E∗(Q−1(dn
l−Rs),Rj+dn0
l0)
=X
Rj
eiR−1k·(Rj+dn0
l0−Q−1dn
l+R−1Rs)Dn(RT)E∗(Q−1dn
l−R−1Rs,Rj+dn0
l0)
=X
Rj
eiR−1k·(Rj+dn0
l0−Q−1dn
l+R−1Rs)Dn(RT)E∗(Q−1dn
l,Rj+dn0
l0+R−1Rs)
Rj+R−1Rs→Rj
============= X
Rj
eiR−1k·(Rj+dn0
l0−Q−1dn
l)Dn(RT)E∗(Q−1dn
l,Rj+dn0
l0)
(A.7)
Compare the last lines of the above two equations and we can find they are equal to each other, i.e.
[H(k)P(Q)]nn0
ll0= [P(Q)H∗(−R−1k)]nn0
ll0⇒H(k)P(Q) = P(Q)H∗(−R−1k)(A.8)
13

It is easy to verify that for A, B not containing Tand C, D containing T,P(Q)obey the following
corepresentation algebra:
P(A)P(B) = P(AB)
P(A)P(C) = P(AC)
P(C)P∗(A) = P(CA)
P(C)P∗(D) = P(CD)
(A.9)
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