PHYSICAL REVIEW B 95, 075135 (2017)
Topological semimetals protected by off-centered symmetries in nonsymmorphic crystals
Bohm-Jung Yang,
1,2,3
Troels Arnfred Bojesen,
4
Takahiro Morimoto,
5
and Akira Furusaki
4,6
1
Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
2
Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, Korea
3
Center for Theoretical Physics (CTP), Seoul National University, Seoul 08826, Korea
4
RIKEN Center for Emergent Matter Science, Wako, Saitama, 351-0198, Japan
5
Department of Physics, University of California, Berkeley, California 94720, USA
6
Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama, 351-0198, Japan
(Received 16 November 2016; revised manuscript received 26 January 2017; published 21 February 2017)
Topological semimetals have energy bands near the Fermi energy sticking together at isolated
points/lines/planes in the momentum space, which are often accompanied by stable surface states and intriguing
bulk topological responses. Although it has been known that certain crystalline symmetries play an important
role in protecting band degeneracy, a general recipe for stabilizing the degeneracy, especially in the presence of
spin-orbit coupling, is still lacking. Here we show that a class of novel topological semimetals with point/line nodes
can emerge in the presence of an off-centered rotation/mirror symmetry whose symmetry line/plane is displaced
from the center of other symmorphic symmetries in nonsymmorphic crystals. Due to the partial translation
perpendicular to the rotation axis/mirror plane, an off-centered rotation/mirror symmetry always forces two
energy bands to stick together and form a doublet pair in the relevant invariant line/plane in momentum space.
Such a doublet pair provides a basic building block for emerging topological semimetals with point/line nodes
in systems with strong spin-orbit coupling.
DOI:
10.1103/PhysRevB.95.075135
I. INTRODUCTION
Dirac particles with a pseudorelativistic energy dispersion
have come to the fore in condensed matter physics research
after the discovery of graphene [
1]. To protect the fourfold
degeneracy at a Dirac point in graphene, two conditions should
be satisfied. One is the simultaneous presence of time-reversal
(T ) and inversion (P ) symmetries, and the other is the absence
of spin-orbit coupling. When these two conditions are satisfied
at the same time, the Berry phase around a Dirac point has a
quantized value of π, which guarantees the stability of the
Dirac point.
Recently, there have been extensive efforts to extend the
physics of the two-dimensional (2D) graphene to three-
dimensional (3D) systems [
2–19]. A natural starting point is
to search for a 3D Dirac point protected by the PT symmetry
and the associated π Berry phase. Interestingly, however, it is
found that the PT symmetry protects a Dirac line node, instead
of a Dirac point, which gives rise to a 3D semimetal with Dirac
line nodes where fourfold band degeneracy occurs along a line
in momentum space [
2–10]. As in the case of graphene, such a
Dirac line node protected by the PT symmetry is unstable in
the presence of the spin-orbit coupling. It is also reported that
a Dirac line node can exist in systems with a mirror symmetry
when two bands with different mirror eigenvalues cross in the
mirror plane [11–16]. However, the resulting line node is also
unstable once the spin-orbit coupling is turned on.
In fact, the existence of a 3D Dirac point in systems with
the spin-orbit coupling requires the introduction of additional
crystalline symmetries other than the time-reversal and the
inversion symmetries [20–26]. Up to now, two different recipes
are known to yield 3D Dirac semimetals with point nodes. One
is to introduce an additional uniaxial rotation symmetry where
3D Dirac points can occur when two bands with different
rotation eigenvalues cross on the rotation axis [
22–24]. Cd
3
As
2
and Na
3
Bi belong to this class [
27–32]. Although a Dirac point
does not carry a nonzero monopole charge which protects
a Weyl point in the case of Weyl semimetals, the rotation
symmetry provides an integer topological charge at the Dirac
point, thus guarantees its stability [25].
The second recipe is to introduce an additional nonsym-
morphic symmetry such as glide mirrors or screw rotations.
When the double point group of a crystal possesses a four-
dimensional irreducible representation, a Dirac point can
appear at the Brillouin zone (BZ) boundary [
20,21]. For sev-
eral representative space groups, projective symmetry group
analysis has been performed, which suggests β-BiO
2
[
20] and
distorted spinel compounds [
21] as candidate systems of 3D
Dirac semimetals belonging to this class. Since each Dirac
point is protected by a different combination of crystalline
symmetries depending on the space group of the crystal,
careful symmetry analysis is required, case by case, to find
the relevant topological charge of each Dirac point.
In this paper, we propose an alternative mechanism to
realize novel 3D semimetals with Dirac point/line nodes in
systems with strong spin-orbit coupling as well as P and
T symmetries. To protect nodal points/lines with fourfold
degeneracy, we find that off-centered crystalline symmetries
play a crucial role. In contrast to the case of ordinary
glide mirror or screw rotation symmetries having a partial
translation in the invariant space of the associated point group
symmetry, an off-centered rotation/mirror symmetry involves
a partial translation that is orthogonal to the invariant space.
In centrosymmetric crystals, such an off-centered symmetry
naturally arises as a combination of a screw/glide symmetry
and inversion symmetry P . An off-centered mirror/rotation
symmetry possesses the characteristics of both the sym-
morphic and nonsymmorphic symmetries. Namely, it has
momentum independent quantized eigenvalues, whereas its
commutation relation with inversion symmetry P depends on
2469-9950/2017/95(7)/075135(16) 075135-1 ©2017 American Physical Society
YANG, BOJESEN, MORIMOTO, AND FURUSAKI PHYSICAL REVIEW B 95, 075135 (2017)
the momentum. Due to such a mixed nature of the off-centered
symmetry, a pair of bands, each with Kramers degeneracy,
form a doublet pair in its invariant space in the first BZ, and
provide a basic building block for nodal points/lines. Similarly,
when the rotation axis (mirror plane) of a screw (glide)
symmetry does not pass the inversion center, an off-centered
screw (glide) symmetry can be defined, which also leads to
doublet pair formation and emerging Dirac points (lines) in
the relevant invariant space. When an external magnetic field
is applied to these semimetals, a Dirac-type point/line node
with fourfold degeneracy splits into two Weyl-type point/line
nodes with twofold degeneracy, with emergent surface states
connecting the split nodes.
The rest of the paper is organized as follows. The nature
of off-centered rotation/mirror symmetries is described in
Sec.
II. In Sec. III (Sec. IV), we explain the basic mechanism
protecting point (line) nodes with fourfold degeneracy by off-
centered rotation (mirror) symmetries. A simple tight-binding
Hamiltonian describing various topological semimetals pro-
tected by off-centered symmetries is proposed in Sec.
V.
The influence of time-reversal breaking on the topological
semimetals is described in Sec. VI, which is followed by
the discussion in Sec. VII. Detailed information about the
tight-binding Hamiltonian is given in Appendix A.The
topological charges of various semimetal phases are defined
in Appendix
B. The stability of nodal and line nodes is further
supported by the Clifford algebras approach described in
Appendix
C. In Appendix D, we perform a low-energy k · p
Hamiltonian analysis. Finally, we explain the properties of
semimetals protected by off-centered screw/glide symmetries
in Appendix E.
II. NATURE OF OFF-CENTERED ROTATION/MIRROR
SYMMETRIES
Generally, a nonsymmorphic symmetry element g ={g|t}
is composed of a point group symmetry operation g and
a partial lattice translation t = t
⊥
+ t
where t
(t
⊥
)is
the component invariant (variant) under the point symmetry
operation g [
33]. For instance, in the case of a nonsymmorphic
mirror symmetry
M ={M|t},wehave
M t
= t
,Mt
⊥
=−t
⊥
. (1)
Since M
2
=−1(M
2
=+1) for particles with a half-integer
(integer) spin, when the nonsymmorphic mirror symmetry
M ={M|t} is operated twice, it should be an element of the
lattice translation group, i.e., {M|t}
2
={M
2
|2t
}∈T, where
T is the group of the pure lattice translation of a given crystal.
Thus 2t
should be a unit lattice translation in the mirror
invariant plane whereas t
⊥
is not influenced by the constraint
above.
In fact, t
⊥
is a fragile quantity whose value depends on
the choice of the reference point of the point group symmetry
operation. For instance, if the reference point for the point
group symmetry operation is shifted by d = d
⊥
+ d
,the
nonsymmorphic mirror symmetry {M|t} also translates to
{M|t − 2d
⊥
}. Thus by choosing 2d
⊥
= t
⊥
, the perpendicular
component of the partial translation can be erased. The
resulting nonsymmorphic mirror symmetry is conventionally
considered as the definition of a glide mirror symmetry
M
≡{M|t
}.
However, t
⊥
can also play a nontrivial role in the presence
of an additional point group symmetry {g|t
′
} centered at a
different reference point with t
′
⊥
= t
⊥
modulo unit lattice
translation. For instance, one can choose the inversion center
as the reference point of the point group symmetry, thus
inversion is given by {P |0} whereas the nonsymmorphic mirror
is {M|t}. Here, the important point is that an additional shift
of the reference point affects the form of the two operators
simultaneously. Namely, under the shift of the reference point
by d = d
⊥
+ d
, the two symmetry operators transform as
{M|t}−→{M|t − 2d
⊥
} and {P |0}−→{P |−2d
⊥
− 2d
},
which indicates that even if t
⊥
is subtracted from the nonsym-
morphic mirror symmetry by choosing 2d
⊥
= t
⊥
, it preserves
its identity in conjunction with the inversion symmetry P .
Therefore in systems with the inversion symmetry, an off-
centered mirror symmetry, defined as
M
⊥
≡{M|t
⊥
}, (2)
deserves a separate consideration.
An off-centered rotation symmetry can also be defined in
a similar way. A generic nonsymmorphic rotation symmetry
element
C
n
={C
n
|t} (n = 2,3,4,6) satisfies
C
n
t
= t
,C
n
t
⊥
= t
′
⊥
, (3)
where C
n
denotes the n-fold rotation symmetry and t
′
⊥
is a par-
tial translation rotated by C
n
satisfying t
⊥
· t
′
⊥
=|t
⊥
|
2
cos
2π
n
.
Since C
n
fulfills C
n
n
=−1(C
n
n
=+1) for particles with a half-
integer (integer) spin, a nonsymmorphic rotation symmetry
{C
n
|t} is under the following constraint, {C
n
|t}
n
={C
n
n
|nt
}∈
T, thus t
should have the form of t
=
p
n
ˆ
a
p = 0,1,...,n − 1,
where
ˆ
a
is the unit translation along the rotation axis. Again,
t
⊥
is not constrained in this case.
If the reference point for the point group symmetry opera-
tion is shifted by d = d
⊥
+ d
, the nonsymmorphic rotation
symmetry {C
n
|t} also transforms to { C
n
|t + C
n
d
⊥
− d
⊥
}.
Thus by choosing d
⊥
to satisfy t
⊥
= d
⊥
− C
n
d
⊥
, t
⊥
can be
removed, leading to a conventional screw rotation symmetry
C
n
≡{C
n
|t
}. However, in the presence of an additional point
group symmetry centered at a different reference point, such
as {P |0}, an off-centered nonsymmorphic rotation symmetry
C
⊥
n
≡{C
n
|t
⊥
} (4)
can be defined, and the partial translation t
⊥
can cause
intriguing physical consequences as shown in the following.
III. POINT NODES PROTECTED BY OFF-CENTERED
ROTATION SYMMETRIES
In electronic systems having both time-reversal and in-
version symmetries, eigenstates are doubly degenerate at
any momentum. Due to level repulsion between degenerate
bands, accidental band degeneracy is lifted unless additional
crystalline symmetry is supplemented [
24]. Here we show that
the presence of an off-centered symmetry creates symmetry-
protected band degeneracy at the BZ boundary. For simplicity,
let us first introduce an off-centered twofold rotation
C
⊥
2z
=
{C
2z
|
1
2
ˆ
x +
1
2
ˆ
y} to an orthorhombic crystal with T and P
075135-2
TOPOLOGICAL SEMIMETALS PROTECTED BY OFF- . . . PHYSICAL REVIEW B 95, 075135 (2017)
symmetries. Here,
ˆ
x,
ˆ
y, and
ˆ
z denote the unit lattice vectors
in the x, y, and z directions, respectively. To understand the
origin of band degeneracy, let us examine how a spatial
coordinate r = (x,y,z) transforms under
C
⊥
2z
,
C
⊥
2z
:(x,y,z) −→
−x +
1
2
, − y +
1
2
,z
,
(
C
⊥
2z
)
2
:(x,y,z) −→ (x,y,z). (5)
One can see that since (
C
⊥
2z
)
2
does not accompany a partial
translation, it is actually equivalent to a symmorphic operation
C
2
2z
, which leads to (
C
⊥
2z
)
2
=−1 independent of the spatial
coordinate. Thus, at the momentum k invariant under
C
⊥
2z
, each
band |(k) can be labeled by the momentum independent
C
⊥
2z
eigenvalue ±i,
C
⊥
2z
|
±
(k)=±i|
±
(k). Since the system is
invariant under
C
⊥
2z
along the four lines k
1
= (0,0,k
z
), k
2
=
(π,0,k
z
), k
3
= (0,π,k
z
), k
4
= (π,π,k
z
) with k
z
∈ [−π,π], a
state |(k) on any of these lines carries a constant
C
⊥
2z
eigenvalue.
Now we consider the combined effect of P and
C
⊥
2z
.From
the combined transformations
P
C
⊥
2z
:(x,y,z) −→
x −
1
2
,y −
1
2
, − z
,
C
⊥
2z
P :(x,y,z) −→
x +
1
2
,y +
1
2
, − z
, (6)
we obtain
C
⊥
2z
P |(k)=e
ik
x
+ik
y
P
C
⊥
2z
|(k). (7)
Thus along the two
C
⊥
2z
invariant lines k
2
= (π,0,k
z
), k
3
=
(0,π,k
z
) with k
z
∈ [−π,π], P and
C
⊥
2z
anticommute, i.e.,
{
C
⊥
2z
,P }=0. Moreover, since the time-reversal symmetry T
commutes with both P and
C
⊥
2z
, we obtain {
C
⊥
2z
,P T }=0,
which gives rise to
C
⊥
2z
[PT|
±
(k)] =−PT[
C
⊥
2z
|
±
(k)]
=−PT[±i|
±
(k)]
=±iPT |
±
(k). (8)
Thus |
±
(k) and PT|
±
(k), which are locally degenerate
at the momentum k,havethesame
C
⊥
2z
eigenvalues of
±i. Therefore when two degenerate bands having different
C
⊥
2z
eigenvalues cross, the resulting band crossing point is
protected and forms a 3D Dirac point with fourfold degeneracy.
In fact, the anticommutation relation between P and
C
⊥
2z
puts a strong constraint on the band structure along the
C
⊥
2z
invariant axis. Considering
C
⊥
2z
[P |
±
(k)] =−P [
C
⊥
2z
|
±
(k)]
=−P [±i|
±
(k)]
=∓iP|
±
(k), (9)
one can find that two energetically degenerate states |
±
(k)
and P |
±
(k), which are located at k and −k, respectively,
have the opposite
C
⊥
2z
eigenvalues. Let us recall that at each
momentum k, a Kramers pair should have the same
C
⊥
2z
eigenvalue. This means that on the
C
⊥
2z
invariant axis where
Y
X
Z
U
T
U
X
U
U
UX
C
2z
~
M
z
II
~
Energy
Energy
R
S
FIG. 1. 3D Dirac points protected by an off-centered twofold
rotation
C
⊥
2z
. (a) A schematic figure describing the distribution of
3D Dirac points in momentum space. The location of four 3D Dirac
points is marked in red dots. The bold blue arrow indicates the axis for
C
⊥
2z
symmetry. The pink square indicates the plane of the glide mirror
symmetry
M
z
, which is dual to
C
⊥
2z
. (b) The band structure along the
U-X-U line on which {P,
C
⊥
2z
}=0. A pair of degenerate bands (a
doublet pair) form a 3D Dirac point at each time-reversal invariant
momentum (TRIM). (c) The band structure when two doublet pairs
cross on the U -X-U line. There are in total 4n (n is an integer) 3D
Dirac points, which are all away from TRIMs. In (b) and (c), the
integers in the bottom indicate the number N(k
z
) defined in Eq. (
B2)
from which the topological charge Q can be found by using Eq. (
B1).
{P,
C
⊥
2z
}=0 is satisfied, there should be a pair of degenerate
bands with different
C
⊥
2z
eigenvalues, which we call a doublet
pair. Since a doublet pair should form a band structure which is
symmetric with respect to a TRIM, they should be degenerate
at the two TRIMs on the
C
⊥
2z
invariant axis as shown in
Figs.
1(a) and 1(b). Here, each of the degenerate points with
fourfold degeneracy represent a 3D Dirac point located at a
TRIM.
Due to the presence of a quantized
C
⊥
2z
eigenvalue, the
band crossing points between two different doublet pairs can
also generate 3D Dirac points. Namely, as long as the two
crossing bands have different
C
⊥
2z
eigenvalues, the crossing
points are symmetry protected. In general, such a crossing
between doublet pairs generates 4n (n is an integer) band
crossing points, and the location of each Dirac point is away
from TRIMs as shown in Fig.
1(c).
For comparison, let us consider a similar problem in sys-
tems with a twofold screw rotation
C
2z
={C
2z
|
1
2
ˆ
z} satisfying
(
C
2z
)
2
:(x,y,z) −→ (x,y,z + 1). (10)
Along the line invariant under
C
2z
, the relevant eigenstates
satisfy
C
2z
|
±
(k)=±ie
ik
z
/2
|
±
(k). Due to the momentum
dependence of the eigenvalues, the two different
C
2z
eigen-
sectors should be interchanged when the momentum k
z
is
shifted by 2π . Moreover, it is straightforward to show that
C
2z
P |(k)=e
−ik
z
P
C
2z
|(k). Then along the line invariant
under
C
2z
, where
C
2z
|
±
(k)=±ie
i
2
k
z
|
±
(k), we obtain
C
2z
[PT|
±
(k)] = e
ik
z
PT[
C
2z
|
±
(k)]
= e
ik
z
PT[±ie
i
2
k
z
|
±
(k)]
=∓ie
i
2
k
z
PT|
±
(k), (11)
which show that the degenerate states |
±
(k) and PT|
±
(k)
belong to different eigensectors of
C
2z
symmetry. Therefore,
075135-3
YANG, BOJESEN, MORIMOTO, AND FURUSAKI PHYSICAL REVIEW B 95, 075135 (2017)
when two bands, each of which is doubly degenerate, touch,
there always is some finite hybridization between degenerate
bands. Thus
C
2z
symmetry cannot protect a stable Dirac point
at a generic momentum. One exception is when the band
crossing happens at the time-reversal invariant momentum
(TRIM) with k
z
= π. In this case, two bands having the same
C
2z
eigenvalues form a Kramers pair, and two Kramers pairs
having different
C
2z
eigenvalues are connected by P , leading
to fourfold degeneracy [
34]. However, such a degeneracy point
does not form a 3D Dirac point. Instead, it becomes a part of
a line node in the k
z
= π plane protected by
M
⊥
z
=
C
2z
P ,as
discussed in Sec.
IV.
IV. LINE NODES PROTECTED BY OFF-CENTERED
MIRROR SYMMETRIES
An off-centered mirror symmetry can create a stable line
node with fourfold degeneracy in systems with P and T
symmetries. For convenience, let us consider
M
⊥
x
={M
x
|
1
2
ˆ
x},
which transforms a spatial coordinate r in the following way:
M
⊥
x
:(x,y,z) →
−x +
1
2
,y,z
,
(
M
⊥
x
)
2
:(x,y,z) → (x,y,z). (12)
From M
2
x
=−1, we obtain (
M
⊥
x
)
2
=−1 independent of a
spatial coordinate. Thus, at the momentum k invariant under
M
⊥
x
, i.e., at any momentum in the 2D plane with k
x
= 0or
k
x
= π, each band |(k) can be labeled by the momentum in-
dependent
M
⊥
x
eigenvalue ±i, i.e.,
M
⊥
x
|
±
(k)=±i|
±
(k).
Now we consider the combined effect of P and
M
⊥
x
.From
P
M
⊥
x
:(x,y,z) −→
x −
1
2
, − y, − z
,
M
⊥
x
P :(x,y,z) −→
x +
1
2
, − y, − z
, (13)
we obtain
M
⊥
x
P |(k)=e
ik
x
P
M
⊥
x
|(k). (14)
Thus in the k
x
= π plane, P and
M
⊥
x
anticommute, i.e.,
{
M
⊥
x
,P }=0. Moreover, since the time-reversal symmetry T
commutes with both P and
M
⊥
x
, we obtain {
M
⊥
x
,P T }=0,
which gives rise to
M
⊥
x
[PT|
±
(k)] =−PT[
M
⊥
x
|
±
(k)]
=−PT[±i|
±
(k)]
=±iPT |
±
(k). (15)
Thus |
±
(k) and PT|
±
(k), which are degenerate at the
momentum k,havethesame
M
⊥
x
eigenvalues of ±i. Therefore
when two degenerate bands having different
M
⊥
x
eigenvalues
cross, the resulting band crossing point is protected and forms
a line node with fourfold degeneracy on the invariant plane
k
x
= π.
In fact, the anticommutation relation between P and
M
⊥
x
puts a strong constraint on the band structure in the
M
⊥
x
invariant plane. Considering
M
⊥
x
[P |
±
(k)] =−P [
M
⊥
x
|
±
(k)]
=−P [±i|
±
(k)]
=∓iP|
±
(k), (16)
we find that two energetically degenerate states |
±
(k) and
P |
±
(k), which are located at k and −k, respectively, have
the opposite
M
⊥
x
eigenvalues. It is worth to remind that a
Kramers pair at each momentum k, which are degenerate due
to PT symmetry, have the same
M
⊥
x
eigenvalue. This means
that in the k
x
= π plane where {P,
M
⊥
x
}=0 is satisfied, two
bands (each with Kramers degeneracy) having different
M
⊥
x
eigenvalues should form a doublet pair again as in the case
of the off-centered rotation symmetry. Since the whole band
structure in the k
x
= π plane is symmetric with respect to a
TRIM, each doublet pair should be degenerate along a line,
which passes two TRIMs as shown in Figs.
2(a) and 2(b).Here
a set of the degenerate points form a line node with fourfold
degeneracy.
Due to the presence of quantized
M
⊥
x
eigenvalues, a band
crossing between two different doublet pairs can also generate
nodal lines. Namely, as long as the two bands have different
M
⊥
x
eigenvalues, their crossing points are symmetry protected.
In general, such crossing between two different doublet pairs
generate 4n (n is an integer) nodal lines, and the location of
each nodal line is away from TRIM as shown in Fig.
2(c).
For comparison, let us consider a similar problem in
systems with a glide mirror
M
x
={M
x
|
1
2
ˆ
y +
1
2
ˆ
z} satisfying
(
M
x
)
2
:(x,y,z) −→ (x,y + 1,z + 1). (17)
Y
Z
U
T
C
2x
~
M
x
~
R
S
X
S
S
X
U
X
S
R
S
S
X
X
U
R
S
II
Energy
Energy
FIG. 2. 3D Dirac lines protected by an off-centered mirror
symmetry
M
⊥
x
. (a) A schematic figure describing the distribution
of nodal lines in momentum space. The location of two line nodes in
the k
x
= π plane is marked in red color. The blue square indicates
the plane of the off-centered mirror symmetry
M
⊥
x
. The bold pink
arrow indicates the axis for the screw rotation
C
2x
symmetry, which
is dual to
M
⊥
x
symmetry. (b) Distribution of the integer N (π,k
y
,k
z
)
defined in Eq. (
B9)inthek
x
= π plane from which the topological
charge Q
′
of the line node can be computed by using Eq. (
B8). The
corresponding band structure along the S-X-S line is shown in the
bottom. The doublet pair are degenerate at each TRIM, which is a
part of line nodes in the k
x
= π plane. (c) Distribution of the integer
N(π,k
y
,k
z
), when two doublet pairs cross in the k
x
= π plane. There
are in total 4n (n is an integer) nodal lines, which are away from
TRIM. The corresponding band structure along the S-X-S line is
shown in the bottom.
075135-4
TOPOLOGICAL SEMIMETALS PROTECTED BY OFF- . . . PHYSICAL REVIEW B 95, 075135 (2017)
FIG. 3. Construction of 3D lattice models by stacking 2D layers.
(a) A schematic figure describing a 3D lattice model obtained by
vertical stacking of 2D square lattices. (b) Structure of a 2D layer
where the B site in a unit cell is shifted along the z direction, thus
the whole system has nonsymmorphic symmetries. (c) An additional
shifting of B sites along the y direction, which breaks the symmetry
C
⊥
2z
(or equivalently,
M
z
). (d) An additional distortion of a unit cell,
which breaks the symmetry
M
⊥
x
and
M
⊥
y
(or equivalently,
C
2x
and
C
2y
).
In a plane invariant under
M
x
, the eigenstates satisfy
M
x
|
±
(k)=±ie
i
2
(k
y
+k
z
)
|
±
(k). Due to the momentum de-
pendence of the eigenvalues, the two different
M
x
eigensectors
should be interchanged when either k
y
or k
z
is shifted by 2π .
Moreover, it is straightforward to show that
M
x
P |(k)=e
i(k
y
+k
z
)
P
M
x
|(k). (18)
Then in a 2D plane invariant under
M
x
, we obtain
M
x
[PT|
±
(k)] = e
−i(k
y
+k
z
)
PT[
M
x
|
±
(k)]
=∓ie
−
i
2
(k
y
+k
z
)
PT|
±
(k), (19)
which shows that |
±
(k) and PT|
±
(k), which are degen-
erate at the momentum k, belong to different eigensectors
of
M
x
symmetry. This means that when two bands, each
doubly degenerate due to the PT symmetry, overlap, there
always is some finite hybridization between them at a generic
momentum, thus a stable line node cannot be protected by
M
x
symmetry in a mirror invariant plane. Instead, stable
Dirac point nodes are protected by an off-centered symmetry
C
⊥
2x
=
M
x
P on its invariant lines (k
x
,π,0) and (k
x
,0,π).
V. M O D E L
To demonstrate the general idea discussed up to now, we
construct a 3D tight-binding Hamiltonian on a tetragonal
lattice, which is composed of 2D square lattices stacked along
the z direction as described in Fig.
3. For a 2D layer, we adopt
the lattice model proposed in Ref. [
34] in which a unit cell
contains two sublattice sites, labeled A and B, where the B
sublattice is displaced by r
AB
= (
1
2
,
1
2
,δ
z
)(0<δ
z
< 1) from
the A sublattice. Here we assume that both the in-plane and
out-of-plane lattice constants to be unity. The vertical shift
δ
z
makes the symmetry of the lattice to be nonsymmorphic.
Explicitly, the Hamiltonian in the real space is given by
ˆ
H
(0)
=
i,j
t(r
ij
)
ˆ
c
†
r
i
ˆ
c
r
j
+
i,j
t
′
(r
ij
)
ˆ
c
†
r
i
ˆ
c
r
j
+
i,j,k
iλ(r
ij
,r
jk
)
ˆ
c
†
r
i
[(r
ij
× r
jk
) · σ ]
ˆ
c
r
k
, (20)
where t(r
ij
)[t
′
(r
ij
)] is the hopping amplitude between same
(different) sublattice sites, and λ(r
ij
,r
jk
) denotes the spin-orbit
induced hopping amplitude between the same sublattice sites i
and k through the site j belonging to the other sublattice. Here,
r
ij
= r
i
− r
j
and the Pauli matrix σ indicates the spin degrees
of freedom. More detailed information about the lattice model
is given in Appendix
A.
Let us note that
ˆ
H
(0)
possesses not only the time-reversal
symmetry T and the inversion symmetry P but also the off-
centered symmetries
C
⊥
2z
={C
2z
|
1
2
ˆ
x +
1
2
ˆ
y},
M
⊥
x
={M
x
|
1
2
ˆ
x},
and
M
⊥
y
={M
y
|
1
2
ˆ
y}, thus the system corresponds to the space
group No. 59. The corresponding screw/glide symmetries can
be defined as
M
z
=
C
⊥
2z
P ={M
z
|
1
2
ˆ
x +
1
2
ˆ
y},
C
2x
=
M
⊥
x
P =
{C
2x
|
1
2
ˆ
x}, and
C
2y
=
M
⊥
y
P ={C
2y
|
1
2
ˆ
y}. By shifting the lo-
cation of the B site relative to the A site in a unit cell, the
symmetry of the Hamiltonian can be systematically lowered,
thus one can examine the role of a particular symmetry to
protect a relevant semimetal phase using a single lattice model.
First, we shift the position of the B site in a unit cell in
the y direction, which makes r
AB
= (
1
2
,δ
y
=
1
2
,δ
z
)asshown
in Fig.
3(c). This distortion breaks
C
⊥
2z
and
M
⊥
y
symmetries,
whereas
M
⊥
x
is preserved as well as the P and T symmetries,
thus the system corresponds to the space group No. 11. The
resulting band structure is shown in Fig.
4(a). One can clearly
see that there are two line nodes in the k
x
= π plane, and
each line node connects two TRIMs, which is consistent with
the prediction of the general theory. To observe the band
crossing between two doublet pairs described in Fig. 2(c),
we construct an eight-band model by adding two copies of the
4 × 4 Hamiltonian in Eq. (20). As the hybridization between
the two 4 × 4 blocks is turned on, each line node passing
two TRIMs splits into two different nodal lines, thus one can
observe four nodal lines, and none of them passes a TRIM as
shown in Figs.
4(b) and 2(c).
The second distortion is achieved by deforming the lattice
along the [110] direction, which breaks
M
⊥
x
and
M
⊥
y
sym-
metries whereas
C
⊥
2z
is preserved as well as the P and T
symmetries, thus the system corresponds to the space group
No. 13. [See Fig. 3(d).] As shown in Fig. 5(a), one can
observe four Dirac points protected by
C
⊥
2z
symmetry located
at TRIMs k = (π,0,0), (π,0,π), (0,π,0), and (0,π,π ). When
the number of bands is doubled by combining two different
4 × 4 Hamiltonians, one can observe four Dirac points on the
line k = (π,0,k
z
) and also k = (0,π,k
z
) with k
z
∈ (−π,π),
respectively, as shown in Fig.
5(b). Here none of Dirac points
is located at a TRIM in agreement with the prediction of the
general theory, and the relevant band structure is also consistent
with Fig.
1(c).
075135-5