Classification of topological quantum matter with reflection symmetries
Andreas P. Schnyder
Max Planck Institute for Solid State Research, Stuttgart
June 14th, 2016
SPICE Workshop on New Paradigms in Dirac-Weyl Nanoelectronics
Outline
0. Introduction: Topological band theory
1. Topological insulators with reflection symmetry
- Ca3PbO, Sr3PbO, Ba3PbO
Mirror plane
arXiv:1606.03456
2. Topological nodal line semi-metals
- Ca3P2, ZrSiS
PRB 93, 205132 (2016)
3. Nodal non-centrosymmetric superconductors
J. Phys.: Condens. Matter 27 (2015) 243201
- CePt3Si
4. Conclusions & Outlook
(a)
E
E
(b)
kz
kx
Topical Review
(c)
kx
kz
E
kx
kz
Figure 1. Energy spectrum of the three different types of topological subgap states that can exist at the surface of nodal noncentrosymmetric
superconductors: (a) helical Majorana cone, (b) arc surface state, and (c) flat-band surface state. Figure adapted from [66].
with codimension p < d (i.e. d > 0), provided that they 3.3. Examples
Review articles: arXiv:1505.03535; J. Phys.:
Condens. Matter 27, 243201 (2015)
are protected by a Z invariant or 2Z invariant. Z numbers,
BZ
n
2
on the other hand, guarantee only the stability of nodes with
dn = 0, i.e. point nodes. These findings are confirmed by
more rigorous derivations based on K theory [58–60] and
minimal Dirac Hamiltonians [61]. The latter approach uses
For the phenomenological model Hamiltonians given in
section 2, we derive in this subsection explicit expressions
for the topological invariants that protect the superconducting
nodes against gap opening. We also use these examples to
2
(2)
H(k) = e−ikrHe+ikr
(3)
Topological band theory
(4)
Consider band structure: H(k) |un (k)⟩ = En (k) |un (k)⟩
(5)
Festk
örperphysik II,
Musterl
ösung
11.
Festk
örperphysik
II,
Musterl
ösung
Festk
örperphysik
II,
Musterl
ösung
11.
Festk
örperphysik
II,
Musterl
ösung
11.örperphysik
Festk
örperphysik
II,
Musterl
ösung
11.
Festk
II, 11.
Musterlös
we have
•
(i) Topological
equivalence for
Prof. M. Sigrist, WS05/06
•Prof.
H(k)WS05/06
k
k
π/aM. Sigrist,
− π/a
Prof.
WS05/06k
M. Sigrist,
we have
we have
majoranas
homotopy
kx
we have
homotopy
ky
π/a
− π/a
kx
Energy
insulators
Energy
Zürich
Prof. M. Sigrist, WS05/06 gap
Prof. M. Sigrist, WS05/06
ETH Zürich
Prof. ETH
M. Sigrist,
WS05/06
∈
Brillouin
Zone
x
y
ETH Zürich
(superconductors):
we have
ky
ky
π/a kx − π/a
π/a
(1)
− π/a
kx
ky
gap
(6)
(1)π/a
ETH Züri
ET
− π/a
(
†= #
=#
k(7)
kxmajoranas
(1) νmomentum
crystal
x
γ1 majoranas
= ψcrystal
+ ψνmomentum
± !
±
"t |dk|
∆
=
∆
±
∆
(2)
∆
= ∆s ±
∆=t |dψk+| ψ†
s
†
k
† k
†
Festk
örperphysik
II,
Musterl
ösung
11.
γ
†
Festk
örperphysik
II,
Musterl
ösung
11.
Festk
örperphysik
II,
Musterl
ösung
11. " ösun
1
γ
=
ψ
+
ψ
(
γ
=
−i
ψ
−
ψ
γ1ösung
= II,
ψ +Musterl
ψ11. 1 ösung
(2)
Festkörperphysik
γ1 = 2 ψII,
+ ψ Musterl
(2)Festk
II,(8)
!Musterl
Festk
örperphysik
! 11. örperphysik
"
!
"
majoranas
majoranas
ψ 1 = 2 γ1 + iγψ2†
ψ † = †γBdG
homotopy
1 − iγ2 H
Lattice
ky
π/a
ψ and =− π/a
γk1xBdG
− kiγ
y 2
we have
Lattice
BdG
homotopy
HBdG
and
majoranas
kx
and
= γ1 − iγ2 ψ † = γ1 − iγ2
(5)
k
k
π/a
− π/a
kx
x
y
(1)
π/a and− π/a
ψ
= (9)
γ − iγ2
(5) 1
(10) − π/a
ky (1)π/a
(
(1
= εmajoranas
αgk ν·momentum
σ= #
(5)
= # 2 kx
crystal
kσ0 +crystal
kxmajoranas
(1)k ν·momentum
h(k)
=
ε
σ
+
αg
σ
k
0
γi = 1
2
2
±
γ
=
1
(6)
γ
=
1
(
±
i
i
and
∆(k) = (∆s σγ0 2+ =∆†t d
·
σ)
iσ
(6)
∆
=
∆
±
∆
|d
|
† 2δ
k
y
∆
=
∆
±
∆
|d
|
(2)
s
t
k
†
†
{γ
,
γ
}
=
k
s
t
k
1 k
(6)
i ψ
j +ψ
ij
∆(k)
= (∆
σ0 ψ+
γ1 s=
+∆ψt d2δ
(2(
γ1 {γ=, γψ}
+=
ψ 2δ
(2)
i
k · σ) iσy γ1 =
. symmetries to consider: γtime-reversal
(7) !
"
1 = ψ+ψ
symmetry,
i j !particle-hole,
ij" {γi , γ(2)
j } !=reflection
ij "
!=
" for
†
∆sγ>
∆t t>
∆
∆Z=
νψ=− ±1
for γ2 ∆=
−i∆ψs − ψ †
{γ
,
γ
}
2δ
(7)
∆s > ∆t
∆s ∼
∆
ν
=
±1
∆
(3)
sψ †∼
t
t >
=
−i
ψ
−
(3)
†
γ
−i
ψ
(3
i
j
ij
2 ∆
t
s
2
mean
field
γ
=
−i
ψ
−
ψ
(3)
%
&
2
2
"#
$ = 1
! kF,−
Lx /2 i
%
mean
field
(11)
meanγfield
#
"
$
!
Lx /2
i
e
dk
and
k
y
µ
and
F,−
†
µ
x
#
and
and
mean
field
and
n
=
F
dk
2
Z
.
top.
equivalence
classes
distinguished
by:
e
dk
γ#
= xγE=0
hex and
Iy ≃
sgn
Hex ρ1 (0, ky )
− t sin ky +
ρn (0,µkyµ) cos
k
(7)
.
y Zλ†
y
E=0
†k )
hex {γiI, yγj }
≃ = 2δij
sgn
Hex ρ1 (0,
−
t
sin
k
+
λ
ρn (0, ky ()
2⇡
! kF,+ 2π
γE=0 =
γE=0
(8)
y
y
†
(12)
γE=0 = γE=0
ψ
=
γ
+
iγ
⇒
γ
=
γ
µ
n=1
1
2
ψ
=
γ
+
iγ
(4)
†
!
2π
1
2
k,E
−k,−E (4
ψ = (8)
γ†q(k)
kF,+
†
1 + iγ2 :(4)
π
[U(2)]
=
∈
U(2)
γ
filled
π3 [U(2)] =ψ = γγ1q(k)
:
∈
U(2)
3
µ
n=1
†
E=0
E=0
+ iγ=
(4)
⇒
γ
=
γ
(9)
∗
γk,E = γ−k,−E
(
2
ψΞ(5)
γ1 −
topological
invariant
†⇒
ψ=
=iγ
τx2ψ−k,−E
ψ † = γ1k,E
− iγ2 −k,−E
Bulk-boundary
correspondence:
+k,+E
†
ψ
=
γ
−
iγ
(5
1
2
∗ states (9)
∗
=γ
and
ψ †⇒= γγk,E
(5) = τx ψ−k,−E
Ξ ψ+k,+E = τx ψ−k,−E
2 (10)
Ξ
ψ
1 − iγ2 −k,−E
+k,+E
Ξ
= +1
Ξ = τx K (1
mean
field
Lattice
BdG
H
and
Lattice BdG HBdG
BdG
'
(
∗
and
#
$
and states
2
Ξ ψ+k,+E
= τsurface
(10)
x ψ−k,−E
and
|nand
edge
states)
Ξ2 = +1
Ξ = τΞ
K
(11)
† (or
†
(
=
+1
Ξ
=
τ
K
(1
x'
Z | = # gapless
x
0
1
jn,ky = −t sin ky 2 cnky ↑ cnky ↑ + cnky ↓ cnky ↓
(8)† $
#
$ †
#
τ
=
2
x
2
Ξ ==
+1ε'kσ0Ξ+=αg
τx K
=(11)
ε
σ
+
αg
·
σ
γ
=
1 1 0
0kh(k)
1 cnk
j
=
−t
sin
c
+
c
c
†
h(k)
·
σ
(5)
k
0
k
γ
=
1
(6)
0
1
i
2
n,k
y
k
y
i
nk
↑
nk
↓
(
nky ↓
y (12)
#
$
τ =
γy ↑ = y 1
(6
•
h(k)
majoranas
Energy
ν! = ±1
fors >
∆
(3)
" ∆
∆=tt >−i∆sψ∆−s ψ∼†γ∆
=−±1
for γ∆
∆s ψ − ψ†
†
−i
t = −iν ψ
2 t=>
†
ψ
(
γ
(3)
2
2
γ2 = −i ψ − ψ
(3)
(ii) Topological equivalence for andand
and
Prof. M. Sigrist, WS05/06 and
ETH
ürich WS05/06 ETH Zürich
Prof. M. Sigrist, WS05/06 and
ETH Züric
and
Prof.
M. Z
Sigrist,
Prof.
M.
Sigrist,
WS05/06
and
band
(nodes in SCs):
Prof. M. Sigrist, WS05/06
E
Prof.
M. crossings
Sigrist, WS05/06
ETH Zürich
ψ (4)
= γ1 + iγ2
π3 [U(2)] =
q(k) : ∈ U(2) ψ =π3 [U(2)]
(4)
(
γ
q(k)
1 + iγ:2 ∈ U(2)
1 + iγ2 = ψ = γ
ψ = γwe +have
iγ
(4)
†
we have
we have
•
∆s ∼ ∆t
Energy
∆s > ∆t
Reflection symmetry
Consider reflection R: x !
x
R
1
with
H( kx , ky , kz )R = H(kx , ky , kz )
z
y
x
R = sx
— w.l.o.g.: eigenvalues of R 2 { 1, +1}
mirror Chern number:
Teo, Fu, Kane PRB ‘08
kx = 0 =) H(0, ky , kz )R
RH(0, ky , kz ) = 0
— project H(0, ky , kz ) onto eigenspaces of
n±
M
1
=
4⇡
Z
F± d2 k
2D BZ
R : H± (ky , kz )
Berry curvature in
+
— total Chern number: nM = nM + nM
— mirror Chern number: nM = n+
M
nM
Bulk-boundary correspondence:
— zero-energy states on surfaces that
are left invariant under the mirror symmetry
± eigenspace
Mirror plane
8
Classification of topological materials with reflection symmetry
TABLE II. Classification of reflection symmetry protected topological insulators and fully gapped superconductors,? ? ? as well as of Fermi
surfaces and nodal points/lines in reflection symmetry protected semimetals and nodal superconductors, respectively. The first row specifies
with protected
T (C or
S) insulators and fully gapped superconductors, while the second and third
+ : Rd commutes
the spatial R
dimension
of reflection symmetry
topological
rows indicate
p = d dFS of with
the reflection
R the :codimension
R anti-commutes
T (Csymmetric
or S) Fermi surfaces (nodal lines) at high-symmetry points [Fig. 3(a)] and
away from high-symmetry points of the Brillouin zone [Fig. 3(b)], respectively.
Reflection sym. class
d=1
d=2
d=3
d=4
d=5
d=6
d=7
d=8
R
A
MZ
0
MZ
0
MZ
0
MZ
0
R+
AIII
0
MZ
0
MZ
0
MZ
0
MZ
R
AIII
MZ Z
0
MZ Z
0
MZ Z
0
MZ Z
0
t
=
0
t
=
3
AI
MZ
0
0
0
2M Z
0
M Z2
M Z2
BDI
M Z2
MZ
0
0
0
2M Z
0
M Z2
D
M Z2
M Z2
MZ
0
0
0
2M Z
0
DIII
0
M Z2
M Z2
MZ
0
0
0
2M Z
R+ ,R++
AII
2M Z
0
M Z2
M Z2
MZ
0
0
0
CII
0
2M Z
0
M Z2
M Z2
MZ
0
0
C
0
0
2M Z
0
M Z2
M Z2
MZ
0
CI
0
0
0
2M Z
0
M Z2
M Z2
MZ
AI
0
0
2M Z
0
T Z2
Z2
MZ
0
BDI
0
0
0
2M Z
0
T Z2
Z2
MZ
t=
t =Z2
D
MZ
0 1
0
0
2M
0
T Z2
Z2
DIII
Z2
MZ
0
0
0
2M Z
0
T Z2
R ,R
AII
T Z2
Z2
MZ
0
0
0
2M Z
0
CII
0
T Z2
Z2
MZ
0
0
0
2M Z
C
2M Z
0
T Z2
Z2
MZ
0
0
0
CI
0
2M Z
0
T Z2
Z2
MZ
0
0
R +
BDI, CII
2Z
0
2M Z
0
2Z
0
2M Z
0
R+
DIII, CI
2M Z
0
2Z
0
2M Z
0
2Z
0
R+
BDI
MZ Z
0
0
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2
R +
DIII
M Z2 Z2 M Z2 Z2 M Z Z
0
0
0
2M Z 2Z
0
R+
CII
2M Z 2Z
0
M Z2 Z2 M Z2 Z2 M Z Z
0
0
0
R +
CI
0
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2 M Z Z
0
“Bott cube”
a
Z2 and
M Z2 invariants
only protect
Fermi
surfaces ofPRB
dimension
zero (dFS = 0) at high-symmetry points of the Brillouin
Morimoto,
Furusaki
PRB 2013;
Chiu,
Schnyder
2014;
b
zone.
FIG. 10 (Color online) The 27 symmetry classes with reflec-
cation
tegers
ber of
by the
t = 0
and (
nian.
the ge
K(s, t
For re
Dk =
a. Bul
system
any b
Classification of topological materials with reflection symmetry
R+ : R commutes with T (C or S)
TI/TSC
Reflection
R
R+
R
R+ ,R++
R ,R
R +
R+
R+
R +
R+
R +
R
: R anti-commutes with T (C or S)
FS1
d=1
p=8
d=2
p=1
d=3
p=2
d=4
p=3
d=5
p=4
d=6
p=5
d=7
p=6
d=8
p=7
FS2
p=2
p=3
p=4
p=5
p=6
p=7
p=8
p=1
A
AIII
AIII
AI
BDI
D
DIII
AII
CII
C
CI
AI
BDI
D
DIII
AII
CII
C
CI
BDI, CII
DIII, CI
BDI
DIII
CII
CI
MZ
0
MZ
0
MZ
0
MZ
0
0
MZ
0
MZ
0
MZ
0
MZ
MZ Z
0
MZ Z
0
MZ Z
0
MZ Z
0
MZ
0
0
0
2M Z
0
M Z2
M Z2
M Z2
MZ
0
0
0
2M Z
0
M Z2
M Z2
M Z2
MZ
0
0
0
2M Z
0
0
M Z2
M Z2
MZ
0
0
0
2M Z
2M Z
0
M Z2
M Z2
MZ
0
0
0
0
2M Z
0 which
MZ
M Z2 class
M and
Z
0
0is
2
For
symmetry
dimension
0
0
2M Z
0
M Z2
M Z2
MZ
0
there
a topological
insulator
or topological
0
0
0
2M Z
0
M Z2
M Z2
MZ
semi-metal
protected
by reflection
symmetry?
0
0
2M Z
0
T Z2
Z2
MZ
0
0
0
0
2M Z
0
T Z2
Z2
MZ
MZ
0
0
0
2M Z
0
T Z2
Z2
Z2
MZ
0
0
0
2M Z
0
T Z2
T Z2
Z2
MZ
0
0
0
2M Z
0
0
T Z2
Z2
MZ
0
0
0
2M Z
2M Z
0
T Z2
Z2
MZ
0
0
0
0
2M Z
0
T Z2
Z2
MZ
0
0
2Z
0
2M Z
0
2Z
0
2M Z
0
2M Z
0
2Z
0
2M Z
0
2Z
0
MZ Z
0
0
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2
M Z2 Z2 M Z2 Z2 M Z Z
0
0
0
2M Z 2Z
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2 M Z Z
0
0
0
0
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2 M Z Z
0
?
Tabelle I Classification of topological insulators and superconductors (“TI/TSC”) as well as of stable Fermi
Classification of topological materials with reflection symmetry
R+ : R commutes with T (C or S)
TI/TSC
Reflection
R
R+
R
R+ ,R++
R ,R
R +
R+
R+
R +
R+
R +
R
: R anti-commutes with T (C or S)
FS1
d=1
p=8
d=2
p=1
d=3
p=2
d=4
p=3
d=5
p=4
d=6
p=5
d=7
p=6
d=8
p=7
FS2
p=2
p=3
p=4
p=5
p=6
p=7
p=8
p=1
A
AIII
AIII
AI
BDI
D
DIII
AII
CII
C
CI
AI
BDI
D
DIII
AII
CII
C
CI
BDI, CII
DIII, CI
BDI
DIII
CII
CI
MZ
0
MZ
0
MZ
0
MZ
0
0
MZ
0
MZ
0
MZ
0
MZ
MZ Z
0
MZ Z
0
MZ Z
0
MZ Z
0
MZ
0
0
0
2M Z
0
M Z2
M Z2
M Z2
MZ
0
0
0
2M Z
0
M Z2
M Z2
M Z2
MZ
0
0
0
2M Z
0
0
M Z2
M Z2
MZ
0
0
0
2M Z
2M Z
0
M Z2
M Z2
MZ
0
0
0
0
2M Z
0
M Z2
M Z2
MZ
0
0
0
0
2M Z
0
M Z2
M Z2
MZ
0
0
0
0
2M Z
0
M Z2
M Z2
MZ
0
0
2M Z
0
T Z2
Z2
MZ
0
0
0
0
2M Z
0
T Z2
Z2
MZ
MZ
0
0
0
2M Z
0
T Z2
Z2
Z2
MZ
0
0
0
2M Z
0
T Z2
T Z2
Z2
MZ
0
0
0
2M Z
0
0
T Z2
Z2
MZ
0
0
0
2M Z
2M Z
0
T Z2
Z2
MZ
0
0
0
0
2M Z
0
T Z2
Z2
MZ
0
0
2Z
0
2M Z
0
2Z
0
2M Z
0
2M Z
0
2Z
0
2M Z
0
2Z
0
MZ Z
0
0
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2
M Z2 Z2 M Z2 Z2 M Z Z
0
0
0
2M Z 2Z
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2 M Z Z
0
0
0
0
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2 M Z Z
0
Tabelle I Classification of topological insulators and superconductors (“TI/TSC”) as well as of stable Fermi
Chiu, Schnyder
PRB 2014
Classification of topological materials with reflection symmetry
R+ : R commutes with T (C or S)
TI/TSC
Reflection
R
R+
R
R+ ,R++
R ,R
R +
R+
R+
R +
R+
R +
R
: R anti-commutes with T (C or S)
FS1
d=1
p=8
d=2
p=1
d=3
p=2
d=4
p=3
d=5
p=4
d=6
p=5
d=7
p=6
d=8
p=7
FS2
p=2
p=3
p=4
p=5
p=6
p=7
p=8
p=1
A
AIII
AIII
AI
BDI
D
DIII
AII
CII
C
CI
AI
BDI
D
DIII
AII
CII
C
CI
BDI, CII
DIII, CI
BDI
DIII
CII
CI
MZ
0
MZ
0
MZ
0
MZ
0
0
MZ
0
MZ
0
MZ
0
MZ
MZ Z
0
MZ Z
0 CaM
Z2Z
0
MZ Z
0
3P
MZ
0
0
0
2M Z
0
M Z2
M Z2
M Z2
M Z CePt
0 3Si
0
0
2M Z
0
M Z2
M Z2
M Z2
MZ
0
0
0
2M Z
0
0
M Z2
M Z2
MZ
0
0
0
2M Z
2M Z
0
M Z2
M Z2
MZ
0
0
0
0
2M Z
0
M Z2
M Z2
MZ
0
0
0
0
2M Z
0
M Z2
M Z2
MZ
0
0
0
0
2M Z
0
M Z2
M Z2
MZ
0
0
2M Z
0
T Z2
Z2
MZ
0
0
0
0
2M Z
0
T Z2
Z2
MZ
MZ
0
0
0 Ca32M
Z
Z2
PbO,
Sr03PbO T Z2
Z2
MZ
0
0
0
2M Z
0
T Z2
T Z2
Z2
MZ
0
0
0
2M Z
0
0
T Z2
Z2
MZ
0
0
0
2M Z
2M Z
0
T Z2
Z2
MZ
0
0
0
0
2M Z
0
T Z2
Z2
MZ
0
0
2Z
0
2M Z
0
2Z
0
2M Z
0
2M Z
0
2Z
0
2M Z
0
2Z
0
MZ Z
0
0
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2
M Z2 Z2 M Z2 Z2 M Z Z
0
0
0
2M Z 2Z
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2 M Z Z
0
0
0
0
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2 M Z Z
0
Tabelle I Classification of topological insulators and superconductors (“TI/TSC”) as well as of stable Fermi
Chiu, Schnyder
PRB 2014
1. Topological insulators with reflection symmetry
Y. Nohara (MPI-FKF)
Yang-Hao Chan (A. Sinica)
Ca3PbO, Sr3PbO
Ching-Kai Chiu (UMD)
Finally, highly entangled bands above the Fermi energy
FULL PAPERS
Remember that there are three Ca atoms in a unit
w-energy
3
he
low-energy
s as a consequence. Since the Pb 6p orbital originated
4
2
Hamiltonian
by configuration Ca2+
onian
e energy,
that by
the expected
low-energy
mi
the
is
3 Pb O
Anti-perovskites:
Ca3PbO,
PbO
paation
of
p-bands
is located
of the Sr
Ca 33d
and
by above
Dirac
Hamiltonian
by the bottom
and
by
Pb
6pthe
shell
iscollaboration
not completely
filled.A.Later
it H.
turns
out
transformation
and bywith
in
Rost,
Takagi
to
momenmomenp-bands
andtothe
of d-bands
is crucial for the
[Thesis]
h
thebottom
momennts.respect
It is
worth
s is
material.
worth
Band structure (without SOC):
T. KARIYADO and M. OGATA
Ca PbO is a reflection symmetry protected TI
Ca
January, 2012
Pb
Dirac
points.wave
It is worth
the basis
nl but
also
the basis wave
asis
(a)
R
(b)
arewave
explicitly
ive model are explicitly
Λ
O
explicitly
ay
the role of
ctions play the role of
M
Σ
chanism
of
the
role
of
Γ
The mechanism of the
Δ X
Ca
3d d
Ca
ial
clarified
m
of
the
his was
material
was clarified
January, 2012
Fig. 1.Fig.(Color
online)
Crystal
structure
PbO.
echanism
quite quite
1. (Color
online)
Crystal
structureof
of Ca
Ca33PbO.
this[Thesis]
mechanism
sain
clarified
of
discussions.
eteness
of discussions.
Pb
6p p
Pb
Fig.
1.
(Color
online)
Crystal structure of CaOrbital
sm
quite
3 PbO. character of bands:
(a)
R
(b)
irac electrons
in general
rons
in general
Figure 2.5: (a) Momentum path in the cubic Brillouin zone on which the band
cussions.
Λ
it should
beOOnoted
that
previous
theoretical
works
showed
2ppthat
it should
be structure
noted
previous
theoretical
showed
is calculated.
(b) Positions
of the works
Dirac
Brillouin
|p
i ,the|pentire
Pb: points
xin
y i , |p
zi
M lattice
term
the Dirac
and
its↵its
that optimized
thezone.
optimized
constants for
for Ca
Ca33 PbO
Σ
nassofgeneral
the of
Dirac
PbO
and
that the
lattice
constants
↵
[Thesis]
Γ
which
wasbriefly
onlyit briefly
family
obtained
in the
first-principles
calculation
agree
well
Δ X
2 , |dx2 z 2 i , dy 2 z 2
dx2 showed
Ca:
ywell
s
only
family
obtained
in
the
first-principles
calculation
agree
should
be
noted
that
previous
theoretical
works
Pb
6s
Pb
s
4)
28,29)
In particular, the roles with the experimental data,
indicating the consistency
28,29)
cular,
the
roles
with
the
experimental
data,
indicating
the3 PbO
consistency
he
Dirac
and its
that
the
optimized
lattice
constants
for
Ca
itals other than Pb-p and between theory and experiments.
(a)
rWe
than
Pb-p
and
between
theory
and
experiments.
(b
y briefly
family
in
the
first-principles
calculation
agree
well
Special
care
should
be
taken
in
the
treatment
of
the
spin–
will also explain
the obtained
at the Fermi energy, the Fermi energy must be at the Dirac point in order to maintain the
Opening
gap:
he
band
structure
(e) of of
Ca3bulk
PbO
obtained
in theshould be taken
Special
care
in thebetreatment
of the spin–
lso
explain
the
28,29)
opological
insulator,
and
orbit
coupling.
Namely,
we
should
careful
in
applying
p
the
roles
with
the
experimental
data,
indicating
the
charge
stoichiometric
condition.
Due
to
the
cubic
symmetry
of the crystal, existence of
Figure
2.5:DOS
(a) Momentum
path in the cubic Brillouin zone on which thexconsistency
band the
a)
shows
the
total
and
partial
for
each
atom,
Ca1
lcalinsulator,
and
orbit
coupling.
Namely,
we
should
be
careful
in
applying
the
insulator.
Finally,
the
second-variational
step
to
heavy
elements
such
as
Pb
in
is calculated.
(b)
Positions
of
the
Dirac
points that
in thethere
entire
Brillouin
acomponents
Dirac
point
on
the
–X
line
implies
exist
six symmetrically equivalent Dirac
hybridisation
w/
Ca
DOS
for each
atomstructure
decomposed
into
|d
i
,
|d
i
,
|d
i
Pb-p
and
between
theory
and
experiments.
xy
xz
yz
pDirac
tor.areFinally,
the
second-variational
stepBrillouin
to
heavy
such
as points
Pb27)inare located
y strong.
O
studied
using
the ofwhich
the
spin–orbit
coupling
expected to
Ca2at (k0 , 0, 0),
zone.
e symmetry.
In (e),
the position
Dirac point
points
in the
whole
zone.iselements
Specifically,
sixbe
27)
Special
care
should
be
taken
in
the
treatment
of
the
spin–
plain
the
is found
that
there
exist
However,
we
do
not
consider
this
to
be
a
serious
problem
in
udied
the
which
the
spin–orbit
coupling
is
expected
to
be
strong.
ote
thatusing
the
energy
is
measured
from
the
Fermi
k0 , 0,gap
0), (0,
+ SOC opens up( bulk
ofk~10
meVk0 , 0), (0, 0, k0 ), and (0, 0, p k0 ). These positions of Dirac points
0 , 0), (0,
z
ondegenerate
and
cannot
our
calculation.
One
of
the
reasons
that
only
theapplying
state
with
that there
However,
weNamely,
do notzone
consider
this to
a serious
problem
ator,
and exist
orbit
coupling.
we
beisbe
careful
thecase of Ca3PbO, we
z y
in the
Brillouin
areshould
schematically
shown
in in
Fig.
2.5(b).
In in
the
jectedand
onto
thesecond-variational
surface
the
total
J¼
important
for
the
have
k0 =angular
0.11875.
Ca3
rate
our calculation.
One
theheavy
reasons
is3=2
thatisonly
the state
nally,
thecannot
stepofmomentum
to
elements
such
as with
Pb in
x
Dirac
electron
in
this
material,
as
will
be
explained
later,
[after
Kariyado
and
Ogata,
JPSJ
‘12]
27)
The
magnified
of the
theJDirac
band
structure
on the
–Xfor
line the
isthe
plotted
in Fig. 2.6 with
Fermi energy,
the Fermi
energy
mustview
be at
pointisinimportant
order
to maintain
to theat the
surface
the total
angular
momentum
¼ 3=2
en the p-bands
d-bands, two
cross the Fermi
using
the and which
thebands
spin–orbit
coupling is expected to be strong.
FULL PAPERS
T. KARIYADO and M. OGATA
ehatCa
low-energy
the3PbO
low-energy
is a reflection symmetry protected TI
irac
Hamiltonian
by
amiltonian
prove
that by
the low-energy
Symmetries:
sformation
and
by
fact a and
Dirac
Hamiltonian
by
tion
by
Ca
1
basis
transformation
and
by
spect
to
the
momenT
=
is
K
T H( k)T = +H(k)
— Time-reversal:
y
o the
momenwith respect
the momencntspoints.
It is toworth
s.the—
ItDirac
is worth
two
: R1 and R2
Pb
points.wave
It issymmetries
worth
R
t also the reflection
basis
iltonian
but
also
the basis wave
he
basis
model
arewave
explicitly
1
effective
model
are
explicitly
R
anti-commutes
with
T
:
T
R
T
= Rj
O
jthe role of
j
explicitly
nsareplay
e functions play the role of
e mechanism
of
the
the The
role
of
model.
mechanism
of the
two
mirror
Chern numbers: nM1 , nM2
=)
material
clarified
anism
of
the
on
in this was
material
was clarified
Fig. 1.Fig.(Color
online)
Crystal
structure
PbO.
his
mechanism
quite quite
1. (Color
online)
Crystal
structureof
of Ca
Ca33PbO.
this mechanism
l explain
was
clarified
ness
of discussions.
completeness
of discussions.
Fig. 1. (Color online) Crystal structure of Ca3R
PbO.
chanism
quite
1 reflection
e electrons
of Dirac electrons
in general
in general
R2 reflection
20,21)
discussions.
e.
it should
be noted
previoustheoretical
theoretical works
works showed
it should
be noted
thatthat
previous
showed
he
mass
term
the Dirac
that optimized
the optimized
latticeconstants
constants for
for Ca
Ca33 PbO
ons
in ofgeneral
term
the of
Dirac
PbO and
anditsits
that the
lattice
which
wasbriefly
onlyit briefly
family
obtained
in the
first-principles
calculation
agree
well
hial,was
only
family
obtained
in
the
first-principles
calculation
agree
well
should
be
noted
that
previous
theoretical
works
showed
14)
28,29)
aper. In particular, the roles with the experimental data,
indicating the consistency
28,29)
particular,
the
roles
with
the
experimental
data,
indicating
the3 PbO
consistency
of
the
Dirac
and its
that
the
optimized
lattice
constants
for
Ca
nd orbitals other than Pb-p and between theory and experiments.
other
than
Pb-p and
between
theory and
experiments.
only
family
theshould
first-principles
agree
Specialincare
be taken in thecalculation
treatment of the
spin–well
ned.
We briefly
will also explain
the obtained
should
be taken
in thebetreatment
of the spin–
willa topological
also explain the andSpecial
nd
orbit care
coupling.
Namely,
we should
careful the
in applying
the
ular, the rolesinsulator,
with the experimental
data,28,29)
indicating
consistency
ogical
and orbit
Namely,step
we to
should
careful in
applying
the
pologicalinsulator,
insulator. Finally,
the coupling.
second-variational
heavybeelements
such
as Pb in
than
Pb-p
and the
between
theory
and
experiments.
nsulator.
Finally,
second-variational
step to
heavyiselements
such
as Pb27)in
Ca
PbO
are
studied
using
the
which
the
spin–orbit
coupling
expected to
be strong.
3
care
should
taken
ofstrong.
the spin–
o explain
the
del.
It is found
that the
thereSpecial
exist
we do be
not
considerinthis
to treatment
be a serious
in27)
re
studied
using
whichHowever,
the spin–orbit
coupling
isthe
expected
to beproblem
are nondegenerate
and
cannot
our calculation.
of should
the reasons
that
onlyin
theapplying
state withinthe
ound
that there
However,
weNamely,
do notOne
consider
this to
a serious
problem
insulator,
and exist
orbit
coupling.
we
beisbe
careful
s projected
onto
thesecond-variational
surface
the total angular
J¼
important
for
the
egenerate
and
our calculation.
One
theheavy
reasons
is3=2
thatisonly
the state
or.
Finally,
thecannot
stepofmomentum
to
elements
such
as with
Pb in
Dirac
electronmomentum
in this material,
as
willis be
explained for
later,
ed
onto
the
surface
the
total
angular
J
¼
3=2
important
the27)
ied
using
the
which
the
spin–orbit
coupling
is
expected
to
be
strong.
s follows. Section 2 is used to while the second-variational step mainly causes problems
FULL PAPERS
T. KARIYADO and M. OGATA
ehatCa
low-energy
the3PbO
low-energy
is a reflection symmetry protected TI
irac
Hamiltonian
by
amiltonian
prove
that by
the low-energy
Symmetries:
sformation
and
by
fact a and
Dirac
Hamiltonian
by
tion
by
Ca
1
basis
transformation
and
by
spect
to
the
momenT
=
is
K
T H( k)T = +H(k)
— Time-reversal:
y
o the
momenwith respect
the momencntspoints.
It is toworth
s.the—
ItDirac
is worth
two
: R1 and R2
Pb
points.wave
It issymmetries
worth
R
t also the reflection
basis
iltonian
but
also
the basis wave
he
basis
wave
model
are
explicitly
1
R
anti-commutes
with
T
:
T
R
T
= Rj
j model are explicitly
j
effective
O
are
explicitly
ns play the role of
e functions play the role of
e mechanism
the The
role
ofof
twothe
mirror
Chern numbers: nM1 , nM2
=)
model.
mechanism
of the
material
clarified
anism
of
the
on
in this was
material
was clarified
Effectivequite
low-energy Hamiltonian
for one
Dirac
cone
Fig. 1.Fig.(Color
online)
Crystal
structure
PbO.
his
mechanism
1.
(Color
online)
Crystal
structureof
of Ca
Ca33PbO.
explain
this
mechanism
quite
l was clarified
R mirror plane:
ness
of within
discussions.
completeness
of 1discussions.
Fig. 1. (Color online) Crystal structure of Ca3R
PbO.
chanism
quite
1 reflection
e electrons
of Dirac
electrons
iny ,general
H± (k
kz ) in
= general
± sin kz x ± sin ky y ± "k z = m± (k) · ~
R2 reflection
20,21)
discussions.
e.
it should
be noted
previoustheoretical
theoretical works
works showed
it should
be noted
thatthat
previous
showed
m±
(k)constants
he
mass
term
of
the Dirac
and
itsits
that optimized
the optimized
lattice
constants for
for Ca
Ca33 PbO
ons
in ofgeneral
term
the
Dirac
PbO
and
that
the
lattice
E = ± |m± (k)|
m̂
=
±
ial,
which
was
only
briefly
family
obtained
in
the
first-principles
calculation
agree
well
|m
(k)|
±
h was
only
briefly
family
obtained
in
the
first-principles
calculation
agree
well
it
should
be
noted
that
previous
theoretical
works
showed
14)
28,29)
aper. In particular, the roles with the experimental data,
indicating the consistency
28,29)
particular,
the
roles
with
the
experimental
data,
indicating
the3 PbO
consistency
of
the
Dirac
and its
that
the
optimized
lattice
constants
for
Ca
trivial
phase
non-trivial
phase
nd orbitals other than Pb-p and between theory and experiments.
z
z
other
than
Pb-p and
between
theory
and
experiments.
mobtained
m
only
family
in
the
first-principles
calculation
agree
Special
care
should
be
taken
in
the
treatment
of the
spin–well
ned.
We briefly
will also explain
the
±
±
|µ|explain
> 6tinsulator,
|µ|
6t should
care
should
be<taken
in thebetreatment
of the spin–
willa topological
also
the andSpecial
28,29)
nd
orbit
coupling.
Namely,
we
careful the
in applying
the
ular, the roles with the experimental data,
indicating
consistency
ogical
and orbit
Namely,
we±1
should
careful in
applying
the
pologicalinsulator,
insulator.
Finally,
the coupling.
second-variational
step
to
heavybeelements
such
as Pb in
n
=
0
n
=
±
±
than
Pb-p
and the
between
theory
and
experiments.
27)in
x strong.
nsulator.
Finally,
second-variational
step to
heavyiselements
such
as
Pb
x
Ca
PbO
are
studied
using
the
which
the
spin–orbit
coupling
expected
to
be
3
m±
m±should be taken in the treatment
27)
Special
care
of
the
spin–
o
explain
the
y
del.studied
It is found
that
there
exist
However,
we
do
not
consider
this
to
be
a
serious
problem
in
re
using
the
which
the
spin–orbit
coupling
is
expected
to
be
strong.
y
m±
m
±
are nondegenerate
and
cannot
of should
the reasons
that
onlyin
theapplying
state withinthe
ound
that there
However,
weNamely,
do notOne
consider
this to
a serious
problem
insulator,
and exist
orbit
coupling.
we
beisbe
careful
Z our calculation.
⇥reasons
⇤ state
1
s
projected
onto
the
surface
the
total
angular
momentum
J
¼
3=2
is
important
for
the
2
µ⌫
egenerate
and
cannot
our
calculation.
One
of
the
is
that
only
the
with
or. Finally, =)
the n±second-variational
step
to
heavy
elements
such
as
Pb in
=
d
k
✏
m̂
·
@
m̂
⇥
@
m̂
±
k
±
k
±
µ
⌫
Dirac
electronmomentum
in this material,
as
willis be
explained for
later,
8⇡
ed
onto
the
surface
the
total
angular
J
¼
3=2
important
the27)
2Dspin–orbit
BZ
ied
using
the
which
the
coupling
is
expected
to
be
strong.
s follows. Section 2 is used to while the second-variational step mainly causes problems
Ca3PbO is a reflection symmetry protected TI
ky
Mirror Chern numbers:
— for Ca3PbO:
nM 1 =
E = 5.6 eV
2, nM2 = +2
Bulk-boundary correspondence:
|nM | = # Dirac cone surface states
kx
Dirac cone surface states on (001) surface:
2
nM2 = +2
Energy
Energy
nM 1 =
X̄
¯
X̄
M̄
¯
M̄
Chiu, Chan, Nohara, Schnyder, arXiv:1606.03456
wever,
type-IIalways
Dirac of
fermions
are therefore
type I. can appear at the surface
Cand
= 2Methods).
andreflection
C x⇡ =We
0, for
allthat
compo
gued
symmetrie
x0 =CC
find
the
surf
x,yx,y(seethat
H
(k
,
k
)
=
Ak
+
k
k
,
(4)
flection symmetric
(andyweak
topox TCIs
y fermions
0can TR
y xsymmetric
y surface
TCI Dirac
However,
type-II
appear
atx the
C x0 = function
C x,y = 2 and
C x⇡ = 0, with
for allnon-ze
compo
topology
Ca
PbO
is
a
reflection
symmetry
protected
TI
3
al
insulators).
The reflection
Hamiltonian
of
eflection
symmetric
TCIs (andsymmetric
weak TR symmetric
topoLet us now discuss in deta
th
A
>
1.
(Without
loss
of
generality
we
have
set
the
Fermi
type-II
surface states
is generically
given Hamiltonian
by
cal
insulators).
The reflection
symmetric
of (i.e., aoftrivial
class
AIIdue
insulator
[3])
A
EO
arises
to
reflecti
3
locities
tosurface
1 and
assumed
that
the reflection
plane is k x = 0.)
Type-II
Dirac
states
on (111)
surface:
e type-II
states
is
generically
given
by
gued
that
reflection
symmetries
give
r
(i.e.,
a
trivial
class
AII
insulator
[3])
surf
that
the
space
group
Pm
3̄m
po
HTCI
(k x , kstate
Akyis 0protected
+ ky x by
k x reflection
(4)
e type-II
Dirac
symmetry
y ) = (4)
y,
function
with
non-zero
mirr
gued symmetries
thattopology
reflection
symmetries
give
surf
R
which
transfor
i
HTCI (k
) =(4)
Akas
kx y ,
(4)
! x, which
acts
x , kyon
y 0 + ky x
Let us topology
now discuss
detail how
function
withinnon-zero
mirr
A > 1. (Without loss of generality we have set the Fermi
of A
due
to reflection
sym
Let
usarises
now discuss
in
detail
how
3 EO
surf
1
surf
andR
the
reflection
plane
is
k
=
0.)
hities
A >to1.1 (Without
of
generality
we
have
set
the
Fermi
R
r
=(
x,
y,
z),
x
A
>xassumed
0 TCI
: loss
type-II
Dirac
state
H
( kthat
,
k
)R
=
H
(k
,
k
),
(5)
x
x y x
TCI x y
that
the
space
group
Pmreflection
3̄m possesses
of A3 EO arises due to
sym
ype-II to
Dirac
state
(4) is protected
by reflection
ocities
1 and
assumed
that the reflection
planesymmetry
is k x = 0.)
Ry r =(x,
y,possesses
z), r = (
symmetries
R
which
transform
that
the
space
group
Pm
3̄m
i
Mirror
x,the
which
actssymmetry:
onoperator
(4)
Dirac
state
(4)as
is protected
reflection
symmetry
thtype-II
reflection
R x = xby
. Since
reflection
flips the
Rz r =(x,
y, z), r = (
symmetries Ri which
transform
!
x, kwhich
acts
on
(4)
as
gn of
,
it
allows
the
linear
term
Ak
but
forbids
k
.
x
x 0
surf
surf y 0
R x r =( x, y, z),
Ry,±
R x HTCI
( k x , ky )R x 1 = HTCI
(k x , ky ),
(5)
The crucial di↵erence
between
type-I
and type-II Dirac sursurf
1
surf
=( x,
y,z),
z),
Rz,±
ByRRFourier
transforming
into
R x HTCI ( k x , ky )R x = HTCI (k x , ky ),
(5)
xrr =(x,
y,±
y,
R
y
ce
is that
the former
closed circular Fermi surhestates
reflection
operator
R x = have
x . Since reflection flips the
there
are
12y,mirror
planes RR
inx,±
t
R
r
=(x,
y,
z),
NB:
Ak
is
forbidden
by
TRS
y
z,±
R
r
=(x,
z),
y
0
z
ces,
latter
open
electron
and
pockof
k xwhereas
, reflection
it allowsthe
the
linearexhibit
forbids
kflips
h the
operator
Rterm
reflection
0 but
x hole
0 . the
x = Ak
x . ySince
ki =Rz0,r ⇡=(x,
andy, kiz),= ±k j forRi,x,±j
which
other.term
As
one
varies
theDirac
Fermi
energy
crucial
di↵erence
type-I
type-II
ne of
k x , ittouch
allowseach
the between
linear
Ak
forbids
k x sury and
0 but
0.
of these
reflection into
planes
we
By
Fourier
transforming
momen
, the
Fermi
surface
of type-I
Diracand
states
can
be shrunk
states
is
that
the
former
have type-I
closed
circular
Fermi
surhe
crucial
di↵erence
between
type-II
Dirac
sur- to
ber12
[4,mirror
5, 22].
However,
due
there
are
planes
in the
Brill
By
Fourier
transforming
into
momen
whereas
the
latter
exhibit
open
electron
and
hole
pockis called
type-Icircular
Dirac Fermi
point. surIn conesingle
statespoint,
is thatwhich
the former
have aclosed
kthere
⇡ and
ki =3 out
±k
forthese
i, the
j =12
x,my
tries,
only
are
12
mirror
planes
in
Brill
i = 0,
j of
hich
touch Dirac
eachlatter
other.
As
one
varies
the and
Fermi
energy
es,
the
exhibit
open
electron
hole
pock-pockst, whereas
type-II
states
give
rise
to electron
and
hole
of
these
reflection
planes
wei, can
defi
k
=
0,
⇡
and
k
=
±k
for
j
=
x,
y
Without
loss
of
generality,
w
i
i
j
he
Fermi
surface
ofother.
type-I
Dirac
states
can
shrunk
to E F
which
touch
each
one
varies
the be
Fermi
, whose
size
depends
onAs
the
Fermi
energy.
At
aenergy
certain
ber
[4,the
5,reflection
22]. However,
due
to the
of these
planesnumbers
we can
defi
mirror
Chern
C xc
gle
point,
which
is
called
a
type-I
Dirac
point.
In
conthe
Fermi
surface
of
type-I
Dirac
states
can
be
shrunk
to
e electron and hole Fermi surfaces touch each other, thereby
tries,
only
322].
out
of these planes
12due
mirror
inv
ber
[4,
5,
However,
to
the
for
the
reflection
k
=
x
type-II
Dirac
states
give
rise
to
electron
and
hole
pockngle
point,
which
is
called
a
type-I
Dirac
point.
In
conalizing a Lifshitz transition. The touching of the electron
Without
loss
of
generality,
choos
tries, tively.
only
3 out
of these
12 we
mirror
inv
Both
first-principles
c
whose
size
depends
on
the
Fermi
energy.
At
a
certain
E
t,
type-II
Dirac
states
give
rise
to
electron
and
hole
pockF
d hole pockets is called a type-II Dirac point. As opposed
0, C
the
mirror
Chern
numbers Cwe
Without
loss
of generality,
choos
x⇡ ,tha
xshow
fective
considerations
ectron
hole
Fermi
touch
other,
thereby
whose and
size
depends
onsurfaces
thedensity
Fermi
energy.
At
aatcertain
E FDirac
type-I
Dirac
points,
the
ofeach
states
type-II
for
reflection
knumbers
=x00,, Ck x⇡ta
thethe
mirror
Chernplanes
numbers
,=
x C
the
mirror
Chern
ing
a
Lifshitz
transition.
The
touching
of
the
electron
electron
and
hole
Fermi
surfaces
touch
each
other,
thereby
ints remains
finite.
In addition,
we observe that inChiu,
type) dense
Landau
level spectrum
tively.
Both
first-principles
for the
reflection
planes
k x =calculati
0, k x =
Chan,
Nohara,
Schnyder,
arXiv:1606.03456
⇡
and C x = 0 (see Methods an
ole pockets
is called
a type-II
point.ofAs
izing
a Lifshitz
transition.
TheDirac
touching
theopposed
electron
2. Topological nodal line semi-metals
Ching-Kai Chiu (UMD)
Yang-Hao Chan (A. Sinica)
Ca3P2
Classification of topological materials with reflection symmetry
R+ : R commutes with T (C or S)
TI/TSC
Reflection
R
R+
R
R+ ,R++
R ,R
R +
R+
R+
R +
R+
R +
R
: R anti-commutes with T (C or S)
FS1
d=1
p=8
d=2
p=1
d=3
p=2
d=4
p=3
d=5
p=4
d=6
p=5
d=7
p=6
d=8
p=7
FS2
p=2
p=3
p=4
p=5
p=6
p=7
p=8
p=1
A
AIII
AIII
AI
BDI
D
DIII
AII
CII
C
CI
AI
BDI
D
DIII
AII
CII
C
CI
BDI, CII
DIII, CI
BDI
DIII
CII
CI
MZ
0
MZ
0
MZ
0
MZ
0
0
MZ
0
MZ
0
MZ
0
MZ
MZ Z
0
MZ Z
0 CaM
Z2Z
0
MZ Z
0
3P
MZ
0
0
0
2M Z
0
M Z2
M Z2
M Z2
MZ
0
0
0
2M Z
0
M Z2
M Z2
M Z2
MZ
0
0
0
2M Z
0
0
M Z2
M Z2
MZ
0
0
0
2M Z
2M Z
0
M Z2
M Z2
MZ
0
0
0
0
2M Z
0
M Z2
M Z2
MZ
0
0
0
0
2M Z
0
M Z2
M Z2
MZ
0
0
0
0
2M Z
0
M Z2
M Z2
MZ
0
0
2M Z
0
T Z2
Z2
MZ
0
0
0
0
2M Z
0
T Z2
Z2
MZ
MZ
0
0
0
2M Z
0
T Z2
Z2
Z2
MZ
0
0
0
2M Z
0
T Z2
T Z2
Z2
MZ
0
0
0
2M Z
0
0
T Z2
Z2
MZ
0
0
0
2M Z
2M Z
0
T Z2
Z2
MZ
0
0
0
0
2M Z
0
T Z2
Z2
MZ
0
0
2Z
0
2M Z
0
2Z
0
2M Z
0
2M Z
0
2Z
0
2M Z
0
2Z
0
MZ Z
0
0
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2
M Z2 Z2 M Z2 Z2 M Z Z
0
0
0
2M Z 2Z
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2 M Z Z
0
0
0
0
0
2M Z 2Z
0
M Z2 Z2 M Z2 Z2 M Z Z
0
Tabelle I Classification of topological insulators and superconductors (“TI/TSC”) as well as of stable Fermi
Chiu, Schnyder
PRB 2014
Topological nodal lines in Ca3P2
see talk by Leslie Schoop
Band structure:
Ca
P
(a)
↵
tR
where
Crystal structure P63/mcm the unit c
at positio
ements (2
factor in
| k↵ i ! e
has a nest
mirror plane
(b)
Dirac ring within reflection plane
Ca3
H(k
kz
Ca6
K'
M
K
where
the
charge balanced:
—
fixed m, n
Orbital character of bands near EF:
Hij repres
(6 Ca atoms, 6 P atoms)
Ca and P
FIG. 1. Crystal structure and electronic bands of Ca3 P2 .
lu
Ca: dz 2 orbitals from 6 Ca atoms
and
(h
(a) Crystal structure of Ca3 P2 , which contains two planes
ij ,
respective
6 P atoms
P: p
with three
Ca
atomsfrom
(blue)
and three P atoms (red) that are
x orbitals
mn
Chiu, Chou,
Phys. Rev. B 93, h
205132 (2016)
is sp
separated by interstitial Ca atoms Chan,
(black).
TheSchnyder,
gray dashed
ij
Ca2+
P3-
Ca2
Ca1
A
Ca5
Ca2
P2
Ca4
M
Ca4
kx
P4
P5
Ca1
Ca5
P1
mirror plane
P2
Ca2
Ca4
K'
P4
px
K
ky
Topological nodal line: Mirror invariant
Reflection ( z !
R
1
z ):
P4
H(kx , ky , kz )R = H(kx , ky , kz )
0
13⇥3
B 0
R(k) = B
@ 0
0
0
13⇥3 e
0
0
ikz
Ca4
C
0
0
(lower
plane)
C
A
13⇥3
0
0
13⇥3 e
P6
Ca1
Ca2
P2
1
mirror
plane
0
0
Ca6
P1
mirror plane
P4
Ca4
Ca6
P6
px
px
ikz
dz2
dz2
Mirror invariant:
— number of occupied states with R = +1
R = +1
0
+,0
NM
=
n
Z
occ (|k| > k0 )
n+,0
occ (k)
=
⇢
1
0
n+,0
occ (|k| < k0 )
|k| < k0
|k| > k0
R=
+k0
Chan, Chiu, Chou, Schnyder, Phys. Rev. B0.4
93, 205132 (2016)
1
M
K
k
k0
Drumhead surface state and Berry phase
Berry phase & charge polarization:
P(kk ) =
i
X Z
j2filled
⇡
⇡
D
Berry phase
(j)
(j)
E
uk? @k? uk? dk?
E
— P(kk ) quantized to ⇡ ) stable line node
— In Ca3P2 Berry phase is quantized due to:
(i) reflection symmetry z !
z
¯
(a)
(b)
(ii) inversion + time-reversal symmetry
k
z
FIG.
3.
Drumhead
surface
states a
K'
+,0
+,⇡
M
face band structureK of Ca3 P2 as
1)nocc (k)+nocc (k) ei@R = eiP(k)
A
binding model (2.2)
for the (001)
(b) Momentum-resolved surface den
nian (2.2) for the (001)K'surface.
Br
ky
M
correspond to
highK and low densit
Ca2
Ca4
kx
ation of the Berry phase (2.7) of
Ca1
P1
P5 Ca5
high-symmetry lines of the (001) s
mirror plane
Fig. 1(d)]. (d) Surface spectrum o
Ca2
Ca4
model (3.2) for the (001) face as a
Chan, Chiu, Chou, Schnyder, Phys. Rev. B 93, 205132 (2016)
menta kx and ky . The bulk states w
Ca3
(
ky
Ca2
Ca5
Ca6
Ca1
Ca4
P2
P4
P2
P4
px
2
Drumhead surface state and Berry phase
Berry phase & charge polarization:
P(kk ) =
i
X Z
j2filled
⇡
⇡
D
(j)
Berry phase
(j)
E
uk? @k? uk? dk?
E
— P(kk ) quantized to ⇡ ) stable line node
— In Ca3P2 Berry phase is quantized due to:
(i) reflection symmetry z !
z
¯
(ii) inversion + time-reversal symmetry
(a)
ase
(b)
ky
5
Surface spectrum
FIG. 3. Drumhead surface states a
face band structure of Ca3 P2 as
( 1)
ei@R = eiP(k)
binding model (2.2) for the (001)
spectrum of
(b) Momentum-resolved surface den
model
(2.2)
Bulk-boundary
correspondence:
nian (2.2) for the (001) surface. Br
se, there apcorrespond to high and low densit
e
— surface
ation of the Berry phase (2.7) of
Figure
3(a) charge: surf = 2⇡ P mod e
high-symmetry lines of the (001) s
001) surface
Fig. 1(d)]. (d) Surface spectrum o
Nearly flat 2D surface states
0 unit cells.connecting
model (3.2) for
the (001) face as a
(c) Dirac ring
(d)
k
(1/a)
y
Chan, Chiu, Chou, Schnyder, Phys. Rev. B 93, 205132 (2016)
h-symmetry
menta kx and ky . The bulk states w
+,⇡
n+,0
(k)+n
occ
occ (k)
)
Drumhead surface state and Berry phase
Nearly
Dirac ring
(a) flat surface states connecting
(b)
Ca3
K'
M
Ca2
D.
kz
Ca6
K
Ca1
D.
A
Ca5
Ca2
P2
Ca4
Ca4
kx
P4
P5
Ca1
Ca5
K'
M
K
Ca2
Ca4
P4
px
dz2
Although t
acterize the
Although the Berry
deeply
related
acterize
the two
di↵
deeply related by the
P1
mirror plane
P2
ky
Relatio
and
D.
Drumhead surface state
( 1)N
Relation between Berry p
R⇡P
and mirrorwhere
invariant
@R
=
where @R = i 0
E
kz . Appendix
kz . Appendix
A show
EE Although the Berry phase
In general,
res
In general,
reflection
and
N
Z indep
k-dependent,
is featu
writte
k-dependent,
acterize the two di↵erent
physical
60], only
k-independe
60],symmetry
only k-in
deeply related by the reflection
considered; the relati
ky
kx
considered;
t
ky
kx
invariants
can
be
wri
0
⇡
NM Z +NM Zinvariants
i@R
iPcan
1
(
1)
e
=
e
Chan, Chiu, Chou, Schnyder, arXiv:1510.02759 ,
0
Low-energy effective theory for Ca3P2
low-energy effective Hamiltonian:
He↵ (k) = (kk2
even in k
k02 )⌧z + kz ⌧y + f (k)⌧0
symmetry operators:
— reflection: R = ⌧z
— time-reversal: T = ⌧0 K
— inversion:
I = ⌧z
Gap-opening term ⌧x is symmetry forbidden:
R
1
— breaks inversion + TRS: (IT )
1
— breaks reflection symmetry:
⌧x R =
⌧x
⌧x IT =
⌧x
) nodal line is stable
Z versus Z2 classification:
He↵ (k) ⌦
0
= (kk2
k02 )⌧z ⌦
0
+ k z ⌧y ⌦
0
+ f (k)⌧0 ⌦
0
— consider gap opening term m̂ = ⌧x ⌦ y :
• (IT )-symmetric:
(⌧z ⌦ 0 K) 1 m̂(⌧z ⌦ 0 K) = m̂
• but breaks R:
(⌧z ⌦ 0 ) 1 m̂(⌧z ⌦ 0 ) 6= m̂
) Z2 classification
) Z classification
3. Nodal non-centrosymmetric
superconductors
R. Queiroz (MPI-FKF)
C. Timm (TU Dresden)
CePt3Si
P. Brydon (U Otago)
International Symposium
Nodal non-centrosymmetric superconductors
Festkörperphysik II, Musterlösung 11.
[E. Bauer et al. PRL ’04]
anti-symmetric SO coupling:
• Lack of inversion causes
X
Festk
örperphysik
II,
Musterl
ösung
11.
†
Prof.
M. Sigrist,
Normal
state: HWS05/06
=
("k 0 + |gk | 3 )
Consider
ETH Z
k
k
k
coupling
for C4v point group: gk = ky x̂ − kx ŷ
rof. M.SO
Sigrist,
WS05/06
!
"
nd
3H
#$
2
ETH Zürich
2
g
=
k
1
+
g
k
x̂
k
x
2
y + kz
1• Lackfrist
chapter
CePt
Si
k
of inversion allows for admixture
of spin-singlet
Festk
örperphysik
II,
Musterl
ösun
3
z
!Festk
" örperphysik
#$
II,
Muster
2
2
+ky 1örperphysik
+ g2 kx + kz ŷ II, Musterlösung
Festk
and spin-triplet pairing components
!
" 2
#$
2
+k
+
g
ẑ
z (1
2 )kx + ky
These arekthe
gaps
g
∥
d
(1)
= ( s 0 + t dk · ~ ) i y
k
k
ν =M.
−2.0
ν M.
=WS05/06
+2.0
νWS05/06
= +4.0
Prof.
Sigrist,
±
Prof.
Sigrist,
Gaps on the two Fermi
surfaces:
Prof.
M. Sigrist,
∆ WS05/06
= ∆ ± ∆ |d |
Superconducting gap
(1,3) k
ginter =
(2,3) s
ginter =
p
−0.05
k
P
H
Y
S
I
C
A
L
PRL
102,
027004
(2009)
∆(k)
=
(∆
σ
+
d
·
σ)
iσ
PRL
102,
027004
(2009)
⇥
>
⇥
⇥
⇤
⇥
⇥
<
⇥
s
0
k
y
⇥
>
⇥
⇤
⇥
<
⇥
⇥
>
⇥
⇥
⇤
⇥
⇥
<
⇥
s
t
s
t
s
t
Particle-hole redundancy:
st
ts
s t
t s
s
s
t t
∆0,t ≫ ∆0,s
(2)
k
x
(3)
R E V(4)I E PWH YL
(5)
⇤ 0 Lx /2κ⇤xy0
L/2
x /2
⇤
L
⇥
⇥
x
⇥
0 ⇥
Abbildung 1:e Panel
(a):
Chiral
surface
bound states for
at the the
(111) faR
κ1xy /T
1⇥
⇥
e
e
1
Iofy superfluid
= Iy 3 He
dE T
(class A). The color scale
is such dE
that dark blue co
=
I
=
dE
!
2~
y· σ
There
is N
akdispersionless
zero mode
subman
† h(k)y= energy.
⇥ in a one-dimensional
cases
fro
2~
N
ε
σ
+
αg
y at
k
0
n=1
k
⇥
y
2~
N
ϕ
=
Ξϕ
,
γ
=
γ
⇒
Majorana
state
E
=
0
y
−E wi cos
E(k ·E
= ∆0,s
Ti ) −E Brillouin zone. Panels
(6)
y⇥n=1
and (c): kBand
structure for
the (111) face of 3
ky(b)n=1
x are l
theoretic
of
surface
momentum
q
and
q
,
respectively.
The
red
(blue)
bands
⇥
{2t
sin
k
⇧
(E,
k
)
⇥
cos
k
⇧
x
y
i
y n
⇥ {2t
sin k y⇧ (E, k ) y ⇥n (E,
cos kk
x
Normal state
negative
helicity FS
(l)
∆s,k
!
y )n
y k ⇧ (E,
(bottom)
surface.
Panel
(d):
Fermi
points
in the⇥three-dimensional
Br
⇥
{2t
sin
k
⇧
(E,
k
cos
(l)
BdG Hamiltonian
l
y
n
y
y
n
lying co
full gap
line
full gapdoubling theorem the lattice
∆t,k = (−1)
∆0,t
wi sin
(knodes
· Ti ) A. Due to the Fermion
(7)Hamiltonian
has
$Note%that the projection of the $
% two Fermi point
and "i
line connecting the
!
1
and #
determin
) helicity basis
absence of parity ) mixed
Nodal non-centrosymmetric superconductors
l us n I
la cigo lopo
T
fo
elba
T
cidoi reP
singlet/triplet, gap line nodes
: seirt e m m y S yr ati n U-it n A
;
;
(
) k
;
H
H
H
• Symmetries:
) k
1
) k(
) k(
1
(
1
) k(
T =
0
C=
1
H
H
H
: l a sr e v e R e miT
-
Time-reversal and particle-hole:
) k(
⌦i
⌦
:
el o H
- el citr a P
-
: yrt e m m y s )l ari h c( yr ati n U
2
2
T =
0
C = +1
2D surface
Brillouin zone
1
3D Brillouin zone
- d n a lt lA
r e u a b nr iZ
m o d n aR
x ir t a M
s e s s a lC
Fermi surface
classinvariant
DIII
no
2 bulk topological
but momentum-dependent
winding number W (kk ) at
1D contour in general not centrosymmetric:
surfaces
P t toB
8002
TRS ,
PHS
• Winding number:
g iw duL
, i ka suruF
8002 , vea ti K
,u yR ,re d ynh c S
W (k
) = ±1: =)
nondegenerate
class AIII
S=TRS
x kPHS
zero-energy flat bands
E
(a)
⇠k± = "k ± |gk |
±
k
=
z
— surface flat bands
s
±
kx
t
1,k (r? )
(c)
Topical Re
E
|dk |
kx
kx
kz
• Surface flat bands have Majorana character:
⇠
1D class AIII Hamiltonian
gap nodes
Spin texture of topological superconductor...
E
(b)
k
• Bulk-boundary correspondence:
k
projected
gap nodes
trivial = 0
J. Phys.: Condens.
Matter
27 1,3
(2015)
243201
Philip M. R.
Brydon,
Andreas
Schnyder,2 and Carsten Timm3
⇣
non−trivial
projected = +/−1
Fermi surface
kz
Schnyder, Ryu, PRB (2012) Schnyder
et al. from
PRL[66].
(2013)
†
(a)†helical Majorana cone, (b) arc surface state, and (c) flat-band
surface state.
Figure adapted
ck,"superconductors:
isgn(k)c
+
(r
)
c
+
isgn(k)c
2,k ?
k,#
Queiroz, Schnyder, PRB (2014)
k,#
k,"
Brydon et al. NJP (2015)
with codimension p < dBZ (i.e. dn > 0), provided that they 3.3. Examples
Queiroz, Schnyder, PRB (2015)
are protected by a Z invariant or 2Z invariant. Z2 numbers,
⌘
⇣
⌘
Figure 1. Energy spectrum of the three different types of topological subgap states that can exist at the surface of nodal noncentrosymme
Conclusions
Outlook
Conclusions& and
Outlook
Mirror plane
• Ca PbO is a topological insulator with reflection symmetry 3
— Two Dirac surface states, type-II Dirac states
arXiv:1606.03456
• Topological nodal line semi-metal Ca P
3
2
— Drumhead surface states
Phys. Rev. B 93, 205132 (2016)
J. Phys.: Condens. Matter 27 (2015) 243201
(a)
E
Topical Review
(b)
E
E
(c)
• Nodal non-centrosymmetric superconductor CePt Si 3
— Majorana flat band surface states
kz
kx
kx
kz
kx
kz
•
Figure 1. Energy spectrum of the three different types of topological subgap states that can exist at the surface of nodal noncentrosymmetric
Topological classification schemes: superconductors: (a) helical Majorana cone, (b) arc surface state, and (c) flat-band surface state. Figure adapted from [66].
with codimension p < dBZ (i.e. dn > 0), provided that they
are protected by a Z invariant or 2Z invariant. Z2 numbers,
on the other hand, guarantee only the stability of nodes with
dn = 0, i.e. point nodes. These findings are confirmed by
more rigorous derivations based on K theory [58–60] and
minimal Dirac Hamiltonians [61]. The latter approach uses
Clifford algebra to classify all possible symmetry-preserving
mass terms that can be added to the Hamiltonian. The
classification of global-symmetry-invariant nodal structures
(and Fermi surfaces) is summarized in table 1, where the first
row indicates the codimension p of the superconducting nodes.
For any codimension p there are three symmetry classes for
which stable superconducting nodes (or Fermi surfaces) exist
that are protected by a Z invariant or 2Z invariant, where the
prefix ‘2’ indicates that the topological number only takes on
even values. Furthermore, in each spatial dimension dBZ there
exist two symmetry classes that allow for topologically stable
point nodes (Fermi points) which are protected by a binary Z2
number.
(i) bring order to the growing zoo of topological materials
3.3. Examples
For the phenomenological model Hamiltonians given in
section 2, we derive in this subsection explicit expressions
for the topological invariants that protect the superconducting
nodes against gap opening. We also use these examples to
illustrate the bulk-boundary correspondence [64, 65], which
links the topological characteristics of the nodal gap structure
to the appearance of zero-energy states at the boundary.
Depending on the case, these zero-energy surface states are
either linearly dispersing Majorana cones, Majorana flat bands,
or arc surface states (see figure 1). We note that in real
superconducting materials the gap nodes are usually positioned
away from the high-symmetry points of the Brillouin zone.
Indeed, this is the case for the three examples of section 2,
which are therefore classified according to section 3.2.
Note that the topological invariants introduced here can be
straightforwardly generalized to more complicated systems.
(ii) give guidance for the search and design of new topological states
(iii) link the properties of the surface states to the bulk wave function topology
Review articles: arXiv:1505.03535; J. Phys.: Condens. Matter 27, 243201 (2015)
3.3.1.
The A phase of
3
He.
The A phase of 3 He is