Anomalous behavior of the superconducting proximity effect

Topological insulators and mathematical science
Harvard University
Topological crystalline insulators
and topological magnetic
crystalline insulators
Chaoxing Liu
Department of Physics, The Pennsylvania State University,
University Park, Pennsylvania 16802, USA
Collaborated with Ruixing Zhang, Xiaoyu
Dong (Tsinghua), and Brian VanLeeuwen
cond-mat/1304.6455, Phys. Rev. B 90, 085304
cond-mat/1401.6922
Outline
•
•
•
•
Introduction: topological insulators and
topological crystalline insulators
Symmetry induced degeneracy and surface
states
Topological crystalline insulators and
topological magnetic crystalline insulators
Summary and outlook
2
Topological states of matters
•
Band theory of metals and insulators
Transport properties of materials are usually
determined by bulk band gap and the position of
Fermi energy in an electronic system.
Ef
Ef
Eg
Insulating
Metallic
3
Topological states of matters
•
Topological phases of free fermions
Topological phases are usually characterized by
insulating bulk states and metallic edge/surface
states in a free fermion system.
e.g. quantum Hall effect
4
Symmetry protected topological phases
•
Time reversal (TR) invariant topological
insulators (TIs)
Topological phases due to the protection of time
reversal symmetry, helical edge states
5
Topological insulators
•
Experimental observation of helical edge
states in TR invariant TIs
Bernevig, et al (2006), Konig,
et al (2007)
Hsieh, et al, (2008), (2009); Roushan, et
al, (2009), H.J. Zhang et al (2009); Xia
et al (2009); Y. L. Chen (2009)
6
Topological insulators
•
Kramers’ theorem
𝑬
𝐸 𝑘, ↑ = 𝐸(−𝑘, ↓)
•
𝑘 = −𝑘 + 𝐺⃗
−𝝅
𝟎
𝝅
Non-trivial surface states protected by
Kramers’ degeneracy
𝒌
7
Topological crystalline insulators
•
•
Can degenerate points be protected by other
symmetries?
Gapless edge/surface states protected by
crystalline symmetry, topological crystalline
insulators (TCIs)
SnTe, mirror symmetry
Fu (2011); Timothy, et al, (2012)
8
Topological crystalline insulators
•
Experimental observations of TCIs
SuYang Xu, et al (2012)
Dziawa, et al (2012)
Tanaka, et al (2012)
9
Topological crystalline insulators
•
How to find a practical and systematical way to
identify which types of crystalline symmetry
group can protect non-trivial surface states?
Taylor, et al (2010), Fang, et al (2012), Slager, et al (2012), Jadaun,
et al (2012)
Most of these works are constructing bulk
topological invariants based on bulk symmetry
directly. However, surfaces can break crystalline
symmetry of a bulk system.
10
Topological crystalline insulators
•
Our strategy: classification based on
symmetry groups of surface states.
Look for gapless surface states
in a semi-infinite system
�
𝒀
𝒌𝒚
�
𝚪
�
𝐗
�
𝑴
𝒌𝒙
11
Topological crystalline insulators
•
How to protect these degeneracies?
•
Degeneracy and symmetry
Wigner, ...
 Different irreducible representations (type I)
E.g. Mirror topological insulators
 Non-commutation relation, high dimensional
irreducible representations (type II)
E.g. non-symmorphic topological insulators
 Anti-unitary symmetry operator (type III)
E.g. Time reversal invariant topological insulators,
magnetic topological insulators
12
Summary of our approach
•
•
•
•
Determine 2D space symmetry group of a
semi-infinite bulk system with one surface
Determine wave vector groups of each
momentum in the surface Brillouin zone.
Determine degeneracies of
high symmetric momenta from
representations of wave
vector group.
Determine possible non-trivial
surface states and bulk
topological invariants.
Surface BZ
Γ
Bulk BZ
13
Topological crystalline insulators
•
Advantages of our approach
 We can guarantee the existence of surface states.
 We show how to use the representation theory of
symmetry groups to classify different surface
states.
 There are only 17 2D space group (wall paper
group). Therefore, it is possible to get a complete
study of all the possible groups.
14
Topological crystalline insulators
•
17 2D space group, wall paper group
Group
P1
P2
P4
P3
P6
Pm
Pg
Cm
Deg
No
No
No
No
No
I
I
I
Group
Pmm
Cmm
Pmg
Pgg
P4m
P4g
P3m1
P31m
P6m
Deg
No
No
I, II
I, II
I, II
I, II
I, II
I, II
I, II
15
Type I TCIs
•
Degeneracy due to different representations
of a symmetry group
Pm group: a line in the Brillouin
zone has symmetry 𝜎𝑦 : 𝑥, 𝑦, 𝑧 →
(𝑥, −𝑦, 𝑧)
𝜎𝑦
𝒌𝒚
�
𝚪
𝒌𝒙
16
Type I TCIs
•
•
Two surface states will not couple to each other if
they belong to different representations
Corresponding to mirror Chern number, mirror
topological insulators
𝐻
0
SnTe system: Timothy, et al, (2012)
𝐻= +
0 𝐻−
𝑬
𝒌𝒚
�
−𝑿
�
−𝑿
𝚪�
�
𝑿
�
𝚪
�
𝑿
𝒌𝒙
17
Topological crystalline insulators
•
17 2D space group, no anti-unitary operators
Group
P1
P2
P4
P3
P6
Pm
Pg
Cm
Deg
No
No
No
No
No
I
I
I
Group
Pmm
Cmm
Pmg
Pgg
P4m
P4g
P3m1
P31m
P6m
Deg
No
No
I, II
I, II
I, II
I, II
I, II
I, II
I, II
18
Type II TCIs
•
Degeneracy due to high dimensional
irreducible representations, non-commutation
between symmetry operations
Eg. zinc-blende semiconductors
19
Type II TCIs
•
A special case, anti-commutation relation
E.g. non-symmorphic symmetry; all the states are
doubly degenerate at some special momenta; nonsymmorphic topological insulators
𝑅, 𝐻 = 0,
𝑆, 𝐻 = 0,
𝑅, 𝑆 = 0
If 𝐻 𝜙 = 𝐸|𝜙〉 and 𝑅 𝜙 = 𝑟|𝜙〉, 𝑆|𝜙〉 and |𝜙〉 are two
orthogonal and degenerate eigen states.
𝐻𝐻 𝜙 = 𝑆𝑆 𝜙 = 𝐸𝐸 𝜙
𝑅𝑅 𝜙 = −𝑆𝑆 𝜙 = −𝑟𝑟 𝜙
→ 𝑆|𝜙〉 is an eigen-state
→ 𝑆|𝜙〉 is different from |𝜙〉.
20
Non-symmorphic symmetry
•
Non-symmorphic symmetry group
Two symmetry operations, 𝜎𝑧 : 𝑥, 𝑦, 𝑧 → 𝑥, 𝑦, −𝑧
𝑐
𝑐
𝑔𝑥 = 𝜎𝑥 𝜏⃗ ∶ 𝑥, 𝑦, 𝑧 → −𝑥, 𝑦, 𝑧 + ,
𝜏⃗ = 0,0,
2
2
𝐴
𝐵
𝑐
𝑎
𝐴
𝑏
21
Non-symmorphic symmetry
•
Non-symmorphic symmetry
𝜎𝑧 𝑔𝑥 = 𝐶2𝑦 −𝝉 ,
𝑔𝑥 𝜎𝑧 = 𝜎𝑧 𝑔𝑥 + 𝒕,
When 𝑘 =
𝜋
0, 𝑘𝑦 ,
𝑐
𝑔𝑥 𝜎𝑧 = 𝐶2𝑦 𝝉 ,
𝒕 = 0,0, 𝑐
𝑜𝑜
𝜋
𝜋
, 𝑘𝑦 ,
𝑎
𝑐
𝐶2𝑦 −𝜏⃗ 𝜙𝑘 = 𝑒 −𝑖𝑘⋅𝜏 𝐶2𝑦 𝜙𝑘
= −𝑖𝐶2𝑦 |𝜙𝑘 〉
𝐶2𝑦 𝜏⃗ 𝜙𝑘 = 𝑖𝐶2𝑦 |𝜙𝑘 〉
{𝜎𝑧 , 𝑔𝑥 } = 0
,
𝑐
𝝉 = 0,0,
2
𝒌𝒛
�
𝒁
�
𝚪
�
𝑼
�
𝐗
𝒌𝒙
22
Non-symmorphic symmetry
•
Non-symmorphic symmetry
•
Anti-commutation relation results in
�.
the degeneracy at 𝑍̅ and 𝑈
Surface states and topological
invariant
𝒌𝒛
�
𝒁
�
𝚪
�
𝑼
�
𝐗
𝒌𝒙
Wannier function
center
R. Yu, PRB (2011)
CXL, RXZ and BV , Phys. Rev. B 90, 085304
23
Non-symmorphic symmetry
•
Eg. Non-symmorphic
symmetry
𝒌𝒛
�
𝒁
�
𝚪
�
𝑼
�
𝐗
𝒌𝒙
TI
24
Topological crystalline insulators
•
17 2D space group, no anti-unitary operators
Group
P1
P2
P4
P3
P6
Pm
Pg
Cm
Deg
No
No
No
No
No
I
I
I
Group
Pmm
Cmm
Pmg
Pgg
P4m
P4g
P3m1
P31m
P6m
Deg
No
No
I, II
I, II
I, II
I, II
I, II
I, II
I, II
25
P4m Group
•
Double degeneracy due to
mirror and rotation symmetry
operation (C4v)
Bernevig’s group (2014)
𝒌𝒚
�
𝒀
�
𝚪
�
𝑴
�
𝐗
𝒌𝒙
26
P4m Group
•
Double degeneracy due to mirror and
rotation symmetry operation (C4v)
Bernevig’s group (2014)
� : 𝐶4𝑣 symmetry
Γ�, M
� line has mirror symmetry
Γ� − M
Halved mirror chirality 𝜒 can be defined
27
Type III magnetic TCIs
•
Degeneracy due to anti-unitary symmetry
operations
 Kramers’ degeneracy, spinful fermions,
Θ2 = −1, TR invariant TIs.
𝐸 𝑘, ↑ = 𝐸(−𝑘, ↓)
𝑘 = −𝑘 + 𝐺⃗
𝒌𝒚
𝒌𝒙
28
Type III magnetic TCIs
•
Other anti-unitary operator, magnetic group
ℳ = 𝒢 + 𝐴𝐴
RMP, 40, 359, (1968)
𝒢 is the unitary sub-group
𝐴 = Θ𝑅 is an anti-unitary element, Θ time reversal
and the unitary operator 𝑅 ∉ 𝒢.
Herring rule
� 𝜒(𝐵2 ) = 𝐺
𝐵∈𝐴𝒢
=−𝐺
Wigner (1932), Herring (1937)
𝑐𝑐𝑐𝑐 𝑎 ;
𝑐𝑐𝑐𝑐 𝑏 ;
= 0 𝑐𝑐𝑐𝑐 𝑐 .
RX Zhang and CXL (arxiv: 1401.6922)
Δ real, reducible
Δ = 𝑃Δ∗ 𝑃−1 , irreducible
Δ ≠ 𝑃Δ∗ 𝑃−1 , irreducible
29
Type III magnetic TCIs
•
Magnetic topological insulators
 𝝉Θ with translation operation 𝝉 and time reversal
symmetry Θ. 𝝉Θ 2 = −1 can exist for both
spinless and spinful fermions.
Mong, Essin, Moore (2010), Fang, et al (2013), Liu (2013)
 Cn Θ with rotation operation 𝐶𝑛 and time reversal
symmetry Θ. 𝐶𝑛 Θ 2 = −1 exists for both spinless
and spinful fermions.
Fu (2011), RX Zhang and CXL (arxiv: 1401.6922)
30
Type III magnetic TCIs
•
𝐶4 Θ model
Eg. 𝐶4 Θ with four-fold rotation symmetry 𝐶4 and
time reversal symmetry Θ.
𝐶4 Θ
𝐶2
2
= 𝜔𝐶2 ,
𝜔 = ±1
𝒌𝒚
𝒌𝒙
31
Type III magnetic TCIs
•
𝐶4 Θ model
RX Zhang and CXL (arxiv: 1401.6922)
32
Type III magnetic TCIs
•
𝐶4 Θ model, surface states and Z2 topological
invariants.
Wannier function center
R. Yu, et al (2011)
RX Zhang and CXL (arxiv: 1401.6922)
33
Type III magnetic TCIs
•
Generalization to other Cn Θ symmetry
Value of n
single group
double group
mirror
no
no
C2 Θ
no
no
C3 Θ
no
no
C4 Θ
yes (Z2)
yes (Z2)
C6 Θ
yes (Z2×Z2)
yes (Z2×Z2)
RX Zhang and CXL (arxiv: 1401.6922)
34
Type III magnetic TCIs
•
C6 Θ symmetry
RX Zhang and CXL (arxiv: 1401.6922)
35
Δ1+Δ2+Δ3=0 mod 2
only two Δi s are
indepedent
{(0,0),(1,0),(0,1),(1,1)}
Z2×Z2 topological
invariant pair
Type III magnetic TCIs
•
C6 Θ symmetry
RX Zhang and CXL (arxiv: 1401.6922)
37
Summary and Outlook
•
•
•
We have presented a theory to explore
different topological crystalline insulators by
constructing non-trivial surface states.
Our approach makes it possible to get a
complete table for possible topological phases
of different crystalline structures in a free
fermion system.
Our approach might be generalized to other
systems, such as Weyl semi-metal systems,
boson systems, superconducting systems, …
38

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