Topological Quantum Phenomena
Nagoya University,
Masatoshi Sato
1
In collaboration with
• Satoshi Fujimoto (Kyoto University)
• Yoshiro Takahashi (Kyoto University)
• Yukio Tanaka (Nagoya University)
• Keiji Yada (Nagoya University)
• Ai Yamakage (Nagoya University)
• Yuji Ueno (Nagoya University)
• Takeshi Mizushima (Okayama University)
• Kazushige Machida (Okayama University)
• Masanori Ichioka (Okayama University)
• Yasumasa Tsutsumi (Riken)
• Takuto Kawakami (NIMS)
• Ken Shiozaki (Kyoto University)
• Shingo Kobayashi (Nagoya University)
Review paper
Y. Tanaka, MS, N. Nagaosa, “Symmetry and Topology in SCs”
Journal of Physical Society of Japan, 81 (2012) 011013 (open access)
2
Outline
Part 1. Topology in quantum mechanics
1.
2.
3.
4.
Vortex and Quantum Hall state
Topological insulators
Topological superconductors
Symmetry and topology
Part 2. Symmetry protected topological phase
3
トポロジーとは
「形」が連続変形でつながるか、つながらないか
マグカップ ≈ ドーナツ
メビウスの輪
ボール
ふつうの輪
4
量子力学のトポロジー
波動関数の「形」が連続変形でつながるか
例）磁束の量子化
磁場x 面積
磁束渦
超伝導渦の磁束量子
プランク定数
電子の電荷
ホログラフィー電子顕微鏡による磁束量子の観察
@日立
5
超伝導体状態の巨視的波動関数(クーパー対)
磁束渦あり
位相 𝜃が0 →2π
磁束渦なし
位相𝜃が 0 → 0
一般に波動関数の一意性から、位相𝜃の変化は2πN (N=0,1,2,..)
6
磁束の量子化
渦の中心から十分離れた領域では
超伝導電流
ベクトルポテンシャル
アブリコソフ
𝑪
したがって
Stokesの定理
位相変化が2πN
磁束
磁束がとびとびの値(N=0,1,2…)をとる
7
トポロジカル不変量
(連続変形で変わらない量)
• トポロジーの違いを区別する便利な量
• トポロジカル不変量が違う場合は、波動関数の連続変形で移りあえない
𝑪
トポロジカル不変量 は、電荷、角運動量などと同じ保存量
量子化
8
例２）整数量子ホール効果
磁場
電場
２次元電子系
ホール電流
グラフェンの整数量子ホール効果
AlGaAs
14
GaAs
ヘテロ接合
フォン・クリッツィング
10
グラフェン
(単層カーボン)
ホ
ー
2 ル
伝
-2 導
率
-6
6
ホール伝導率
-10
-14
キャリア密度
9
磁場中の電子
一様磁場のベクトルポテンシャル
結晶構造による
周期ポテンシャル
ブロッホの定理
中野(久保)公式
電流
10
TKNN(サウレス・甲元ら)公式 (1982)
サウレス
甲元
チャーン数 (トポロジカル不変量)
占有バンドのベリー位相
運動量空間での”ベクトルポテンシャル”
Nchは数学的な定理から整数値となる
11
占有バンドのトポロジー
チャーン数 1
チャーン数 0
𝐸
𝐸
(𝑘𝑥 , 𝑘𝑦 )
𝑢(𝑘) ≈
チャーン数は保存量
(𝑘𝑥 , 𝑘𝑦 )
占有バンドの状態ベクトル
量子化
12
磁束の量子化
量子ホール効果
14
10
6
2
-2
-6
-10
-14
ホ
ー
ル
伝
導
率
キャリア密度
実空間
運動量空間
13
ノーベル賞
トポロジカル量子現象
1985 von Klitzing 整数量子ホール効果
1998 Laughlin, Störmer, Tsui 分数量子ホール効果
2003 Abrikosovら 超伝導・超流動の理論
脳磁図
量子化ホール抵抗による直流抵抗標準
定義
超伝導量子干渉計
フォン・クリッツィング定数
14
トポロジカルエッジ状態の発見
(2008 ~)
トポロジカル絶縁体： 表面が金属となる絶縁体
角度分解型光電子分光
Bi1-xSbx
電子
Bi2Se3
光子
伝導帯
フェルミ準位
エ
ネ
ル
ギ
ー
表面状態
(eV)
価電子帯
波数(Å-1)
15
普通の絶縁体との違い
トポロジカル絶縁体(TI)
𝐸
𝑢(𝑘) ≈
普通の絶縁体
𝐸
•
価電子帯の電子のつまり方のトポロジーが普通の絶縁体と違う
•
外部磁場がゼロであるため、量子ホール状態とも電子のつまり
方のトポロジーが違う (= トポロジカル不変量が違う)
16
トポロジカルエッジ状態
真空
(eV)
トポロジカル絶縁体 ギ
ャ
ッ
プ
レ
ス
状
態
エ
ネ
ル
ギ
ー
波数(Å-1)
ひねり1
ひねり0
メビウスの輪
単なる輪
リボン
17
• 表面状態由来の特異な輸送現象を示す
例) 表面状態による純粋スピン流
エ
ネ
ル
ギ
ー
(eV)
表面状態の電流は全体でゼロ
しかし、スピン流は有限に残る
波数(Å-1)
• 不純物の下でも安定
• 多数のトポロジカル絶縁体が発見されている
Bi2Se3, Bi2Te3,
TIBi(S1-xSex)2, Bi2Te2Se, (Bi1-xSbx)2(Te1-ySey)3,
Pb(Bi1-xSbx)2Te4, …..
18
Simplest model of TI
= Massive Dirac Hamiltonian
𝜎𝜇 : Pauli matrices in orbital space (two pz-orbitals of Se)
𝑠𝜇 : Pauli matrices in spin space
Bi2Se3
Topological # = Z2 invariant
TR-invariant momentum
occupied state
19
Surface bound state
z
Top.Insulator
Dirac fermion
It satisfies b.c if
The surface state obeys 2+1 D Dirac equation
20
トポロジカルエッジ状態は超伝導体にも現れる
トポロジカル超伝導体
超伝導体: クーパー対の形成
電子
≈
クーパー対
ホール
基底状態では、フェルミエネルギー以下の状態が詰まっている
21
トポロジカル絶縁体と同様に、色々な「形」で状態をつめるやり方
がある。
𝑢(𝑘) ≈
トポロジカル不変量という「新しい保存量」を持つ超伝導状態
トポロジカル超伝導体
Qi et al, PRB (09), Schnyder et al PRB (08),
佐藤, PRB 79, 094504 (09), 佐藤・藤本, PRB79, 214526 (09)
22
トポロジカル超伝導体のトポロジカルエッジ状態
トポロジカル超伝導
ギ
ャ
ッ
プ
レ
ス
状
態
真空
単なる輪
メビウスの輪
トポロジカル絶縁体 ギ
ャ
ッ
プ
レ
ス
状
態
(eV)
エ
ネ
ル
ギ
ー
波数(Å-1)
真空
23
Topological SCs/SFs
T-breaking topological SC
Energy
Sr2RuO4
[Kashiwaya et al (11)]
T-invariant topological SC
3He-B
CuxBi2Se3
CuxBi2Se3
E
chiral
k
[Murakawa, Nomura et al (09)]
[Sasaki et al (09)]
helical
Experiment
[Sasaki-Kriener-Segawa-YadaTanaka-MS –Ando (11)]
[Fu-Berg (10)]
CuxBi2Se3
Theory
[MS (10)]
[Yamakage-Yada-MS-Tanaka (12)]
24
S-wave SCs can host topological superconductivity if a spinless
system is realized effectively
•
Dirac fermion + s-wave condensate
•
S-wave superconducting state with Rashba SO + Zeeman field
[MS(03), Fu-Kane (08)]
[MS-Takahashi-Fujimoto (09), J. Sau et al (10)]
Hsieh et al
Non spin-degenerate
single Fermi surface
Fermi
Level
MF
Topolgiocal SC
Zeeman field
nanowire
[Lutchyn et al (10), Oreg et al (10)]
Zeeman field
Zero modes
[Mourik et al (12)]
B
25
Why such new topological phases can be found ?
The key is symmetry
Time-reversal symmetry (TRS)
Kramers theorem
• No back scattering
• topologically stable
26
Particle-hole symmetry (PHS)
• Spectrum is symmetric between E and –E
• Quasiparticles can be their own antiparticles
Majorana condition
27 (09)]
[Wilczek , Nature
PH symmetry also provides topological stability
nanowire
PHS
PHS
• Single isolated zero mode is topologically stable due to PH
symmetry
• It realizes Majorana zero mode in condensed matter physics
28
Topological Periodic Table
IQHS
[Schnyder-Ryu-Furusaki-Ludwig
(12)]
[Avron-Seiler-Simon (83)]
A
AIII
AI
BDI
D
DIII
AII
CII
C
CI
TRS
PHS
CS
d=1
d=2
d=3
0
0
0
0
1
0
Z
0
Z
0
Z
0
0
Z
Z2
Z2
0
2Z
0
0
0
0
Z
Z2
Z2
0
2Z
0
1
1
0
-1
-1
-1
0
1
0
1
1
1
0
-1
-1
-1
0
1
0
1
0
1
0
1
0
Z
Z2
Z2
0
2Z
0
0
Majorana nanowire
p+ip chiral p
Sr2RuO4, 3He-A
3He-B
CuxBi2Se3
3D TI
QSH
Taking into account TRS, PHS and their combinations, nine new
topological classes are found
29
Is there any possibility to extend topological phases by using
other symmetries ?
ex.) Inversion symmetry
Topological Insulator
[Fu-Kane (06)]
Z2 number
Inversion sym
TR-invariant momentum
occupied state
• Non-local
• Difficult to evaluate
Bi1-xSbx
Parity of occupied state
• Local
• Easy to evaluate
30
Topological odd parity SCs
[MS (09, 10), Fu-Berg (10)]
If the number of TRI momenta enclosed by the Fermi surface
is odd, the spin-triplet SC is (strongly) topological.
Even
Odd
(001)
BW gap fn.
Majorana fermion
(001)
CuxBi2Se3
31
However, inversion symmetry gives no additional gapless
surface state beyond the topological periodic table
Broken on surface
bulk-edge
correspondence
New bulk top. #
by inversion
No additional
state
Idea
If we use symmetry that is not broken near the surface, we can
obtain new gapless states beyond the topological periodic table
Symmetry Protected Topological Surface State
32
Topological Crystalline Insulator
[L. Fu (11), Hsieh et al (12)]
Point group symmetry provide a topological surface state beyond
topological periodic table
SnTe
(110)
Mirror reflection
surface BZ
BZ
33
Idea
Using the eigen value of mirror operator, ky=0 plane can be
separated into two QH states.
Two Dirac fermions
[Y. Tanaka et al (12) ]
Not ordinal TI
(Top Crystalline Insulator)34
Question
Can we generalize the same idea to obtain new topological
SCs ?
YES
Majorana fermions protected by additional symmetry
35
Symmetry Protected Majorana fermions
•
•
•
•
•
MS, Fujimoto, Phys. Rev. B 79, 094504 (09)
Mizushima, MS, Machida, Phys. Rev. Lett. 109, 165301 (12)
Mizushima, MS, New J. Phys. 15, 075010 (13)
Ueno, Yamakage, Tanaka, MS, Phys. Rev. Lett. 111, 087002 (13)
MS, Yamakage, Mizushima, arXiv: 1307.1264, invited paper in Physica E
•
•
•
•
Chui-Yao-Ryu, Phys. Rev. B88, 074142 (13)
Zang-Kane-Mele, Phys. Rev. Lett. 111, 056403 (13)
Morimoto-Furusaki, arXiv: 1306.2505
Fang-Gilbert-Bernevig, arXiv:1308.2424
36
Now we know that MFs can be realized in SCs.
But spinless systems are often needed to realized MFs.
Hsieh et al
Dirac fermion + s-wave condensate
MS(03), Fu-Kane (08)
S-wave superconducting state with Rashba SO + Zeeman field
MS-Takahashi-Fujimoto (09), J. Sau et al (10)
Non spin-degenerate
single Fermi surface
Fermi
Level
Zeeman field
MF
nanowire
Zeeman field
Mourik et al (12)
Lutchyn et al (10), Oreg et al (10)
37
Why Majorana Fermions favor spinless SCs ?
For spinless SCs, we have the Majorana condition (self-antiparticle
property) naturally.
However, the spin degrees of freedoms obscure the Majorana
condition
Nitta, JPSJ talk
Majorana condition
Majorana condition
38
Moreover, spinful SCs support MFs in pairs because of the spin
degeneracy.
They can be considered as Dirac fermions as well as MFs
The Dirac fermions are easily gapped away by the Dirac mass
term
No topologically stable MFs
39
Question
Is it possible to realize Majorana fermions in spinful
SCs ?
Key observation
If there is an additional symmetry such as time-reversal
symmetry, Majorana fermions can be realized in spinful SCs
Sasaki-Kriener-SegawaYada-Tanaka-MS -Ando(11)
CuxBi2Se3
Fu-Berg (10)
CuxBi2Se3
Yamakage--Yada-MS-Tanaka(12)
MS (10)
40
Ex.) 1D spinful px-wave superconductor
A pair of MFs
px-wave SC
Kramers theorem
No scattering between
and
Thus, they naturally can be considered as two independent
particles, not as a single Dirac fermion.
Actually, the Dirac mass term is forbidden by the time-reversal
symmetry.
Topologically stable MF
41
Can we use symmetries other than time-reversal
symmetry?
Topological crystalline SC
Ueno-Yamakage-Tanaka-MS (13)
Chui-Yao-Ryu (13)
Zhang-Kane-Mele (13), …
42
Topological Crystalline SCs
[Ueno, Yamakage, Tanaka, MS (13)]
mirror reflection symmetry
Sr2RuO4
UPt3
BZ
43
Like topological crystalline insulators, kz=0 plane can be
separated into two mirror subsectors
mirror Chern #
for Mxy=i
mirror Chern #
for Mxy=-i
When the mirror Chern numbers are nonzero, we have gapless
surface states
44
However, there is an important difference between TCIs and TCSCs
Particle-hole symmetry = Majorana condition
PH symmetry
The problem is how the particle-hole symmetry is realized in the
mirror subsectors.
45
Key point
Two different mirror symmetries are possible in SCs.
a)
S-wave SC
Spin-triplet SC
with 𝒅 ∥ 𝒛
U(1) gauge sym
b)
Spin-triplet SC
with 𝒅 ⊥ 𝒛
46
Even
Class A
Class A
Dirac fermion
• Mirror subsector does not support its own particle-hole
symmetry.
• Mirror subsector is topologically the same as quantum Hall
states.
47
Odd
Class D
Class D
Majorana fermion
• Mirror subsector supports its own particle-hole symmetry .
• Mirror subsector is topologically the same as spinless SCs.
• Majorana zero mode can exit in a vortex or in a dislocation
Schnyder et al (08)
Teo-Kane (10)
class D
1D
Z2
2D
Z
3D
-
48
Stable MFs are predicted for various spinful SCs/SFs
• Sr2RuO4
[Ueno, Yamakage, Tanaka, MS (13)]
• Thin film of 3He-A
• UPt3
[MS, Yamakage, Mizushima (13)]
[Tsutsumi-Yamamoto-Kawami-Mizushima-MS-Ichioka-Machida (13)]
integer vortex
[MS, Yamakage, Mizushima (13)]
3He-A
LDOS at core of integer vortex
Majorana zero modes exist
in integer vortex when
mirror odd
49
Summary (1)
1. In general, spinful SCs support a pair of Majorana
fermions that can be identified with a single Dirac fermion.
2. With symmetry, unconventinal spinful SCs can host
intrinsic Majorana fermions
In particular, a pair of Majorana zero modes in a vortex
can be stable by additional SCs
Is it possible to generalize topological periodic table
with additional symmetry ?
50
Topological Periodic Table with Mirror Symmetry
10 classes
27 classes
[Chui-Yao-Ryu, PRB(13) ,
Morimoto-Furusaki, PRB (13)]
MF protected by mirror symmetry
Sr2RuO4, UPt3
Topological crystalline insulator
SnTe
Still not enough …..
51
There are many symmetries other than mirror
reflection
spin-flip, mirror reflection, rotation, inversion …..
Unitary symmetry
magnetic point group, hidden time-reversal symmetry
Anti-unitary symmetry
52
Anti-Unitary case
Anti-unitary symmetries are often realized as a hidden timereversal symmetry
Time-reversal + Mirror reflection
magnetic
field
T-invariant
Hidden time-reversal symmetry
These hidden time-reversal symmetries also provide symmetry
protected MFs
53
Using the hidden time-reversal symmetry,
we can define a new topological number
Combining with particle-hole symmetry,
we obtain chiral symmetry
Then, we can define new topological number
New topological phase
MS-Fujimoto (09)
54
MFs protected by the hidden time-reversal symmetry can be
found in various system under magnetic fields
• Rashba SC under magnetic filed
[MS-Fujimoto (09), Tewari-Sau (12), Wong-Liu-Law-Lee(13), Mizushima-MS (13),
Zhang-Kane-Mele (13)]
nanowire
MF
Hx
•
3He-B
s-wave Rashba SF tubes
under parallel filed
[Mizushima-MS-Machida (12) ]
𝜋
Topological QPT
with SSB
Hy
●
3He-B
55
New Topological Periodic Tables
[Shiozaki-MS, in preparation (14)]
We have complete the topological classification with order-two
additional symmetry
10 classes
[Schnyder et al (08)]
Hx
27 classes
[Chui-Yao-Ryu (13) ,
Morimoto-Furusaki (13)]
(27+10)x4=148 classes
[Shiozaki-MS (14)]
56
Summary
• The idea of topological phase has been established now with
many experimental supports.
• While time-reversal invariance and particle-hole symmetry
has been used to extend topological phase, other
symmetries specific to material structures are also useful to
have new topological phases.
• Many undiscovered topological materials can be expected by
combining various symmetries in nature.
57