Topological Quantum Phenomena Nagoya University, Masatoshi Sato
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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) In
collaboration with 2 Review paper Y. Tanaka, MS, N. Nagaosa,
Symmetry and Topology in SCs Journal of Physical Society of Japan,
81 (2012) 011013 (open access)
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Outline Part 1. Topology in quantum mechanics 1.Vortex and
Quantum Hall state 2.Topological insulators 3.Topological
superconductors 4.Symmetry and topology Part 2. Symmetry protected
topological phase 3
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4
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5 @ x
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6 ( )
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7 2N Stokes (N=0,1,2)
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8 ( )
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9 14 10 6 2 -2 -6 -10 -14 ( ) AlGaAs GaAs
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( ) 10
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11 TKNN( ) ( ) (1982) N ch
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12 1 0
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13 14 10 6 2 -2 -6 -10 -14
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14 1985 von Klitzing 1998 Laughlin, Strmer, Tsui 2003
Abrikosov
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15 Bi 1-x Sb x Bi 2 Se 3 ( -1 ) (2008 ~)
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16 (= ) (TI)
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17 1 0 ( -1 )
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18 ( -1 ) ) Bi 2 Se 3, Bi 2 Te 3, TIBi(S 1-x Se x ) 2, Bi 2 Te
2 Se, (Bi 1-x Sb x ) 2 (Te 1-y Se y ) 3, Pb(Bi 1-x Sb x ) 2 Te
4,..
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19 Simplest model of TI = Massive Dirac Hamiltonian Bi 2 Se 3
Topological # = Z 2 invariant TR-invariant momentum occupied
state
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20 Surface bound state Top.Insulator z It satisfies b.c if
Dirac fermion The surface state obeys 2+1 D Dirac equation
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21 :
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22 Qi et al, PRB (09), Schnyder et al PRB (08), , PRB 79,
094504 (09), , PRB79, 214526 (09)
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23 ( -1 )
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24 Topological SCs/SFs 3 He-B Sr 2 RuO 4 Cu x Bi 2 Se 3
[Fu-Berg (10)] [Sasaki-Kriener-Segawa-Yada- Tanaka-MS Ando (11)]
[Yamakage-Yada-MS-Tanaka (12)] [MS (10)] Cu x Bi 2 Se 3 Energy E k
T-invariant topological SC T-breaking topological SC chiral helical
[Kashiwaya et al (11)] [Murakawa, Nomura et al (09)] [Sasaki et al
(09)] Experiment Theory
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25 Dirac fermion + s-wave condensate S-wave superconducting
state with Rashba SO + Zeeman field Zeeman field Fermi Level Non
spin-degenerate single Fermi surface [MS(03), Fu-Kane (08)]
[MS-Takahashi-Fujimoto (09), J. Sau et al (10 )] Zeeman field MF
nanowire [Lutchyn et al (10), Oreg et al (10)] [Mourik et al (12)]
S-wave SCs can host topological superconductivity if a spinless
system is realized effectively Hsieh et al Topolgiocal SC Zero
modes B
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26 Why such new topological phases can be found ? Time-reversal
symmetry (TRS) Kramers theorem No back scattering topologically
stable The key is symmetry
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Particle-hole symmetry (PHS) Majorana condition [ Wilczek,
Nature (09) ] 27 Spectrum is symmetric between E and E
Quasiparticles can be their own antiparticles
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28 PH symmetry also provides topological stability nanowire
Single isolated zero mode is topologically stable due to PH
symmetry It realizes Majorana zero mode in condensed matter physics
PHS
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29 A AIII AI BDI D DIII AII CII C CI TRS PHS CS 00 0 d=1 d=2
d=3 0 Z 0 0 0 1 Z 0 Z 1 0 0 1 0 1 0 0101010101010101 0 Z Z 2 0 2Z 0
Z Z 2 0 2Z 0 Z Z 2 0 2Z IQHS p+ip chiral p Sr 2 RuO 4, 3 He-A 3
He-B 3D TI Cu x Bi 2 Se 3 Majorana nanowire QSH
[Schnyder-Ryu-Furusaki-Ludwig (12)] [Avron-Seiler-Simon (83)]
Topological Periodic Table Taking into account TRS, PHS and their
combinations, nine new topological classes are found
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30 Is there any possibility to extend topological phases by
using other symmetries ? ex.) Inversion symmetry [Fu-Kane (06)]
Parity of occupied state TR-invariant momentum Topological
Insulator Non-local Difficult to evaluate Local Easy to evaluate
Inversion sym Z 2 number occupied state Bi 1-x Sb x
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31 [MS (09, 10), Fu-Berg (10)] Topological odd parity SCs 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 Cu x Bi 2 Se 3
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32 However, inversion symmetry gives no additional gapless
surface state beyond the topological periodic table Idea If we use
symmetry that is not broken near the surface, we can obtain new
gapless states beyond the topological periodic table bulk-edge
correspondence New bulk top. # by inversion Broken on surface No
additional state Symmetry Protected Topological Surface State
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33 Topological Crystalline Insulator Point group symmetry
provide a topological surface state beyond topological periodic
table SnTe [L. Fu (11), Hsieh et al (12)] Mirror reflection BZ
surface BZ (110)
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34 Idea Using the eigen value of mirror operator, ky=0 plane
can be separated into two QH states. [Y. Tanaka et al (12) ] Two
Dirac fermions Not ordinal TI (Top Crystalline Insulator)
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35 Can we generalize the same idea to obtain new topological
SCs ? Question Majorana fermions protected by additional symmetry
YES
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36 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
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37 Now we know that MFs can be realized in SCs. But spinless
systems are often needed to realized MFs. Dirac fermion + s-wave
condensate MS(03), Fu-Kane (08) Hsieh et al S-wave superconducting
state with Rashba SO + Zeeman field Zeeman field Fermi Level Non
spin-degenerate single Fermi surface MS-Takahashi-Fujimoto (09), J.
Sau et al (10) Zeeman field MF nanowire Lutchyn et al (10), Oreg et
al (10) Mourik et al (12)
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38 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 Majorana condition Nitta, JPSJ talk
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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
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40 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 Cu x Bi 2 Se 3 Fu-Berg (10) Sasaki-Kriener-Segawa-
Yada-Tanaka-MS -Ando(11) Yamakage--Yada-MS-Tanaka(12) MS (10)
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Thus, they naturally can be considered as two independent
particles, not as a single Dirac fermion. 41 Ex.) 1D spinful p x
-wave superconductor A pair of MFs No scattering between and
Kramers theorem Actually, the Dirac mass term is forbidden by the
time-reversal symmetry. Topologically stable MF p x -wave SC
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42 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),
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43 mirror reflection symmetry Topological Crystalline SCs Sr 2
RuO 4 UPt 3 BZ [Ueno, Yamakage, Tanaka, MS (13)]
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44 Like topological crystalline insulators, k z =0 plane can be
separated into two mirror subsectors When the mirror Chern numbers
are nonzero, we have gapless surface states mirror Chern # for M xy
=i mirror Chern # for M xy =-i
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45 However, there is an important difference between TCIs and
TCSCs Particle-hole symmetry = Majorana condition The problem is
how the particle-hole symmetry is realized in the mirror
subsectors. PH symmetry
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46 Key point Two different mirror symmetries are possible in
SCs. S-wave SC a)a) b)b) U(1) gauge sym
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47 Even Dirac fermion Mirror subsector does not support its own
particle-hole symmetry. Mirror subsector is topologically the same
as quantum Hall states. Class A
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48 Odd 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
Zclass D 1D 2D 3D Z2Z2 - Class D Schnyder et al (08) Teo-Kane (10)
Majorana fermion
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49 Thin film of 3 He-A [Ueno, Yamakage, Tanaka, MS (13)] Stable
MFs are predicted for various spinful SCs/SFs LDOS at core of
integer vortex Majorana zero modes exist in integer vortex when 3
He-A integer vortex Sr 2 RuO 4 UPt 3
[Tsutsumi-Yamamoto-Kawami-Mizushima-MS-Ichioka-Machida (13)] [MS,
Yamakage, Mizushima (13)] mirror odd [MS, Yamakage, Mizushima
(13)]
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50 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
?
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51 Topological Periodic Table with Mirror Symmetry [
Chui-Yao-Ryu, PRB(13), Morimoto-Furusaki, PRB (13) ] MF protected
by mirror symmetry Topological crystalline insulator 10 classes 27
classes Sr 2 RuO 4, UPt 3 SnTe Still not enough ..
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52 Anti-unitary symmetry spin-flip, mirror reflection,
rotation, inversion .. magnetic point group, hidden time-reversal
symmetry There are many symmetries other than mirror reflection
Unitary symmetry
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53 Anti-Unitary case Anti-unitary symmetries are often realized
as a hidden time- reversal symmetry T-invariant magnetic field
These hidden time-reversal symmetries also provide symmetry
protected MFs Time-reversal + Mirror reflection Hidden
time-reversal symmetry
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54 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
MS-Fujimoto (09) New topological phase
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55 Rashba SC under magnetic filed MFs protected by the hidden
time-reversal symmetry can be found in various system under
magnetic fields [MS-Fujimoto (09), Tewari-Sau (12),
Wong-Liu-Law-Lee(13), Mizushima-MS (13), Zhang-Kane-Mele (13)] 3
He-B under parallel filed 3He-B [Mizushima-MS-Machida (12) ] MF
nanowire s-wave Rashba SF tubes HyHy HxHx Topological QPT with
SSB
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56 New Topological Periodic Tables 10 classes 27 classes
(27+10)x4=148 classes [ Chui-Yao-Ryu (13), Morimoto-Furusaki (13) ]
[Shiozaki-MS (14)] We have complete the topological classification
with order-two additional symmetry [Shiozaki-MS, in preparation
(14)] [ Schnyder et al (08) ] HxHx
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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