Slide 1

Topological Quantum Phenomena Nagoya University, Masatoshi Sato 1

Slide 2

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)

Slide 3

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

Slide 4

4

Slide 5

5 @ x

Slide 6

6 ( )

Slide 7

7 2N Stokes (N=0,1,2)

Slide 8

8 ( )

Slide 9

9 14 10 6 2 -2 -6 -10 -14 ( ) AlGaAs GaAs

Slide 10

( ) 10

Slide 11

11 TKNN( ) ( ) (1982) N ch

Slide 12

12 1 0

Slide 13

13 14 10 6 2 -2 -6 -10 -14

Slide 14

14 1985 von Klitzing 1998 Laughlin, Strmer, Tsui 2003 Abrikosov

Slide 15

15 Bi 1-x Sb x Bi 2 Se 3 ( -1 ) (2008 ~)

Slide 16

16 (= ) (TI)

Slide 17

17 1 0 ( -1 )

Slide 18

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,..

Slide 19

19 Simplest model of TI = Massive Dirac Hamiltonian Bi 2 Se 3 Topological # = Z 2 invariant TR-invariant momentum occupied state

Slide 20

20 Surface bound state Top.Insulator z It satisfies b.c if Dirac fermion The surface state obeys 2+1 D Dirac equation

Slide 21

21 :

Slide 22

22 Qi et al, PRB (09), Schnyder et al PRB (08), , PRB 79, 094504 (09), , PRB79, 214526 (09)

Slide 23

23 ( -1 )

Slide 24

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

Slide 25

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

Slide 26

26 Why such new topological phases can be found ? Time-reversal symmetry (TRS) Kramers theorem No back scattering topologically stable The key is symmetry

Slide 27

Particle-hole symmetry (PHS) Majorana condition [ Wilczek, Nature (09) ] 27 Spectrum is symmetric between E and E Quasiparticles can be their own antiparticles

Slide 28

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

Slide 29

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

Slide 30

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

Slide 31

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

Slide 32

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

Slide 33

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)

Slide 34

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)

Slide 35

35 Can we generalize the same idea to obtain new topological SCs ? Question Majorana fermions protected by additional symmetry YES

Slide 36

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

Slide 37

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)

Slide 38

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

Slide 39

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

Slide 40

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)

Slide 41

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

Slide 42

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),

Slide 43

43 mirror reflection symmetry Topological Crystalline SCs Sr 2 RuO 4 UPt 3 BZ [Ueno, Yamakage, Tanaka, MS (13)]

Slide 44

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

Slide 45

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

Slide 46

46 Key point Two different mirror symmetries are possible in SCs. S-wave SC a)a) b)b) U(1) gauge sym

Slide 47

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

Slide 48

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

Slide 49

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)]

Slide 50

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 ?

Slide 51

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 ..

Slide 52

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

Slide 53

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

Slide 54

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

Slide 55

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

Slide 56

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

Slide 57

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