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Topological Quantum Phenomena Nagoya University, Masatoshi Sato

11 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 paperY. Tanaka, MS, N. Nagaosa, Symmetry and Topology in SCsJournal of Physical Society of Japan, 81 (2012) 011013 (open access)

2OutlinePart 1. Topology in quantum mechanics Vortex and Quantum Hall stateTopological insulatorsTopological superconductorsSymmetry and topologyPart 2. Symmetry protected topological phase334

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2NStokes(N=0,1,2)

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1985 von Klitzing 1998 Laughlin, Strmer, Tsui Abrikosov

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15Bi1-xSbx

Bi2Se3(eV)(-1)(2008 ~)1516 (= )

(TI)

1617

10

(eV)(-1)1718

(eV)(-1)

) Bi2Se3,Bi2Te3,TIBi(S1-xSex)2,Bi2Te2Se,(Bi1-xSbx)2(Te1-ySey)3,Pb(Bi1-xSbx)2Te4,..1819Simplest model of TI

= Massive Dirac HamiltonianBi2Se3Topological # = Z2 invariant

TR-invariant momentum

occupied state

1920Surface bound state

Top.Insulator

z

It satisfies b.c if

Dirac fermion

The surface state obeys 2+1 D Dirac equation2021:

2122Qi et al, PRB (09), Schnyder et al PRB (08), , PRB 79, 094504 (09), , PRB79, 214526 (09)

2223

(eV)(-1)

2324Topological SCs/SFs 3He-BSr2RuO4CuxBi2Se3

CuxBi2Se3[Fu-Berg (10)][Sasaki-Kriener-Segawa-Yada-Tanaka-MS Ando (11)]

[Yamakage-Yada-MS-Tanaka (12)][MS (10)]CuxBi2Se3

EnergyEkT-invariant topological SCT-breaking topological SC

chiralhelical[Kashiwaya et al (11)][Murakawa, Nomura et al (09)][Sasaki et al (09)]ExperimentTheory2425 Dirac fermion + s-wave condensate S-wave superconducting state with Rashba SO + Zeeman field Zeeman field

Fermi Level

Non spin-degeneratesingle Fermi surface[MS(03), Fu-Kane (08)][MS-Takahashi-Fujimoto (09), J. Sau et al (10)]Zeeman fieldMFnanowire[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 alTopolgiocal SC

Zero modesB2526Why such new topological phases can be found ? Time-reversal symmetry (TRS)Kramers theorem

No back scatteringtopologically stable

The key is symmetry 26Particle-hole symmetry (PHS)Majorana condition

[Wilczek , Nature (09)]27Spectrum is symmetric between E and EQuasiparticles can be their own antiparticles2728PH 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

PHSPHS2829AAIIIAIBDIDDIIIAIICIICCITRSPHSCS000d=1d=2d=30Z0001Z0Z110-1-1-10101110-1-1-1010101010ZZ2Z202Z0000ZZ2Z202Z0000ZZ2Z202ZIQHSp+ip chiral pSr2RuO4, 3He-A3He-B3D TICuxBi2Se3Majorana nanowireQSH[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 found2930Is there any possibility to extend topological phases by using other symmetries ?ex.) Inversion symmetry[Fu-Kane (06)]

Parity of occupied stateTR-invariant momentumTopological InsulatorNon-localDifficult to evaluateLocalEasy to evaluateInversion sym

Z2 numberoccupied state

Bi1-xSbx3031[MS (09, 10), Fu-Berg (10)] Topological odd parity SCsIf the number of TRI momenta enclosed by the Fermi surface is odd, the spin-triplet SC is (strongly) topological.

EvenOdd

(001)(001)BW gap fn.

Majorana fermion

CuxBi2Se33132However, inversion symmetry gives no additional gapless surface state beyond the topological periodic tableIdeaIf 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 inversionBroken on surfaceNo additional stateSymmetry Protected Topological Surface State3233Topological Crystalline InsulatorPoint 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)3334

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)3435Can we generalize the same idea to obtain new topological SCs ?QuestionMajorana fermions protected by additional symmetry YES3536Symmetry Protected Majorana fermionsMS, 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 EChui-Yao-Ryu, Phys. Rev. B88, 074142 (13) Zang-Kane-Mele, Phys. Rev. Lett. 111, 056403 (13)Morimoto-Furusaki, arXiv: 1306.2505Fang-Gilbert-Bernevig, arXiv:1308.242437Now 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-degeneratesingle Fermi surfaceMS-Takahashi-Fujimoto (09), J. Sau et al (10)Zeeman fieldMFnanowireLutchyn et al (10), Oreg et al (10)Mourik et al (12)

3738Why 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 conditionMajorana condition

Nitta, JPSJ talk38Moreover, spinful SCs support MFs in pairs because of the spin degeneracy.

They can be considered as Dirac fermions as well as MFsThe Dirac fermions are easily gapped away by the Dirac mass termNo topologically stable MFs

393940Question 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 SCsCuxBi2Se3

CuxBi2Se3Fu-Berg (10)Sasaki-Kriener-Segawa-Yada-Tanaka-MS -Ando(11)

Yamakage--Yada-MS-Tanaka(12)MS (10)40Thus, they naturally can be considered as two independent particles, not as a single Dirac fermion.41

Ex.) 1D spinful px-wave superconductor

A pair of MFsNo scattering between and

Kramers theoremActually, the Dirac mass term is forbidden by the time-reversal symmetry.

Topologically stable MFpx-wave SC4142Can we use symmetries other than time-reversal symmetry? Topological crystalline SC Ueno-Yamakage-Tanaka-MS (13)Chui-Yao-Ryu (13)Zhang-Kane-Mele (13), 43mirror reflection symmetry

Topological Crystalline SCs Sr2RuO4

UPt3

BZ [Ueno, Yamakage, Tanaka, MS (13)]43

44

Like topological crystalline insulators, kz=0 plane can be separated into two mirror subsectors

When the mirror Chern numbers are nonzero, we have gapless surface statesmirror Chern #for Mxy=imirror Chern #for Mxy=-i

45However, there is an important difference between TCIs and TCSCsParticle-hole symmetry = Majorana condition

The problem is how the particle-hole symmetry is realized in the mirror subsectors.

PH symmetry4546Key pointTwo different mirror symmetries are possible in SCs.

S-wave SC

a)b)U(1) gauge sym

4647Even Dirac fermion

Mirror subsector does not support its own particle-hole symmetry.Mirror subsector is topologically the same as quantum Hall states.Class AClass A

4748Odd 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 D1D2D3DZ2-Class D

Class DSchnyder et al (08)Teo-Kane (10)

Majorana fermion

4849Thin film of 3He-A

[Ueno, Yamakage, Tanaka, MS (13)]Stable MFs are predicted for various spinful SCs/SFs

LDOS at core of integer vortexMajorana zero modes exist in integer vortex when

3He-A

integer vortex

Sr2RuO4UPt3[Tsutsumi-Yamamoto-Kawami-Mizushima-MS-Ichioka-Machida (13)][MS, Yamakage, Mizushima (13)]

mirror odd[MS, Yamakage, Mizushima (13)]4950Summary (1) In general, spinful SCs support a pair of Majorana fermions that can be identified with a single Dirac fermion.

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 SCsIs it possible to generalize topological periodic table with additional symmetry ?5051Topological Periodic Table with Mirror Symmetry [Chui-Yao-Ryu, PRB(13) ,Morimoto-Furusaki, PRB (13)]

MF protected by mirror symmetryTopological crystalline insulator10 classes 27 classesSr2RuO4, UPt3 SnTe Still not enough ..5152Anti-unitary symmetryspin-flip, mirror reflection, rotation, inversion .. magnetic point group, hidden time-reversal symmetry There are many symmetries other than mirror reflectionUnitary symmetry5253Anti-Unitary case Anti-unitary symmetries are often realized as a hidden time-reversal symmetryT-invariant magnetic fieldThese hidden time-reversal symmetries also provide symmetry protected MFsTime-reversal + Mirror reflection

Hidden time-reversal symmetry

5354Using the hidden time-reversal symmetry,

we can define a new topological number

Combining with particle-hole symmetry,we obtain chiral symmetryThen, we can define new topological numberMS-Fujimoto (09)

New topological phase5455Rashba SC under magnetic filedMFs 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)]3He-B under parallel filed3He-B

[Mizushima-MS-Machida (12) ]MF

nanowires-wave Rashba SF tubes HyHxTopological QPT with SSB5556New Topological Periodic Tables10 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)]Hx56SummaryThe 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.

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