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Topological Quantum Phenomena

Topological Quantum Phenomena . Nagoya University , Masatoshi Sato. In collaboration with. Satoshi Fujimoto (Kyoto University) Yoshiro Takahashi (Kyoto University) Yukio Tanaka (Nagoya University) Keiji Yada (Nagoya University) Ai Yamakage (Nagoya University)

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Topological Quantum Phenomena

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  1. Topological Quantum Phenomena Nagoya University, Masatoshi Sato

  2. In collaboration with • Satoshi Fujimoto(Kyoto University) • Yoshiro Takahashi (Kyoto University) • Yukio Tanaka (Nagoya University) • KeijiYada (Nagoya University) • Ai Yamakage (Nagoya University) • Yuji Ueno (Nagoya University) • Takeshi Mizushima (Okayama University) • Kazushige Machida (Okayama University) • Masanori Ichioka (Okayama University) • YasumasaTsutsumi (Riken) • Takuto Kawakami (NIMS) • Ken Shiozaki (Kyoto University) • Shingo Kobayashi (Nagoya University) Review paperY. Tanaka, MS, N. Nagaosa, “Symmetry and Topology in SCs”Journal of Physical Society of Japan, 81 (2012) 011013 (open access)

  3. Outline Part 1. Topology in quantum mechanics Vortex and Quantum Hall state Topological insulators Topological superconductors Symmetry and topology Part 2. Symmetry protected topological phase

  4. 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 ボール マグカップ　　ドーナツ ふつうの輪 メビウスの輪

  5. 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 磁束渦 超伝導渦の磁束量子 磁場x 面積 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

  6. 超伝導体状態の巨視的波動関数(クーパー対) 磁束渦あり 磁束渦なし 位相が 0 →0 位相 が0 →2π 一般に波動関数の一意性から、位相の変化は2πN(N=0,1,2,..)

  7. 磁束の量子化 渦の中心から十分離れた領域では アブリコソフ 超伝導電流 ベクトルポテンシャル したがって Stokesの定理 磁束 位相変化が2πN 磁束がとびとびの値(N=0,1,2…)をとる

  8. 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 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

  9. 例２）整数量子ホール効果 磁場 ２次元電子系 フォン・クリッツィング 電場 グラフェンの整数量子ホール効果 14 10 ホール伝導率 6 ホール電流 グラフェン(単層カーボン) 2 ヘテロ接合 AlGaAs -2 ホール伝導率 -6 GaAs -10 -14 キャリア密度

  10. 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

  11. 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

  12. 占有バンドのトポロジー チャーン数 1 チャーン数 0 占有バンドの状態ベクトル 量子化 チャーン数は保存量

  13. 磁束の量子化 量子ホール効果 14 10 ホール伝導率 6 2 -2 -6 -10 -14 キャリア密度 実空間 運動量空間

  14. 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 • 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

  15. トポロジカルエッジ状態の発見 (2008~) 表面が金属となる絶縁体 トポロジカル絶縁体： Bi1-xSbx 角度分解型光電子分光 電子 Bi2Se3 光子 伝導帯 フェルミ準位 エネルギー(eV) 表面状態 価電子帯 波数(Å-1)

  16. 普通の絶縁体との違い トポロジカル絶縁体(TI) 普通の絶縁体 • 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)

  17. トポロジカルエッジ状態 エネルギー(eV) トポロジカル絶縁体 ギャップレス状態 真空 波数(Å-1) ひねり1 ひねり0 リボン メビウスの輪 単なる輪

  18. 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) 波数(Å-1) 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 • 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 (Bi1-xSbx)2(Te1-ySey)3, Bi2Se3, TIBi(S1-xSex)2, Bi2Te3, Bi2Te2Se, Pb(Bi1-xSbx)2Te4, …..

  19. Simplest model of TI = Massive Dirac Hamiltonian Bi2Se3 : Pauli matrices in orbital space (two pz-orbitals of Se) : Pauli matrices in spin space Topological # = Z2 invariant occupied state TR-invariant momentum

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

  21. 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

  22. 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。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。 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)

  23. 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体のトポロジカルエッジ状態 エネルギー(eV) ギャップレス状態 ギャップレス状態 真空 真空 トポロジカル絶縁体 トポロジカル超伝導 波数(Å-1) 単なる輪 メビウスの輪

  24. Topological SCs/SFs T-breaking topological SC Sr2RuO4 [Kashiwaya et al (11)] chiral T-invariant topological SC 3He-B [Murakawa, Nomura et al (09)] CuxBi2Se3 [Sasaki et al (09)] helical CuxBi2Se3 Experiment [Sasaki-Kriener-Segawa-Yada-Tanaka-MS –Ando (11)] Energy E Theory [Fu-Berg (10)] [MS (10)] CuxBi2Se3 [Yamakage-Yada-MS-Tanaka (12)] k

  25. S-wave SCs can host topological superconductivity if a spinless system is realized effectively Non spin-degeneratesingle Fermi surface • Dirac fermion + s-wave condensate [MS(03), Fu-Kane (08)] Fermi Level • S-wave superconducting state with Rashba SO + Zeeman field [MS-Takahashi-Fujimoto (09), J. Sau et al (10)] Hsieh et al Topolgiocal SC nanowire Zeeman field Zeeman field MF [Mourik et al (12)] Zero modes [Lutchyn et al (10), Oreg et al (10)] B

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

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

  28. 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

  29. Topological Periodic Table IQHS [Schnyder-Ryu-Furusaki-Ludwig (12)] [Avron-Seiler-Simon (83)] d=3 d=2 d=1 TRS PHS CS 0 Majoranananowire 0 Z A 0 0 0 Z 0 1 Z 0 0 AIII p+ip chiral p Sr2RuO4, 3He-A 0 0 0 Z Z2 Z2 0 2Z 0 Z Z2 Z2 0 2Z 0 0 0 1 1 1 0 -1 -1 -1 1 1 0 -1 -1 -1 0 1 0 0 Z Z2 Z2 0 2Z 0 0 1 0 1 0 1 0 1 AI BDI D DIII AII CII C CI 3He-B CuxBi2Se3 3D TI QSH Taking into account TRS, PHS and their combinations, nine new topological classes are found

  30. 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 Parity of occupied state Bi1-xSbx • Local • Easy to evaluate • Non-local • Difficult to evaluate

  31. 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) Majoranafermion (001) BW gap fn. CuxBi2Se3

  32. However, inversion symmetry gives no additional gapless surface state beyond the topological periodic table Broken on surface bulk-edge correspondence No additional state New bulk top. # by inversion 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

  33. 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

  34. 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)

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

  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

  37. Now we know that MFs can be realized in SCs. But spinless systems are often neededto realized MFs. Non spin-degeneratesingle Fermi surface Hsieh et al Dirac fermion + s-wave condensate MS(03), Fu-Kane (08) Fermi Level S-wave superconducting state with Rashba SO + Zeeman field MS-Takahashi-Fujimoto (09), J. Sau et al (10) nanowire Zeeman field Zeeman field MF Mourik et al (12) Lutchyn et al (10), Oreg et al (10)

  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 Nitta, JPSJ talk Majorana condition Majorana condition

  39. Moreover, spinfulSCs 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

  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 Sasaki-Kriener-Segawa-Yada-Tanaka-MS -Ando(11) CuxBi2Se3 Fu-Berg (10) Yamakage--Yada-MS-Tanaka(12) CuxBi2Se3 MS (10)

  41. 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

  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), …

  43. Topological Crystalline SCs [Ueno, Yamakage, Tanaka, MS (13)] mirror reflection symmetry Sr2RuO4 UPt3 BZ

  44. 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

  45. 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.

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

  47. 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.

  48. 3D 1D 2D - class D Z Z2 Odd Class D Class D Majoranafermion • 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)

  49. Stable MFs are predicted for various spinful SCs/SFs • Sr2RuO4 [Ueno, Yamakage, Tanaka, MS (13)] • Thin film of 3He-A [MS, Yamakage, Mizushima(13)] • UPt3 [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

  50. Summary (1) In general, spinful SCs support a pair of Majorana fermions that can be identified with a single Dirac fermion. With symmetry, unconventinalspinful SCs can host intrinsic Majorana fermionsIn 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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