Une image / un lien ci-dessous est fourni (tel quel) pour télécharger la présentation
Politique de téléchargement : le contenu du site vous est fourni TEL QUEL pour votre information et votre usage personnel ; il ne peut être vendu, concédé sous licence ni partagé sur d’autres sites sans l’accord de son auteur.Le contenu vous est fourni TEL QUEL, pour votre information et votre usage personnel uniquement.Téléchargez la présentation en cliquant sur ce lien.Lors du téléchargement, si vous ne parvenez pas à télécharger une présentation, l’éditeur a peut-être supprimé le fichier de son serveur.Pendant le téléchargement, si vous ne parvenez pas à obtenir une présentation, le fichier a peut-être été supprimé par l’éditeur.
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)
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
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
Surface bound state z Top.Insulator Dirac fermion It satisfies b.c if The surface state obeys 2+1 D Dirac equation
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
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
Particle-hole symmetry (PHS) • Spectrum is symmetric between E and –E • Quasiparticles can be their own antiparticles Majoranacondition [Wilczek , Nature (09)]
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
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
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
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
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
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
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)
Question Can we generalize the same idea to obtain new topological SCs ? YES Majorana fermions protected by additional symmetry
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
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)
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
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)
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
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
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.
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)
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
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 ?
Connexion au compte
Connectez-vous pour publier votre contenu sur SlideServe.
