1 / 18

Majorana Fermions and Topological Insulators

Charles L. Kane, University of Pennsylvania. Majorana Fermions and Topological Insulators. Topological Band Theory - Integer Quantum Hall Effect - 2D Quantum Spin Hall Insulator - 3D Topological Insulator - Topological Superconductor

emily
Télécharger la présentation

Majorana Fermions and Topological Insulators

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Charles L. Kane, University of Pennsylvania Majorana Fermions and Topological Insulators Topological Band Theory - Integer Quantum Hall Effect - 2D Quantum Spin Hall Insulator - 3D Topological Insulator - Topological Superconductor Majorana Fermions - Superconducting Proximity Effect on Topological Insulators - A route to topological quantum computing? Thanks to Gene Mele, Liang Fu, Jeffrey Teo

  2. The Insulating State Characterized by energy gap: absence of low energy electronic excitations Covalent Insulator Atomic Insulator The vacuum e.g. intrinsic semiconductor e.g. solid Ar electron Dirac Vacuum 4s Egap ~ 10 eV Egap = 2 mec2 ~ 106 eV 3p Egap ~ 1 eV positron ~ hole Silicon

  3. + - + - + - + + - + - + + - + - + - + The Integer Quantum Hall State 2D Cyclotron Motion, Landau Levels E Hall Conductance sxy = n e2/h IQHE without Landau Levels (Haldane PRL 1988) Graphene in a periodic magnetic field B(r) Band structure B(r) = 0 Zero gap, Dirac point B(r) ≠ 0 Energy gap sxy = e2/h k Egap

  4. Topological Band Theory The distinction between a conventional insulator and the quantum Hall state is a topological property of the manifold of occupied states The set of occupied Bloch wavefunctions defines a U(N) vector bundle over the torus. Classified by the first Chern class (or TKNN invariant) (Thouless et al, 1984) Berry’s connection Berry’s curvature 1st Chern class Trivial Insulator: n = 0 Quantum Hall state: sxy = n e2/h The TKNN invariant can only change at a phase transition where the energy gap goes to zero

  5. Edge States Gapless states must exist at the interface between different topological phases IQHE state n=1 Vacuum n=0 n=0 n=1 y x Smooth transition : gap must pass through zero Edge states ~ skipping orbits Gapless Chiral Fermions : E = v k Band inversion – Dirac Equation M>0 E Egap Egap M<0 Domain wall bound state y0 ky K’ K Jackiw, Rebbi (1976) Su, Schrieffer, Heeger (1980) Haldane Model Bulk – Edge Correspondence :Dn = # Chiral Edge Modes

  6. E E k*=0 k*=p/a k*=0 k*=p/a J↓ J↑ E Time Reversal Invariant 2Topological Insulator Time Reversal Symmetry : All states doubly degenerate Kramers’ Theorem : 2 topological invariant (n = 0,1) for 2D T-invariant band structures n=1 : Topological Insulator n=0 : Conventional Insulator Edge States Kramers degenerate at time reversal invariant momenta k* = -k* + G n is a property of bulk bandstructure. Easiest to compute if there is extra symmetry: 1. Sz conserved : independent spin Chern integers : (due to time reversal) Quantum spin Hall Effect : 2. Inversion (P) Symmetry : determined by Parity of occupied 2D Bloch states

  7. 2D Quantum Spin Hall Insulator ↑ I. Graphene Kane, Mele PRL ‘05 Eg ↓ • Intrinsic spin orbit interaction •  small (~10mK-1K) band gap • Sz conserved : “| Haldane model |2” • Edge states : G = 2 e2/h 2p/a p/a 0 ↑ ↓ ↓ ↑ II. HgCdTe quantum wells HgTe HgxCd1-xTe Theory: Bernevig, Hughes and Zhang, Science ’06 Experiement: Konig et al. Science ‘07 d HgxCd1-xTe d < 6.3 nm Normal band order d > 6.3 nm: Inverted band order G ~ 2e2/h in QSHI E E Normal G6 ~ s G8 ~ p k Inverted G8 ~ p G6 ~ s Conventional Insulator QSH Insulator

  8. L3 L4 E E L1 L2 k=La k=Lb k=La k=Lb 3D Topological Insulators There are 4 surfaceDirac Points due to Kramers degeneracy ky kx OR 2D Dirac Point How do the Dirac points connect? Determined by 4 bulk 2 topological invariantsn0 ; (n1n2n3) Surface Brillouin Zone n0 = 1 : Strong Topological Insulator EF Fermi circle encloses odd number of Dirac points Topological Metal : 1/4 graphene Robust to disorder: impossible to localize n0 = 0 : Weak Topological Insulator Fermi circle encloses even number of Dirac points Related to layered 2D QSHI

  9. Theory: Predict Bi1-xSbx is a topological insulator by exploiting inversion symmetry of pure Bi, Sb (Fu,Kane PRL’07) Experiment: ARPES (Hsieh et al. Nature ’08) Bi1-xSbx • Bi1-x Sbx is a Strong Topological • Insulator n0;(n1,n2,n3) = 1;(111) • 5 surface state bands cross EF • between G and M Bi2 Se3 ARPES Experiment : Y. Xia et al., Nature Phys. (2009). Band Theory : H. Zhang et. al, Nature Phys. (2009). • n0;(n1,n2,n3) = 1;(000) : Band inversion at G • Energy gap: D ~ .3 eV : A room temperature • topological insulator • Simple surface state structure : • Similar to graphene, except • only a single Dirac point EF Control EF on surface by exposing to NO2

  10. Topological Superconductor, Majorana Fermions BCS mean field theory : Bogoliubov de Gennes Hamiltonian Particle-Hole symmetry : Quasiparticle redundancy : (Kitaev, 2000) 1D 2 Topological Superconductor : n = 0,1 Discrete end state spectrum : END n=0 “trivial” n=1 “topological” E Majorana Fermion bound state D D E E=0 0 0 -E -D -D “half a state”

  11. Periodic Table of Topological Insulators and Superconductors Kitaev, 2008 Schnyder, Ryu, Furusaki, Ludwig 2008 Anti-Unitary Symmetries : - Time Reversal : - Particle - Hole : Unitary (chiral) symmetry : Complex K-theory Altland- Zirnbauer Random Matrix Classes Real K-theory Bott Periodicity

  12. Majorana Fermion : spin 1/2 particle = antiparticle ( g = g† ) Potential Hosts : • Particle Physics : • Neutrino (maybe) Allows neutrinoless double b-decay. • Condensed matter physics : Possible due to pair condensation • Quasiparticles in fractional Quantum Hall effect at n=5/2 • h/4e vortices in p-wave superconductor Sr2RuO4 • s-wave superconductor/ Topological Insulator ... among others • Current Status : NOT OBSERVED Topological Quantum ComputingKitaev, 2003 • 2 Majorana bound states = 1 fermion bound state • - 2 degenerate states (full/empty) = 1 qubit • 2N separated Majoranas = N qubits • Quantum Information is stored non locally • - Immune from local sources of decoherence • Adiabatic Braiding performs unitary operations • - Non Abelian Statistics

  13. Proximity effects : Engineering exotic gapped states on topological insulator surfaces m Dirac Surface States : Protected by Symmetry 1. Magnetic : (Broken Time Reversal Symmetry) Fu,Kane PRL 07 Qi, Hughes, Zhang PRB (08) • Orbital Magnetic field : • Zeeman magnetic field : • Half Integer quantized Hall effect : M. ↑ T.I. 2. Superconducting : (Broken U(1) Gauge Symmetry) proximity induced superconductivity Fu,Kane PRL 08 • S-wave superconductor • Resembles spinless p+ip superconductor • Supports Majorana fermion excitations S.C. T.I.

  14. Majorana Bound States on Topological Insulators 1. h/2e vortex in 2D superconducting state E D h/2e 0 SC -D TI Quasiparticle Bound state at E=0 Majorana Fermion g0 2. Superconductor-magnet interface at edge of 2D QSHI M S.C. m>0 Egap =2|m| QSHI m<0 Domain wall bound state g0

  15. 1D Majorana Fermions on Topological Insulators 1. 1D Chiral Majorana mode at superconductor-magnet interface E M SC kx TI : “Half” a 1D chiral Dirac fermion 2. S-TI-S Josephson Junction f = p f 0 f  p SC SC TI Gapless non-chiral Majorana fermion for phase difference f = p

  16. f1 f2 + - 0 Manipulation of Majorana Fermions Control phases of S-TI-S Junctions Majorana present Tri-Junction : A storage register for Majoranas Create A pair of Majorana bound states can be created from the vacuum in a well defined state |0>. Measure Fuse a pair of Majoranas. States |0,1> distinguished by • presence of quasiparticle. • supercurrent across line junction Braid A single Majorana can be moved between junctions. Allows braiding of multiple Majoranas E E E 0 0 0 f-p f-p f-p 0 0 0

  17. A Z2 Interferometer for Majorana Fermions A Signature for Neutral Majorana Fermions Probed with Charge Transport N even g2 e e g1 N odd • Chiral electrons on magnetic domain wall split • into a pair of chiral Majorana fermions • “Z2 Aharonov Bohm phase” converts an • electron into a hole • dID/dVs changes sign when N is odd. g2 -g2 e h g1 Fu and Kane, PRL ‘09 Akhmerov, Nilsson, Beenakker, PRL ‘09

  18. Conclusion • A new electronic phase of matter has been predicted and observed • - 2D : Quantum spin Hall insulator in HgCdTe QW’s • - 3D : Strong topological insulator in Bi1-xSbx , Bi2Se3, Bi2Te3 • Superconductor/Topological Insulator structures host Majorana Fermions • - A Platform for Topological Quantum Computation • Experimental Challenges • - Charge and Spin transport Measurements on topological insulators • - Superconducting structures : • - Create, Detect Majorana bound states • - Magnetic structures : • - Create chiral edge states, chiral Majorana edge states • - Majorana interferometer • Theoretical Challenges • - Further manifestations of Majorana fermions and non-Abelian states • - Effects of disorder and interactions on surface states

More Related