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New platforms

for topological quantum computing. New platforms. Netanel Lindner (Caltech -> Technion ). Jerusalem, July 2013. Lessons from Yosi. Useful. Elegant. Simple. Quantum Hall Effect. Topological Quantum Computing. Non- abelian fractional quantum states.

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New platforms

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  1. for topological quantum computing New platforms Netanel Lindner (Caltech -> Technion) Jerusalem, July 2013

  2. Lessons from Yosi Useful Elegant Simple

  3. Quantum Hall Effect

  4. Topological Quantum Computing

  5. Non-abelianfractional quantum states Miller et. al, Nature Physics3, 561 - 565 (2007)  R. L. Willett et. al., arXiv:1301.2639

  6. Topological 1D superconductor Semiconductor wire Superconductor Two degenerate ground states: ●The two states correspond to a total even or odd number of electrons in the system.● Ground state degeneracy is “topological”: no local measurement can distinguish between the two states! Read and Green (2000), Kitaev(2002), Sau et al. (2010), Oreg et al. (2010)

  7. Topological 1D superconductor “Majorana fermion edge modes” Superconductor

  8. Topological SC in 1D Majorana Fermions: Superconductor

  9. Possible solid-state realizations Spin orbit coupled semiconductor wires Quantum Spin Hall Effect Superconductor

  10. Majorana based TQC • Advantages • Energy gap induced by external SC and not by interactions. • Control • Problems • Not universal: • Gapless electrons in the environment

  11. SC Fractionalized zero modes Consider counter propagating edge states of a FQH state, coupled to superconductivity Backscattering FM FQH =1/m FQH =1/m FM Backscattering Zero modes at SC/FM interfaces: Read Green (2000), Fu and Kane (2009)

  12. Ground state degeneracy FM FTI FM Effectively, the ferromagnet “stitches” the two annuli into a torus

  13. Ground state degeneracy Spin on outer edge (el. spin=1)Sout= 2n/m, n = 0,...,m-1 Assuming no q.p. in the bulk:Sin = - Sout G.S. Degeneracy = m

  14. Ground state degeneracy • FM: Spins, S1 Q1 Q2 SC: Charges S3 S2 Q3

  15. Ground state degeneracy • Spins, Charges S1 Q1 Q2 S3 S2 Q3 2N domains, fixed = Qtot, Stot (2m)N-1 ground states

  16. Non-abelian statistics: 1) Degenerate number of ground states, depending on the number of particles. 2) Exchanging two particles, yields a topologically protected unitary transformation in the ground state manifold.

  17. Braiding • Result is independent of the details of the path (topological) • Obeys braiding relations.

  18. Braiding Some Properties of • Coupling two zero modes: • Same ground state degeneracy when two or three zero modes are coupled. • Degeneracy is lifted when four are coupled.

  19. Braiding Braiding interfaces : S2 Q1 Q2 S1 S3 Q3 • Coupling two zero modes: • Same ground state degeneracy when two or three zero modes are coupled. • Degeneracy is lifted when four are coupled.

  20. Braiding Properties of the path • Fixed g.s. degeneracy for all • Charge doesn’t change • Therefore acquired phase be a function of • Overall phase is non universal

  21. Braiding Braiding interfaces 3 and 4: Braiding 2 and 3: etc…

  22. Braiding Relations The group generated by (Yang-Baxter equation) Both equations hold (up to a global phase)

  23. Decomposition of braid matrices new non-abelian“anyon” Isinganyons

  24. Two types of particles: -q X • M. Barkeshli, C-M. Jian, X-L. Qi (2013) • D. Clarke, J. Alicea, K. Shtengel, (2013) • M. Cheng, PRB 86, 195126 (2012) • NHL, E. Berg, G. Refael, A. Stern, (2012) • A. Kapustin,  N. Sauling, (2011) X X X q

  25. Point particles vs. line objects F(a) a a

  26. Twist Defects in SET’s • SET: Top. Phase with onsite finite symmetry group G • Local Hamiltonian: • L. Bombin (2010) • A. Kitaev and L. Kong (2012) • M. Barkeshli,, X-L. Qi (2012) • Y.-Z. You and X.-G. Wen (2012)

  27. Braiding defects with anyons g defect a

  28. Braiding defects with anyons Different SETs with symmetry G, characterized by Permutations have to be consistent with the top. order: fusion, braiding, and with the group structure. g defect b

  29. point particles vs. defects a gha d c b g a h = gha d c a gh

  30. Local G action Suppose that G has trivial permutation of the anyons: (c) (b) (a)

  31. Projective local G action

  32. Projective local G action Constraints from associativity: Mathematical terminology:

  33. Algebraic theory of defect braiding • Group action on anyons

  34. Algebraic theory of defect braiding • Group action on anyons • Projective G- charges carried by anyons

  35. Algebraic theory of defect braiding P. Etingof, et. al. (2010) • Group action on anyons • Projective G- charges carried by anyons • Fractional charges carried by defects

  36. Example 1: • Group action on anyons: • Projective G- charges carried by anyons • Fractional charges carried by defects Stack a non trivial SPT

  37. “Toric Code” Example 2: • Group action on anyons • Projective G- charges carried by anyons • Fractional charges carried by defects

  38. Collaborators • ErezBerg, Gil Refael, AdyStern • PRX 2, 041002 (2012) • Lukasz Fidkowski, Alexei Kitaev • (to be published soon) • Jason Alicea, David Clarke, Kirril Stengel • M. Barkeshli, C-M. Jian, X-L. Qi (2013) • D. Clarke, J. Alicea, K. Shtengel, (2013) • M. Cheng, PRB 86, 195126 (2012) • M. Lu, A. Vishwanath, arXiv:1205.3156v3 • M. Levin and Z.-C. Gu. PRB 86, 115109 (2013) • A M. Essin and M.Hermele, PRB 87, 104406 (2013) • X. Chen, Z-C. Gu, Z-X. Liu, and X-G. Wen, PRB, 87, 155114 (2013)

  39. Summary • Zero modes yielding non-Abelian statistics emerge on abelian FQH edges coupled to a superconductor. • The braiding rules are akin to those of defects in a symmetry enriched topological phase: a route for engineering new non-Abelian systems. • Projective quantum numbers carried by anyons lead to a modified braiding theory for defects. • Finite number of consistent braiding theories, classified by three physically measurable invariants: each theory corresponds to a different class of SETs. • Advantages to TQC: Braid universality*,enhanced robustness.

  40. Happy Birthday!!!

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