1 / 35

Jung Hoon Han (SKKU, Korea)

Topological Numbers and Their Physical Manifestations. Jung Hoon Han (SKKU, Korea). “Topological Numbers”. Numbers one can measure that do not depend on sample , level of purity, or any kind of details as long as they are minor. Examples of Topological Numbers.

ilori
Download Presentation

Jung Hoon Han (SKKU, Korea)

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. Topological Numbers and Their Physical Manifestations Jung Hoon Han (SKKU, Korea)

  2. “Topological Numbers” Numbers one can measure that do not depend on sample, level of purity, or any kind of details as long as they are minor

  3. Examples of Topological Numbers • Quantized circulation in superfluid helium • Quantized flux in superconductor • Chern number for quantized Hall conductance • Skyrmionnumber for anomalous Hall effect • Z2 number for 3D topological insulators Each TN has been worth a NP

  4. Condensates and U(1) Phase • Quantized circulation in superfluid helium • Quantized flux in superconductor Despite being many-particle state, superfluid and superconductor are described by a “wave function” Y(r) Singularity must be present for nonzero winding number Y(r)=| Y(r)|eif(r)is single-valued, and has amplitude and phase Singularity means vanishing | Y(r)|, or normal core

  5. Wavefunction around a Singularity • Near a singularity one can approximate wavefunction by its Taylor expansion • Employing radial coordinates, • b/a is a complex number, for simplicity choose b/a=1 • Indeed a phase winding of 2p occurs

  6. “Filling in” of DOS as vortex core is approached

  7. “flux quantization”

  8. Singularity in real space Flux/circulation quantization are manifestations of real-space singularities of the complex (scalar) order parameter

  9. Quantized Hall Conductance in 2DEG • Discovery of IQHE by Klitzing in 1980 • 2D electron gas (2DEG) • Hall resistance a rational fraction of h/e2

  10. Hall Conductance from Linear Response Theory • Kubo formulated a general linear response theory • Longitudinal and transverse conductivities as • current-current correlation function • Works for metals, insulators, whatever

  11. Hall Conductance for Insulators • Thouless, Kohmoto, Nightingale, den Nijs (TKNN) • considered band insulator with an energy gap • formulated a general linear response theory • TKNN formula works for any 2D band insulator

  12. TKNN on the Go • Integral over 2D BZ of Bloch eigenfunctionyn(k) for • periodic lattice • Define a “connection” • Using Stokes’, bulk integral becomes line integral • As with the circulation, this number is an integer • sxy is this integer (times e2/h)

  13. Singularity in real vs. momentum space • Magnetic field induces QHE by creating singularities in the Bloch wave function • In both, relevant variable is a complex scalar

  14. Haldane’s Twist • Haldane devised a model with quantized Hall conductance without external B-field (PRL, 88) • His model breaks T-symmetry, but without B-field which topological invariant is related to sxy ? A graphene model with real NN, complex NNN hopping

  15. Skyrmion Number in Momentum Space • By studying graphene, Haldane doubles the wave function size to two components • (Dirac Hamiltonian in 2D momentum space) • Hall conductance of H can be derived as • an integral over BZ “Skyrmion number”

  16. QAHE & QSHE • If two-component electronic system carries nonzero Skyrmion number in momentum space, you get QHE effect without magnetic field (QAHE) • If sublattice as well as spin are involved (4-component), you might get QSHE (Kane&Mele, PRL 05)

  17. Momentum vs. Real-space Skyrmions

  18. Presence of Gapless Edge States • Gapless edge states occur at the 1D boundary of these models (charge and/or spin transport)

  19. “QSHI” Kramers pair Kramers pairs not mixed by T-invariant perturbations Zero charge current Quantized spin current • BULK BULK Kramers pair

  20. “QAHI” Partner change due to large perturbation Zero spin current Quantized charge current Zero magnetic field • BULK BULK

  21. “BI” Partner change due to large perturbation Zero spin current Zero charge current Counterpropagating edge modes mix • BULK BULK

  22. ALL discussions were limited to 2D 2D quantized flux 2D flux lattice 2D quantized Hall effect 2D quantized anomalous Hall effect 2D quantized spin Hall effect Extension of topological ideas to 3D has been a long dream of theorists

  23. Z2 Story of Kane, Mele, Fu (2005-2007) • For generic SO-coupled systems, spin is not a good • quantum number, then is there any meaning to • “quantized spin transport”? • Kane&Mele came up with Z2 concept for arbitrary • SO-coupled 2D system • The concept proved applicable to 3D • Z2 number was shown to be related to parity of • eigenfunctions in inversion-symmetric insulators • -> Explosion of activity on TI

  24. Surface States of Band Insulator • Take a band insulator in 2D or 3D • Introduce a boundary condition (surface), and • as a result, some midgap states appear CB Ly Lx kz ky VB kx (Lx,Ly)

  25. TRIMs and Kramers Pairs • Band Hamiltonian in Fourier space H(k) is related by • TimeReversal (TR) operation to H(-k) • Q H(k) Q-1 = H(-k) • IIf k is half the reciprical lattice vector G, k=G/2, • Q H(G/2) Q-1 = H(-G/2) = H(+G/2) • These are special k-vectors in BZ • called TimeReversalInvariantMomenta • (TRIM)

  26. TRIMs and Kramers Pairs • At these special k-points, ka, H(ka) commutes with Q • By Kramers’ theorem all eigenstates of H(ka) are pairwise degenerate, i.e. • H(ka) |y(ka)> = E(ka) |y(ka)>, • H(ka) (Q |y(ka)>) = E(ka) (Q |y(ka)>)

  27. To Switch Partners or Not to Switch Partners (Either-Or, Z2 question) k1 k2 (Lx,Ly) k1 k2 (Lx,Ly) Charlie and Mary gets a divorce. A year later, they re-marry. (Boring!) Charlie and Mary gets a divorce. A year later, Charlie marries Jane, Mary marries Chris. (Interesting!)

  28. Protection of Gapless Surface States EF k1 k2 (Lx,Lx) k1 k2 (Lx,Lx) No guarantee of surface states crossing Fermi level Guarantee of surface states This is the TBI

  29. Kane-Mele-Fu Proposal : Kramers partner switching is a way to guarantee existence of gapless edge (surface) states of bulk insulators

  30. 4 TRIMs in 2D bands • Each TRIM carries a number, da=+1 or -1 • Projection to a given surface (boundary) results • in surface TRIMs, and surface Z2 numbers pi ky Band Insulator Gapless Edge? d3 d4 p2=d3d4 p1=d1d2 d1 d2 kx

  31. If the product of a pair of pi numbers is -1, • the given pair of TRIMs show partner-switching • -> gapless states • In 2D, p1p2=d1d2d3d4 • Z2 number n0 defined from (-1)n0=d1d2d3d4 ky d3 d4 p2=d3d4 p1p2=-1 p1=d1d2 d1 d2 kx

  32. In 3D, projection to a particular surface gives four surface numbers p1, p2, p3 , p4 d8 d7 p4=d7d8 d5 p3=d5d6 d6 d3 d4 p2=d3d4 d1 d2 p1=d1d2

  33. p1p2 p3 p4 =-1 • p1p2 p3 p4 = d1d2 d3 d4 d5d6 d7 d8=-1 • Gapless surface state on every surface d8 d7 -1 d5 1 d6 Dirac Circle Strong TI d3 -1 d4 d1 d2 -1

  34. So What is d ? • For inversion-symmetric insulator, d is a product of the parity numbers of all the occupied eigenstates at a given TRIM • For general insulators, d is the ratio of the square root of the determinant of some matrix divided by its Pfaffian

  35. Summary

More Related