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

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

3. Examples of Topological Numbers Quantized circulation in superfluid helium Quantized flux in superconductor Chern number for quantized Hall conductance Skyrmion number for anomalous Hall effect Z 2 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”  (r)  (r) =|  (r)| e  ( r ) is single-valued, and has amplitude and phase Singularity must be present for nonzero winding number Singularity means vanishing |  r  or normal core

5. Wavefunction around a Singularity Near a singularity one can approximate wavefunction by its Taylor expansion Employing radial coordinates,  is a complex number, for simplicity choose  Indeed a phase winding of 2  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. Discovery of IQHE by Klitzing in 1980 2D electron gas (2DEG) Hall resistance a rational fraction of h/e 2 Quantized Hall Conductance in 2DEG

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

11. 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 Hall Conductance for Insulators

12. TKNN on the Go Integral over 2D BZ of Bloch eigenfunction  n (k) for periodic lattice Define a “connection” Using Stokes’, bulk integral becomes line integral As with the circulation, this number is an integer  xy is this integer (times e 2 /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 acomplex scalar Topological ObjectSpacePhysical Manifestation U(1) vortexRFlux quantization in 2D SC Circulation quantization in 2D SF U(1) vortexKQHE in 2DEG under B-field

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  xy ? 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 Momentum-spaceReal-space Looks like Physical Role Quantized Hall response in two-component electronic systems Anomalous Hall effect by coupling to conduction electrons

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

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

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

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

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. Z 2 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 Z 2 concept for arbitrary SO-coupled 2D system The concept proved applicable to 3D Z 2 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 kyky kxkx kzkz LxLx LyLy CB VB (L x, L y ) Introduce a boundary condition (surface), and as a result, some midgap states appear

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) I If 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, k a, H(k a ) commutes with Q By Kramers’ theorem all eigenstates of H(k a ) are pairwise degenerate, i.e. H(k a ) | y (k a )> = E(k a ) |y (k a )>, H(k a ) ( Q | y (k a )>) = E(k a ) (Q |y (k a )>)

27. To Switch Partners or Not to Switch Partners (Either-Or, Z 2 question) (L x, L y ) Charlie and Mary gets a divorce. A year later, they re-marry. (Boring!) k1k1 k2k2 (L x, L y ) k1k1 k2k2 Charlie and Mary gets a divorce. A year later, Charlie marries Jane, Mary marries Chris. (Interesting!)

28. Protection of Gapless Surface States (L x, L x ) No guarantee of surface states crossing Fermi level k1k1 k2k2 (L x, L x ) k1k1 k2k2 Guarantee of surface states This is the TBI EFEF

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, d a =+1 or -1 Projection to a given surface (boundary) results in surface TRIMs, and surface Z 2 numbers p i kxkx kyky d1d1 d2d2 d4d4 d3d3 p 2 =d 3 d 4 p 1 =d 1 d 2 Gapless Edge? Band Insulator

31. If the product of a pair of p i numbers is -1, the given pair of TRIMs show partner-switching -> gapless states In 2D, p 1 p 2 = d 1 d 2 d 3 d 4 Z 2 number n 0 defined from (- 1) n 0 = d 1 d 2 d 3 d 4 kxkx kyky d1d1 d2d2 d4d4 d3d3 p 1 p 2 = -1 p 2 =d 3 d 4 p 1 =d 1 d 2

32. In 3D, projection to a particular surface gives four surface numbers p 1, p 2, p 3, p 4 p 3 =d 5 d 6 p4=d7d8p4=d7d8 p 2 =d 3 d 4 p 1 =d 1 d 2 d1d1 d2d2 d4d4 d8d8 d7d7 d5d5 d6d6 d3d3

33. p 1 p 2 p 3 p 4 =-1 1 Dirac Circle p 1 p 2 p 3 p 4 = d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 =-1 Gapless surface state on every surface Strong TI d1d1 d2d2 d4d4 d8d8 d7d7 d5d5 d6d6 d3d3

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 Topological ObjectSpacePhysical Manifestation U(1) vortexRFlux quantization in 2D SF Circulation quantization in 2D SF U(1) vortexKQHE in 2DEG under B-field SkyrmionRAHE in 2D metallic magnet AHE of magnons SkyrmionKAHE, QSHE in 2D band insulator Z2Z2 K2D&3D TBI