Jung Hoon Han (SKKU, Korea)

Topological Numbersand

Their Physical Manifestations

Numbers one can measure that do not depend on sam-ple, level of purity, or any kind of details as long as

they are minor

“Topological Numbers”

Examples of Topological Numbers Quantized circulation in superfluid helium

Quantized flux in superconductor

Chern number for quantized Hall conduc-tance

Skyrmion number for anomalous Hall effect

Z2 number for 3D topological insulators

Each TN has been worth a NP

Condensates and U(1) Phase Quantized circulation in superfluid helium Quantized flux in superconductor

Despite being many-particle state, superfluid and superconduc-tor are described by a “wave function” Y(r)Y(r) =| Y(r)| eif(r) is single-valued, and has amplitude and phase

Singularity must be present for nonzero winding number

Singularity means vanishing | (Y r)|, or normal core

Wavefunction around a Singularity Near a singularity one can approximate wavefunc-

tion by its Taylor expansion

Employing radial coordinates,

/ b a is a complex number, for simplicity choose / =1b a

Indeed a phase winding of 2p occurs

“Filling in” of DOS as vortex core is ap-proached

“flux quantiza-tion”

Singularity in real space

Flux/circulation quantization are manifestations of real-space singularities of the complex (scalar) order parame-ter

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

Quantized Hall Conductance in 2DEG

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 Re-sponse Theory

Thouless, Kohmoto, Nightingale, den Nijs (TKNN)considered band insulator with an energy gap

formulated a general linear response theoryTKNN formula works for any 2D band insulator

Hall Conductance for Insulators

TKNN on the Go

Integral over 2D BZ of Bloch eigenfunction yn(k) for periodic lattice

Define a “connection”

Using Stokes’, bulk integral becomes line integralAs with the circulation, this number is an integersxy is this integer (times e2/h)

Singularity in real vs. momentum space

Magnetic field induces QHE by creating singulari-ties in the Bloch wave function

In both, relevant variable is a complex scalar

Topological Ob-ject

Space Physical Manifestation

U(1) vortex R Flux quantization in 2D SCCirculation quantization in 2D SF

U(1) vortex K QHE in 2DEG under B-field

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

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”

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-compo-nent), you might get QSHE (Kane&Mele, PRL 05)

Momentum vs. Real-space Skyrmions

Momentum-space Real-space

Looks like

Physical Role

Quantized Hall response in two-component elec-tronic systems

Anomalous Hall effect by cou-pling to conduc-tion electrons

Presence of Gapless Edge States

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

BULK

BULK

Kramers pairs not mixed by T-invariant perturbations

Zero charge current

Quantized spin current

Kramers pair

Kramers pair

“QSHI”

BULK

BULK

Partner change due tolarge perturbation

Zero spin current

Quantized charge current

Zero magnetic field

“QAHI”

BULK

BULK

Partner change due tolarge perturbation

Zero spin current

Zero charge current

Counterpropagating edge modes mix

“BI”

ALL discussions were limited to 2D

2D quantized flux2D flux lattice

2D quantized Hall effect2D quantized anomalous Hall

effect2D quantized spin Hall effect

Extension of topological ideas to 3D has been a long dream

of theorists

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

Surface States of Band Insulator Take a band insulator in 2D or 3D

ky

kx

kz

Lx

Ly

CB

VB

(Lx,Ly)

Introduce a boundary condition (surface), and as a result, some midgap states appear

TRIMs and Kramers PairsBand 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)

TRIMs and Kramers Pairs

At these special k-points, ka, H(ka) commutes with Q By Kramers’ theorem all eigenstates of H(ka) are pair-

wise degenerate, i.e.

H(ka) |y(ka)> = E(ka) |y(ka)>, H(ka) (Q |y(ka)>) = E(ka) (Q |y(ka)>)

To Switch Partners or Not to Switch Partners (Either-Or, Z2 question)

(Lx,Ly)

Charlie and Mary gets a di-vorce. A year later, they re-marry. (Boring!)

k1 k2(Lx,Ly)k1 k2

Charlie and Mary gets a di-vorce. A year later, Charlie marries Jane, Mary marries Chris.(Interesting!)

Protection of Gapless Surface States

(Lx,Lx)

No guarantee of surface states crossing Fermi level

k1 k2(Lx,Lx)k1 k2

Guarantee of surface statesThis is the TBI

EF

Kane-Mele-Fu Proposal :Kramers partner switching is

a way to guarantee exis-tence of gapless edge (sur-

face) states of bulk insulators

4 TRIMs in 2D bandsEach TRIM carries a number, da =+1 or -1Projection to a given surface (boundary) re-sults

in surface TRIMs, and surface Z2 numbers pi

kx

ky

d1 d2

d4d3 p2=d3d4

p1=d1d2

Gap

less E

dg

e?

Ban

d In

su

lato

r

If the product of a pair of pi numbers is -1, the given pair of TRIMs show partner-switch-ing -> gapless statesIn 2D, p1p2=d1d2d3d4

Z2 number n0 defined from (-1)n0=d1d2d3d4

kx

ky

d1 d2

d4d3

p1p2=-1

p2=d3d4

p1=d1d2

In 3D, projection to a particular surface gives four surface numbers p1, p2, p3 , p4

p3=d5d6

p4=d7

d8

p2=d3d4

p1=d1d2d1 d2

d4

d8d7

d5

d6

d3

p1p2 p3 p4 =-1

1

-1

-1

-1

Dira

c Circle

p1p2 p3 p4 = d1d2 d3 d4 d5d6 d7 d8=-1Gapless surface state on every sur-face

Strong TI

d1 d2

d4

d8d7

d5

d6

d3

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

Summary

Topological Ob-ject

Space Physical Manifestation

U(1) vortex R Flux quantization in 2D SFCirculation quantization in 2D SF

U(1) vortex K QHE in 2DEG under B-field

Skyrmion R AHE in 2D metallic magnetAHE of magnons

Skyrmion K AHE, QSHE in 2D band insulator

Z2 K 2D&3D TBI