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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
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