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
 Quantized circulation in superfluid helium
 Quantized flux in superconductor
 Chern number for quantized Hall conductance
 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
superconductor are
described by a
“wave function” Y(r)
eif(r)
Y(r) =| Y(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 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
“Filling in” of DOS as vortex core is approached
“flux quantization”
Singularity in real space
Flux/circulation quantization are
manifestations of
real-space singularities of the
complex (scalar) order parameter
Quantized Hall Conductance in 2DEG
 Discovery of IQHE by Klitzing in 1980
 2D electron gas (2DEG)
 Hall resistance a rational fraction of h/e2
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
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
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 integral
As with the circulation, this number is an integer
sxy is this integer (times e2/h)
Singularity in real vs. momentum space
Topological Object
Space
Physical Manifestation
U(1) vortex
R
Flux quantization in 2D SC
Circulation quantization in 2D SF
U(1) vortex
K
QHE in 2DEG under B-field
Magnetic field induces QHE by creating singularities
in the Bloch wave function
In both, relevant variable is a complex scalar
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-component),
you might get QSHE (Kane&Mele, PRL 05)
Momentum vs. Real-space Skyrmions
Momentum-space
Real-space
Quantized Hall response in
two-component electronic
systems
Anomalous Hall
effect by coupling
to conduction
electrons
Looks like
Physical
Role
Presence of Gapless Edge States
 Gapless edge states occur at the 1D boundary of these
models (charge and/or spin transport)
“QSHI”
Kramers pair
Kramers pairs not mixed by
T-invariant perturbations
BULK
Zero charge current
Quantized spin current
Kramers pair
BULK
“QAHI”
Partner change due to
large perturbation
BULK
Zero spin current
BULK
Quantized charge current
Zero magnetic field
“BI”
Partner change due to
large perturbation
BULK
Zero spin current
Zero charge current
Counterpropagating
edge modes mix
BULK
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
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
Introduce a boundary condition (surface), and
as a result, some midgap states appear
CB
Ly
Lx
kz
ky
VB
kx
(Lx,
Ly)
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)
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)>)
To Switch Partners or Not to Switch Partners
(Either-Or, Z2 question)
(Lx,
1
Ly )
Charlie and Mary gets a divorce.
A year later, they re-marry.
(Boring!)
k
k2
k
1
k2
(Lx,
Ly )
Charlie and Mary gets a divorce.
A year later, Charlie marries
Jane, Mary marries Chris.
(Interesting!)
Protection of Gapless Surface States
EF
(Lx,
1
Lx )
No guarantee of surface states
crossing Fermi level
k
k2
k
1
k2
(Lx,
Lx )
Guarantee of surface states
This is the TBI
Kane-Mele-Fu Proposal :
Kramers partner switching is a way
to guarantee existence of gapless
edge (surface) states of
bulk insulators
 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
d4
d1
d2
p2=d3d4
kx
p1=d1d
2
Band Insulator
d3
Gapless Edge?
ky
 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=d1kd
d3d4
y 2
d3
d1
d4
d2
p2=d3d4
kx
p1=d1d
2
p1p2=
-1
In 3D, projection to a particular
surface gives four surface numbers
p1, p2, p3 , p4
d8
d7
d5
p4=
d7d
p3=d5d6
d6
8
d3
d4
d1
d2
p2=d3d4
p1=d1d2
p1p2 p3
p4 p
=-1 p p = d d d d
p
1 2
3
4
1 2
3
4
d5d6 d7 d8=-1
Gaplessdsurface
state ondevery
surface
8
7
1
d6
Strong TI
d3
-1
d4
d1
d2
Dirac Circle
d5
-1
-1
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 Object
Space
Physical Manifestation
U(1) vortex
R
Flux quantization in 2D SF
Circulation quantization in 2D SF
U(1) vortex
K
QHE in 2DEG under B-field
Skyrmion
R
AHE in 2D metallic magnet
AHE of magnons
Skyrmion
K
AHE, QSHE in 2D band insulator
Z2
K
2D&3D TBI