2. Berry curvature: Symmetry Consideration
Time reversal (i.e. “ motion reversal)
Inversion Symmetry:

7. Eq. (a) is gauge invariant although q depends on
This Eq. represents an important topological
feature of the systems.
Consider an arrow whose directional angles are
given by the phase of the wave function. The
arrow rotates p times as we go around the boun.
This gives a topological constraint to the wave fn.
Consider a zero of the wave function. If we go
around clockwise a small circle which contains the
0, the corresponding arrow rotates once either
clwise or anticlwise.

8. Therefore we can regard a 0 of a wvfn as a vor-
tex which has vorticity either 1 or -1 correspon-
ding clwise or antclwise rotns of arrow,resp.
Cases where we have a multiple rotation are consi-
dered to be specail one of having several vortices
at the same point.
The magnetic fd. forces a wave function to have –p
vorticity in the magnetic unit cell.
This is topological constraint because the total vorti
city in the magnetic unit cell is independent of parti-
cular potential chosen.

9. Define vector potential (Berry Connection)
Hall Conductivity as Curvature

10. Observation:
The magnetic BZ is topologically a Torus T2.
Application of Stoke’s thm to Eq. would give
cond. 0 if A(k1,k2) is uniquely defined on the
entire torus.
A possible non 0 value of cond. Is a consequences
of a non-trivial topology of A.
In order to understand non-trivial topology of A,
let us first discuss a ‘guage transformation’ of a
special kind.
Introduce a transformation

11. Non-trivial topology arises when the phase of wavefn
can not determined uniquely over entire MBZ
The previous transformation implies that over all pha
se factor can be chosen arbitrary.
The phase can be determined by demanding that a
a component of the state vector u(x0,y0) is real.
This convention is not enough to fix the phase on the
entire MBZ, since u(x0,y0) vanishes for some value
of (k1,k2)
Consider a simple case when u(x0,y0) vanishes only
at one point (k10,k20) in MBZ.

12. Phase can be fixed by deman
ding that a component of the
State vector u(x0,y0) is real.
However, this is not enough to
fix the phase over the MBZ,
when u vanishes at some pt.
Divide Torus in 2 pieces H1 &
H2 such that H1 contains
(k10,k20).
Adopt different convention in
H1 so that another component
u(x1,y1) real. The overall phase
Is uniquely determined.

13. The Chern no is topological in the sense that it is
invariant under small deformation of the Hamiltonian
Small changes of the Hamiltonian result a small change
of the Berry Curvature (adiabatic curvature),one might
think small change in chern no, but chern no is invaria
nt. Therefore, we observe plateau.
But how chern no change from one plateau to the next?
Large deformation of the Hamiltonian can cause the
ground state to cross over other eigenstates. When such
Level crossing happens in QHS, the adiabatic curvature
diverges and the chern no is no longer defined. The
transition between chern nos plateau take place at level
crossing

16. Z2 Invariants
T symmetry identifies two important subspaces of Bloch
Hmailtonian H(k) and the corresponding occupied band
wave function |u_i(k)>.
Even subspace:
T symmetry requires that H(k) belong to the even
subspace at the G point k=0 and as well as three M
points.

26. Insulators a material in which no electrons that are
not bound with their respective place inside material
happens in specific type of materials, and depends on thing such as no of electrons per atom and how they are arranged in solid

28. Effect of the sample boundary
In 1960, Kohn characterized the insulating state in terms of the sensitivity of electron inside the material to effect on the sample boundary.
The presence of a bulk gap does not
guarantee that electrons will always
show insensitivity to boundary.

30. Electrons feels force perp. to it motion and
Applied field. Cause to move in circular orbit
radius depending on the field.

33. Berry curvature: Symmetry Consideration
Time reversal (i.e. “ motion reversal”)
Inversion Symmetry:

35. Define vector potential (Berry Connection)
Hall Conductivity as Curvature

36. Topological invariance

38. What happens if

39. Gauss-Bonnet

42. Chern number and Magnetic monopole

51. Observation:
The magnetic BZ is topologically a Torus T2.
Application of Stoke’s thm to Eq. would give
cond. 0 if A(k1,k2) is uniquely defined on the
entire torus.
A possible non 0 value of cond. Is a consequences
of a non-trivial topology of A.
In order to understand non-trivial topology of A,
let us first discuss a ‘guage transformation’ of a
special kind.
Introduce a transformation

52. Non-trivial topology arises when the phase of wavefn
can not determined uniquely over entire MBZ
The previous transformation implies that over all pha
se factor can be chosen arbitrary.
The phase can be determined by demanding that a
a component of the state vector u(x0,y0) is real.
This convention is not enough to fix the phase on the
entire MBZ, since u(x0,y0) vanishes for some value
of (k1,k2)
Consider a simple case when u(x0,y0) vanishes only
at one point (k10,k20) in MBZ.

53. Phase can be fixed by deman
ding that a component of the
State vector u(x0,y0) is real.
However, this is not enough to
fix the phase over the MBZ,
when u vanishes at some pt.
Divide Torus in 2 pieces H1 &
H2 such that H1 contains
(k10,k20).
Adopt different convention in
H1 so that another component
u(x1,y1) real. The overall phase
Is uniquely determined.

54. The Chern no is topological in the sense that it is
invariant under small deformation of the Hamiltonian
Small changes of the Hamiltonian result a small change
of the Berry Curvature (adiabatic curvature),one might
think small change in chern no, but chern no is invaria
nt. Therefore, we observe plateau.
But how chern no change from one plateau to the next?
Large deformation of the Hamiltonian can cause the
ground state to cross over other eigenstates. When such
Level crossing happens in QHS, the adiabatic curvature
diverges and the chern no is no longer defined. The
transition between chern nos plateau take place at level
crossing

55. What are topological band insulators?
Topology characterizes the identity of objects up to deformation, e.g. genus of surfaces
Similarly, band insulator can be classified up to the deformation of band structure. Modify smoothly preserving gap.
Figure courtesy C. Kane

56. Imagine an interface when a crystal slowly interpolates
as a function of x between a QHS (n=1) and a trivial
Insulator (n=0). Somewhere along the way the energy
Gap has to go to zero, because otherwise it is impossib
le for the topological invariant to change. There will be
low energy electronic state bound to the region where
Energy gap passes through zero.

58. Can one realize a quantum hall like insulator WITHOUT a magnetic field?
Yes: Kane and Mele; Bernevig & Zhang (2005),
Spin-orbit interaction » spin-dependent magnetic field
Spin-orbit interaction is Time Reversal symmetric:
“Spin-Hall Effect”

61. Two Dimensional Topological Insulator (Quantum Spin Hall Insulator)
Time reversal symmetry =>two counter-propagating edge modes
Requires spin-orbit interactions
Protected by Time Reversal.
Only Z2 (even-odd) distinction.
(Kane-Mele)

63. Special features of 2D T-I edge states
Single Dirac node – impossible in 1D with time reversal symmetry
Stable to (non-magetic) disorder:
(no Anderson localization though 1D)
Single Dirac Node
Experiments: transport on HgTe quantum well
(Bernevig et al., Science 2006; Konig etal. Science 2007)

67. If number edge states pair are even, then all right
movers would hybridized with left movers with
Exception their partner, hence no transport.
On other hand odd pair after hybridization there
Will be still edge state connecting the bands.
Which of these two alternative occures is determined
by the topological class of bulk band structure.

68. In strong topological insulator , the Fermi surface for
the surface state encloses an odd number of
degeneracy points.