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
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.
Electrons feels force perp. to it motion and
Applied field. Cause to move in circular orbit
radius depending on the field.
Berry curvature: Symmetry Consideration
•Time reversal (i.e. “ motion reversal”)
T rT
1
 r , T vT
1
  v . T  n ( k )T
1
 n ( k )   n (k )
•Inversion Symmetry:
I rI
1
  r , I vI
1
  v . I n ( k ) I
 n ( k )   n (k )
1
  n (k )
  n (k )
 KdS
 2 n  2 ( 2  g )
•Define vector potential (Berry Connection)
s
•Hall Conductivity as Curvature
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
•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.
•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.
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
What are topological band insulators?

Topology characterizes the identity of objects up to
deformation, e.g. genus of surfaces
Figure courtesy C. Kane

Similarly, band insulator can be classified up to the
deformation of band structure. Modify
smoothly
preserving gap.
H
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.
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”
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)
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)
•Experiments: transport on HgTe quantum well
(Bernevig et al., Science 2006; Konig etal. Science
2007)
Single Dirac Node
•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.
In strong topological insulator , the Fermi surface for
the surface state encloses an odd number of
degeneracy points.