InsulatorsInsulators

a material in which no electrons that are a material in which no electrons that are not bound with their respective place not bound with their respective place

inside material inside material happens in specific type of materials, and happens in specific type of materials, and

depends on thing such as no of electrons depends on thing such as no of electrons per atom and how they are arranged in per atom and how they are arranged in solidsolid

Effect of the sample boundaryEffect of the sample boundary

In 1960, Kohn characterized the In 1960, Kohn characterized the insulating state in terms of the insulating state in terms of the sensitivity of electron inside the sensitivity of electron inside the material to effect on the sample material to effect on the sample boundary. boundary.

The presence of a bulk gap does notThe presence of a bulk gap does not

guarantee that electrons will alwaysguarantee that electrons will always

show insensitivity to boundary. show insensitivity to boundary.

Electrons feels force perp. to it motion andApplied field. Cause to move in circular orbitradius depending on the field.

Berry curvature: Symmetry Consideration

•Time reversal (i.e. “ motion reversal”)

)()( . , 111 kTkTvTvTrTrT nn

)()( kk nn

)()( . , 111 kIkIvIvIrIrI nn

)()( kk nn •Inversion Symmetry:

•Define vector potential (Berry Connection)

•Hall Conductivity as Curvature

)2(22 gnKdSs

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 wavefncan 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 demanding that a component of theState 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 inH1 so that another componentu(x1,y1) real. The overall phaseIs uniquely determined.

The Chern no is topological in the sense that it isinvariant under small deformation of the HamiltonianSmall changes of the Hamiltonian result a small changeof the Berry Curvature (adiabatic curvature),one mightthink small change in chern no, but chern no is invariant. 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 suchLevel crossing happens in QHS, the adiabatic curvaturediverges and the chern no is no longer defined. The transition between chern nos plateau take place at levelcrossing

What are topological band insulators?What are topological band insulators? Topology characterizes the identity of objects up to Topology characterizes the identity of objects up to

deformation, e.g. genus of surfacesdeformation, e.g. genus of surfaces

Similarly, band insulator can be classified up to the Similarly, band insulator can be classified up to the deformation of band structure. Modify smoothly deformation of band structure. Modify smoothly preserving gap.preserving gap.

H

Figure courtesy C. Kane

Imagine an interface when a crystal slowly interpolatesas a function of x between a QHS (n=1) and a trivialInsulator (n=0). Somewhere along the way the energyGap has to go to zero, because otherwise it is impossible for the topological invariant to change. There will below energy electronic state bound to the region whereEnergy gap passes through zero.

Can one realize a quantum hall like Can one realize a quantum hall like insulator WITHOUT a magnetic field?insulator WITHOUT a magnetic field?

Yes: Yes: Kane and Mele; Bernevig & Zhang (2005),Kane and Mele; Bernevig & Zhang (2005), – Spin-orbit interaction Spin-orbit interaction »» spin-dependent spin-dependent

magnetic fieldmagnetic fieldSpin-orbit interaction is Time Reversal symmetric:

“Spin-Hall Effect”

Two Dimensional Topological Insulator Two Dimensional Topological Insulator (Quantum Spin Hall Insulator)(Quantum Spin Hall Insulator)

•Requires spin-orbit interactions•Protected by Time Reversal.•Only Z2 (even-odd) distinction.(Kane-Mele)

Time reversal symmetry =>two counter-propagating edge modes

Single Dirac node – Single Dirac node – impossible in 1D with time impossible in 1D with time reversal symmetryreversal symmetry

Stable to (non-magetic) Stable to (non-magetic) disorder:disorder:

(no Anderson localization (no Anderson localization though 1D)though 1D)

Single Dirac Node

Special features of 2D T-I edge statesSpecial features of 2D T-I edge states

•Experiments: transport on HgTe quantum well

(Bernevig et al., Science 2006; Konig etal. Science 2007)

•If number edge states pair are even, then all rightmovers would hybridized with left movers with Exception their partner, hence no transport.

•On other hand odd pair after hybridization thereWill be still edge state connecting the bands.•Which of these two alternative occures is determinedby the topological class of bulk band structure.

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