2D electron gas under magnetic field: formation of landau level

Let us consider a free electron gas assuming it as a simple charge particle with no other internal degree of freedom (e.g spin etc) in 2D dimension under magnetic field along z direction.

According to classical mechanics the electron will perform Larmor precession with cyclotron frequency ()does not depend on radius and give a special characteristic time to this problem.

But solving the problem through Quantum mechanics gives an interesting result:  
The quantum Hamiltonian is H= 1/2m(P+eA/C)^2= Q.Q/2m ....(eq1)


Just to be carefull the total momentum has two contribution, one is mechanical origin another is cannonical and it is from the magnetic field (C kitel book page no. 246)

now poison bracket of different component of mechanical moment does not commute with each other e.g
{mx_i ,mx_j}=-e A_ijk B_k A_ijk is levicivita symbol
 this finally give 

{Q_x, Q_y}=-iehB

Now there is twist in the story , we introduce a set of new variable namly quantm mehanical eaising and lowering operator a and a' and these can be written interms of Q_x and Q_Y as following
a=1/sqrt(ehB)(Q_x-iQ_y)

a'=1/sqrt(ehB)(Q_x+iQ_y)


So now hamiltonian H=Q.Q/2m=C(aa'+1/2)  where C=heB/m

Waooo..!! now the hamiltonian looks like a harmonic oscillator

So energy of nth state is E_n=(n+1/2)hw is quantize, equispaced and these level are called Landau level. Each of these  level are highly degenerate. 

In the above figure x axis is density of states.

So what we got is an incredible strong point and is starting to make CMP more interesting !!

    If we do an experiment where quantum mechanics work i.e, low temperature and we switched a magnetic field then electron are able to occupy the state only in landau level. If we now can manipulate to move the Fermi energy E_f then if it is in between levels where no state is available then the 2D free electron gas will be behave like an insulator !! Remember these are different from the band insulator where Fermi level lie in between valence and conduction band. 
* From now onwards these phase of material i.e a 2D electron gas with high magnetic field and having such beautiful Lanadau level  will be called as "Landau Phase"

Integer Quantum Hall Effect:
Let's measure the transport property in Landau phase. For that we have apply an electric field with in the x-y plane let say along x direction E_y and remember magnetic field is already applied in z direction.

Now we can measure the resistivity along x direction or in y direction. Such measurement is known as Hall measurement. Whenn electric field is applied alonf x direction and resistivity is also measuring along x then the resistivity is called longitudinal R_xx and when it is along y it is called transverse R_xy. 

The above figure show the experimental result. Where R_xy show discrete steps !! This is known as integer quantum hall effect. For this discovery Klitzing got nobel prize in 1985 https://www.nobelprize.org/nobel_prizes/physics/laureates/1985/klitzing.pdf  .

In classical picture e.g at high temperature when so called Landau phase is not there, this should be a st line instead of  discrete jump.

Explaining integar quantum hall effect through topology: 

One spectacular phenomena about this quantized hall resistance is that they extremely robust with respect to sample quality, defect, geometry etc. Another important aspect is this effect is highly accurate e.g R_xy has been measured to 1 part in 10^9 !!

The hall resistivity is expressed as

                    R_xy=ne^2/h

Now TKNN show that this integar n can be written as

Where c/n is called Chern number, it is the total Berry flux in the Brillouin zone. Chern number c/n is a topological invariant in the sense that it cannot change when the Hamiltonian varies smoothly. This helps to explain the robust quantization of R_xy.

The Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula relates the topological invariant called the Chern number with the Hall conductivity.Topology plays an intriguing role in quantum physics under the name of the Berry phase (Rev. Mod. Phys. 82, 1959)

.The TKNN formula tells us that the Hall conductivity is proportional to the Berry phase of a closed path encompassing the entire Brillouin zone.The integer in IQE hall effect is measuring something really special i.e topological inverient. The topology of valance band as function of momentum globally could be imagined as some sort  hole in different objects.

2D surfaces can be topologically classified by their genus g, which counts the number of holes. For instance, a sphere or drinking glass has g=0, while a donut/ tea cup  has g=1. A theorem in mathematics due to Gauss and Bonnet (Nakahara, 1990) states that the integral of the Gaussian curvature over a closed surface is a quantized topological invariant, and its value is related to g. The Chern number is an integral of a related curvature.

The longitudinal resistance becomes zero since all electrons can flow freely at the edge of system without any back-scattering. .The edge state is protected since it is the boundary between two phases, i.e., inside and outside the system, with different topological numbers.