Quantum hall effect without magnetic field

For quantum hall effect the interior of the material does not carry a current.The electron have full circular motion and has net contribution is zero. Rather the boundary has skipping orbits of electron and carry current. Now this current can flow in one direction.This is classical analogue. 

In quantum mechanics these are expressed as edge state connecting between valance and conduction band (Phys. Rev. B 25, 2185). These are edge states are call called chiral edge state which are topologically protected. Now in this edge state electric current flow without dissipation. In IQH measurement we measure the transport of this chiral edge state. Now this does not depend on the edge of the sample, geometry etc. But this edge state are entirely determine  by the bulk bands  So this is called bulk-boundary correspondence. Unless therer is no phase transition in bulk there is no change in he edge state so the edge states are topologically protected by the bulk and these edge states are robust.

It is to be noted that chiral edge state  breaks time reversal symmetry. As edge current flowing left or right direction is different. So in conclusion IQH demands to break time revarsal symmetry. 

 Can we realize IQE with out magnetic field???.

Lets go back and discuss a bit about toy model by haldane og Graphene. Because these has led letter a clear answer of the above question. Realizing quantum hall effect without magnetic field and landau level.These are known now to be as Chern insulator/ Qunatum anomalous hall effect/ Time reversal inverient topological insulator.
   

The inversion symmetry can be broken by replacing graphene and B site by dissimilar atom e.g Boron nitride.

Now the time reversal symmetry with out breaking any other internal symmetry ,can be broken by considering a NNN hopping

The second nearest neighbor hopping are described in above figure by dotted line. Such hopping can be clockwise and anti clockwise and they are not equivalent !! So time reversal symmetry is broken.

Now by the above mentioned two method a gap is opened then now an interesting concepts come that is "chirality".

When a gap is open then the Dirac equation has a mass term. Now introduction of mass term give rise to chirality i,e handeness. When gap is opened by breaking the space inversion symmetry then the two Dirac point are still connected by time reversal symmetry (it basically reverse the handiness) and the net chirality is zero. But when gap is opened by breaking time inversion symmetry, then the dirac ppint preserve space inversion symmetry and the handiness is preserved. So this situation is really special.

Now let's hink about the edge when a gap is opened . Now Graphene has interesting particle hole symmetry. Because of this there is exact zero energy edge state. This zero energy state connect the two Dirac points of of graphene from bulk to the edge.

This edge state appear because of excess number either A or B site at the edge of graphene. Because of particle- hole symmetry the system require an equal number of A and B sites. Now when on the edge there is excess of any local site then it will give rise to such zero energy edge state. 

   Now when the gap is open the edge states has to be connected among the valence or conduction band of the Dirac cone

Now the breaking inversion symmetry make these edge state connected either with in the valence and conduction band. This case is trivial.But for broken time reversal symmetry an special situation appear when the edge connect in between valence and conduction band. This is a nontrivial case.

For graphene it is semimetal and valvce, conduction band touches at Dirac point. If we break inversion /Time reversal symmetry then a gap is opened. 

Yes we can by spin orbit coupling (SOC). The SOC can give rise an fictitious magnetic field in side a material and give rise a landau level.So the system is time reversal inverient unlike IQE system!! 
If we consider 2 copies of quantum hall system 1 with spin up-field and another spin down-field down and combine them together now total hall conductance is zero. But the whole system is time reversal. Then what is other way out??

These class of material known as Quantum hall insulator.

Under SOC  the up and spin down will fill different force (SOC give velocity dependent transport i.e two spin Chanel )  and time reversal symmetry  is conserve by kramers theorem (T^2=-1) !!. Theoretically predicted Phys. Rev. Lett. 95, 226801 (2005).

Phys. Rev. Lett. 96, 106802 (2006)

Phys. Rev. Lett. 95, 136602 (2005)

Experimentally observed by Konig et al. Science  02 Nov 2007:Vol. 318, Issue 5851, pp. 766-770 !! 

So what we got is the interior is insulator but surface is conducting. It is a 2D topological insulator.

Now the question is can we have a topological insulator???