Dear Qingtian,

First, I would like  to precise that I am not a specialist of Graphene or spin hall effect (neither Kwant) so you need to take the

solution I give you with precaution and you need to do tests knowing the expected results.

The amazing thing with Kwant is that the process of defining the Hamiltonan becomes simpler once you precise the Tight

binding model.

So, when you have the lattice expression of the Hamiltonian forget the x and y directions and work as in a graph  (sites

neighbors and links).

The starting point to your problem is may be the simpler case of electrons with spin in honeycomb lattice (without  SO interaction).

You should understand that the state | n,m, sigma> =|n,m> |sigma>     ,where   is the the tensorial product. Your

hamiltonian becomes H=h 1     . h is the spinless hamiltonian in graphene and 1 is the 2x2 identity matrix. So now, with the

definition of the tensorial product (matricial version) you can understand why we need to change in the spinless hamiltonian all

the elements (sites and hopping) by hij*1  (hij---->identity *hij) to obtain the whole Hamiltonian with spins.

With the interaction presented in the model of Kane & Mele,  you need to do the same and understand how is the Hamiltonian is

written as a Tensorial product.

you will understand that in the Hamiltonian with the second nearest neighbors you need to change the elements hij by matrices

as follows:

 hij ------>  1j*t2*Vij*Sigma_z

you need just to be careful with the hopping  Vij  which are site-dependent .

For this you need to just define the Vij in the clockwise: Vij=+1  by choosing the directions for the next nearest neighbors as

follows: (1,0)  (-1,-1)  (0,-1)

we do this for the two sublattices a, b.

The other anticlockwise hoppings Vij are directly insured by hermeticity.

The same procedure is done for the leads.

The results at the end seems to me at least "not bizarre" but you need to check this with specialists.

The program following these remarks is included with this mail.

Regards

Adel Abbout

www.abboutadelhomepage.weebly.com

On Sun, Feb 23, 2014 at 1:03 AM, Qingtian Zhang <qtzhang@mail.ustc.edu.cn> wrote:

Dear all,
I note that we can easily consider the next-nearest-neighbor hopping in honeycomb,  however, what is the model (Hamiltonian) for this code? In graphene, the intrinsic SOI is included in the reference:PRL 95, 226801 (2005) and we also have that in silicene: PRL 110, 026603 (2013). Is the model for Kwant the same as the two reference papers?  Moreover,can we consider the Rashba SOI using the Tinyarray package in honeycomb? the hoppings along x and y direction are more complicated in honeycomb.
Regards,
Qingtian Zhang

--
Abbout Adel