Dear Subhadeep,
I think it is better to do it without continuous hamiltonians. Defining a
function for the hopping will be enough.
If you still want to do it with the discretization of a continuous
hamiltonian, you can proceed this way:
Define a hamiltonian: "k_x*A(x)* k_x". #check docomentation
This will induce a hopping equal to A(x+1/2) but at the same time, you will
have an onsite potential A(x+1/2)+A(x-1/2).
What you have to do in this case is to cancel this latter with a potential
V(x)= -A(x+1/2)-A(x-1/2)
So the hamiltonian becomes "k_x*A(x)* k_x+V(x)".
The function A(x)=x-1/2 (in order to have t=x).
The following program summarizes all this.
I hope this helps,
Adel
####################################################################
import kwant
import kwant.continuum
import scipy.sparse.linalg
import scipy.linalg
import numpy as np
# For plotting
import matplotlib as mpl
from matplotlib import pyplot as plt
L=30
def Line(site):
x=site.pos[0]
return abs(x)<L
def Hopping(x):
return x-0.5
hamiltonian= "k_x*A(x)*k_x-A(x+0.5)-A(x-0.5)"
template=kwant.continuum.discretize(hamiltonian)
print(template)
syst=kwant.Builder()
syst.fill(template,Line,(0,))
syst=syst.finalized()
ham=syst.hamiltonian_submatrix(params=dict(A=Hopping),sparse=True)
vals,evecs=scipy.sparse.linalg.eigsh(ham,k=36,which='SM')
plt.plot(vals,'-o')
On Thu, Jun 9, 2022 at 4:43 PM Subhadeep Chakraborty <
sc20rs...@iiserkol.ac.in> wrote:
> Hi Kwant Users !
>
> I was trying to plot the band structure of the SSH model using
> kwant.continuum. I could do it when the hopping terms are constant. Now let
> say the hoppings are linearly varying along x direction (t*x) or say I want
> to add a chemical potential like (\mu*x). In that case, how should I
> introduce this potential in the code ?
>
> I have attached my code below. It will be a great help if anyone kindly
> tells me where I need to modify the code and how to insert this (\mu*x
> term) in the code.
>
> The code is attached herewith.
> _______________________________________
> import kwant
> import kwant.continuum
> import scipy.sparse.linalg
> import scipy.linalg
> import numpy as np
>
> # For plotting
> import matplotlib as mpl
> from matplotlib import pyplot as plt
>
>
> def ssh():
> hamiltonian = "m*sigma_x + alpha * k_x * sigma_y"
>
>
>
>
> def plot(ax, a=1):
>
> params = dict(m=.5, alpha=.5)
>
> h_k = kwant.continuum.lambdify(hamiltonian, locals=params)
> k_cont = np.linspace(-1, 1, 201)
> e_cont = [scipy.linalg.eigvalsh(h_k(k_x=ki)) for ki in k_cont]
>
> template = kwant.continuum.discretize(hamiltonian, grid=a)
> syst = kwant.wraparound.wraparound(template).finalized()
>
> def h_k(k_x):
> p = dict(k_x=k_x, **params)
> return syst.hamiltonian_submatrix(params=p)
>
>
>
>
> ax.plot(k_cont, e_cont, 'r-')
>
> ax.plot([], [], 'r-', label='continuum')
>
>
> ax.set_xlim(-1, 1)
> ax.set_ylim(-2, 2)
> ax.set_title('a={}'.format(a))
>
> ax.set_xlabel('Momentum k_x')
> ax.set_ylabel('energy E')
> ax.grid()
> ax.legend()
>
> _, (ax1, ax2) = plt.subplots(1, 2, sharey=True, figsize=(12, 4))
>
> plot(ax1, a=1)
> plot(ax2, a=.25)
> plt.show()
>
>
>
>
> def main():
> ssh()
>
>
> if __name__ == '__main__':
> main()
> _______________________________________________________
>
>
>
>
> *_Thanks in advance,*
>
>
>
--
Abbout Adel
