Commit a4b05839 authored by psd's avatar psd
Browse files

Exercise-13

parent 44cbb98f
%% Cell type:code id: tags:
``` python
import numpy as np
from matplotlib.patches import Rectangle
```
%% Cell type:markdown id: tags:
## Theoretical background
$\Psi_k = \frac{1}{\sqrt{N}} \sum_j e^{(j\,k\cdot r_j)} \, \phi_j = \frac{1}{\sqrt{N}} \sum_j c_k(r_j) \, \phi_j$
Notation:
$[\,]_{nm}$ = matrix of dimensions $n\times m$
- h = $[\phi_0]_{nn}$
- t = $[\phi_{+1}]_{mnn}, \ m=1$ for first nn hopping
For first nn:
- $c(k) = e^{(j\,k\cdot R)} \, ,k=(0,0,0)...(2\pi,0,0) \, , R=(1,0,0) $
- $[h(k)]_{nn} = [h]_{nn} + c(k) \cdot [t]_{0nn} + (c(k) \cdot [t]_{0nn})^H$
- $E(k)_n=eigvalsh([h(k)]_{nn})$
%% Cell type:code id: tags:
``` python
def bands(h,t,K):
'''
input:
h = [\phi_0]_nn
t = [\phi_1]_mnn
variables:
'''
kpts = np.linspace(0,K,300)
E = np.zeros((len(kpts),len(h)))
R = [1,0,0]
for k, kpt in enumerate(kpts):
c_k = np.exp(1j*2*np.pi*np.dot(R,[kpt,0,0]))
h_k = h + np.sum(c_k*t,axis=0) + np.conj(c_k)*np.sum(np.swapaxes(t,1,2),axis=0)
E[k] = np.linalg.eigvalsh(h_k)
return kpts,E
```
%% Cell type:code id: tags:
``` python
def make_ht(e,tau):
dim = e.ndim
if dim>0:
dim = len(e)
h = np.zeros((dim,dim),complex)
t = np.zeros((dim,dim),complex)
h.flat[::dim+1]=e
h.flat[1::dim+1]=tau[0]
h.flat[dim::dim+1]=np.conj(tau[0])
t[-1,0]=tau[1]
else:
h = np.array(e).reshape(1,1)
t = np.array(tau).reshape(1,1)
return h,np.expand_dims(t,0)
```
%% Cell type:code id: tags:
``` python
def visulize(ax,h,t,replica=5):
import matplotlib.cm as cm
colors = cm.hot(np.diag(h).real)
colors = np.tile(colors,(replica,1))
dim = len(h)
if dim>1:
uc = np.arange(dim)*1/abs(h[0,1].real)
colors = cm.hot(np.diag(h).real)
colors = np.tile(colors,(replica,1))
a = 1/abs(t[0,-1,0].real)+1/abs(h[0,1].real)
atoms = [uc[i]+(j*a) for j in range(replica) for i in range(dim)]
else:
uc = 0
a = 1/abs(t[0,-1,0].real)
atoms = [uc+(j*a) for j in range(replica)]
rect = Rectangle((-1/abs(2*t[0,-1,0].real),-0.25),a,0.5,
linewidth=1,edgecolor='r',facecolor='none')
ax.add_patch(rect)
ax.set_ylim((-1,1))
ax.scatter(atoms,np.zeros(dim*5),100,c=colors)
```
%% Cell type:code id: tags:
``` python
tau = np.array((0.2,-0.2))
e = np.array((-0.25,-0.1))
h, t = make_ht(e,tau)
kpts,E = bands(h,t,1)
```
%% Cell type:code id: tags:
``` python
fig, axs = plt.subplots(1,2,figsize=(15,5))
visulize(axs[0],h,t)
axs[1].plot(kpts,E)
```
%% Output
[<matplotlib.lines.Line2D at 0x7fd41a787278>,
<matplotlib.lines.Line2D at 0x7fd418708e80>]
%% Cell type:code id: tags:
``` python
```
......@@ -180,7 +180,7 @@
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......
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