The weak coupling electron model

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%matplotlib inline
from itertools import product
from functools import wraps

import numpy as np
from lll import lll
import matplotlib.pyplot as plt
import plotly.plotly as py
import scipy
import plotly.graph_objs as go
from plotly.offline import init_notebook_mode, iplot, iplot_mpl
from ipywidgets import interact

init_notebook_mode()


    

    




/opt/environments/python2_2016_03_04/local/lib/python2.7/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.
  warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')
/opt/environments/python2_2016_03_04/local/lib/python2.7/site-packages/requests/packages/urllib3/util/ssl_.py:315: SNIMissingWarning: An HTTPS request has been made, but the SNI (Subject Name Indication) extension to TLS is not available on this platform. This may cause the server to present an incorrect TLS certificate, which can cause validation failures. For more information, see https://urllib3.readthedocs.org/en/latest/security.html#snimissingwarning.
  SNIMissingWarning
/opt/environments/python2_2016_03_04/local/lib/python2.7/site-packages/requests/packages/urllib3/util/ssl_.py:120: InsecurePlatformWarning: A true SSLContext object is not available. This prevents urllib3 from configuring SSL appropriately and may cause certain SSL connections to fail. For more information, see https://urllib3.readthedocs.org/en/latest/security.html#insecureplatformwarning.
  InsecurePlatformWarning






    

In [33]:

    

def ham_1d(k, V, n):
    k_array = k*np.ones(n)
    r_momenta = np.linspace(-np.pi, np.pi, n + 2)
    ham_free = np.diag(np.subtract(k_array, r_momenta[1:-1])**2)
    ham_coupling = V*(np.ones((n, n)) - np.eye(n))
    ham = ham_free + ham_coupling
    eig = scipy.linalg.eigvalsh(ham)
    return eig
 
    
def ham_2d(kx, ky, V, n):
    k1 = np.outer(np.array([1, 0]), np.arange(-n, n+1))
    k2 = np.outer(np.array([0, 1]), np.arange(-n, n+1))
    k_offsets = (k1.reshape(2, 1, -1) + k2.reshape(2, -1, 1)).reshape(2, -1)
    k = np.array(([kx, ky]))
    arr = np.sum((k_offsets - k.reshape(2, 1))**2, axis=0)
    ham_free = np.diag(arr)
    ham_coupling = V*(np.ones((len(arr), len(arr))) - np.eye(len(arr)))
    ham = ham_free + ham_coupling
    eig = scipy.linalg.eigvalsh(ham)
    return eig


def energy_spectrum_1d(V, n):
    k_range = np.linspace(-1, 1, 200)
    energies = [ham_1d(k, V, n) for k in k_range]
    energy_data = np.array(energies).reshape((len(k_range), n))
    return energy_data


def energy_spectrum_2d(V, n):
    k_rangex = np.linspace(-np.pi, np.pi, 100)
    k_rangey = np.linspace(-np.pi, np.pi, 100)
    energies = []
    for kx in k_rangex:
        for ky in k_rangey:
            energies.append(ham_2d(kx, ky, V, n))
    data_2d = np.array(energies).reshape((len(k_rangex), len(k_rangey), (2*n + 1)**2))
    #if V is not 0:
    #    data_2d = np.sort(data_2d)
    return data_2d


def plot_1d_bands(data_1d, n):
    k_range = np.linspace(-1, 1, 200)
    layout = go.Layout(showlegend=False,
            autosize=False,
            width=700,
            height=500,
        xaxis=dict(
            title = 'k',
            titlefont = dict(size=20),
            showgrid=True,
            zeroline=False,
            showline=True,
            autotick=True,
            ticks='',
            showticklabels=True
        ),
        yaxis=dict(
            title = 'energy',
            titlefont = dict(size=20),
            autorange=True,
            showgrid=True,
            zeroline=False,
            showline=True,
            autotick=True,
            ticks='',
            showticklabels=True
        )
    )
    data = [go.Scatter(x=k_range, y=data_1d[:, i]) for i in range(n)]
    fig = go.Figure(data=data, layout=layout)
    iplot(fig, show_link=False)  
    
    
def plot_2d_bands(data_2d, n):
    k_rangex = np.linspace(-np.pi, np.pi, 100)
    k_rangey = np.linspace(-np.pi, np.pi, 100)
    mband = ((2*n + 1)**2)/2
    data = [go.Contour(z=data_2d[:, :, mband], x=k_rangex, y=k_rangey)]
    layout = go.Layout(showlegend=False,
                        autosize=False,
                        width=600,
                        height=600,
                        xaxis=dict(title='k_x',
                            autorange=True,
                            showgrid=False,
                            zeroline=False,
                            showline=True),
                        yaxis=dict(title='k_y',
                            autorange=True,
                            showgrid=False,
                            zeroline=False,
                            showline=True))
    fig = go.Figure(data=data, layout=layout)
    iplot(fig, show_link=False)


    

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n = 5
V = 0.

data_1d = energy_spectrum_1d(V, n)


    

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plot_1d_bands(data_1d, n)


    

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n = 1
V = 0

data_2d = energy_spectrum_2d(V, n)


    

In [37]:

    

plot_2d_bands(data_2d, n)


    

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