# Chirality of a Weyl point¶

In this example, a k•p model for a Weyl point is set up by explicitly specifying its hamiltonian.

Using a spherical surface around the Weyl point (k=[0, 0, 0]), the chirality of the point (Chern number of the surface) can be calculated.

#!/usr/bin/env python
# -*- coding: utf-8 -*-

import numpy as np
import matplotlib.pyplot as plt

import z2pack

# defining pauli matrices
pauli_x = np.array([[0, 1], [1, 0]], dtype=complex)
pauli_y = np.array([[0, -1j], [1j, 0]], dtype=complex)
pauli_z = np.array([[1, 0], [0, -1]], dtype=complex)
pauli_vector = list([pauli_x, pauli_y, pauli_z])

def hamilton0(k):
"""simple 2-band hamiltonian k.sigma with a Weyl point at k=0"""
res = np.zeros((2, 2), dtype=complex)
for kval, p_mat in zip(k, pauli_vector):
res += kval * p_mat
return res

def hamilton1(k):
"""2-band hamiltonian k.sigma with k_y -> -k_y"""
k[2] = -k[2]
res = np.zeros((2, 2), dtype=complex)
for kval, p_mat in zip(k, pauli_vector):
res += kval * p_mat
return res

# creating the two systems
system0 = z2pack.hm.System(hamilton0, bands=1)
system1 = z2pack.hm.System(hamilton1)  # bands=1 is default (#orbitals / 2)

# the surface is a sphere around the Weyl point
surface = z2pack.shape.Sphere([0., 0., 0.], 0.01)
res0 = z2pack.surface.run(system=system0, surface=surface)
res1 = z2pack.surface.run(system=system1, surface=surface)

# plotting
fig, ax = plt.subplots(1, 2)

# plot styling
fs = 15
ax[0].set_xlabel(r'$\theta$', fontsize=fs)
ax[0].set_ylabel(r'$\bar{x}$', rotation='horizontal', fontsize=fs)
ax[1].set_xlabel(r'$\theta$', fontsize=fs)
ax[0].set_xticks([0, 1])
ax[1].set_xticks([0, 1])
ax[0].set_xticklabels([r'$0$', r'$\pi$'])
ax[1].set_xticklabels([r'$0$', r'$\pi$'])
ax[0].set_title(r'$\vec{k}.\vec{\sigma}$', fontsize=fs)
ax[1].set_title(r'$(k_x, -k_y, k_z).\vec{\sigma}$', fontsize=fs)

# plotting the evolution of polarization
z2pack.plot.chern(res0, axis=ax[0])
z2pack.plot.chern(res1, axis=ax[1])

plt.savefig('plot.pdf', bbox_inches='tight')