.. seealso::
The complete source code of this example can be found in :jupyter-download:script:`kdotp_generator`.
A Jupyter notebook can be found in :jupyter-download:notebook:`kdotp_generator`.
.. jupyter-kernel::
:id: kdotp_generator
.. jupyter-execute::
:hide-code:
import numpy as np
import sympy
import qsymm
In the :ref:`Bloch generator tutorial <tutorial_bloch_generator>` we saw how to generate tight-binding Hamiltonians using Qsymm. In this tutorial we will explore how to generate k \cdot p Hamiltonians.
We reproduce the effective Hamiltonian derived in Phys. Rev. Lett. 103, 266801 (2009) to explain hexagonal warping effects in the surface dispersion of a topological insulator.
The symmetry group is generated by threefold rotation symmetry, mirror symmetry and time-reversal symmetry.
.. jupyter-execute::
# C3 rotational symmetry - invariant under 2pi/3
C3 = qsymm.rotation(1/3, spin=1/2)
# Time reversal
TR = qsymm.time_reversal(2, spin=1/2)
# Mirror symmetry in x
Mx = qsymm.mirror([1, 0], spin=1/2)
symmetries = [C3, TR, Mx]
The Hamiltonian includes the momenta k_x and k_y, and contains terms up to third order in momenta.
.. jupyter-execute::
dim = 2 # Momenta along x and y
total_power = 3 # Maximum power of momenta
We use ~qsymm.hamiltonian_generator.continuum_hamiltonian to generate the Hamiltonian family:
.. jupyter-execute::
family = qsymm.continuum_hamiltonian(symmetries, dim, total_power, prettify=True)
qsymm.display_family(family)
We can then verify that the Hamiltonian family satisfies the symmetries:
.. jupyter-execute::
qsymm.hamiltonian_generator.check_symmetry(family, symmetries)
We reproduce the Hamiltonian for the quantum spin Hall effect derived in Science, 314, 1757 (2006).
The symmetry group is generated by spatial inversion symmetry, time-reversal symmetry and fourfold rotation symmetry.
.. jupyter-execute::
# Spatial inversion
pU = np.array([
[1.0, 0.0, 0.0, 0.0],
[0.0, -1.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0],
[0.0, 0.0, 0.0, -1.0],
])
pS = qsymm.inversion(2, U=pU)
# Time reversal
trU = np.array([
[0.0, 0.0, -1.0, 0.0],
[0.0, 0.0, 0.0, -1.0],
[1.0, 0.0, 0.0, 0.0],
[0.0, 1.0, 0.0, 0.0],
])
trS = qsymm.time_reversal(2, U=trU)
# Rotation
phi = 2.0 * np.pi / 4.0 # Impose 4-fold rotational symmetry
rotU = np.array([
[np.exp(-1j*phi/2), 0.0, 0.0, 0.0],
[0.0, np.exp(-1j*3*phi/2), 0.0, 0.0],
[0.0, 0.0, np.exp(1j*phi/2), 0.0],
[0.0, 0.0, 0.0, np.exp(1j*3*phi/2)],
])
rotS = qsymm.rotation(1/4, U=rotU)
symmetries = [rotS, trS, pS]
The Hamiltonian includes the momenta k_x and k_y, with terms up to second order.
.. jupyter-execute::
dim = 2
total_power = 2
.. jupyter-execute::
family = qsymm.continuum_hamiltonian(symmetries, dim, total_power, prettify=True)
qsymm.display_family(family)
We reproduce the Hamiltonian for a three-dimensional topological insulator introduced in Nature Physics 5, 438–442 (2009).
The symmetry group is generated by threefold rotation symmetry around z, inversion symmetry and time-reversal symmetry.
.. jupyter-execute::
# Spatial inversion
pU = np.diag([1, -1, 1, -1])
pS = qsymm.inversion(3, pU)
# Time reversal
trU = np.array([
[0.0, 0.0, -1.0, 0.0],
[0.0, 0.0, 0.0, -1.0],
[1.0, 0.0, 0.0, 0.0],
[0.0, 1.0, 0.0, 0.0],
])
trS = qsymm.time_reversal(3, trU)
# Rotation
phi = 2.0 * np.pi / 3.0 # Impose 3-fold rotational symmetry
rotU = np.array([
[np.exp(1j*phi/2), 0.0, 0.0, 0.0],
[0.0, np.exp(1j*phi/2), 0.0, 0.0],
[0.0, 0.0, np.exp(-1j*phi/2), 0.0],
[0.0, 0.0, 0.0, np.exp(-1j*phi/2)],
])
rotS = qsymm.rotation(1/3, axis=[0, 0, 1], U=rotU)
symmetries = [rotS, trS, pS]
The model includes the momenta k_x, k_y and k_z, up to second order.
.. jupyter-execute::
dim = 3
total_power = 2
.. jupyter-execute::
family = qsymm.continuum_hamiltonian(symmetries, dim, total_power, prettify=True)
qsymm.display_family(family)
Rotation in real and spin space with spin 1/2. Should have linear Rashba-term as there is no inversion symmetry.
.. jupyter-execute::
# 3D real space rotation generators
L = qsymm.groups.L_matrices(3)
# Spin-1/2 matrices
S = qsymm.groups.spin_matrices(1/2)
# Spin-3/2 spin matrices
J = qsymm.groups.spin_matrices(3/2)
# Continuous rotation generators
symmetries = [qsymm.ContinuousGroupGenerator(l, s) for l, s in zip(L, S)]
dim = 3
total_power = 2
family = qsymm.continuum_hamiltonian(symmetries, dim, total_power, prettify=True)
qsymm.display_family(family)
Also impose inversion symmetry, which removes the linear Rashba term.
.. jupyter-execute::
symmetries = [qsymm.ContinuousGroupGenerator(l, s) for l, s in zip(L, S)]
# Add inversion
symmetries.append(qsymm.PointGroupElement(-np.eye(3), False, False, np.eye(2)))
dim = 3
total_power = 2
family = qsymm.continuum_hamiltonian(symmetries, dim, total_power, prettify=True)
qsymm.display_family(family)
Rotations in real and spin space with spin 3/2.
.. jupyter-execute::
symmetries = [qsymm.ContinuousGroupGenerator(l, s) for l, s in zip(L, J)]
dim = 3
total_power = 2
family = qsymm.continuum_hamiltonian(symmetries, dim, total_power, prettify=True)
qsymm.display_family(family)
Reproduce the k \cdot p model for SnTe used in Phys. Rev. Lett. 122, 186801 (2019)
The symmetry group is generated by three-fold rotation symmetry, mirror symmetry in x, inversion symmetry, and time-reversal symmetry.
.. jupyter-execute::
# Double spin-1/2 representation
spin = [np.kron(s, np.eye(2)) for s in qsymm.groups.spin_matrices(1/2)]
# C3 rotational symmetry
C3 = qsymm.rotation(1/3, axis=[0, 0, 1], spin=spin)
# Time reversal
TR = qsymm.time_reversal(3, spin=spin)
# Mirror x
Mx = qsymm.mirror([1, 0, 0], spin=spin)
# Inversion
IU = np.kron(np.eye(2), np.diag([1, -1]))
I = qsymm.inversion(3, IU)
.. jupyter-execute::
dim = 3
total_power = 2
family = qsymm.continuum_hamiltonian([C3, TR, Mx, I], dim, total_power, prettify=True)
qsymm.display_family(family)
There are 3 combinations proportional to the identity. Generate all terms proportional to the identity and subtract them.
.. jupyter-execute::
identity_terms = qsymm.continuum_hamiltonian([qsymm.groups.identity(dim, 1)], dim, total_power)
identity_terms = [
qsymm.Model({
key: np.kron(val, np.eye(4))
for key, val in term.items()
})
for term in identity_terms
]
family = qsymm.hamiltonian_generator.subtract_family(family, identity_terms, prettify=True)
qsymm.display_family(family)
Removing the C3 symmetry, we find 8 additional terms, or 11 after removing terms proportional to the identity.
.. jupyter-execute::
dim = 3
total_power = 2
no_c3_family = qsymm.continuum_hamiltonian([TR, Mx, I], dim, total_power, prettify=True)
no_c3_family = qsymm.hamiltonian_generator.subtract_family(no_c3_family, identity_terms, prettify=True)
The new terms are given in the following.
.. jupyter-execute::
new_terms = qsymm.hamiltonian_generator.subtract_family(no_c3_family, family, prettify=True)
qsymm.display_family(new_terms)
Breaking inversion symmetry as well yields 22 additional terms.
.. jupyter-execute::
dim = 3
total_power = 2
broken_family = qsymm.continuum_hamiltonian([TR, Mx], dim, total_power, prettify=True)
broken_family = qsymm.hamiltonian_generator.subtract_family(broken_family, identity_terms, prettify=True)
new_terms = qsymm.hamiltonian_generator.subtract_family(broken_family, no_c3_family, prettify=True)
qsymm.display_family(new_terms)