Qsymm Basics¶

Defining a Qsymm model¶

Let’s start by defining a 3D Rashba Hamiltonian symbolically as a Python string:

ham = ("hbar^2 / (2 * m) * (k_x**2 + k_y**2 + k_z**2) * eye(2) +" +
        "alpha * sigma_x * k_x + alpha * sigma_y * k_y + alpha * sigma_z * k_z")

We can then create a Qsymm Model directly from this symbolic Hamiltonian:

H = qsymm.Model(ham)

We can then directly inspect the contents by printing the Model:

print(H)

{alpha*k_z:
[[ 1.+0.j  0.+0.j]
 [ 0.+0.j -1.+0.j]],

hbar**2*k_x**2/m:
[[0.5+0.j 0. +0.j]
 [0. +0.j 0.5+0.j]],

hbar**2*k_y**2/m:
[[0.5+0.j 0. +0.j]
 [0. +0.j 0.5+0.j]],

hbar**2*k_z**2/m:
[[0.5+0.j 0. +0.j]
 [0. +0.j 0.5+0.j]],

alpha*k_x:
[[0.+0.j 1.+0.j]
 [1.+0.j 0.+0.j]],

alpha*k_y:
[[0.+0.j 0.-1.j]
 [0.+1.j 0.+0.j]],

}

We can also extract a more readable representation by using the tosympy method, which converts the Model to a sympy expression:

H.tosympy(nsimplify=True)

\[\begin{split}\displaystyle \left[\begin{matrix}\alpha k_{z} + \frac{\hbar^{2} k_{x}^{2}}{2 m} + \frac{\hbar^{2} k_{y}^{2}}{2 m} + \frac{\hbar^{2} k_{z}^{2}}{2 m} & \alpha k_{x} - i \alpha k_{y}\\\alpha k_{x} + i \alpha k_{y} & - \alpha k_{z} + \frac{\hbar^{2} k_{x}^{2}}{2 m} + \frac{\hbar^{2} k_{y}^{2}}{2 m} + \frac{\hbar^{2} k_{z}^{2}}{2 m}\end{matrix}\right]\end{split}\]

The argument nsimplify=True makes the output more readable by forcing sympy to elide factors of 1.0 that multiply each term. Note that Qsymm automatically interprets the symbols sigma_x, sigma_y and sigma_z as the Pauli matrices, and eye(2) as the 2x2 identity matrix.

Model as a momenta attribute that specifies which symbols are considered the momentum variables:

H.momenta

(k_x, k_y, k_z)

By default Qsymm assumes that your model is written in 3D (even if it does not include all 3 momenta). To define a lower-dimensional model you must explicitly specify the momentum variables, e.g:

ham2D = ("hbar^2 / (2 * m) * (k_x**2 + k_z**2) * eye(2) +" +
         "alpha * sigma_x * k_x + alpha * sigma_y * k_z")
H2D = qsymm.Model(ham2D, momenta=['k_x', 'k_z'])

H2D.tosympy(nsimplify=True)

\[\begin{split}\displaystyle \left[\begin{matrix}\frac{\hbar^{2} k_{x}^{2}}{2 m} + \frac{\hbar^{2} k_{z}^{2}}{2 m} & \alpha k_{x} - i \alpha k_{z}\\\alpha k_{x} + i \alpha k_{z} & \frac{\hbar^{2} k_{x}^{2}}{2 m} + \frac{\hbar^{2} k_{z}^{2}}{2 m}\end{matrix}\right]\end{split}\]

H2D.momenta

(k_x, k_z)

Defining group elements¶

Qsymm is all about finding and generating symmetries of models, so it is unsurprising that it contains utilities for defining group elements.

Below are a few examples of the sorts of things you can define with Qsymm:

# Identity in 3D
E = qsymm.identity(3)
# Inversion in 3D
I = qsymm.inversion(3)
# 4-fold rotation around the x-axis
C4 = qsymm.rotation(1/4, [1, 0, 0])
# 3-fold rotation around the [1, 1, 1] axis
C3 = qsymm.rotation(1/3, [1, 1, 1])
# Time reversal
TR = qsymm.time_reversal(3)
# Particle-hole
PH = qsymm.particle_hole(3)

The documentation page of the qsymm.groups module contains an exhaustive list of what can be generated.

As with other Qsymm objects we can get a readable representation of these group elements:

C4

\[R\left(\frac{\pi}{2}, [1 0 0]\right)\]

TR

\[ \mathcal{T}\]

Given a set of group generators we can also generate a group:

cubic_gens = {I, C4, C3, TR, PH}
cubic_group = qsymm.groups.generate_group(cubic_gens)
print(len(cubic_group))

Group elements can be multiplied and inverted, as we would expect:

C3 * C4

\[R\left(\pi, [1 1 0]\right)\]

C3**-1

\[R\left(-\frac{2\pi}{3}, [1 1 1]\right)\]

We can also apply group elements to the Model that we defined in the previous section:

H_with_TR = TR.apply(H)
H_with_TR.tosympy(nsimplify=True)

\[\begin{split}\displaystyle \left[\begin{matrix}- \alpha k_{z} + \frac{\hbar^{2} k_{x}^{2}}{2 m} + \frac{\hbar^{2} k_{y}^{2}}{2 m} + \frac{\hbar^{2} k_{z}^{2}}{2 m} & - \alpha k_{x} - i \alpha k_{y}\\- \alpha k_{x} + i \alpha k_{y} & \alpha k_{z} + \frac{\hbar^{2} k_{x}^{2}}{2 m} + \frac{\hbar^{2} k_{y}^{2}}{2 m} + \frac{\hbar^{2} k_{z}^{2}}{2 m}\end{matrix}\right]\end{split}\]

Defining continuous group generators¶

In addition to the group elements we can also define generators of continuous groups using qsymm.groups.ContinuousGroupGenerator:

sz = qsymm.ContinuousGroupGenerator(None, np.array([[1, 0], [0, -1]]))

The first argument to ContinuousGroupGenerator is the realspace rotation generator; by specifying None we indicate that we want the rotation part to be zero. The second argument is the unitary action of the generator on the Hilbert space as a Hermitian matrix.

Applying a ContinuousGroupGenerator to a Model calculates the commutator:

sz.apply(H).tosympy(nsimplify=True)

\[\begin{split}\displaystyle \left[\begin{matrix}0 & - 2 i \alpha k_{x} - 2 \alpha k_{y}\\2 i \alpha k_{x} - 2 \alpha k_{y} & 0\end{matrix}\right]\end{split}\]

For the 3D Rashba Hamiltonian we defined at the start of the tutorial spin-z is not conserved, hence the commutator is non-zero.

Finding symmetries¶

The function symmetries allows us to find the symmetries of a Model. Let us find out whether the 3D Rashba Hamiltonian defined earlier has cubic group symmetry:

discrete_symm, continuous_symm = qsymm.symmetries(H, cubic_group)
print(len(discrete_symm), len(continuous_symm))

48 0

It has 48 discrete symmetries (cubic group without inversion and time-reversal) and no continuous symmetries (conserved quantities).

For more detailed examples see Finding Symmetries and Finding symmetries of the Kekule-Y continuum model.

Generating Hamiltonians from symmetry constraints¶

The qsymm.hamiltonian_generator module contains algorithms for generating Hamiltonians from symmetry constraints.

For example let us generate all 2-band \(k \cdot p\) Hamiltonians with the same discrete symmetries as the Rashba Hamiltonian that we found in the previous section:

family = qsymm.continuum_hamiltonian(discrete_symm, dim=3, total_power=2, prettify=True)
qsymm.display_family(family)

\[\begin{split}\displaystyle \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]\end{split}\]

\[\begin{split}\displaystyle \left[\begin{matrix}k_{z} & k_{x} - i k_{y}\\k_{x} + i k_{y} & - k_{z}\end{matrix}\right]\end{split}\]

\[\begin{split}\displaystyle \left[\begin{matrix}k_{x}^{2} + k_{y}^{2} + k_{z}^{2} & 0\\0 & k_{x}^{2} + k_{y}^{2} + k_{z}^{2}\end{matrix}\right]\end{split}\]

It is exactly the Hamiltonian family we started with.

For more detailed examples see Generating k \cdot p models, Generating tight-binding models and Finding symmetries of the Kekule-Y continuum model.