Defining custom models

The library has been built with a particular focus on Slater-Koster models. However, sometimes it is useful to define instead a specific model instead of using a Slater-Koster parametrization. For instance, we may have a toy model whose band structure can be obtained analitically, or the Bloch Hamiltonian can be written directly. In those cases, it is more convenient to hard-code directly the Bloch Hamiltonian instead of constructing it following the SK approach. The library currently implements several models, which can be seen in the tightbinder.models section. Here, we show how to define one of this models.

Any new model has to extend the tightbinder.system.System class. Then, the minimum requirements to be able to instanciate the new class is to implement the constructor, __init__() and the Bloch Hamiltonian via hamiltonian_k(). For instance, next we show the implementation of the BHZ model, which is a toy model known to describe time-reversal topological insulators [Bernevig2013]

from tightbinder.system import System, FrozenClass
import numpy as np

class BHZ(System, FrozenClass):

    def __init__(self, g, u, c):
        super().__init__(system_name="BHZ model",
                        crystal=Crystal([[1, 0, 0], [0, 1, 0]],
                                        motif=[[0, 0, 0, 0]]))
        self.norbitals = [4]
        self.filling = 2
        self.basisdim = self.norbitals[0]
        self.g = g
        self.u = u
        self.c = c

        self._freeze()

    def hamiltonian_k(self, k):

        sigma_x = np.array([[0, 1],
                            [1, 0]], dtype=np.complex_)
        sigma_y = np.array([[0, -1j],
                            [1j, 0]], dtype=np.complex_)
        sigma_z = np.array([[1, 0],
                            [0, -1]], dtype=np.complex_)

        coupling = self.c * sigma_y

        id2 = np.eye(2)
        kx, ky = k[0], k[1]

        hamiltonian = (np.kron(id2, (self.u + np.cos(kx) + np.cos(ky)) * sigma_z + np.sin(ky) * sigma_y) +
                    np.kron(sigma_z, np.sin(kx) * sigma_x) + np.kron(sigma_x, coupling) +
                    self.g * np.kron(sigma_z, sigma_y) * (np.cos(kx) + np.cos(7 * ky) - 2))

        return hamiltonian

Note that the attributes of the model have to comply with those of tightbinder.system.System, so we refer to the documentation for details on the expected values. This is the very minimum needed to define a new model, but one can also define more complex models where the Bloch Hamiltonian is not known from the beginning, but has to be built with a call to tightbinder.system.System.initialize_hamiltonian(). This is useful if as with a SK model, before getting the spectrum we want to modify the unit cell in different ways.

[Bernevig2013]

Bernevig, B. A. (2013). Topological insulators and topological superconductors. Princeton university press.