from pythtb import TBModel, Lattice
import numpy as np
[docs]
def kane_mele(delta, t, soc, rashba) -> TBModel:
r"""Kane-Mele tight-binding model.
.. versionadded:: 2.0.0
This function creates a Kane-Mele tight-binding model with the specified
parameters. The model is defined on a 2D honeycomb lattice with two sublattices.
The lattice vectors are given by:
.. math::
\mathbf{a}_1 = a(1, 0), \quad \mathbf{a}_2 = a\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right),
and the orbital positions are given by:
.. math::
\mathbf{r}_1 = \frac{1}{3} \mathbf{a}_1 + \frac{1}{3} \mathbf{a}_2,
\quad \mathbf{r}_2 = \frac{2}{3} \mathbf{a}_1 + \frac{2}{3} \mathbf{a}_2
The Hamiltonian in second-quantized form is given by:
.. math::
H = \Delta \sum_{i} c_i^\dagger c_i +
t \sum_{\langle i,j \rangle} ( c_i^\dagger c_j + h.c.) +
\lambda_{SO} \sum_{\langle \langle i,j \rangle \rangle} ( c_i^\dagger \sigma_z c_j + \text{h.c.}) + \\
\lambda_{R} \sum_{\langle i,j \rangle} ( c_i^\dagger \mathbf{\sigma} \times
\mathbf{\hat{d}}_{\langle i,j \rangle} c_j + \text{h.c.})
Parameters
----------
onsite : float
On-site energy.
t : float, complex
Hopping parameter.
soc : float, complex
Spin-orbit coupling strength.
rashba : float, complex
Rashba coupling strength.
Returns
-------
TBModel
An instance of the model.
Notes
-----
The Kane-Mele model describes a two-dimensional topological insulator with spin-orbit coupling.
It is defined on a honeycomb lattice and includes both intrinsic and Rashba spin-orbit coupling [kane-mele]_.
References
----------
.. [kane-mele] Kane, C. L., & Mele, E. J. (2005). Quantum Spin Hall Effect in Graphene. *Physical Review Letters*, 95(22), 226801.
"""
# define lattice vectors
lat_vecs = [[1, 0], [1 / 2, np.sqrt(3) / 2]]
# define coordinates of orbitals
orb_vecs = [[1 / 3, 1 / 3], [2 / 3, 2 / 3]]
lat = Lattice(lat_vecs, orb_vecs, periodic_dirs=[0, 1])
# make two dimensional tight-binding Kane-Mele model
ret_model = TBModel(lattice=lat, spinful=True)
# set on-site energies
ret_model.set_onsite([delta, -delta])
# useful definitions
sigma_x = np.array([0, 1, 0, 0])
sigma_y = np.array([0, 0, 1, 0])
sigma_z = np.array([0, 0, 0, 1])
# set hoppings (one for each connected pair of orbitals)
# (amplitude, i, j, [lattice vector to cell containing j])
# spin-independent first-neighbor hoppings
ret_model.set_hop(t, 0, 1, [0, 0])
ret_model.set_hop(t, 0, 1, [0, -1])
ret_model.set_hop(t, 0, 1, [-1, 0])
# second-neighbour spin-orbit hoppings (s_z)
nnn_hop = 1j * soc * sigma_z
ret_model.set_hop(-nnn_hop, 0, 0, [0, 1])
ret_model.set_hop(nnn_hop, 0, 0, [1, 0])
ret_model.set_hop(-nnn_hop, 0, 0, [1, -1])
ret_model.set_hop(nnn_hop, 1, 1, [0, 1])
ret_model.set_hop(-nnn_hop, 1, 1, [1, 0])
ret_model.set_hop(nnn_hop, 1, 1, [1, -1])
# Rashba first-neighbor hoppings: (s_x)(dy)-(s_y)(d_x)
# bond unit vectors are (np.sqrt(3) / 2, 1/2) then (0,-1) then (-np.sqrt(3) / 2, 1/2)
ret_model.set_hop(
1j * rashba * ((1 / 2) * sigma_x - (np.sqrt(3) / 2) * sigma_y),
0,
1,
[0, 0],
mode="add",
)
ret_model.set_hop(1j * rashba * -sigma_x, 0, 1, [0, -1], mode="add")
ret_model.set_hop(
1j * rashba * ((1 / 2) * sigma_x + (np.sqrt(3) / 2) * sigma_y),
0,
1,
[-1, 0],
mode="add",
)
return ret_model
