import numpy as np
from pythtb import TBModel, Lattice
[docs]
def fu_kane_mele(t, soc, dt=[0, 0, 0, 0]):
r"""Fu-Kane-Mele tight-binding model.
This function creates a Fu-Kane-Mele tight-binding model on a diamond
lattice. The lattice vectors are given by,
.. math::
\mathbf{a}_1 = (0, 1, 1), \quad \mathbf{a}_2 = (1, 0, 1),
\quad \mathbf{a}_3 = (1, 1, 0)
and the orbital positions are given by,
.. math::
\mathbf{\tau}_1 = (0, 0, 0),
\quad \mathbf{\tau}_2 = \frac{1}{4} \mathbf{a}_1 + \frac{1}{4} \mathbf{a}_2
+ \frac{1}{4} \mathbf{a}_3
The second-quantized Hamiltonian can be written as:
.. math::
H = t \sum_{\langle ij \rangle} c_i^{\dagger} c_j
+ i \lambda_{SO} \sum_{\langle\langle ij \rangle\rangle} c_i^{\dagger}
\vec{\sigma} \cdot (\mathbf{d}_{ij}^{1} \times \mathbf{d}_{ij}^{2}) c_j
where the first term is a nearest-neighbor hopping term connecting the two fcc sublattices
of the diamond lattice, and the second term is a spin-orbit coupling term connecting
second-neighbor sites within the same sublattice. Here, :math:`\mathbf{d}_{ij}^{1,2}`
are the two nearest-neighbor bond vectors connecting sites :math:`i` and :math:`j`.
Due to inversion symmetry, each band is doubly degenerate. The degeneracy is lifted by symmetry
lowering perturbations of the four nearest-neighbor hoppings :math:`t \rightarrow t + \delta t_p`
with :math:`p = 1, 2, 3, 4` indexing the four bonds connected to each site.
.. versionadded:: 2.0.0
Parameters
----------
t : float
Spin-independent nearest-neighbor hopping amplitude.
soc : float
Spin-orbit coupling strength. Modulates next-nearest neighbor
hopping amplitudes.
dt : list[float, float, float, float], optional
Offsets added to the four nearest-neighbor hoppings along the
bonds connected to each site. The entries are applied in the
following order:
- `dt[0]` : bond along ``R = [0, 0, 0]``
- `dt[1]` : bond along ``R = [-1, 0, 0]``
- `dt[2]` : bond along ``R = [0, -1, 0]``
- `dt[3]` : bond along ``R = [0, 0, -1]``
The default is ``[0, 0, 0, 0]``, which corresponds to uniform
hopping amplitudes. This parameter allows for symmetry-lowering
perturbations to the nearest-neighbor hoppings.
Returns
-------
TBModel
An instance of the model.
Notes
-----
- The Fu-Kane-Mele model describes a three-dimensional topological insulator with a
non-trivial band structure. It is characterized by a strong :math:`\mathbb{Z}_2` invariant
and exhibits surface Dirac cones that are protected by time-reversal and inversion
symmetry [1]_.
References
----------
.. [1] \ L. Fu, C. L. Kane, and E. J. Mele, *Phys. Rev. Lett.*, **98**, 106803
(2007).
"""
lat_vecs = [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
orb_vecs = [[0, 0, 0], [0.25, 0.25, 0.25]]
lat = Lattice(lat_vecs, orb_vecs, periodic_dirs=[0, 1, 2])
model = TBModel(lattice=lat, spinful=True)
# spin-independent first-neighbor hops
for idx, lvec in enumerate([[0, 0, 0], [-1, 0, 0], [0, -1, 0], [0, 0, -1]]):
model.set_hop(t + dt[idx], 0, 1, lvec)
# spin-dependent second-neighbor hops
lvec_list = ([1, 0, 0], [0, 1, 0], [0, 0, 1], [-1, 1, 0], [0, -1, 1], [1, 0, -1])
dir_list = ([0, 1, -1], [-1, 0, 1], [1, -1, 0], [1, 1, 0], [0, 1, 1], [1, 0, 1])
for j in range(6):
spin = np.array([0.0] + dir_list[j])
model.set_hop(1j * soc * spin, 0, 0, lvec_list[j])
model.set_hop(-1j * soc * spin, 1, 1, lvec_list[j])
return model
