from pythtb import TBModel, Lattice
import numpy as np
[docs]
def haldane(delta: float, t1: float, t2: float, phi: float = np.pi / 2) -> TBModel:
r"""Haldane tight-binding model.
.. versionadded:: 2.0.0
This function creates a Haldane tight-binding model with the specified
hopping parameters and on-site energy. The model is defined on a 2D honeycomb
lattice with two sublattices. The lattice vectors are given by,
.. math::
\mathbf{a}_1 = (1, 0), \quad \mathbf{a}_2 = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)
and the orbital positions are given by,
.. math::
\mathbf{\tau}_1 = \frac{1}{3} \mathbf{a}_1 + \frac{1}{3} \mathbf{a}_2,
\quad \mathbf{\tau}_2 = \frac{2}{3} \mathbf{a}_1 + \frac{2}{3} \mathbf{a}_2
The second-quantized Hamiltonian can be written as:
.. math::
H = \Delta \sum_i (-)^i c_i^\dagger c_i + t_1 \sum_{\langle i,j \rangle} (c_i^\dagger c_j
+ \text{h.c.}) + t_2 \sum_{\langle\langle i,j \rangle\rangle} (ic_i^\dagger c_j + \text{h.c.})
Parameters
----------
delta : float
Onsite mass term. Opposite sign for the two sublattices.
t1 : float
Nearest neighbor hopping amplitude.
t2 : float
Next-nearest neighbor hopping amplitude. Peierls phase is included.
Returns
-------
TBModel
An instance of the model.
Notes
-----
The Haldane model describes a two-dimensional topological insulator with a
non-trivial band structure. It is characterized by a finite Chern number
and exhibits edge states that are protected by time-reversal symmetry [haldane]_.
References
----------
.. [haldane] Haldane, F. D. M. (1988).
O(3) Nonlinear :math:`\sigma` Model and the Quantum Hall Effect in Two Dimensions.
*Physical Review Letters*, 61(20), 2015–2018.
"""
lat_vecs = [[1, 0], [1 / 2, np.sqrt(3) / 2]]
orb_vecs = [[1 / 3, 1 / 3], [2 / 3, 2 / 3]]
lat = Lattice(lat_vecs, orb_vecs, periodic_dirs=[0, 1])
model = TBModel(lattice=lat, spinful=False)
model.set_onsite([-delta, delta], mode="set")
for lvec in ([0, 0], [-1, 0], [0, -1]):
model.set_hop(t1, 0, 1, lvec, mode="set")
for lvec in ([1, 0], [-1, 1], [0, -1]):
model.set_hop(t2 * np.exp(1j * phi), 0, 0, lvec, mode="set")
for lvec in ([-1, 0], [1, -1], [0, 1]):
model.set_hop(t2 * np.exp(1j * phi), 1, 1, lvec, mode="set")
return model
