Download 1.1 Construction of two band models

1.1 Construction of two band models
Tuesday, July 28, 2015
9:42 AM
We first illustrate the form of our two band models and how to construct it from a simple square lattice with ,
and orbitals. The two band model takes the following form
where
. is the 2 by 2 Pauli matrix and the basis for this matrix depends on
the details of the problem and will be illustrated below for our simple case. For the continuous limit (
), the
above Hamiltonian can be simplified as
This is nothing but Dirac Hamiltonian in the 2+1D, but with a momentum dependent mass
Now let's consider a square lattice with ,
the figure below.
and
.
orbitals. The lattice, as well as orbitals on each site, is shown in
Figure 1.1.1: A tight-binding model for a square lattice with both s and p orbitals.
The tight-binding Hamiltonian of this system can be written as
where is for sites,
parameters for
is for the distance of hopping,
:
With the Fourier transform
are for orbitals (
and
where
For our case, we have
Or in a matrix form, the Hamiltonian is written as
Quantum anomalous Hall effe Page 1
). Here we consider the following
, we have
Or in a matrix form, the Hamiltonian is written as
For this Hamiltonian, we can see that linear coupling appears naturally between and
orbitals. However,
and orbitals are degenerate with each other at
and there are three bands appearing in low energy
physics for this system. To remove this degeneracy, we consider a coupling to the angular momentum of
orbitals. We may consider the superposition of these two orbitals,
which
possess the angular momentum
along the z direction. Here we have set to be . Let's denote the angular
momentum operator along the z direction to be and we introduce a Zeeman type of Hamiltonian for the
coupling
on the basis
. We can also transform the Hamiltonian
and the corresponding Hamiltonian is written as
at
The above Hamiltonian is degenerate when
two orbitals. Let's consider the limit
the low energy physics is dominated by
Hamiltonian is written as
from the basis
for
and
orbitals. Thus, we can neglect
to the basis
orbitals, but
will split these
, in which
orbital and the total
This recover the Hamiltonian for the two band model except that we need to redefine the Pauli matrices and
neglect the identity matrix term. One can further show that in the above limit, the contribution from the
orbital can be included in the second order perturbation theory, which is not essential. The above model can
be realized in the optical lattice of cold atom systems and the Hamiltonian
can be generated by rotating
each lattice site by a laser. See the following references for more details.
Congjun Wu, Phys. Rev. Lett. 101, 186807 (2008).
W Zheng, H. Zhai, Phys. Rev. A 89, 061603(R) (2015) .
Before we go to the detailed discussion of chiral edge mode and bulk topology of the above model, we first take
a look at the bulk energy dispersion. The two band model can be written as
where
energy is given by
and
, or explicitly,
. The eigen
. Again let's consider the
limit
, in which the energy dispersion is
. For this energy dispersion, energy gap
closes when
, corresponding to a topological phase transition between a normal insulator with
and
a so-called quantum anomalous Hall insulator ( or Chern insulator) with
. Here we always assume
.
Intuitively, this transition is also called "band inversion", as illustrated in the figure below. The concept of band
inversion plays a central role in searching for topological materials. We will show explicitly how band inversion
determines topological structures of electronic states later.
Quantum anomalous Hall effe Page 2
Figure 1.1.2: Band inversion.
Problems:
1. Complete the derivation of the tight-binding model of sp orbitals in the momentum space.
2. Complete the derivation of effective two band model from our three orbital model up to the second order terms
with the perturbation theory.
Quantum anomalous Hall effe Page 3