WOLFRAM|DEMONSTRATIONS PROJECT

Wannier Representation for Tight-Binding Hamiltonian of a Periodic Chain with N Sites

​
N sites
1
ϵ
t
Hamiltonian Matrix and Discrete Fourier Transform Intensity Pattern
This Demonstration shows the construction of the tight-binding Hamiltonian matrix for a periodic chain with
N
sites within the Wannier representation. The Hamiltonian in second quantization form is given by
H=ϵ
N
∑
i=1
†
c
i
c
i
+t
N
∑
i=1

†
c
i
c
i±1
+
†
c
i±1
c
i

, where
†
c
i
and
c
i
are the fermionic creation and destruction operators of electrons at each site
i
, respectively. Periodic boundary conditions at chain ends are expressed as
†
c
N+1
=
†
c
1
and
c
N+1
=
c
1
. The tight-binding on-site energy parameter ϵ gives the on-diagonal matrix elements, the hopping parameter
t
gives the off-diagonal matrix elements. Both
ϵ
and
t
are expressed in electron-volts. This representation, unlike the reciprocal space-based Bloch representation, works in real space. However, physically, it is fully equivalent, since with
N
sites one can sample
N
k
-points in the reciprocal space of the first Brillouin zone (BZ). Thus the same energy eigenvalues are expected from exact diagonalization of the Hamiltonian matrix. The information about the
κ
quantum numbers (
κ=0,1,…,N-1
or equivalently
κ=-N/2+1,…,N/2
in the reduced BZ scheme) and the related
k
-points (
k=2πκ/(Na)
with
a
lattice parameter of the chain) can be extracted by performing a discrete Fourier transform on each of the obtained eigenvectors and subsequently by inspecting the frequency components with nonzero intensity. The electronic energy eigenvalues associated to the
k
-points thus obtained are plotted and superimposed onto the analytical Bloch dispersion relation
E(k)=ϵ+2tcos(ka)
in order to show the full equivalence of the Wannier result with the one for the reciprocal space.