Download 1D LATTICE GREEN`S FUNCTIONS Some preliminaries

1
1D LATTICE GREEN'S FUNCTIONS
Some preliminaries
Green's functions arise as useful tools for the solution of operator equations. We shall consider the eigenequation
commonly referred to in physics as Schrödinger's equation:
E Iˆ − Ĥ |ψi = 0,
(1)
here Ĥ is a hermitian operator, or as we usually call it the Hamiltonian. Note that depending on the problem at
hand this operator can be a dierential operator or a matrix operator. Later on we shall consider a concrete system
and write down an explicit form for it, but let us remain in the abstract form for a little bit longer.
The Green's equation dening the Green's operator Ĝ, or more commonly referred to as Green's function, associated
to the Schrödinger's equation above is
ˆ
z Iˆ − Ĥ Ĝ = I.
(2)
Here z is an arbitrary complex number. Formally by inverting z Iˆ − Ĥ we can obtain the Green's function simply
as
−1
Ĝ = z Iˆ − Ĥ
.
(3)
If the eigensystem of the Hamiltonian is known, that is we have obtained eigenvectors|ni and the corresponding
eigenvalues En satisfying
Ĥ |ni = En |ni ,
(4)
X
Ĝ = f Ĥ =
f (En ) |ni hn| ,
(5)
We can express the Green's function as
n
=
X |ni hn|
.
z − En
n
(6)
The above equation tells us that if we want to evaluate the Green's function exactly at one of the (real) eigenenerges
of the system we have to be careful. To illustrate what sort of problems we might face let us imagine a pathologically
simple system where the Hilbert space consists only of a single state: |0i. The Hamiltonian in this case is also
embarrassingly simple:
Ĥ1site = ε0 |0i h0| .
(7)
Hence the Green's function is simply
Ĝ1site =
|0i h0|
.
z − ε0
(8)
Now recall the mathematical identity
1
=P
δ→0 E ± iδ − ε0
lim
1
E − ε0
1
∓ i δ (E − ε0 ) .
π
(9)
This means that if the Green's function has a dierent limiting behavior for E = ε0 if we are approaching it from the
positive or negative imaginary direction! The two limits are referred to as retarded (ĜR ) and advanced (ĜA ) Green's
functions, that is
h
i−1
ĜR/A = lim (E ± iδ) Iˆ − Ĥ
.
δ→0
(10)
The name convention, as we shall elaborate on later in a bit more in detail, comes from the fact that retarded Green's
functions describe situations where we describe the response of a system to a perturbation in a causal manner. For
example a localized perturbation will give a response that propagates away from it, like dropping a stone in a pond
and in response waves rush towards the shore from the point of the impact. While advanced Green's functions describe
the time reversed situation where the waves, in anticipation of the stone jumping out of the pond run towards the
point where it will emerge.
2
The one dimensional chain
We shall now introduce a basis of localized and orthogonal states associated to sites of a simple one dimensional
lattice denoted by |rp i. This basis element of the Hilbert space corresponds to a localized site and we assume it is
orthogonal to other lattice sites that is
(11)
hrp |rq i = δpq .
The Hamiltonian that we shall consider in these following sections describes particles hopping along this one dimensional lattice:
Ĥ = γ
X
(12)
[|rn+1 i hrn | + |rn i hrn+1 |] .
n
Sometimes it is customary to include an on-site term ε0 |rn i hrn | to all constituent sites of the lattice but since it
only serves to dene the origin of the energy we can safely assume it to be zero. The matrix elements of the above
Hamiltonian
(13)
hrp | Ĥ |rq i = γ (δp+1,q + δp,q+1 )
describe a tridiagonal matrix with γ on the rst upper and lower o-diagonal. One usually obtains the spectrum of
this Hamiltonian by guessing the eigenstates. The ansatz we shall take is that of plane-waves indexed by wavenumber
k:
|Ek i =
X
e kn |rn i .
(14)
i
n
Multiplying the Schrödinger's equation
(15)
Ĥ |Ek i = Ek |Ek i
from the left with hrp | we get
hrp | γ
X
[|rm+1 i hrm | + |rm i hrm+1 |]
m
X
e kn |rn i = hrp | Ek
i
n
X
e kn |rn i
i
(16)
n
after some straight forward algebraic steps
γ
X
hrp | [|rm+1 i hrm | + |rm i hrm+1 |] eikn |rn i = Ek eikp
(17)
mn
γ
X
h
i
hrp | eik(m−1) + eik(m+1) |rm i = Ek eikp
(18)
h
i
γ eik(p−1) + eik(p+1) = Ek eikp
(19)
m
we get the spectrum, or as it is also called the dispersion relation:
(20)
Ek = 2γ cos (k)
It is common practice to chose γ to be a negative number and if there is no other energy scale in the system one
might as well chose it to be γ = −1. The consequence of this choice is that the group velocity
(21)
vk = ∂k Ek = −2γ sin (k)
will be positive for k > 0 and negative at k < 0. As the spectrum is symmetric with respect to k at each energy we
will have two states, one at k and one at −k. Note that this is true even at energies |E| > |2γ| with the caveat that
in this energy range the dispersion relation is only satised by imaginary k-s!
The central quantity of this section is the calculation of the matrix elements of the Green's function between two
real space points, that is the quantity
Gpq (z) = hrp | Ĝ(z) |rq i = hrp |
X |Ek i hEk |
k
z − Ek
|rq i
(22)
3
The more rigorous way
The denition of the Green's function for a 1D chain with nearest neighbor hopping γ = −1 between sites p and q :
ˆ
π
dk exp(ik|p − q|)
−π
2π z + 2 cos(k)
Gpq (z) =
.
(23)
Since the spectrum is symmetric in k only the symmetric portion of the numerator matters, hence the did we take
|p − q| instead of just p − q ,thus making the above expression symmetric also in p and q . Transforming the integral
from the Brillouin zone to a complex unit circle with the parametrization:
w = eik = cos(k) + i sin(k),
dw
= ieik = iw,
dk
dw
dk =
,
iw
cos(k) = (w + w−1 )/2,
we have
˛
dw
w|p−q|
2π iw(z + w + w−1 )
˛
1
w|p−q|
.
=
dw 2
2π i
w + wz + 1
Gpq (z) =
(24)
(25)
(26)
(27)
(28)
(29)
The roots of the denominator are
p
xin,out = −z/2 ± sgn(Re(z)) z 2 /4 − 1
(30)
where xin /xout is the root inside/outside of the unit circle for any z . Cauchy theorem
f (a) =
implies
1
Gpq (z) =
2π i
1
2π i
˛
˛
dw
f (z)
dz,
(z − a)1
w|p−q|
(w − xin )(w − xout )
|p−q|
=
xin
,
xin − xout
(31)
(32)
(33)
hence the Green's function is given
|p−q|
p
−z/2 + sgn(Re(z)) z 2 /4 − 1
p
.
Gpq (z) =
2sgn(Re(z)) z 2 /4 − 1
(34)
lim Gpq (ω ± δ) = GR/A
pq (ω),
(35)
The side limits
δ→0
are called the retarded and advanced Green's functions.