PhD Thesis - Universität Augsburg

12 1. Models
elements t bb′
ij
are expressed as
1 ∑
ij = −δ bb ′ e −i(R j−R i )·k ǫ
L
kb
. (1.17)
t bb′
k∈BZ
Thus, the Hamiltonian (1.2) describing electrons on a lattice reads
H (e)
tot = − ∑
t b ij c† ibσ c jbσ +
∑
i,j,b,σ
i,j,l,m
b,b ′ ,b ′′ ,b
σσ ′ V bb′ b ′′ b ′′′
ijlmσσ ′ c† ibσ c† jb ′ σ
c ′ lb ′′ σ
c ′ mb ′′′ σ
(1.18)
′′′
Unfortunately, even after restricting the problem of a quantum-mechanical description of
a solid to a description of its electronic properties and making all these assumptions and
approximations, the obtained Hamiltonian describes a many-particle problem that is still
technically not tractable in most cases. Therefore further approximations are needed to
reduce its complexity, which will be described in the following section.
1.2 Hubbard model
It follows from Eq. (1.17) that the position of the bands relative to each other is determined
by the matrix elements t b ii . For bands that lie energetically far away from the Fermi
energy E F
, V bb′ b ′′ b ′′′
ijlmσσ /(E ′ F − tb ii) should be a small parameter, so that the electron-electron
interaction between these bands may be neglected. In case there is only one band b F
near
E F
, it may be justified to omit the band indices and focus only on the band that is closest
to the Fermi energy. This assumption leads to an effective one-band model.
The second approximation is to take into account only the maximum term in the
Coulomb interaction. Since the Coulomb interaction decreases as 1/r with distance r, the
maximum term is the local interaction of two electrons residing in the same orbital. For a
single band, the local Coulomb interaction matrix element (“Hubbard U”) reads
∫
U = 2V b Fb F b F b F
iiii↑↓
= e 2 d 3 r d 3 r ′ |ψ i↑b F
(r)| 2 |ψ i↓bF (r ′ )| 2
. (1.19)
|r − r ′ |
The omission of all other contributions of the electron-electron interaction is motivated
by strong screening of the electron-electron interaction, so that the effective interaction
between the electrons is not really long-range and decays stronger than 1/r. This is justified
if the band at the Fermi energy is only partially filled, i.e., if the Fermi energy lies in the
band, as it is the case for metals. However, since the Hubbard model [115, 116, 117,
118] is the conceptually simplest model incorporating the full many-body electron-electron
interaction, from a pragmatic point of view it is justified to use it also to study insulating
systems, if one does not expect a fundamental change of the physics by incorporating