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Hubbard Model Interactive Calculator for 1D Systems

​
number of sites
2
3
4
5
6
7
8
9
10
quantum operator
O
Hamiltonian
velocity
source state
i 〉
spin up ↑
spin down ↓
target state
〈 j
spin up ↑
spin down ↓
boundary conditions
open
closed
Contents cannot be rendered at this time; please try again later
〈j
H
i〉
=
0
This Demonstration shows the basics of the fermion Hubbard model applied to one-dimensional systems such as chains with
N
sites with either open or closed boundary conditions and any phase
ϕ
at the ends. The Hubbard model provides a very simple but physically powerful description of electronic many-body effects in quantum mechanical systems. This can be understood by considering the form of its Hamiltonian, expressed in second quantization formalism:
H=-t
∑
<i,j>,σ
†
c
i,σ
c
j,σ
+
†
c
j,σ
c
i,σ
+U
∑
i
n
i,
n
i,
, where
i
and
j
are site indices,
<i,j>
are all pairs of first nearest neighbors sites,
n
i,σ
=
†
c
i,σ
c
i,σ
gives the number of electrons on site
i
with spin
σ
,
†
c
i,σ
and
c
i,σ
are electron creation and annihilation operators, respectively, and
t
and
U
are positive interaction constants, respectively. The
t
term is a single-particle (tight-binding) term (thus it contains no many-body features) and describes the hopping of electrons localized on atomic-like orbitals between nearest neighbor sites and models the kinetic energy of the system. The
U
term gives a potential energy contribution to the Hamiltonian and models the Coulomb repulsion between two electrons with opposite spin in the same orbital, hence it is the effective many-body (two-body) term. Another related operator is the velocity operator
v=-
iat
ℏ
∑
<i,j>,σ
†
c
i,σ
c
j,σ
-
†
c
j,σ
c
i,σ
, where
a
is the lattice parameter of the chain and
ℏ
is the Planck constant (since these are normalization constants, they have been set equal to 1 in the program). The term
∑
<i,j>,σ
†
c
i,σ
c
j,σ
-
†
c
j,σ
c
i,σ
represents the current operator, which is relevant for investigating optical conductivity properties.
This Demonstration lets you interactively calculate either the Hubbard
<m|H|l>
or velocity matrix elements
<m|V|l>
between any two many-electron states defined in occupation number formalism. By selecting the checkboxes in the respective panels you can set up the desired electronic occupation configuration for the source
|l>
and the target
<m|
states. This is also displayed in the respective graphics together with the chosen boundary conditions (either open or closed chain).
For example, if
|l>
and
<m|
have the same occupation pattern with only doubly occupied sites, the Hamiltonian matrix element is a multiple of
U
(e.g.,
), whereas the velocity matrix element is zero (no fermionic current). If they differ by a hopping of one electron to the nearest neighbor site,
<m|H|l>
is equal to
t
(e.g.,
). This holds also in the case of hopping between chain ends, when periodic boundary conditions (up to a given phase factor
exp(iϕ)
) are considered (e.g.,
).
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