Hubbard Model Interactive Calculator for 1D Systems

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〈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.,

).