This Demonstration shows the basics of the fermion Hubbard model applied to one-dimensional systems such as chains with

sites with either open or closed boundary conditions and any phase

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:

, where

and

are site indices,

are all pairs of first nearest neighbors sites,

gives the number of electrons on site

with spin

,

and

are electron creation and annihilation operators, respectively, and

and

are positive interaction constants, respectively. The

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

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

, where

is the lattice parameter of the chain and

is the Planck constant (since these are normalization constants, they have been set equal to 1 in the program). The term

represents the current operator, which is relevant for investigating optical conductivity properties.

This Demonstration lets you interactively calculate either the Hubbard

or velocity matrix elements

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

and the target

states. This is also displayed in the respective graphics together with the chosen boundary conditions (either open or closed chain).

For example, if

and

have the same occupation pattern with only doubly occupied sites, the Hamiltonian matrix element is a multiple of

(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,

is equal to

(e.g.,

). This holds also in the case of hopping between chain ends, when periodic boundary conditions (up to a given phase factor

) are considered (e.g.,

).