# Hubbard Model Interactive Calculator for 1D Systems

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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.

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Contributed by: Jessica Alfonsi (University of Padova, Italy) (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: Hamiltonian matrix element between two equal states with one doubly occupied site

Snapshot 2: velocity matrix element between a doubly occupied state and a state with one electron on each site

Snapshot 3: velocity matrix element between two equal states with one doubly occupied site

Part of the initialization is taken from the following book: W. Kinzel and G. Reents, *Physics by Computer: Programming Physical Problems Using Mathematica and C*, New York: Springer, 1998.

H. Q. Lin and J. E. Gubernatis, "Exact Diagonalization Method for Quantum Systems," *Computers in Physics *7(4), 1993 p. 400.

J. Alfonsi, "Small Crystal Models for the Electronic Properties of Carbon Nanotubes," Ph.D. thesis, University of Padova, 2009, Chapter 6 and references therein.

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