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Hubbard Model Interactive Calculator for 1D Systems

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., ).

SNAPSHOTS

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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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