Phy5670/HubbardModel

The Hubbard Model

Introduction

The Hubbard Model is the simplest model used to describe interacting particles on a lattice. Proposed in 1963 by John Hubbard, it consists of a seemingly small correction to the tight-binding model of electrons on a lattice. In Hubbard's model, a term is included to account for the interaction energy between particles on a the same lattice site. The Hamiltonian is written thusly:

$\mathbb {H} =-2t\sum _{\langle {\vec {r}},{\vec {r'}}\rangle }{({\hat {c}}_{{\vec {r}},\sigma }^{\dagger }{\hat {c}}_{{\vec {r'}},\sigma }+{\hat {c}}_{{\vec {r'}}\sigma }^{\dagger }{\hat {c}}_{{\vec {r}}\sigma })}+U\sum _{\vec {r}}{{\hat {n}}_{{\vec {r}},\sigma '}{\hat {n}}_{{\vec {r}},\sigma }}$

The operator ${\hat {c}}_{{\vec {r}},\sigma }^{\dagger }$ creates a particle of spin $\sigma$ at lattice position ${\vec {r}}$. Note that the sum over possible spins is implied. The summation in the kinetic energy (tight-binding) term is carried out over all nearest-neighbor bonds, or over all pairs $({\vec {r}},{\vec {r}}+{\vec {\delta }})$ where ${\vec {\delta }}$ runs over half of the nearest-neighbor vectors. (For the square lattice, ${\vec {\delta }}\in \{+a{\hat {x}},+a{\hat {y}}\}$. If the other two nn's were included, the sum as written would overcount the bonds.) The hopping integral $t$ is the familiar term from the tight-binding model, and may in general depend on ${\vec {\delta }}$.

There exist multiple variations of this Hamiltonian, such as the Extended Hubbard Model, which takes into account the Coulomb interaction between particles on distant sites, with a term like

$\mathbb {H} _{ex}=\sum _{{\vec {r}}\neq {\vec {r'}}}{{\hat {n}}_{{\vec {r}},\sigma }{\hat {n}}_{{\vec {r'}},\sigma '}}U({\vec {r}}-{\vec {r'}})$

As the length scales involved are can be on the order of a few angstroms, this interaction is not entirely trivial, and must be included to carefully look for charge density fluctuations. The undecorated model has proven complicated enough to resist complete solution, however, and will remain the main focus of this article.

For the last 40 years or so, this model has been studied intensively in hopes of gaining insight into the problem of interacting electrons, which is needed to better understand many strange properties of solids, such as superconductivity. The Hubbard Model also successfully fulfilled the prediction of Sir Neville Mott that the strange behavior of the class of materials known as Mott Insulators was due to electron-electron interactions [ref]. These materials were expected to be conductors according to conventional band theory (which does not account for interactions), but were found instead to be insulating.

Symmetry in the Hubbard Model

It is also possible to write this Hamiltonian in such a way that the spin symmetry of the system becomes more apparent. (FOLLOW THE NOTES IN CLASS HERE!!) Conserved Quantities! (AGAIN, CLASS NOTES...) Charge and Spin! Two exitations: Spinons and Holons!

The One-Dimensional Hubbard Model

Main focus, since cannot be solved exactly in Higher D. Solution by Bethe Ansatz - Lieb and Wu. Thermodynamic Properties - Use the review by L & W! Approximations! Perhaps use the class notes? Maybe RPA? From the paper. MonteCarlo FTW! Reduced Dimensionality? Electrons cannot pass each other!? Broken Symmetry? Non! This results in luttinger liquid? But... there are excitations. Spinons and Holons.

Higher Dimensions

Broken Symmetry Effects? Magnetic Properties! - Mahan! Short discussion of usefulness of numerical results/methods - Manousakis Review Paper?

Related Physical Systems

Mott Insulators Ultra-Cold atoms.