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 in a Wannier state 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!

Also, particle-hole symmetry of the Hamiltonian.

The One-Dimensional Hubbard Model

It is a frustrating feature of this problem that it seems insoluble (though it only seems so, this has not been proven) in more than one dimension. In a single dimension, however, it is possible to find exactly the ground state energy and wavefunction. The simplest excitations of the one-dimensional system can also be described, which in turn leads to an approximate description of the thermodynamics of the model.

The solution was obtained in 1968 by Eliot Lieb and Fa-Yueh Wu, using the method of Bethe Ansatz.

To begin, the problem most simply consists of solving the Schrodinger equation:

${\hat {\mathbb {H} }}|M,M'\rangle =E|M,M'\rangle$

Where $(M,M')$ gives the number of spin down and spin up particles, respectively. Thanks to the particle-hole symmetry of the Hamiltonian, we need only consider states in which the total number of particles, $N$ is less than or equal to the number of lattice sites, $N_{l}$. Additionally, thanks to the spin-symmetry, it is only necessary to consider solutions for $M\leq M'$.

Now, writing the state as:

$|M,M'\rangle =\sum _{1\leq x_{j}\leq N_{a}}{f(x_{1}...x_{N})|x_{1}...x_{N}\rangle }$

The above state is written so that the spin down electrons are at sites $\{x_{1}...x_{M}\}$ and the spin up electrons are at sites $\{x_{M+1}...x_{N}\}$.

Applying this to the Schrodinger equation gives the following:

$-2t\sum _{i=1}^{N}{f(x_{1}...x_{i+1}...x_{N})+f(x_{1}...x_{i-1}...x_{N})}+U\sum _{i<j}{\delta (x_{i}-x_{j})}f(x_{1}...x_{N})=Ef(x_{1}...x_{N})$

Now, the Bethe Ansatz consists of properly choosing the form of $f(x_{1}...x_{N})$.

$f(x_{1}...x_{N})=\sum _{P}{[Q,P]e^{i\sum _{j=1}^{N}{k_{Pj}x_{Qj}}}}$

$P$ and $Q$ are two permutations of sets of the numbers $(1...N)$, and the $\{k_{j}\}$ is a set of $N$ unequal numbers. $[Q,P]$ represents a set of $N!\times N!$ coefficients to be determined. Determination of these coefficients and the $\{k_{j}\}$ will give the final wavefunction.

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.