Hubbard Model

1. The Hubbard Model

The Hubbard model was introduced by John Hubbard in 1963 to describe the behavior of electrons in a lattice, especially their interaction and collective motion. It is a cornerstone in the study of strongly correlated electron systems.

The model captures two key physical effects: the kinetic energy of electrons hopping between sites and the on-site Coulomb repulsion when two electrons occupy the same site.

\[ \hat{H} = -t \sum_{\langle i,j \rangle, \sigma} c^\dagger_{i\sigma} c_{j\sigma} + U \sum_i n_{i\uparrow} n_{i\downarrow} \]

With:

\( t \): Hopping parameter (kinetic term)
\( U \): On-site Coulomb repulsion
\( c^\dagger_{i\sigma}, c_{i\sigma} \): Fermionic creation and annihilation operators for electrons with spin \( \sigma \) at site \( i \)
\( n_{i\sigma} \): Number operator for electrons with spin \( \sigma \) at site \( i \)

One can also use the extended Hubbard Hamiltonian, given by:

\[ \hat{H} = -\sum_{\langle i,j \rangle, \sigma} t_{ij} c^\dagger_{i\sigma} c_{j\sigma} + U \sum_i n_{i\uparrow} n_{i\downarrow} + V \sum_{\langle i,j \rangle} n_i n_j + K \sum_{\langle i,j \rangle} \vec{S}_i \cdot \vec{S}_j \]

With:

\( V \): Direct Coulomb interaction
\( K \): Exchange interaction

The previous Hamiltonian corresponds to the one-band extended Hubbard model, which is sufficient to describe some correlated systems.

However, in more complex materials such as transition metal oxides or systems with strong crystal field splitting, multi-band Hubbard models become necessary to accurately capture the low-energy physics.

The extended multi-band Hubbard Hamiltonian takes the form:

\[ \hat{H} = - \sum_{R,R',\sigma} \sum_{i,i'} t_{Ri,R'i'} \hat{c}_{Ri,\sigma}^\dagger \hat{c}_{R'i',\sigma} + \frac{1}{2} \sum_{R_i, R_j, R_k, R_l} \sum_{i,j,k,l} \hat{c}^\dagger_{R_i, i} \hat{c}_{R_j, j} U_{R_i,i; R_j, j; R_k, k; R_l, l} \hat{c}_{R_k, k}^\dagger \hat{c}_{R_l, l} \]

With:

\( R, R', R_i \): Different lattice sites
\( i, j, k, l \): Different bands or Maximally Localized Wannier Functions (MLWFs)

The interaction parameter \( U_{R_i,i;\,R_j,j;\,R_k,k;\,R_l,l} \) is given by:

\[ U_{R_i,i;\,R_j,j;\,R_k,k;\,R_l,l} = \int d\mathbf{r} \, d \mathbf{r'} \,\phi_{R_i,i}^*(\mathbf{r})\,\phi_{R_j,j}^*(\mathbf{r'})\,\hat{H}_U\, \phi_{R_k,k}(\mathbf{r})\,\phi_{R_l,l}(\mathbf{r'}) \]

With:

\( \phi_{R_i,i}, \phi_{R_j,j}, \phi_{R_k,k}, \phi_{R_l,l} \): Different Maximally Localized Wannier Functions (MLWFs)

These MLWFs are intuitive localized orbitals that will describe the strongly correlated electrons in our system.

The two-body operator \( \hat{H}_U \) encapsulates more than just the bare Coulomb repulsion. Due to renormalization effects, the screening by high-energy electrons, the effective interaction between low-energy degrees of freedom is modified. This results in a screened Coulomb interaction, often denoted as \( W_r(\mathbf{r}, \mathbf{r}', \omega) \), this can be computed using the constrained Random Phase Approximation (cRPA).

Welcome to the Hubbard Model Explorer

1. The Hubbard Model

2. Search for Material Parameters

3. Computed Results