Welcome to hubbard’s documentation!

The hubbard Python package allows to find the self-consistent solution for the mean-field Hubbard Model for a certain tight-binding Hamiltonian and a certain Coulomb repulsion parameter \(U\).

\[\begin{split}H &= -\sum_{ij\sigma}t_{ij\sigma}c^{\dagger}_{i\sigma}c_{j\sigma} + \sum_{i}U_in_{i\uparrow}n_{i\downarrow} + \frac{1}{2}\sum_{i\neq j\sigma\sigma^\prime}U_{ij}n_{i\sigma}n_{j\sigma^\prime} \approx\\ & -\sum_{ij\sigma}t_{ij\sigma}c^{\dagger}_{i\sigma}c_{j\sigma} + \sum_{i\sigma} U_i \left\langle n_{i\sigma}\right\rangle n_{i\bar{\sigma}} + \frac{1}{2}\sum_{i\neq j\sigma}\left(U_{ij} + U_{ji}\right)\left(\langle n_{i\uparrow}\rangle + \langle n_{i\downarrow}\rangle\right)n_{j\sigma} + E_U\end{split}\]

Where it has been sepparated into intra- (\(U_{i}\)) and inter-atomic (\(U_{ij}\)) Coulomb repulsion terms, \(\langle n_{i\sigma}\rangle\) is the \(\sigma=\uparrow,\downarrow\)-spin density on site \(i\) and \(E_U\) is a constant term:

\[E_U = -\sum_i U_i \langle n_{i\uparrow}\rangle\langle n_{i\downarrow}\rangle - \frac{1}{2}\sum_{i\neq j}U_{ij}\left(\langle n_{i\uparrow}\rangle+\langle n_{i\downarrow}\rangle\right)\left(\langle n_{j\uparrow}\rangle + \langle n_{j\downarrow}\rangle\right)\]

which can be directly added to the total electronic energy.

This package allows for:

  • Easy calculations of spin-resolved quantities.

    It takes advantage of many routines from sisl as well as numpy and scipy, which makes it very efficient when handling with thousands of atoms, given the usage of sparse matrices. The goal of this package is to include electron correlations in the tight-binding Hamiltonian by solving self-consistently the mean-field Hubbard model. Given the simplicity of the model one can find the solution in short time to problems that are typically adressed with DFT and obtain similar accuracy, especially for sp2 carbon systems. Here it is also very easy to manipulate the spin configuration to obtain different magnetic solutions, e.g., obtain the approximated energy difference between the singlet and the triplet states, etc. This package is fully implemented in Python, which makes it very easy and comfortable to use.

  • It provides with nice plotting functions to visualize the different physical quantities

    that are obtained with the hubbard package, such as the spin-polarization, wavefunctions for each spin-channel, density of states maps, etc.

User Guide

Publications

Funding

Contents

Contributing

Contributions are highly appreciated.

Indices and tables

The complete package index can be found below: