Introduction
Tight binding is a model used to describe the behavior of electrons in a solid, particularly in insulators and semiconductors. It is based on the that the electrons in a solid are strongly bound to the atoms but can jump between atoms with some probability. In other words, the electrons are localised on the atomic lattice site and hop between sites.
This model model is particularly useful to study the electronic properties of large sized materials with a simple crystal structure, where more sophisticated Denisty Functional Theory (DFT) methods become too expensive in terms of computation time.
Solving a tight binding model, always narrows down to diagonalizing the Hamiltonian. Resulting in eigenvalues and eigenvectors problem. The former produces band structure of the solid, and the latrer econdes topological properties. Explict examples of such Hamiltonians are presented in the example section.
The eigenvectors represent the probability amplitudes for the electrons to be found at each site in the crystal lattice. Symmetry of eigenvectors, tells us how the electronic states of a crystal are is distributed in real space. By studying topology, we analyse how symmetry evolves across momentum space.
The concepts and tools of group theory, allows us to systematically analyse the symmetries of the crystal lattice and their impact on the electronic structure. Namely spatial and time-reversal symmetries.
Most researchers in the field of twistronics, prefer to have a framework that not only diagonalises tight binding Hamiltonians, but also allows them to implement and investigate more topological and symmetry properties of the eigenvectors.
LTB-Symm started as a handy tool for our own research, but grew into a general and flexible code that worth sharing with others!