# An Example of Symmetry Exploitation for Energy-related Eigencomputations

## TL;DR

This paper discusses exploiting symmetries in Hamiltonian matrices to improve eigenvalue and eigenvector computations in material simulations, leading to more accurate band structure analysis.

## Contribution

It introduces a symmetry-aware approach for computing eigenvectors that commute with the Hamiltonian, enhancing the accuracy of band structure calculations.

## Key findings

- Symmetry exploitation simplifies eigenvector computation.
- Eigenvalues are expressed in terms of lattice momentum components.
- Improved accuracy in material property simulations.

## Abstract

One of the most used approaches in simulating materials is the tight-binding approximation. When using this method in a material simulation, it is necessary to compute the eigenvalues and eigenvectors of the Hamiltonian describing the system. In general, the system possesses few explicit symmetries. Due to them, the problem has many degenerate eigenvalues. The ambiguity in choosing a orthonormal basis of the invariant subspaces, associated with degenerate eigenvalues, will result in eigenvectors which are not invariant under the action of the symmetry operators in matrix form. A meaningful computation of the eigenvectors needs to take those symmetries into account. A natural choice is a set of eigenvectors, which simultaneously diagonalizes the Hamiltonian and the symmetry matrices. This is possible because all the matrices commute with each other. The simultaneous eigenvectors and the corresponding eigenvalues will be in a parametrized form in terms of the lattice momentum components. This functional dependence of the eigenvalues is the dispersion relation and describes the band structure of a material. Therefore it is important to find this functional dependence in any numerical computation related to material properties.