# SeeK-path

## Explanation of the extended Bravais symbol

In the input example selector of the SeeK-path website, as well as in the output of SeeK-path, you will find a short string called extended Bravais symbol.
Here we briefly explain its meaning.

### First two characters

The first two characters are the Bravais symbol identifying the type of Bravais lattice of the system.

The first letter denotes the crystal family, and can have the following values:

• c: cubic
• h: hexagonal
• t: tetragonal
• o: orthorhombic
• m: monoclinic
• a: triclinic

The second letter described the centering of the Bravais lattice, and (after standardization) it can assume the following values:

• P: primitive
• F: face-centered
• I: body-centered
• A: base-centered (A face)
• C: base-centered (C face)
• R: rhombohedrally-centered

### Third character

The third character is an integer (the possible values are 1, 2 or 3).

Indeed, in some cases, in order to describe the Brillouin zone, different cases must be considered. These are indexed by an integer. The algorithm to establish the value of this third character is explained in Table 94 of the SeeK-path publication.

This symbol can depend on the spacegroup of the crystal, and/or on the values of the (a, b, c, α, β, γ) lattice parameters. In particular, we mention that there are two possible cases that need distinction:

• Depending on the spacegroup, additional lines might be needed. This is the case for instance for cP (cubic primitive) lattices.
• The topology of the Brillouin zone (and therefore the coordinates of the corners of the Brillouin zone) depend on the relative ratio of the lattice parameters. This is for instance the case for tI (tetragonal body-centered) where the topology is different depending on whether a is larger or smaller than c.

### A note on inversion symmetry

For systems that do not have inversion symmetry, the band energies typically differ at k and -k if the Hamiltonian does not have time-reversal symmetry. In this case, an additional set of lines is taken by inverting the path about the origin (Γ point).

This is the reason why we provide examples both for systems with and without inversion symmetry. In the python code, you should specify if your Hamiltonian has time-reversal symmetry as an input flag to the SeeK-path functions.