Pulgon-tools: A toolkit for analysing and harnessing symmetries in quasi-1D systems

Abstract

Pulgon-tools is an open-source software package providing building blocks for the analysis and modeling of quasi-one-dimensional (quasi-1D) periodic systems based on line-group theory. While mature libraries exist for space-group detection in three-dimensional crystals, an automated and structure-based identification of line groups has so far been lacking. We present software that integrates four complementary components within a consistent line-group framework: (i) structure generation, (ii) symmetry detection, (iii) irreducible representations (irreps) and character table and (iv) harmonic interatomic force constants (IFCs) correction. This paper introduces the general code structure and several examples that illustrate some relevant applications of the program.

Figures (7)

• Figure 1: Data flow within Pulgon-tools.
• Figure 2: Geometric definition of chiral indices $(n,m)$ for an MoS2-type nanotube. (a) Top view of the two-dimensional hexagonal parent lattice, showing the primitive lattice vectors $\bm{a}_{1}$ and $\bm{a}_2$, the chiral vector $\bm{C}_h = n\bm{a}_1 + m\bm{a}_2$, and the translational vector $\bm{T}$ parallel to the nanotube axis. (b) Side view of the S-Mo-S sandwich structure, showing the interlayer spacing $dz$ between the Mo layer and each S layer.
• Figure 3: Illustration of the general symmetry-based construction of a quasi-1D structure. (a) A minimal set of atomic positions is specified in cylindrical coordinates. (b) The monomer is generated by applying the axial-point-group operations defined by the line group. (c) The full quasi-1D structure is obtained by applying the generalized translational symmetry, combining rotation and fractional translation along the symmetry axis.
• Figure 4: Illustration of the chiral roll-up construction of quasi-1D nanotubes from a two-dimensional hexagonal lattice. For each set of chiral indices $(n,m)$, the nanotube is generated by geometrically rolling the parent lattice along the chiral vector $C_{h} = n\bm{a}_{1} + m\bm{a}_{2}$. Representative examples of armchair $(8,8)$, zigzag $(8,0)$, and chiral $(8,4)$ nanotubes are shown.
• Figure 5: Symmetry detection workflow for a $(5,5)$ single-wall carbon nanotube (SWCNT).