CITE THIS NOTEBOOK: Group Theory Package (GTPack) and Symmetry Principles in Condensed Matter by Matthias Geilhufe. Wolfram Community 30 OCT 2022.
Symmetry and symmetry-breaking are fundamental concepts, with wide-ranging applications from the Higgs mechanism of elementary particles to ordered states in many-particle physics. Group theory is the mathematical framework for symmetry. With the aim of making accessible group theory algorithms for studying symmetry principles in the solid state, we develop the Mathematica group theory package GTPack. GTPack contains more than 200 additional commands for the Mathematica language, covering basic group and representation theory as well as applications in electronic/phononic/photonic structure calculations, quantum chemistry, and phenomenological theories. Beisdes active research, GTPack is a valuable tool for teaching, as demonstrated in the corresponding book: Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica (Wiley-VCH).
Please see https://gtpack.org/download/ to download GTPack. Installation guide can be found here: https://gtpack.org/tutorials/install-gtpack/
Load GTPack
In[]:=
Needs["GroupTheory`"]
In[]:=
?GT*
Out[]=
Symbols
Symbols
Transformation symbols
Representation matrices
SU(2) matrices?
Groups
Groups
GTPack focus is on crystallographic groups, i.e., point and space groups
Install a group from generators
Install a pre-defined point group
Install a pre-defined space group
Install a pre-defined double group
Representation theory
Representation theory
Install a more interesting group
Calculate the character table
Representation matrices
Play with representations - superconductivity
Play with representations - superconductivity
symmetry-adapted free energy expansion (available with GTPack 1.4)
Calculate bilinears
Calculate bilinears
Bilinear basis
Decompose into Clebsch-Gordan sum
can emerge as a nematic vestigial order within the above free energy
[Fernandes, et al., Annu. Rev. Condens. Matter Phys., 10:133-154]
[Fernandes, et al., Annu. Rev. Condens. Matter Phys., 10:133-154]
Chemistry - crystal field theory
Chemistry - crystal field theory
Phenomenological theory
Phenomenological theory
Install octahedral group
Characters of the angular momentum representation for d-electrons
Quantitative crystal field theory
Quantitative crystal field theory
Derive the crystal field expansion in cubic symmetry
Use e.g. Buckmaster-Smith-Thornley operator equivalents
Alternatively, one can include values for r^l from databases
Molecular symmetry-adapted spin space (mSASS)
Molecular symmetry-adapted spin space (mSASS)
R. M. Geilhufe, J. D. Rinehart, arXiv: arXiv:2209.03123
Compute the representation matrices of SU (2) , for J=2.
Generate a generic 5x5 Hamiltonian matrix
To match the right hand side with the left-hand side, we solve for the symmetry invariant terms:
Imposing H to be Hermitian, the final Hamiltonian is given by:
Graphene - band structure calculations
Graphene - band structure calculations
Define the structure in the GTPack style
Generate an atomic cluster and neighbor shells
Construct 2-center tight-binding Hamiltonian
Plot the band structure