GroupTheory`
GroupTheory`

GTSpinCharacters

GTSpinCharacters[classes]

gives the character of the spinor representation for each class.

Details and Options

  • For each double group a representation using rotation matrices in spin space can be found. This representation is not faithful in general. For a rotation of an angle about the axis , the rotation matrices are given by . are the Pauli matrices and is the 2-dimensional identity matrix. For the representation of the inversion the Pauli-gauge is used, where the inversion is represented by .
  • The following option can be given:
  • GOFast GOFastValueSkips the input validation
  • See: S. L. Altmann, Rotations, Quaternions and Double Groups, Dover Publications, Inc., 2005
  • W. Hergert, M. Geilhufe, Group Theory in Solid State Physics and Photonics. Problem Solving with Mathematica, chapter 8.6.

Examples

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Basic Examples  (1)

First, load the package:

As an example, the characters of the spinor representation for the double group are calculated.

Options  (2)

GOFast  (2)

The evaluation time can be decreased, using GOFastTrue.

If GOFastFalse is used, a check of the input will be performed.

Applications  (1)

Within this example the level splitting of an irreducible representation due to spin-orbit coupling is calculated. As an example the double group is used.

To get an overview about the irreducible representations, the character table is calculated via GTCharacterTable.

The class contains the 4-fold rotations about the z-axis, whereas contains the related double group elements. By comparison of the characters for both classes, the irreducible representations , , and can be identified as the extra representations of , since the character changes sign.

The irreducible representation , which is also an irreducible representation of the ordinary point group , will be discussed in more detail. is a 2-dimensional irreducible representation and represents a 4-fold degenerated energy level, if the 2-fold degeneracy due to the spin is included. To estimate the influence of spin orbit coupling, the characters of the direct product representation have to be calculated. First, the characters of the spinor representation are calculated via GTSpinCharacters.

Now the characters of are calculated via GTDirectProductChars.

Now, the Clebsch-Gordan sum of is calculated via GTIrep.