mSASS part 1 - d¹ configuration in an octahedral field

Expected level splitting	The mSASS Hamiltonian for the point group O
Lowering the symmetry from octahedral to square symmetry	Lowering the symmetry to D₄

The tutorial shows the construction of the mSASS-Hamiltonian for a d¹ configuration in octahedral and square fields. More information can be found in the following reference:

R. M. Geilhufe, J. D. Rineart, arXiv:2209.03123

Expected level splitting

The octahedral symmetry is described by the point group O. First, we compute the character system of the SO(3) representation with l=2 and analyze its decomposition into irreducible representations of O.

GTInstallGroup	Installs point and space groups
GTCharacterTable	Comutes the character table
GTAngularMomentumChars	Gives the characters of representations of O(3) or SU(2)
GTIrep	Computes the decomposition of a reducible representation into irreducible representations of a group

Install the point group O

Compute the character table of O

Calculate the characters of the SO(3) representation with l=2.

Compute the decomposition into irreducible representations of O.

Lowering the symmetry from octahedral to square symmetry

GTCrystalFieldSplitting

Compute the decomposition of representations upon lowering the symmetry

The symmetry of a square is described by the point group D_4.

We compute the character table of D₄.

Compute the decomposition of irreducible representation of O into irreducible representations D₄.

The mSASS Hamiltonian for the point group O

GTAngularMomentumRep

Computes representation matrices for the irreducible representations of O(3) or SU(2)

Use the point group O, installed above

Compute the representation matrices of the SO(3) representation with l=2.

Generate a generic 5x5 Hamiltonian matrix

The mSASS Hamiltonian is projected using the representation matrices of the angular momentum representation,

The right hand-side of the above equation is evaluated as follows.

Solve for the symmetry invariant terms.

Imposing H to be Hermitian, the final Hamiltonian is given by:

Lowering the symmetry to D₄

GTGroupFromGenerators

Install a group from group generators

To obtain the proper symmetry axis (z-axis), we install the group from its generators.

Compute the representation matrices of SO(3).

Generate a generic 5x5 Hamiltonian matrix.

Compute the projected Hamiltonian.

Solve for the symmetry invariant terms.

Construct the final Hermitian Hamiltonian.