Time-Reversal Symmetry and Matrix Block Structure

Time-Reversal Symmetry and Matrix Block Structure 

We now return to the question of how time-reversal symmetry relates to doublegroup symmetry and the block structure of operator matrices. For case 2 above, with the two components of the Kramers pair belonging to different irreps, there are no matrix elements of totally symmetric operators between a fermion function and 

It remains to discuss case 1, where both functions in a Kramers pair belong to the same irrep. For this case, ordinary point group symmetry does not promise any a priori blocking. We will therefore look for a unitary transformation on the Kramers pair basis that can block-diagonalize a matrix constructed over functions belonging to a case 1 irrep. The problem can be reduced to the diagonalization of a general 2x2 matrix spanned by the components of a Kramers pair. For the operator this is 

We have previously shown that in a Kramers basis we have the following relations  

Using these relations, we see that the matrix takes the form 

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