Time-reversal, and T-signature or T-simplex

For operators $\hat{\cal{O}}$, for which both Eq. (11) and (15) hold, matrix ${\cal{O}}$ in Eq. (13) can be additionally simplified, and reads

$$\left(\begin{array}{cc} A & Y \\ -\epsilon_T Y & \epsilo... ...& Y' \\ \epsilon_T Y' & -\epsilon_T A' \end{array} \right),$$ (23)

for $\epsilon_Z$=+1 and $\epsilon_Z$=-1, respectively, where all submatrices are real, A is symmetric, A' is antisymmetric, $\widetilde{Y}$= $-\epsilon_T Y$, and $\widetilde{Y}'$= $-\epsilon_T Y'$.

In particular, with $\epsilon_T$= $\epsilon_Z$=+1, the matrix from Eq. (23) reduces to

$${\cal{O}}= \left(\begin{array}{cc} A & Y \\ -Y & A \end{array} \right),$$ (24)

where A is symmetric, Y antisymmetric, and both are real. In order to diagonalize such matrix, one can consider a smaller problem, by constructing a complex matrix ${\cal{O}}_C$=A-iY which has the size twice smaller than the original matrix ${\cal{O}}$. After diagonalizing ${\cal{O}}_C$, and separating real and imaginary parts of its complex eigenvectors,

$${\cal{O}}_C({u}+i{v}) = \omega({u}+i{v}),$$ (25)

one gets two degenerate real eigenvectors of ${\cal{O}}$: $\left(\begin{array}{c} {u}\\ {v} \end{array} \right)$and $\left(\begin{array}{c} {v}\\ {-u} \end{array} \right)$. If $\hat{\cal{O}}$ is the mean-field Hamiltonian, such a form of eigenvectors simplifies expressions for densities.