Canonical representation

The canonical-basis single-particle wave functions,

$$\chi_{\mu}({\bm r},\sigma )= \sum_{n} W_{n\mu}~ \psi_{n}({\bm r},\sigma),$$ (75)

are defined by the unitary matrix $W$ which diagonalizes the density matrices,

$$\begin{array}{llr} {\sum\limits_{n'} {\rho _{nn'} W_{n'\mu } ... ...{\rho}_{nn'} W_{n'\mu} & = & u_\mu v_\mu W_{n\mu }, \end{array}$$ (76)

where $v_\mu^2$ are the occupation probabilities $0 \leq v_\mu^2 \leq 1$ and $v_\mu^2 + u_\mu^2=1$. In the canonical representation, the gauge-angle-dependent matrices become diagonal with the diagonal matrix elements given by:

$$C_{\mu}(\phi_{q}) = \frac{e^{2\imath \phi_{q}}}{u_{\mu q}^{2}+e^{2\imath \phi_{q}}v_{\mu q}^{2}},$$ (77)

$$\rho_{\mu q}(\phi_{q}) = \frac{e^{2\imath \phi_{q}}v_{\mu q}^{2}}{u_{\mu q }^{2}+e^{2\imath \phi }v_{\mu q}^{2}},$$ (78)

$$\tilde{\rho}_{\mu q}(\phi_{q}) = \frac{e^{\imath \phi_{q}}u_{\mu q}v_{\mu q}}{u_{\mu q}^{2}+e^{2\imath \phi_{q}}v_{\mu q}^{2}},$$ (79)

and the determinant of matrix $C(\phi_{q})$, needed in Eq. (40), becomes a product of the diagonal values (77). The use of the canonical representation significantly simplifies calculations of the projected fields.