Explicit inversion of the symmetry-broken saddle-point matrix

Derive an explicit expression for the inverse of the matrix d_\pm(\varepsilon) arising in the saddle-point calculation for the spin-symmetry-breaking coupling vectors, beyond the perturbative expansion in \varepsilon d_1.

Background

In the symplectic case, the corresponding matrix can be diagonalized up to a unitary transformation, which permits an explicit evaluation of its inverse. Introducing the spin preference through the coupling vectors breaks this simplifying structure: the relevant transformation does not commute with the modified metric, so the matrix d_\pm(\varepsilon) is no longer diagonal up to a unitary transformation.

The authors therefore replace the unavailable explicit inverse by an expansion in powers of \varepsilon d_1 and restrict the subsequent calculation to first order in the spin-preference parameter. An explicit inverse would remove or reduce this perturbative restriction and could enable systematic treatment of the symmetry-breaking distributions beyond the calculated order.