partner Hamiltonian of ${\cal H}^+$ is: $${\cal H}^- = {\cal D} \, {\cal D}^\dagger= \left( \begin{array}{cccc}
-\Delta + |\psi|^2 & 0 & 0 & 0 \\
0 & -\Delta +|\psi|^2 & 0 & 0 \\
0 & 0 & -\Delta +\frac{1}{2} (|\psi|^2+1)+V_kV_k & -2 V_k \partial_k -\partial_k V_k \\
0 & 0 & 2V_k \partial_k + \partial_k V_k & -\Delta +\frac{1}{2} (|\psi|^2+1) + V_kV_k
 \end{array} \right)$$ ${\cal H}^+$ and ${\cal H}^-$ are isospectral operators (although the spectral densities in the continuous spectra may differ). Thus, the index of ${\cal D}$, regularized by means of the spectral heat functions of ${\cal H}^\pm$, $${\rm ind}\,{\cal D}= {\rm dim}\,{\rm Ker}\,{\cal D}-{\rm dim}\,{\rm Ker}\,{\cal D}^\dagger={\rm Tr}_{L^2}\, e^{-\beta {\cal H}^+} - {\rm Tr}_{L^2}\, e^{-\beta {\cal H}^-} \ \, ,$$ where $\beta$ is a fictitious inverse temperature, is independent of $\beta$. It is possible to evaluate the difference between the functional traces in the $\beta=0$ limit having in mind that the operators ${\cal H}^\pm$ have the structure of Schr$\ddot{\rm o}$dinger operators: ${\cal H}^\pm = {\cal H}_0+ \vec{\mathbf{Q}}(\vec{x})\cdot \vec{\nabla} + \mathbf{U}^\pm(\vec{x})$, where ${\cal H}_0$ is the Helmoltz operator times the $4\times 4$ unit matrix and the matrix potentials read: [$$\mathbf{U}^\pm(\vec{x})= \left(\begin{array}{cccc}\vert\psi\vert^2-1 & 0 & -(D_1\psi_2\pm D_1 \psi_2) 