Measurement phase transitions in the no-click limit as quantum phase transitions of a non-hermitean vacuum

\\1. We have moved the analytic results in Sec. 2 to the Appendix. We kept in Sec 2 only the strict necessary to make the rest of the manuscript clear to the reader without necessarily going through the appendices. In particular in the revised version we write: "All the details concerning the diagonalization can be found in Appendix A . For both models the diagonalized Hamiltonian takes the form
\begin{equation}
\hat{H}=\sum_{k>0} \lambda_k \hat{\gamma}^*_k\hat{\gamma}_k - \lambda_k \hat{\gamma}_{-k}\hat{\gamma}_{-k}^*=\sum_{k>0} \lambda_k (\hat{\gamma}^*_k\hat{\gamma}_k + \hat{\gamma}_{-k}^*\hat{\gamma}_{-k})-\Lambda_0,\end{equation}
where $\Lambda_0=\sum_{k>0}\lambda_k$, $\hat{\gamma}$ are the non-hermitian quasiparticle annihilation operators and $\lambda_k$ are the ( complex ) eigenvalues which are specific of the model considered. We find that for the Quantum Ising model
\begin{equation}
\lambda_k= \pm \sqrt{4\bigg(h-J\cos{k}+i\frac{\gamma}{4}\bigg)^2+ 4 J^2\sin^2{k}},
\end{equation}
while for the Long Range Kitaev model
\begin{equation}
\lambda_k= \pm \sqrt{4\bigg(h-J\cos{k}+i\frac{\gamma}{4}\bigg)^2+ J^2 g_d(k)^2},
\end{equation}
where $g_d(k)=\sum_{r=1}^{L-1} \frac{\sin(k r)}{l^d}$. The sign is chosen in such a way that the sign of the imaginary part of $\lambda_k$ is negative [49]. Thus the non-hermitian Hamiltonian right vacuum
\begin{align}
\hat{\gamma}_k |0_\gamma\big>=0,\hspace{1 cm}
\hat{\gamma}_{-k}|0_\gamma\big>=0,
\end{align}
can always be construct as the state with largest imaginary part (not lowest real part). Because of the normalization factor in Eq.(4), the stationary states of the dynamics will be a linear combination of the vacuum and of quasi-particle states such that $\Gamma_k \equiv \Im[\lambda_k]=0$, while all amplitudes of the other quasi-particle states will decay to zero at long times.". In the Appendix the diagonalization of the Hamiltonian as reported in section 2 of the previous version is presented.\\
\\ 2. Figure 3 has been changed as requested: the quantity reported in the graph is the relative difference $(S_{zc}-S_{v})/S_{v}$. \\
\\3. In figure 5(c) we clarified how the parameter $c$ was computationally obtained by specifying in the caption of the figure the number of trajectories considered: "$N_{tr}=20$". In figure 5(c) we inserted a comparison with the result from the zero click trajectories and the vacuum. We commented on this comparison in the text by highlighting that "the transition point is however unchanged by the presence of rare jumps".\\