Broken Hermiticity phase transition in Bose-Hubbard model

A new version of the change of the "phase" (i.e., of the set of observable characteristics) of a quantum system is proposed. In a general scenario the evolution is assumed generated, before the phase transition, by some standard Hermitian Hamiltonian $H^{(before)}$, and, after the phase transition, by one of the recently very popular non-standard, non-Hermitian (but hiddenly Hermitian, i.e., still unitarity-guaranteeing) Hamiltonians $H^{(after)}$. For consistency, a smoothness of matching between the two operators as well as between the related physical Hilbert spaces must be guaranteed. The feasibility of the idea is illustrated via the two-mode $(N-1)-$bosonic Bose-Hubbard Hamiltonian. In $H^{(before)}=H^{(BH)}(\varepsilon)$ we use the decreasing real $\varepsilon^{(before)} \to 0$. In the hiddenly Hermitian continuation $H^{(after)}=H^{(BH)}(\tilde{\varepsilon})$ the imaginary part of the purely imaginary $\tilde{\varepsilon}^{(after)}$ grows. The smoothness of the transition occurring at the interface $\varepsilon=\tilde{\varepsilon}=0$ is then guaranteed by an {\it ad hoc\,} amendment of the inner product in Hilbert space "after". The trivial Hilbert-space metric $\Theta^{(before)}=I$ must match $\Theta^{(after)} \neq I$ smoothly. This is confirmed and illustrated by the explicit constructions of a few $\Theta^{(after)}$s in closed form.