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) → 0. In the hiddenly Hermitian continuation H^(after)=H^(BH)\({varepsilon}\) the imaginary part of the purely imaginary {varepsilon}^(after) grows. The smoothness of the transition occurring at the interface varepsilon={varepsilon}=0 is then guaranteed by an {\it ad hoc\,} amendment of the inner product in Hilbert space "after". The trivial Hilbert-space metric Θ^(before)=I must match Θ^(after) neq I smoothly. This is confirmed and illustrated by the explicit constructions of a few Θ^(after)s in closed form.