Year: 2018
Type: article
Abstract: For the two-mode and $(N\ensuremath{-}1)\ensuremath{-}\mathrm{bosonic}$ Bose-Hubbard quantum system a less usual phase transition controlled by the parameter $\ensuremath{\varepsilon}$ representing the on-site energy difference is studied. In the literature the parameter is considered either real ($\ensuremath{\varepsilon}>0$) or purely imaginary (with, say, $\ensuremath{\gamma}=\mathrm{Im}\phantom{\rule{4pt}{0ex}}\ensuremath{\varepsilon}>0$), so the phase transition is analyzed here at the interface $\ensuremath{\varepsilon}=\ensuremath{\gamma}=0$. The evolution in the $\ensuremath{\gamma}\ensuremath{-}\mathrm{controlled}$ phase is required unitary so that the main task for the theory is found in the (quasi-)Hermitization of the Hamiltonian, achieved by a suitable amendment of the inner product in Hilbert space, $\ensuremath{\langle}\ifmmode\cdot\else\textperiodcentered\fi{}|\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\rangle}\ensuremath{\rightarrow}\ensuremath{\langle}\ifmmode\cdot\else\textperiodcentered\fi{}|\mathrm{\ensuremath{\Theta}}|\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\rangle}$. In the most relevant domain of small $\ensuremath{\gamma}$ the linearized Hilbert-space metric $\mathrm{\ensuremath{\Theta}}(\ensuremath{\gamma})$ [constrained by the requirement ${lim}_{\ensuremath{\gamma}\ensuremath{\rightarrow}0}\mathrm{\ensuremath{\Theta}}(\ensuremath{\gamma})=I$ of the smoothness of the change of the Hilbert space at the phase transition] is constructed in closed form. Beyond the phase-transition instant, several forms of the systematic non-numerical recurrent construction of the exact metrics $\mathrm{\ensuremath{\Theta}}(\ensuremath{\gamma})$ are also shown user friendly and feasible, at the not too large matric dimensions $N$ at least.
Source: Physical Review A
Author: Miloslav Znojil
Institution: Czech Academy of Sciences, Nuclear Physics Institute
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2897096372