Broken-Hermiticity phase transition in the Bose-Hubbard model

For the two-mode and $(N-1)-\mathrm{bosonic}$ Bose-Hubbard quantum system a less usual phase transition controlled by the parameter $\varepsilon$ representing the on-site energy difference is studied. In the literature the parameter is considered either real ($\varepsilon >0$) or purely imaginary (with, say, $\gamma =\mathrm{Im}\varepsilon >0$), so the phase transition is analyzed here at the interface $\varepsilon =\gamma =0$. The evolution in the $\gamma -\mathrm{controlled}$ 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, $\langle ·|·\rangle \to \langle ·|\mathrm{\Theta }|·\rangle$. In the most relevant domain of small $\gamma$ the linearized Hilbert-space metric $\mathrm{\Theta }\left(\gamma \right)$ [constrained by the requirement ${lim}_{\gamma \to 0}\mathrm{\Theta }\left(\gamma \right)=I$ 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 $\mathrm{\Theta }\left(\gamma \right)$ are also shown user friendly and feasible, at the not too large matric dimensions $N$ at least.

Figure 1

The decrease of the three lowest eigenvalues ${\theta }_j\left(\gamma \right)$ of the $N=4$ matrix (14) to zero for the growth of $\gamma$ to its maximal physical value ${\gamma }^{\left(\text{EP}\right)}=1$. The fourth root is much larger and lies out of the picture.

Figure 2

The $\gamma \mathrm{dependence}$ of the eigenvalues of the $N=5$ matrix (21) and their positivity in the interval of $\gamma \in \left[0,{\gamma }_{max}\right)$ with ${\gamma }_{max}=1/2\sqrt{\sqrt{5}-1}$.