$\mathcal{PT}$-symmetry phase transition in a Bose-Hubbard model with localized gain and loss

We study the dissipative dynamics of a one-dimensional bosonic system described in terms of the bipartite Bose-Hubbard model with alternating gain and loss. This model exhibits the $\mathcal{P}\mathcal{T}$ symmetry under some specific conditions and features a $\mathcal{P}\mathcal{T}$-symmetry phase transition. It is characterized by an order parameter corresponding to the population imbalance between even and odd sites,similar to the continuous phase transitions in the Hermitian realm. In the noninteracting limit, we solve the problem exactly and compute the parameter dependence of the order parameter. The interacting limit is addressed at the mean-field level, which allows us to construct the phase diagram for the model. We find that both the interaction and dissipation rates induce $\mathcal{PT}$-symmetry breaking. On the other hand, periodic modulation of the dissipative coupling in time stabilizes the $\mathcal{PT}$-symmetric regime. Our findings are corroborated numerically on a tight-binding chain with gain and loss.

Figure 1

Schematic representation of the bipartite Bose-Hubbard model coupled in an alternating way to the environment. At sites $A$, the particles are injected into the systems while at the sites $B$, the system is losing particles. The hopping integral between two nearest neighbor sites is denoted by $J$, while $U$ represents the on-site Coulomb interaction, and $\mathrm{\Gamma }$ is the coupling strength between system and environment and is considered to be constant for all sites.

Figure 2

Comparison between the analytical results for ${\psi }_{\alpha }\left(t\right)$ given in Eqs. (16) and (15) (solid lines) and the numerical solution obtained by integrating numerically the homogeneous equation Eq. (9) (symbols). The upper panel displays the periodic time dependence of ${\psi }_{\alpha }\left(t\right)$ in the $\mathcal{PT}$-symmetric regime with $\mathrm{\Gamma }<{\mathrm{\Gamma }}_C$, while the lower panel shows the behavior in the $\mathcal{PT}$-broken phase where the ${\psi }_{\alpha }\left(t\right)$ diverges in the long-time limit when $\mathrm{\Gamma }>{\mathrm{\Gamma }}_C$. The right panels display the occupation of each site $n_{\alpha }\left(t\right)$ as a function of time.

Figure 3

Evolution of the order parameter $\mathrm{\Delta }$ as function of $\mathrm{\Gamma }$ across the $\mathcal{PT}$-phase transition. The symbols are the numerically computed values, while the solid line represents the analytical result in Eq. (20). (Inset) The scaling of $\mathrm{\Delta }$ close to the transition point capturing the mean field critical behavior. The red solid line corresponds to the power-law behavior $\sim {(\mathrm{\Gamma }-{\mathrm{\Gamma }}_C)}^{0.5}$.

Figure 4

Thermodynamic entropy as a function of time. (a) In the $\mathcal{PT}$-symmetric phase, $S\left(t\right)$ has logarithmic growth as function of time, $S\left(t\right)\propto logt$. (b) In the $\mathcal{PT}$-broken phase, $S\left(t\right)$ grows linearly in time $S\left(t\right)\propto t$. In both cases, we present results for three system sizes $L$ (the number of unit cells) and two rates $\mathrm{\Gamma }$.

Figure 5

The real and imaginary parts of the numerical solution for Eqs. (24) for a finite $U$. Because of the cubic powers, the solutions are always complex. In the long-time limit, the solutions remain periodic irrespective of $U$ or $\mathrm{\Gamma }$ strength, but their amplitude grows in time.

Figure 6

(a) Order parameter $\mathrm{\Delta }\left(\mathrm{\Gamma }\right)$ as function of $\mathrm{\Gamma }$ for different values of the interaction strength $U$. The solid lines are the analytical results. The plot corresponding to $U=0$ is added for completeness. (b) Mean-field phase diagram in the $(\mathrm{\Gamma },U)$ parameter space. The separation line corresponds to the equation ${\mathrm{\Gamma }}_C\left(U\right)=\sqrt{{\mathrm{\Gamma }}_C^2-{(U/2)}^2}$.

Figure 7

The real and imaginary parts of the numerical solution for Eqs. (24) for a large driving frequency $\mathrm{\Omega }$. When $\mathrm{\Omega }\gg \mathrm{\Gamma }$, irrespective of the $\mathrm{\Gamma }$ value, the order parameter vanishes, and the system remains in the $\mathcal{PT}$-symmetric regime. Correspondingly the occupation grows linearly in time. The strength of the interaction is fixed to $U=0$, while $\mathrm{\Gamma }=2.5$ in both cases. In the absence of interactions, the wave functions are either real or imaginary.