Critical behavior at dynamical phase transition in the generalized Bose-Anderson model

Critical properties of the dynamical phase transition in the quenched generalized Bose-Anderson impurity model are studied in the mean-field limit of an infinite number of channels. The transition separates the evolution toward ground state and toward the branch of stable excited states. We perform numerically exact simulations of a close vicinity of the critical quench amplitude. The relaxation constant describing the asymptotic evolution toward ground state, as well as asymptotic frequency of persistent phase rotation and number of cloud particles at stable excited state are power functions of the detuning from the critical quench amplitude. The critical evolution (separatrix between the two regimes) shows a non-Lyapunov power-law instability arising after a certain critical time. The observed critical behavior is attributed to the irreversibility of the dynamics of particles leaving the cloud and to memory effects related to the low-energy behavior of the lattice density of states.

Figure 1

Upper panel: time evolution of the real part of the order parameter for three different quench amplitudes. The dotted line depicts the time evolution for the critical quench amplitude (separatrix); the black and red lines correspond to quenches with detuning $\delta {\varepsilon }_0=\pm 0.002$ from the critical value. The black line corresponds to a solution relaxing to the new equilibrium state, while the red line corresponds to a stable excited state (rotating phase). Lower panel: critical quenches plotted on an equilibrium phase diagram of the model. The dashed line and dots depict positions on the phase diagram from which the critical quench takes place.

Figure 2

Upper panel: exponential fit of equilibration asymptotic dynamics. Red curve shows the dynamics of the real part of the order parameter, while the black dashed one depicts exponential fit. Lower panel: power-law dependencies of relaxation parameter $\sigma$ (blue dots) and frequency $\omega$ (red dots) as functions of quench amplitude $\mathrm{\Delta }{\varepsilon }_0$. In this picture the critical quench amplitude is $\mathrm{\Delta }{\varepsilon }_0^{\mathrm{crit}}=0.8719$. As seen from the picture, $\sigma \propto {\left(\delta {\varepsilon }_0\right)}^{1/2}$ and $\omega \propto {\left(\delta {\varepsilon }_0\right)}^{1/3}$.

Figure 3

The dependence of $|\delta a(t-t^{*})|=|〈a(t-t^{*})〉-〈a_{\mathrm{crit}}(t-t^{*})〉|$ on $t-t^{*}$, where $|〈a〉{|}_{\mathrm{crit}}$ corresponds to critical quench with $\delta {\varepsilon }_0=0$. One can consider this quantity as a divergence of trajectories in phase space (as is commonly considered in nonlinear dynamics), which shows the kind of instability of a critical point. Numbers of curves correspond to numbers of critical quenches shown in Fig. 1. Two lines with the same color (almost on top of each other) depict curves for quenches with $\delta {\varepsilon }_0=\pm 0.0005$. The dashed line shows a power-law fit with power $4/3$ and the dash-dotted line depicts a power-law fit with power 1.3 for these dependencies. Inset: dependence of critical time $t^{*}$ on $V$.