Quench Dynamics and Nonequilibrium Phase Diagram of the Bose-Hubbard Model

We investigate the time evolution of correlations in the Bose-Hubbard model following a quench from the superfluid to the Mott insulator. For large values of the final interaction strength the system approaches a distinctly nonequilibrium steady state that bears strong memory of the initial conditions. In contrast, when the final interaction strength is comparable to the hopping, the correlations are rather well approximated by those at thermal equilibrium. The existence of two distinct nonequilibrium regimes is surprising given the nonintegrability of the Bose-Hubbard model. We relate this phenomenon to the role of quasiparticle interactions in the Mott insulator.

Figure 1

Short time behavior of the nearest-neighbor correlation functions for $U_i=2J$ and $U_f=40J$. (a) $t$-DMRG and ED results for 1D chains. For comparison, the thin full line without symbols shows a relaxation for $U_f=80J$. (b) ED results for 2D square lattices. In both cases there are clear oscillations with a period $2\pi /U_f$ (dotted vertical lines for $U_f=40J$), which relax on a time scale of $1/J$. Revival of the correlations at $\approx 1.5/J$ in (b) is due to finite size effects. The two insets show the Fourier transform of the oscillations. The main weight lies in a broad band at $\omega \approx U_f$, with some smaller bands at higher multiples of $U_f$.

Figure 2

Decay of the correlations $⟨b_i^{†}{b}_{i+r}⟩$ with distance $r$ after a quench from $U_i=2J$ to $U_f=40J$. □ (ED) and ▵ ($t$-DMRG) show the averaged results of the quasisteady state. The average value is determined fitting a linear function to the results between $t_1\approx J^{-1}$ and $t_2=20J^{-1}$. ⋄ show equilibrium QMC results at finite temperature (see text for details), and the filled and open circles display the $T=0$ correlations in the ground state for $U_i$ and $U_f$, respectively.

Figure 3

Left panel: decay of the correlations $⟨b_i^{†}{b}_{i+r}⟩$ with distance $r$ after a quench from $U_i=J$ to $U_f=4J$. □ show the averaged results for times between $t_1\approx J^{-1}$ and $t_2=20J^{-1}$ determined as in Fig. 2. ⋄ show equilibrium QMC results at finite temperature ($T\approx 0.8J$) and $U=4J$, and the filled and open circles display the $T=0$ correlations in the ground state for $U_i$ and $U_f$, respectively. Right panel: particle number distribution $P(n)$. The labeling is the same as in the left panel.

Figure 4

Left panel: nonequilibrium phase diagram in the space of initial and final interaction strengths. Two regions are found, one where the steady state is distinctly nonthermal and the other where correlations do appear to thermalize within the numerical error bounds. □ and ○ mark points in the respective regions where numerical results for a one-dimensional system were obtained. The full line follows $U_i=U_f$. Small arrows mark the equilibrium critical value for an infinite one-dimensional system. Right panel: difference between the correlation $⟨b_i^{†}{b}_{i+r}⟩$ at steady state and its nominal value for the ground state of the final Hamiltonian. Data are shown for $r=1$ (○) and $r=2$ (□) along the cut $U_i/J=2$. The crossover from slower than ground state decay ($\delta >0$) to faster than ground state is seen at $U_f\approx 6J$.