Quench dynamics in the one-dimensional mass-imbalanced ionic Hubbard model

Using the time-dependent Lanczos method, we study the nonequilibrium dynamics of the one-dimensional ionic-mass imbalanced Hubbard chain driven by a quantum quench of the on-site Coulomb interaction, where the system is prepared in the ground state of the Hamiltonian with a different Hubbard interaction. The exact diagonalization method (Lanczos algorithm) is adopted to study the zero temperature phase diagram in equilibrium, which is shown to be in good agreement with previous studies using density matrix renormalization group (DMRG). We then study the nonequilibrium quench dynamics of the spin and charge order parameters by fixing the initial and final Coulomb interactions while changing the quenching time protocols. Our study shows that the time evolution of the charge and spin order parameters strongly depend on the quenching time protocols. In particular, the effective temperature of the system will decrease monotonically as the quenching time is increased. By taking the final Coulomb interaction strength to be in the strong coupling regime, we find that the oscillation frequency of the charge order parameter increases monotonically with the Coulomb interaction. By contrast, the frequency of the spin order parameter decreases monotonically with increasing Coulomb interaction. We explain this result using an effective spin model and a two-site Hubbard model in the strong coupling limit. Finally, we take the final Coulomb interaction strength to be in the weak coupling regime and find that the oscillation frequency of both the charge and spin order parameters increases monotonically with decreasing Coulomb interaction. Our study suggests strategies to engineer the relaxation behavior of interacting quantum many-particle systems.

Figure 1

The equilibrium phase diagram of the one-dimensional mass imbalanced ionic Hubbard chain at half-filling and zero temperature, calculated using the exact diagonalization method (10 sites or 14 sites with periodic boundary condition). The density matrix renormalization group (DMRG) and mean-field theory (MFT) data are obtained from the references [59, 61]. (a) The charge gap ${\mathrm{\Delta }}_c$ as a function Coulomb interaction are plotted to characterize the phase transition ($U_c=6.8$), where the mass imbalance and crystal field are $\eta =0.75$ and $\mathrm{\Delta }=3.0$, respectively. (b) Critical points are plotted in the plane of Coulomb interaction $U$ and crystal field $\mathrm{\Delta }$ with fixed mass imbalance $\eta =V_{\downarrow }/V_{\uparrow }=0.90$. (c) Critical points are plotted in the plane of Coulomb interaction $U$ and mass imbalance $\eta$ with fixed crystal field $\mathrm{\Delta }=2.0$. The lines are guides to the eye.

Figure 2

Time evolution of the charge and spin order parameter $\delta {\rho }_c$ and $\delta {\rho }_s$ after the Coulomb interaction quench protocol $U\left(t\right)=U_{\mathrm{i}}+(U_{\mathrm{f}}-U_{\mathrm{i}})t/t_q$ for $t<t_q$ and $U\left(t\right)=U_{\mathrm{f}}$ for $t\ge t_q$, where the initial and final Coulomb interaction strength is set as $U_{\mathrm{i}}=6.0$ and $U_{\mathrm{f}}=8.0$, respectively. (a) $t_q=0.0^{+}$, (b) 1.0, (c) 2.0, (d) 4.0, (e) 8.0, and (f) 16.0. The expectation value of $\delta {\rho }_c$ and $\delta {\rho }_s$ at zero temperature is represented by a dashed line with Coulomb interaction $U=U_{\mathrm{f}}$ in the equilibrium calculation. The arrows indicate the thermal values of the two order parameters at effective temperature $T_{\mathrm{eff}}=0.632,0.420,0.296,0.173,0.104,0.085$ for $t_q=0.0^{+},1.0,\cdots ,16.0$, respectively. The insets in (e) plot the order parameters as a function of time in the long-time regime $t\in [150,200]$.

Figure 3

Time evolution of charge and spin order parameter $\delta {\rho }_c$ (a) and $\delta {\rho }_s$ (b) for Coulomb quenches from $U_{\mathrm{i}}=6.5$ to $U_{\mathrm{f}}=6.7,6.9,\cdots ,7.5$, (c) Comparison of $\delta {\rho }_c$ and $\delta {\rho }_s$ for quench from $U_{\mathrm{i}}=6.5$ to $U_{\mathrm{f}}=7.5$.

Figure 4

Time evolution of the charge and spin order parameters $\delta {\rho }_c$ and $\delta {\rho }_s$ for the quench protocol with (a) $U_{\mathrm{i}}=2.0\to U_{\mathrm{f}}=8.0$, where the quench time $t_q=8.0$. The dashed line represents the equilibrium $\delta {\rho }_c$ and $\delta {\rho }_s$ at zero temperature with $U=U_{\mathrm{f}}$. (b) $U_{\mathrm{i}}=2.0\to U_{\mathrm{f}}=8.0,12.0,16.0,\text{and}20.0$, and the periods of the charge density order parameter are $3.27,1.15,0.69,\text{and}0.50$. The inset shows the oscillation behavior at $195<t<200$ for $U_{\mathrm{f}}=12,16,\text{and}20$. (c) $U_{\mathrm{i}}=2.0\to U_{\mathrm{f}}=8.0,12.0,16.0,\text{and}20.0$, and the periods of the spin density order parameter are $12.40,24.32,43.72,\text{and}63.53$.

Figure 5

Time evolution of charge and spin order parameter $\delta {\rho }_c$ and $\delta {\rho }_s$ for Coulomb quenches from $U_{\mathrm{i}}=12.0$ to $U_{\mathrm{f}}=8.0,6.0,\text{and}4.0$. This goes from strong coupling to weak coupling.

Figure 6

The initial and final Coulomb interaction are fixed as $U_{\mathrm{i}}=6.0$ and $U_{\mathrm{f}}=8.0$. (a) The time evolution of $\delta X$ in Eq. (A1) are plotted as a function of time with the quenching time $t_q=8.0$ and $t_q=16.0$, respectively. [(b) and (c)] The probability distribution of each Fock basis for $|c_I{|}^2={\left|\langle I|{\mathrm{\Psi }}_g^f\rangle \right|}^2$ and ${\left|\langle I|\mathrm{\Psi }\left(t\right)\rangle \right|}^2$ are plotted with quenching time $t_q=8.0$ (b) and $t_q=16.0$ (c), respectively, where $|{\mathrm{\Psi }}_g^f\rangle$ is the ground state of equilibrium Hamiltonian with $U=U_{\mathrm{f}}, \left|\mathrm{\Psi }\right(t)\rangle$ is the time evolved wave function from nonequilibrium time evolution.

Figure 7

Time evolution of the charge (a) and spin (c) order parameter $\delta {\rho }_c$ and $\delta {\rho }_s$ after the Coulomb interaction quench where the initial and final Coulomb interactions are set as $U_{\mathrm{i}}$ = 2.0 and $U_{\mathrm{f}}$ = 12.0, respectively. The quench protocols with different quenching time $t_q=8.0,12.0,16.0,\text{and}20.0$ are applied, respectively. Time evolution of $\delta {\rho }_c$ (b) and $\delta {\rho }_s$ (d) for different quench protocols with fixed final Coulomb interaction $U_{\mathrm{f}}=12.0$ and quenching time $t_q=8$ while the initial Coulomb interaction is changed with $U_{\mathrm{i}}=2.0,4.0,6.0,\text{and}8.0$.

Figure 8

[(a) and (b)] The charge, spin and excitation gap of the ionic mass imbalanced Hubbard model as a function Coulomb interaction $U$. The mass imbalance and the ionic potential are $\eta =0.75$ and $\mathrm{\Delta }=3.00$. The calculation is done with 10 and 14 sites, respectively. (c) The Von Neumann block entropy as a function of Coulomb interaction is plotted.

Figure 9

Time evolution of the charge and spin order parameter $\delta {\rho }_c$ and $\delta {\rho }_s$ after the Coulomb interaction quench where the initial and final Coulomb interactions are set as $U_{\mathrm{i}}$ = 2.0 and $U_{\mathrm{f}}$ = 8.0, respectively. The quenching time is fixed at $t_q=0.0$ [(a) and (b)] and 8.0 [(c) and (d)]. The order parameters are calculated with 6, 10, and 14 sites with periodic boundary condition, respectively.