Quench dynamics of two-component dipolar fermions subject to a quasiperiodic potential

Motivated by recent experiments in fermionic polar gases, we study the nonequilibrium dynamics of two-component dipolar fermions subject to a quasiperiodic potential. We investigate the localization of charge and spin degrees of freedom time evolving with a long-range spin-SU(2) symmetric fermionic Hamiltonian by calculating experimentally accessible dynamical observables. To study the nonequilibrium dynamics, we start the time evolution with two initial states at half-filling: (i) product state with doublons $|\uparrow \downarrow 0\uparrow \downarrow 0\uparrow \downarrow 0\uparrow \downarrow 0\uparrow \downarrow \rangle$ and (ii) product state with singlons $|\uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \rangle$. We carried out the real-time evolution via the fermionic Hamiltonian using exact diagonalization (ED) and the time-dependent variational principle (TDVP) for finite matrix product states (MPS), within experimentally relevant timescales. For the product state with doublons, we observe a delocalized to localized phase transition varying disorder strengths by monitoring the decay of charge imbalance with time. For the long-range interacting Hamiltonian of our focus, and in the presence of strong enough disorder, starting the time evolution with singlons we find a strong reduction in the spin delocalization, contrary to results of previous studies using the disordered short-range (on-site) Hubbard model with SU(2) symmetry. In addition to the quench dynamics, we also demonstrate the localization of charge and spin in the full energy spectrum of the long-range spin-SU(2) symmetric Hamiltonian by monitoring spin and charge autocorrelation functions. Our predictions for localization of both charge and spin should be observable in ultracold experiments with fermionic dipolar atoms subject to a quasiperiodic potential.

Figure 1

(a) Time evolution of the average charge imbalance $I\left(t\right)$ vs time $t$ (in log-log scale and time in units of $\hslash /t_{\text{hop}}$) at fixed interaction couplings $U=2V=8$ and for the various disorder strengths $\mathrm{\Delta }$ shown in the inset. The red curves between $5t$ to $80t$ represent power-law fits to the $I\left(t\right)$ decay. (b) Relaxation exponent $\alpha$ of the charge imbalance decay vs $\mathrm{\Delta }$, for $L=8$, $L=10$, and $L=12$, showing the delocalized to localized transition increasing $\mathrm{\Delta }$. The size of data point symbols (circle, square, triangle) represent the fitting error bars. The delocalized region is denoted as DEL.

Figure 2

Time evolution of the entanglement entropy $S\left(t\right)$ with increasing time $t$ (in log scale and in units of $\hslash /t_{\text{hop}}$). We work at $U=2V=8$ and use several values of $\mathrm{\Delta }$. The red curves show a power-law fit of $S\left(t\right)$ at $\mathrm{\Delta }=6$ and a logarithmic fit at $\mathrm{\Delta }=8$.

Figure 3

Comparison of decay exponent $\alpha$ obtained by power-law fit to the average charge imbalance $I\left(t\right)$ for (a) disordered extended-Hubbard $\left[H\right(U,V,\mathrm{\Delta }\left)\right]$ at $U=2V=4$ and (b) disordered Hubbard model $\left[H\right(U,\mathrm{\Delta }\left)\right]$ at $U=4$. Gray color (triangle) data point symbol for long-range interacting Hamiltonian with same interaction parameter. Insets shows decay of charge imbalance vs time $t$ (in log scale and in units of $\hslash /t_{\text{hop}}$) near the transition region of delocalize to localize transition. These results were obtained by using TDVP1 for system sizes $L=24$ and 48 and ED for $L=12$.

Figure 4

(a) Time evolution of the spin imbalance $I_S\left(t\right)$, with $t$ in units of $\hslash /t_{\text{hop}}$. The red curve represents the fit to $I_S\left(t\right)$ by using the fitting functions described in the text. (b) Entanglement entropy $S\left(t\right)$ vs time $t$. The red curve contains the power-law fit to $S\left(t\right)$. These results are obtained by ED at $\mathrm{\Delta }=16$, using system sizes $L=12$ [for $I_S\left(t\right)]$ and $L=10$ [for $S\left(t\right)]$, in the large interaction $U$ limit.

Figure 5

Time evolution of the spin imbalance $I_S\left(t\right)$, with $t$ in units of $\hslash /t_{\text{hop}}$, in the small interaction limit $U=V=1$ and with strong disorder $\mathrm{\Delta }=12$. (a) Results using ED for a system size $L=12$. (b) Results using TDVP1 for a system size $L=64$. The red curve represents the power-law fit to $I_S\left(t\right)$ using the fitting function described in the text. The inset shows the time evolution of the entanglement entropy $S\left(t\right)$ vs time $t$. The red curve displays the linear fit to $S\left(t\right)$.

Figure 6

(a) Time evolution of the spin imbalance $I_S\left(t\right)$, with $t$ in units of $\hslash /t_{\text{hop}}$, for the disordered on-site Hubbard model at small $U=1$ and $V=0$. The red curve represents the power-law fit to $I_S\left(t\right)$, using the fitting function described in the text. (b) Entanglement entropy $S\left(t\right)$ vs $t$. The red curve shows the power-law fit to $S\left(t\right)$. These result were obtained using TDVP1 at $\mathrm{\Delta }=12$ and for system size $L=64$.

Figure 7

Averaged spin autocorrelation $R_S\left(t\right)$ vs time $t$ (in units of $\hslash /t_{\text{hop}}$) for the cases of (a) large interaction limit $U=8$ and (b) small interaction limit $U=1$. The red curves represent power-law fittings of $R_S\left(t\right)$ with exponent $\eta$. The inset shows the charge autocorrelation $R_C\left(t\right)$ vs $t$. These results were obtained using the full-diagonalization method of a system size $L=8$ at $\mathrm{\Delta }=16$.

Figure 8

Time evolution of the site average double occupancy $d\left(t\right)$ vs time $t$ (in units of $\hslash /t_{\text{hop}}$) (a) long-range Hubbard model (b) Hubbard model, for different values of interaction parameters. In both cases $L=12$, and $\mathrm{\Delta }=0$ (no disorder).

Figure 9

Time evolution of the bipartite fluctuations: (a) charge-fluctuation $F_N\left(t\right)$ and (b) spin-fluctuation $F_S\left(t\right)$ vs time $t$ (in log-scale and in units of $\hslash /t_{\text{hop}}$). We work at $U=2V=8$ and use several values of $\mathrm{\Delta }$. The red curves show a power-law fit to the charge and spin fluctuations near the transition point $\mathrm{\Delta }=6$.

Figure 10

Comparison of the charge imbalance $I\left(t\right)$ near the transition region: using ED (solid lines) and TDVP (broken red lines) time evolution methods using system size $L=12$. (a) For an extended disordered Hubbard model at fixed value of $U=2V=4$ and for different values of $\mathrm{\Delta }$. (b) For a disordered Hubbard model at $U=4$ and for different values of $\mathrm{\Delta }$. The charge imbalance $I\left(t\right)$ is averaged over sites followed by averaging over ten different values of phase factor $\varphi$.

Figure 11

(a) Time evolution of the spin imbalance $I_S\left(t\right)$ vs time $t$ (in units of $\hslash /t_{\text{hop}}$) for different interaction strengths. (b) Average doublons occupancy vs time $t$. (c) Time-averaged spin imbalance $I_S$ vs $V$ at $U=2$, showing that the spin imbalance acquires maximum values when $U=V=2$. These results are obtained using ED at disorder strength $\mathrm{\Delta }=16$, $\varphi =0$, and for a system size $L=12$.

Figure 12

Spin imbalance $I_S\left(t\right)$ vs time $t$ (in units of $\hslash /t_{\text{hop}}$) working at $\mathrm{\Delta }=12$, and for different interaction strength when $U=V$ (where $V$ is the prefactor of long-range interaction and $U$ is on-site interaction): (a) $U=V=1$, (b) $U=V=2$, (c) $U=V=3$, and (d) $U=V=4$. The red curves represent the power-law fit to $I_S\left(t\right)$ using the fitting function described in the text, while the cyan curve represents a linear fit to $I_S\left(t\right)$.

Figure 13

Spin imbalance $I_S\left(t\right)$ vs time $t$ (in units of $\hslash /t_{\text{hop}}$) working at $U=V=1$, and for different values of disorder strength $\mathrm{\Delta }$: (a) $\mathrm{\Delta }=4$, (b) $\mathrm{\Delta }=8$, (c) $\mathrm{\Delta }=12$, and (d) $\mathrm{\Delta }=16$. The red curve represents the fit to $I_S\left(t\right)$ using the fitting function described in the text. (e) Bipartite charge fluctuation $F_N\left(t\right)$ vs time $t$ for $\mathrm{\Delta }=4.0$ and 12.0.

Figure 14

Long-time dynamics of charge and spin imbalance using TDVP1 method for larger system size $L=64$. (a) The charge imbalance $I_C\left(t\right)$ vs time $t$ (in units of $\hslash /t_{\text{hop}}$) for product state with doublons at $U=2V=4$ and $\mathrm{\Delta }=12$. (b) The spin imbalance $I_S\left(t\right)$ vs time $t$ for product state with Néel state, working at $\mathrm{\Delta }=16$ and $U=1,V=1$.