Prethermalization and entanglement dynamics in interacting topological pumps

We investigate the formation of quasisteady states in one-dimensional pumps of interacting fermions at noninteger filling fraction, in the regime where the driving frequency and the interaction strength are small compared to the instantaneous single-particle band gap throughout the driving cycle. The system rapidly absorbs energy from the driving field and approaches a quasisteady state that locally resembles a maximal entropy state subject to the constraint of a fixed particle number in each of the system's single-particle Floquet bands. We explore the nature of this quasisteady state through one-body observables including the pumped current and natural orbital occupations, as well as the (many-body) entanglement spectrum and entropy. Potential disorder significantly reduces the amplitude of fluctuations of the quasisteady-state current around its universal value, while the lifetime of the quasisteady state remains nearly unaffected for disorder strengths up to the scale of the single-particle band gap. Interestingly, the natural orbital occupations and the entanglement entropy display patterns signifying the periodic entangling and disentangling of the system's degrees of freedom over each driving cycle. Moreover, prominent features in the system's time-dependent entanglement spectrum reveal the emergence of long timescales associated with the equilibration of many-particle correlations.

Figure 1

(a) Schematic of a one-dimensional quantum pump consisting of a periodically modulated one-dimensional lattice with a two-site unit cell. The sublattice potential $V\left(t\right)$ and the intracell and intercell hopping amplitudes, $J_{+}\left(t\right)$ and $J_{-}\left(t\right)$, are modulated with a driving period $T$ [see Eqs. (1) and (2)]. (b) Driving protocol represented in the plane of dimerization $\delta J\left(t\right)\equiv 0.5[J_{+}\left(t\right)-J_{-}\left(t\right)]$ and sublattice potential $V\left(t\right)$. The instantaneous band gap closes when $V=0$ and $\delta J=0$ (red X). The number of times that the drive trajectory encircles the origin of this plane over one cycle (i.e., the winding number) labels topologically distinct driving protocols. (c) Time-averaged spectral function of the system in the absence of interactions, for the topologically nontrivial driving protocol indicated by the blue curve in panel (b). Lower panel: Zoom-in of the lower (right-moving, “R”) Floquet-band. Each Floquet sideband shifts by $\omega$ as $k$ traverses the Brillouin zone, indicative of a quantized, nonzero value of the band-averaged group velocity.

Figure 2

Transient- and quasisteady-state behavior of single-particle observables. (a) Period-averaged current for clean and disordered systems, at quarter filling. The system is initialized with the right-moving Floquet band half-filled. The current starts from an initialization-dependent value and then quickly approaches the expected universal value [see discussion below Eq. (5)]. Inset: Zoom-in showing deviation of the period-averaged current from the quasisteady-state value. Fluctuations are significantly suppressed by disorder. (b) The populations of the 16 natural orbitals with highest occupation (approximately corresponding to the right-moving Floquet band) quickly converge around the expected value of 1/2 (red curves); the remaining 16 nearly empty natural orbitals are shown in blue, magnified by a factor of 10 for visibility. (c) Total occupation of the initially empty band, measured by the sum of the 16 smallest natural orbital populations. The timescale for interband excitation defined by the slopes of these curves (${\tau }_{\mathrm{inter}}^{-1}\approx 3\times {10}^{-4}/T$) is nearly unaffected by disorder. (d) Effect of disorder strength $\eta$ on the interband excitation rate ${\tau }_{\mathrm{inter}}^{-1}$. Each point is an average taken over 50 disorder realizations in a system of $L=28$ sites. The red dashed line indicates where the disorder bandwidth $2\eta$ is equal to the minimal instantaneous single-particle band gap over the driving cycle. Drive parameters [also corresponding to Fig. 1]: $\mathrm{\Omega }=0.23, J_0=1.0, \lambda =0.85, \delta J_0=0.0$, and $U=2J_0$. The modulations of the hopping and sublattice potentials are set by $\delta J_1=\lambda J_0$ and $V_1=3\lambda J_0$.

Figure 3

Time evolution of the entanglement entropy $S_{\mathrm{vN}}$ [Eq. (7)] and scaling of its saturation value in the quasisteady state with the number of sites, $L$. (a) Entanglement entropy $S_{\mathrm{vN}}$ for a half-system cut for a clean system in the topologically trivial (gray and orange curves) and nontrivial (blue curve) driving regimes. The black dashed and red solid lines indicate the entanglement entropies of the band-projected infinite-temperature state, $S_{\mathrm{vN}}^{\left(\infty \right)}$ [Eq. (8)], and its Page-corrected value, respectively. When the entanglement partition cuts through the tails between Wannier centers in the populated band, the quasisteady-state entanglement entropy of the topologically trivial system sits just above the Page-corrected value (orange curve). For an entanglement partition that cuts through the maximum of the Wannier function, an additional contribution to $S_{\mathrm{vN}}$ is observed in the quasisteady state (gray curve). Periodic spikes observed for the topologically nontrivial drive (blue curve) arise due to polarization pumping across the entanglement cut during the driving cycle. (b) Scaling of the absolute difference between the average of the time-averaged entropy $S_{\mathrm{vN}}$ in the numerically obtained quasisteady state (between the 10th and 20th period) and $S_{\mathrm{vN}}^{\left(\infty \right)}$, with system size $L$ (at quarter filling). (c) Extrapolation of the entanglement entropy density $S_{\mathrm{vN}}/L$ to the thermodynamic limit is consistent with the expected value of $ln\left(2\right)/4$ for an infinite-temperature state. Error bars show the standard deviation of the period-averaged entanglement entropy for periods 10–20. Driving parameters for the topologically trivial regime: $\mathrm{\Omega }=0.1, J_0=3.5, \lambda =0.25, \delta J_0=-2.2J_0$, and $U=1.0J_0$. Parameters for the topologically nontrivial driving regime are the same as those in Fig. 2.

Figure 4

Histogram of the time-dependent entanglement spectrum, $\left\{{\xi }_i\right\}$ [see text below Eq. (6)]. (a) Evolution of the stroboscopic ES histogram at integer multiples of the driving period. Bright streaks showing accumulation of entanglement levels slowly converge towards ${\xi }_{\mathrm{qs}}$, signifying the emergence of long-relaxation timescales associated with the relaxation of many-body correlations. (b) Interperiod evolution of the ES histogram. Sloped lines running between the bright streaks are associated with winding of the ES in the pumping regime [64]. Driving parameters are the same as those in Fig. 2, with $\eta =0$.

Figure 5

One-body observables in the topologically trivial driving regime. (a) As expected for the trivial phase, the current decays to 0 over the intraband equilibration timescale ${\tau }_{\mathrm{intra}}$. (b) Similar to the behavior in the topologically nontrivial driving regime shown in the main text, the 16 highest natural orbital populations converge to values around 0.5 on the timescale ${\tau }_{\mathrm{intra}}$. The remaining 16 natural orbital populations show periodic spikes (magnified by a factor of 10), with a structure that correlates well with the size of the instantaneous single-particle band gap. Simulation parameters: $\mathrm{\Omega }=0.1, J_0=3.5, \lambda =0.25, \delta J_0=-2.2J_0$, and $U=1.0J_0$.