Flux quench in a system of interacting spinless fermions in one dimension

We study a quantum quench in a one-dimensional spinless fermion model (equivalent to the XXZ spin chain), where a magnetic flux is suddenly switched off. This quench is equivalent to imposing a pulse of electric field and therefore generates an initial particle current. This current is not a conserved quantity in the presence of a lattice and interactions, and we investigate numerically its time evolution after the quench, using the infinite time-evolving block decimation method. For repulsive interactions or large initial flux, we find oscillations that are governed by excitations deep inside the Fermi sea. At long times we observe that the current remains nonvanishing in the gapless cases, whereas it decays to zero in the gapped cases. Although the linear response theory (valid for a weak flux) predicts the same long-time limit of the current for repulsive and attractive interactions (relation with the zero-temperature Drude weight), larger nonlinearities are observed in the case of repulsive interactions compared with that of the attractive case.

Figure 1

Dynamics of the current after the quench for ${\theta }_0=\pi /2$ (upper left), ${\theta }_0=\pi /6$ (upper right), and ${\theta }_0=\pi /10$ (bottom left and right).

Figure 2

Fitting of numerical data for initial flux ${\theta }_0=\pi /6$. Roughly up to first oscillation, the empirical fitting works well. The black arrow indicates the LR prediction of the long-time limit of the current for $\mathrm{\Delta }=\pm 0.5$ [Eq. (8)].

Figure 3

Frequency $\omega$ from numerical fitting. Error bar is estimated by error of fitting.

Figure 4

Time evolution of the momentum distribution $n_q$ for (left) ${\theta }_0=\pi /3,\mathrm{\Delta }=-0.5$ and (right) ${\theta }_0=\pi /6,\mathrm{\Delta }=0.5$. An oscillating dip/peak structure is visible in the left panel (marked by arrows), whereas such structure is absent in the right case. See also the Supplemental Material [17].

Figure 5

Left: Momentum distribution for $\mathrm{\Delta }=-0.5$ with different ${\theta }_0$. The time is chosen to match the first minimum of the current after the quench. Right: Momentum of dip, $p^{\mathrm{dip}}$, versus ${\theta }_0$.

Figure 6

Numerical data for (left) $J(t=\infty )/D$ and (right) normalized deviation from LR theory $1-J(t=\infty )/\left(D{\theta }_0\right)$.

Figure 7

Time evolution of the momentum distribution in the gapped phase. Left panel: ${\theta }_0=\pi /2,\mathrm{\Delta }=-1.2$. Right panel: ${\theta }_0=\pi /6,\mathrm{\Delta }=-1.2$. In both cases, the whole momentum distribution moves towards the center $\left(q=0\right)$ and the current decays to zero.

Figure 8

Relaxation time $\tau$ versus the interaction parameter $\mathrm{\Delta }$. Left panel: semi-log plot. Right panel: log-log plot. The data points where the decay is too slow to be reliably quantified with our calculations are omitted.

Figure 9

Growth of entanglement entropy of a half chain for ${\theta }_0=\pi /6$ (left) and ${\theta }_0=\pi /2$ (right) and various values of $\mathrm{\Delta }$. One can check that the (weak) oscillations of the entanglement entropy are in phase with the oscillations of the current.

Figure 10

Expectation value of the current, computed with several values of the bond dimension $\chi$. In this case we retain the data for $\chi =1000$, and only up to $t=20$ (indicated by arrow).