Strong quenches in the one-dimensional Fermi-Hubbard model

The one-dimensional Fermi-Hubbard model is used as test bed for strong global parameter quenches. With the aid of iterated equations of motion in combination with a suitable scalar product for operators, we describe the dynamics and the long-term behavior in particular of the system after interaction quenches. This becomes possible because the employed approximation allows for oscillatory dynamics avoiding spurious divergences. The infinite-time behavior is captured by an analytical approach based on stationary phases; no numerical averages over long times need to be computed. We study the most relevant frequencies in the dynamics after the quench and find that the local interaction $U$ as well as the bandwidth $W$ dominate. In contrast to former studies, a crossover instead of a sharp dynamical transition depending on the strength of the quench is identified. For weak quenches, the bandwidth is more important while for strong quenches the local interaction $U$ dominates.

Figure 1

Sketch of the one-dimensional Fermi-Hubbard model in real space. Electron hopping is determined by the matrix element $J$, double occupancy of a site costs an additional energy $U$.

Figure 2

Combined effect of hopping and interaction: green (red) colored arrows stand for particle (hole) creation operators, the spin direction is described by the arrow orientation.

Figure 3

Local particle number computed for the 3-basis and the $3^{+}$-basis, for two interaction strengths $U$ at half-filling $n=\frac{1}{2}$. The horizontal dashed infinite-time averages $n_{\infty }$ are depicted for orientation.

Figure 4

Infinite-time averages $n_{\infty }$ for the 3-basis and $3^{+}$-basis depending on $U$. The dashed black lines denote the analytically correct result for half-filling. Especially the $3^{+}$-basis is able to keep this value regardless of $U$ except for a dip around $U\approx J$ while the 3-basis appears not to be very reliable. Calculations in the upper (or lower) panel are computed using the Lanczos algorithm for a maximum Krylov space dimension $f$ of a third of the overall Hilbert space dimension (or $f\le 1000$). A lattice size of $N=20$ was examined for the 3-basis only.

Figure 5

Momentum distribution for a lattice of $N=12$ sites after being quenched to $U=4J$ vs time. Only the range of $k\in [0,\pi ]$ (lattice constant is set to unity) is shown due to symmetry. Finite-size features are the washed-out transition between occupied and unoccupied states at $t=0$. Starting from the Fermi sea, an out-of-phase oscillation of the initially occupied and unoccupied states occurs with almost featureless distributions in-between.

Figure 6

Momentum distribution for a lattice of $N=12$ sites after being quenched to $U=20J$ vs time. Qualitatively, the same behavior as in Fig. 5 can be observed. Compared to results of $U=4J$, the time period of the oscillations is much shorter and there is a huge increase in their amplitudes. At about $t\approx 1.7J^{-1}<t_{\mathrm{WA}}$, a dip in the momentum distribution for excitations of long wavelengths, i.e., $k\approx 0$, is visible.

Figure 7

Infinite-time average of the momentum distribution $n_k^{\infty }$ for two different values of $U$ for a periodic chain of $N=12$ sites. The infinite-time averaged distributions are very similar and featureless. They show comparatively highly occupied momenta for $k>\frac{\pi }{2}$ which were initially completely empty. This calculation is based on a complete diagonalization and does not make use of the Lanczos algorithm.

Figure 8

Comparison of results obtained using the scalar product for operators (solid lines) and results calculated for normal-ordered operators without scalar product [44, 73] (dashed lines). The latter results are calculated in a highly accurate approach, but are only converged up to short times. The respective maximum times are given by the three different plot symbols. Triangles at the $x$ axis mark predictions for zeros of the curve based on the Rabi oscillations (46).

Figure 9

Jump at the Fermi surface for two values of $U$ (upper and lower panels) in dependence on the lattice size $N$. Finite-size revivals occur the earlier the smaller the lattice is. Once a finite-size revival has occurred, the data are cropped by hand. The corresponding end of curve is denoted by the four different glyphs. Dashed vertical lines mark $t_{\mathrm{WA}}$, i.e., the time up to which results are estimated to be independent of $N$, colored triangles mark the instants $2t_{\mathrm{WA}}$ which indeed almost coincide with the onset of the spurious finite-size revivals.

Figure 10

Spectra as obtained by Fourier transform from the time-dependent symmetrized Fermi jump for weaker interaction quenches. The symbols with error bars indicate spectral features which we read off.

Figure 11

Spectra as obtained by Fourier transform from the time-dependent symmetrized Fermi jump for stronger interaction quenches. The symbols with error bars indicate spectral features which we read off.

Figure 12

Frequencies of the most important spectral features in the spectra of $\mathrm{\Delta }n\left(t\right)$ plotted as functions of the interaction strength. The latter is given in a compactified form with $U_c=-8E_{\mathrm{kin}}/N$ where $E_{\mathrm{kin}}/N=-W/\pi$ is the kinetic energy per site of the half-filled noninteracting system. This allows us to show the full range from $U=0$ to $\infty$. The error bars are determined approximately by the half-widths at half-maximum of the peaks. The frequency of local Rabi oscillations, i.e., $\omega =U$, and the bandwidth, i.e., $\omega =W$, are depicted for comparison as dashed lines.

Figure 13

Infinite-time averages of the jump $\mathrm{\Delta }{n}_{\infty }$ calculated from (38). No pronounced dependence on $U$ is found except for weaker quenches $U<2W=8J$ (see discussion in main text). The calculation shown is performed using the Lanczos algorithm with a maximum Krylov space dimension $f\le 1000$.

Figure 14

Mean values $\overline{\mathrm{\Delta }{n}_{\infty }}$ of the infinite-time jump $\mathrm{\Delta }{n}_{\infty }$ averaged between $U=15J$ and $U=40J$ vs the inverse lattice size. The linear fit 1 (2) includes (excludes) the data point for $N=4$.