Dynamical transition in interaction quenches of the one-dimensional Hubbard model

We show that the nonequilibrium time evolution after interaction quenches in the one-dimensional, integrable Hubbard model exhibits a dynamical transition in the half-filled case. This transition ceases to exist upon doping. Our study is based on systematically extended equations of motion. Thus it is controlled for small and moderate times; no relaxation effects are neglected. Remarkable similarities to the quench dynamics in the infinite-dimensional Hubbard model are found, suggesting that dynamical transitions are a general feature of quenches in such models.

Figure 1

Upper panel: Jump $\Delta n$(t) for the half-filled Hubbard model for various loop numbers. Lower panel: $\Delta n\left(t\right)$ for increasing $U$ (from top to bottom at small $t$: $U/W=0.125,0.25,0.5,0.8,1.0,1.25,1.5,2.0,5.0$) in 11 loops.

Figure 2

Jump $\Delta n\left(t\right)$ for the half-filled Hubbard model in 1D (black, solid lines) and on the Bethe lattice with infinite coordination number (gray, dashed lines from Ref. 12). Dotted line: Second-order result $\Delta n=1-U^2{t}^2/4$.

Figure 3

Oscillations in $\Delta n\left(t\right)$ for various dopings. In the left panel, the dopings are (from top to bottom at the first minimum) $n=0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0$. In the right panel, the dopings are (from top to bottom at the minimum) $n=0.8,0.82,0.84,0.86,0.88,0.9,0.92,0.94,0.96,0.98,1.0$. Clearly, zeros occur only at half filling.

Figure 4

Period of the oscillations in $\Delta n\left(t\right)$ as a function of $U$ ($U_c=8W/\pi$ in 1D); $n$ denotes the total filling factor per site. The dotted curve depicts the semiclassical Gutzwiller result (2).[27]

Figure 5

Absolute difference of the jump $\Delta n_m\left(t\right)$ at various numbers of loops $m$ relative to the 11-loop result $\Delta n_{11}\left(t\right)$ for the half-filled Hubbard model. Dashed line: Threshold for the determination of the runaway time.

Figure 6

Double logarithmic plot of the inverse runaway time vs the inverse number of loops of the corresponding calculation. Dashed line: Power-law fit of the data with an exponent of about $1.8$.

Figure 7

Period of the oscillations in $\Delta n\left(t\right)$ as a function of $U$ ($U_c=8W/\pi$ in 1D); $n$ denotes the total filling factor per site.

Figure 8

Period of the oscillations for the fillings $n=1.0,0.8,0.5$. The error bars depict the error of the calculated period for some exemplary values of the interaction. They are estimated as described in the main text.