Dynamics of doublon-holon pairs in Hubbard two-leg ladders

The dynamics of holon-doublon pairs is studied in Hubbard two-leg ladders using the time-dependent density matrix renormalization group method. We find that the geometry of the two-leg ladder, which is qualitatively different from a one-dimensional chain due to the presence of a spin gap, strongly affects the propagation of a doublon-holon pair. Two distinct regimes are identified. For weak interleg coupling, the results are qualitatively similar to the case of the propagation previously reported in Hubbard chains, with only a renormalization of parameters. More interesting is the case of strong interleg coupling where substantial differences arise, particularly regarding the double occupancy and properties of the excitations such as the doublon speed. Our results suggest a connection between the presence of a spin gap and qualitative changes in the doublon speed, indicating a weak coupling between the doublon and the magnetic excitations.

Figure 1

Creation of a doublon-holon pair in a two-leg Hubbard ladder.

Figure 2

Charge $\langle n_i\rangle$ at each site for different times for the cases $t_y=0$ [(a)–(c)] and $t_y=3$ [(d)–(f)]. Charge values in the upper leg are shifted by $0.2$ for better visualization. Times are $\tau =0$ [(a), (d)], $\tau =1$ [(b), (e)], and $\tau =2$ [(c), (f)].

Figure 3

Charge $\langle n_i\rangle$ on sites close to the holon-doublon creation sites (8 and 10) for $t_y=0$ [(a), (b)] and $t_y=3$ [(c), (d)].

Figure 4

Double occupancy $\langle n_{i\uparrow }{n}_{i\downarrow }\rangle$ at each site for different times for the cases $t_y=0$ [(a)–(c)] and $t_y=3$ [(d)–(f)]. Double occupancy values in the upper leg are shifted by $0.2$ for better visualization. Times are $\tau =0$ [(a), (d)], 1 [(b), (e)], and 2 [(c), (f)].

Figure 5

Double occupancy vs time at sites close to the holon-doublon creation sites (8 and 10) for $t_y=0$ [(a), (b)] and $t_y=3$ [(c), (d)].

Figure 6

Excess double occupancy $\Delta N_d\left(\tau \right)$ vs $\tau$ for $U=10$ and different values of $t_y$. Inset: $\Delta N_d$ at fixed times ($\tau =0$ and $\tau =6$) vs $t_y$.

Figure 7

Double occupancy at a site at the end of the ladder for (a) $t_y\le 1$ and (b) $t_y\ge 1$ (no shift added). The position of the peak marks the “arrival time” ${\tau }_a$ of the doublon. Inset: Inverse doublon arrival time ${\tau }_a^{-1}$ (circles) and spin gap calculated for a $2\times 32$ ladder (squares) vs $t_y$. Error bars are given by $\Delta \tau /{\tau }^2$ and indicate the uncertainty in the determination of the arrival time given the time step $\Delta \tau$.

Figure 8

(a) Average energy $E\left(\tau \right)$ as a function of $\tau$ for $U=10$ with $m=400$ kept states and different values of $t_y$. In (b), $E\left(\tau \right)/E\left(0\right)$ is plotted, with dashed and dotted horizontal lines marking the 5% and 10% “confidence ranges,” respectively.

Figure 9

Comparison between ladder and chain calculations: Double occupancy at a site located near the end of the system vs time for a chain with $m=400$ kept states and a ladder with $t_y=0$ and $m=200,400$ [same data as in Fig. 7a], and 600 kept states. Inset: $E\left(\tau \right)$ in the ladder for $t_y=0$ and different numbers of kept states during the time evolution.