Fast trimers in a one-dimensional extended Fermi-Hubbard model

We consider a one-dimensional two-component extended Fermi-Hubbard model with nearest-neighbor interactions and mass imbalance between the two species. We study the binding energy of trimers, various observables for detecting them, and expansion dynamics. We generalize the definition of the trimer gap to include the formation of different types of clusters originating from nearest-neighbor interactions. Expansion dynamics reveal rapidly propagating trimers, with speeds exceeding doublon propagation in the strongly interacting regime. We present a simple model for understanding this unique feature of the movement of the trimers, and we discuss the potential for experimental realization.

Figure 1

Energy of a single trimer. (a) $V=V^{\prime }=-0.5t_{\uparrow },U=0$, (b) $V=V^{\prime }=-0.5t_{\uparrow },U=-0.5t_{\uparrow }$, (c) $V=V^{\prime }=-0.5t_{\uparrow },U=0.5t_{\uparrow }$, having the same legends, and (d) $V=-0.5t_{\uparrow },V^{\prime }=0, U=0$ and $U=-0.5t_{\uparrow }$. Shown are the energies of a trimer (configuration $N_{\uparrow }=1,N_{\downarrow }=2$), an $\uparrow -\downarrow$ dimer separated from a single atom (configurations $N_{\uparrow }=1,N_{\downarrow }=1$ and $N_{\uparrow }=0,N_{\downarrow }=1$), and a $\downarrow \downarrow$ dimer separated from a single atom (configurations $N_{\uparrow }=0,N_{\downarrow }=2$ and $N_{\uparrow }=1,N_{\downarrow }=0$), except for the last case in which the intraspecies dimer is unstable in absence of interactions. The separation of the two lowest lines is the trimer gap. In order to better show the effect of interactions, the total energy of the corresponding noninteracting system (roughly equal to $-t_{\uparrow }-2t_{\downarrow }$) has been subtracted from all energies.

Figure 2

Average size of a trimer. (a) $V=V^{\prime }=-0.5t_{\uparrow },U=0$, (b) $V=V^{\prime }=-0.5t_{\uparrow },U=-0.5t_{\uparrow }$, (c) $V=V^{\prime }=-0.5t_{\uparrow },U=0.5t_{\uparrow }$, and (d) $V=-0.5t_{\uparrow },V^{\prime }=0, U=0$.

Figure 3

Expansion of atoms for $V=-2,U=0$. Top left: density of the minority atom ($\uparrow$) as a function of time shows the expansion of the gas with a speed of roughly 13 sites in time $50\hslash /t_{\uparrow }$, i.e., 0.$26t_{\uparrow }/\hslash$. Top right: the density of the majority atoms ($\downarrow$) shows the same expansion. Bottom left: The same minority atom density in logarithmic scale shows much smaller but also faster single-particle wave front that propagates at speed of roughly 20 sites in time $10\hslash /t_{\uparrow }$, i.e., speed of $2t_{\uparrow }/\hslash$. Bottom right: The majority atom density in the logarithmic scale shows single-particle propagation speed of roughly $1t_{\uparrow }/\hslash$. Here hopping ratio is $t_{\downarrow }/t_{\uparrow }=0.5$, which matches well with the observed single-particle propagation speeds.

Figure 4

The $\text{ln}\langle n_{\downarrow ,i}{n}_{\downarrow ,j}\rangle$ (left) and $\text{ln}\langle n_{\uparrow ,i}{n}_{\downarrow ,j}\rangle$ (right) density matrices for $U=0.0,V=-10,t_{\mathrm{ratio}}=0.3$ at time $T=80$ show the broad diagonal contribution from trimer propagation and the much weaker dimer propagation appearing as a diagonal fork in the $\uparrow \downarrow$ correlator.

Figure 5

Trimer dynamics: the diagonal elements of the $\uparrow -\downarrow$ correlator, $\langle n_{\uparrow ,i}{n}_{\downarrow ,i}\rangle$ for $U=0.0,t_{\mathrm{ratio}}=0.7,V=-10.0$ (left) and $V=-20.0$ (right) at different times after the release from the initial strong confinement, shows the trimer propagation wave front.

Figure 6

Trimer propagation speed as a function of the hopping ratio $t_{\downarrow }/t_{\uparrow }$ for nearest-neighbor interaction strengths $V=-10$ and $V=-20$. Quadratic fits show the clear quadratic dependence on the hopping ratio, but only a weak dependence on the interaction strength $V$.

Figure 7

Doublon propagation speed as a function of the hopping ratio $t_{\downarrow }/t_{\uparrow }$ for nearest-neighbor interaction strengths $V=-10$ and $V=-20$. The doublon speed depends linearly on the hopping ratio, but it also depends strongly on the interaction strength. For strong interactions, the speed is inversely proportional to $V$.

Figure 8

Left: the average distance between the two $\downarrow$ atoms shows oscillations corresponding to transitions between different trimer states for $U=0.0,V=-20.0$ and $t_{\mathrm{ratio}}=0.7$. Right: Fourier transform of the average distance shows strong peaks at frequencies $\omega =3.0t_{\uparrow }$ and $\omega =1.3t_{\uparrow }$, yielding energy separations of the various trimer states.

Figure 9

Degenerate states of deeply bound trimers for equal nearest-neighbor interactions between like and opposite spins ($V=V^{\prime }$) and in absence of on-site interactions ($U=0$). All configurations have the total interaction energy of $2V$. The topmost state is connected to the next state through hopping of the $\downarrow$ atom, but the rest of the states are connected through the hopping of the $\uparrow$ atom. In addition to the states shown here, the mirror images of the four lower configurations are also degenerate in a symmetric way, and translationally symmetric states have been neglected (i.e., states with the same configuration as the ones shown here, but shifted along the lattice).

Figure 10

Left: The near degeneracy of the trimer configurations shown in Fig. 9 allows the trimers to propagate without intermediate virtual states involving high energy cost. The starting configuration is the same as the final configuration, except for a single lattice site shift to the right along the lattice. Right: For sufficiently large interactions $V\gg t_{\uparrow }$, the propagation of dimers is slower than for trimers, since dimer propagation necessarily involves virtual states with a broken nearest-neighbor dimer.