Impurity and soliton dynamics in a Fermi gas with nearest-neighbor interactions

We study spinless fermions with repulsive nearest-neighbor interactions perturbed by an impurity particle or a local potential quench. Using the numerical time-evolving block decimation method and a simplified analytic model, we show that the perturbations create a soliton-antisoliton pair. If solitons are already present in the bath, the two excitations have a drastically different dynamics: The antisoliton does not annihilate with the solitons and is therefore confined close to its origin while the soliton excitation propagates. We discuss the consequences for experiments with ultracold gases.

Figure 1

(a) Model: the bath contains spinless fermions (gray). At half filling, the ground state has a particle every two sites. Above half filling, solitons consisting of two neighboring occupied sites exist, as schematically shown. At time $t=0$, either an impurity particle (dark) or a static potential barrier is created at site $j_0$ at the center. (b) For incommensurate filling, the ground-state density of the bath shows two neighboring maxima or minima at the most probable locations of the solitons. Only the left half of the lattice with $j\le j_0$ is drawn; the other half is symmetric. The analytic result of our simplified domain-wall (DW) model agrees well with the TEBD solution. The number of fermions is 31 for the MI and 33 for the incommensurate filling, $L=61$, and $V=50J$ (see text).

Figure 2

Density distribution of the bath $\langle n_j^b\rangle$ and the impurity $\langle n_j\rangle$ in the first half of the lattice. Panel (a) [(b)] is for the commensurate (incommensurate) filling. The bath density is shown at $t=0$ and the bath and impurity densities at (a) $t=6/J$ and (b) $t=8/J$. The analytic result of the DW model in panel (a) agrees well with the TEBD result.

Figure 3

(a),(b) Soliton distribution $\langle n_j^b{n}_{j+1}^b\rangle$ as a function of position and time for the MI initial state when the system is perturbed with a static barrier or an impurity, respectively. The distributions are almost identical. (c),(d) The corresponding antisoliton distributions $\langle n_j^h{n}_{j+1}^h\rangle$. The antisoliton can form a bound state with the impurity (d), and a maximum remains at the center. (e),(f) The antisoliton distributions for incommensurate filling. The motion of the antisoliton is restricted because of the solitons present in the initial state.

Figure 4

(a) Number of antisolitons $N_{\text{AS}}$ as a function of time for both perturbations and initial states. A dashed line is used for the impurity since the lines for the static barrier and impurity overlap. When the initial state is a MI (incommensurate), $N_{\text{AS}}$ saturates at a value close to 1 (0.8). (b) The difference between the soliton distributions at times $t=6/J$ and $t=0,\delta \langle n_j^b{n}_{j+1}^b\rangle =\langle \psi \left(t\right)|n_j^b{n}_{j+1}^b|\psi \left(t\right)\rangle -\langle \psi \left(0\right)|n_j^b{n}_{j+1}^b|\psi \left(0\right)\rangle$, for incommensurate filling. The distributions are very close to each other for the two perturbations. (c)–(f) The antisoliton distribution $\langle n_j^h{n}_{j+1}^h\rangle$ at time $t=6/J$ for (c),(e) commensurate and (d),(f) incommensurate filling. For the impurity, the distribution has a maximum at the center whereas, for the static barrier, there is a minimum. In panels (a)–(d), $U=V=50J$ and in panels (e) and (f), $U=V=10J$.

Figure 5

Spinless fermions with nearest-neighbor interactions off half filling can be mapped to an $XXZ$ chain with a nonzero magnetization. The solitons of two neighboring occupied sites are mapped to two neighboring up spins.

Figure 6

Indexing of the lattice sites and bonds of the spin model.

Figure 7

Instead of lattice sites, the system can be represented as bonds. The bonds are either empty or occupied by a spinless fermion, corresponding to a domain wall in the spin chain.

Figure 8

Density distribution has one domain wall when $L=60$ and the number of fermions is $N_b=\frac{L}{2}+1$. The result of the DW model for $N=1$ agrees very well with the many-body TEBD solution.

Figure 9

Upper panel: ground-state density distribution in the MI and incommensurate phases. The nearest-neighbor interaction is $V=50J$ and the size of the lattice is $L=61$. The number of fermions is $N_b=31$ in the MI case and $N_b=33$ in the incommensurate case, which gives $N_{\text{DW}}=4$. Lower panel: the density distribution of free fermions $\langle n_j\rangle$ as in Eq. (A24) with $N=4$.

Figure 10

Correlator $\langle \psi \left(t\right)|\mathrm{\Delta }{n}_i^b\mathrm{\Delta }{n}_{-i}^b|\psi \left(t\right)\rangle$ at different time steps in the case of (a),(c) a static potential barrier and (b),(d) an impurity. Panels (a) and (b) correspond to the MI and panels (c) and (d) to the incommensurate phase.

Figure 11

Density distribution of the bath fermions and the impurity in (a) the commensurate and (b) the incommensurate phase. The bath density is shown for both perturbations at times $t=0$ and (a) $t=6/J$, (b) $t=8/J$. The nearest-neighbor and on-site interactions are $V=U=10J$.

Figure 12

Panels (a) and (b) show the antisoliton distribution $\langle n_j^{\text{h}}{n}_{j+1}^{\text{h}}\rangle$ as a function of position and time for the MI initial state when the system is perturbed with a static barrier or an impurity, respectively. Panels (c) and (d) show the corresponding soliton distributions $\langle n_j^{\text{b}}{n}_{j+1}^{\text{b}}\rangle$. Panels (e) and (f) show the antisoliton distribution for incommensurate filling.

Figure 13

Number of antisolitons in the MI and incomensurate states for $U=V=10J$. The initial value is larger than for $U=V=50J$ since the energy cost of neighboring occupied sites in the ground state is smaller.

Figure 14

Density of particles in a harmonic trap, in the ground state of Hamiltonian (D1). Upper: for a sufficiently shallow trap, the central region of the system is in the MI state. Lower: for a larger $V_0$ or $N_b$, the central region of higher average density is surrounded by Mott insulator regions. Here, $L=80$, and the density distributions extend over approximately 60 sites.

Figure 15

Density distribution in the ground state of Hamiltonian (D1) calculated by TEBD. The strength of the harmonic potential $V_{\text{trap}}\left(j\right)=V_0{(j-j_0)}^2$ is $V_0=0.005J$. The filling is incommensurate with $N_b=33$ and $L=61$, which leads to four domain walls as in Fig. 9. The height of the potential at the edges of the lattice is $4.5J$ and the nearest-neighbor repulsion is $V=10J$.

Figure 16

Antisoliton distributions as in Figs. 12–12 in the case of a harmonic trapping potential with $V_0=0.005J$. Here, $L=61,N_b=33$, and the nearest-neighbor repulsion is $V=10J$. At commensurate filling, the distribution is not significantly modified from the case of a uniform potential, whereas at incommensurate filling, the antisoliton is confined closer to the center of the lattice.