Dynamics and evaporation of defects in Mott-insulating clusters of boson pairs

Repulsively bound pairs of particles in a lattice governed by the Bose-Hubbard model can form stable incompressible clusters of dimers corresponding to finite-size $n=2$ Mott insulators. Here we study the dynamics of hole defects in such clusters corresponding to unpaired particles which can resonantly tunnel out of the cluster into the lattice vacuum. Due to bosonic statistics, the unpaired particles have different effective mass inside and outside the cluster, and “evaporation” of hole defects from the cluster boundaries is possible only when their quasimomenta are within a certain transmission range. We show that quasithermalization of hole defects occurs in the presence of catalyzing particle defects which thereby purify the Mott-insulating clusters. We study the dynamics of a one-dimensional system using analytical techniques and numerically exact time-dependent density-matix renormalization-group simulations. We derive an effective strong-interaction model that enables simulations of the system dynamics for much longer times. We also discuss a more general case of two bosonic species which reduces to the fermionic Hubbard model in the strong interaction limit.

Figure 1

Physical models studies in this paper. (a) The Bose-Hubbard model [Eq. (1)]. (b) Monomer (hole defect) effective hopping (Sec. 3). (c) Single-defect effective theory (Sec. 3). (d) Trimer (particle defect) effective hopping (Sec. 4). (e) Single hole defect in two-species Bose-Hubbard model (Sec. 6).

Figure 2

Transmission probability $T\left(k\right)$ [cf. Eq. (A9)] for various $\alpha =J_B/J_A$.

Figure 3

Dynamics of the quasimomentum distribution for the monomer (left column) and trimer (right column) in a lattice of $L=64$ sites. The initial quasimomenta are $k_a=\frac{13}{16}\pi$ and $k_t=-\frac{9}{16}\pi$. Upper panels correspond to periodic boundary conditions, where the markers on the right indicate multiples of the revival time $t_{\mathrm{c}}\approx 46.63/J_t$. Lower panels are obtained for open boundary conditions. Dashed vertical lines mark the transmission regions for monomer quasimomenta as per Eq. (4).

Figure 4

Density of monomers (left column) and trimers (right column) in the $n=2$ MI cluster of 24 sites surrounded by empty lattice, ${|·\rangle }_{32}^0$, on both sides. In the top panels, the initial state of the cluster ${|-\pi /2\rangle }_8^{2-}{|·\rangle }_8^2{|\pi /2\rangle }_8^{2-}$ corresponds to two monomers at the center of the band moving to the left and right. In the central panel, the initial state ${|\pi \rangle }_8^{2-}{|·\rangle }_8^2{|0\rangle }_8^{2-}$ corresponds to two monomers at the upper and lower band edges. In the bottom panel, the initial state ${|\pi \rangle }_8^{2-}{|\pi /2\rangle }_8^{2+}{|0\rangle }_8^{2-}$ is the same as in the central panels plus a particle defect at the center of the band, moving to the right. The interaction strength is $U=100J$. The density of monomers (trimers) corresponds to the probability of finding exactly one (three) particles at a given site. A TEBD [18] algorithm with bond dimension $\chi =200$ is used for the time evolution with a fourth-order Trotter decomposition and time-step size $1/50J$, with particle number conservation explicitly included in the MPS [19].

Figure 5

Same as Fig. 4 but for localized initial states. Top panel, $|4_{-}{\rangle }_8^2{|·\rangle }_8^2{|4_{-}\rangle }_8^2$, two localized monomers; bottom panel, $|4_{-}{\rangle }_8^2|4_{+}{\rangle }_8^2{|4_{-}\rangle }_8^2$, same plus a localized trimer.

Figure 6

Total population (integrated density, ${\sum }_{j=1}^{31}\langle {\widehat{a}}_j^{†}{\widehat{a}}_j\rangle +{\sum }_{j=58}^{88}\langle {\widehat{a}}_j^{†}{\widehat{a}}_j\rangle$) of monomers outside the dimer cluster. (a) Initial states of the cluster are as follows: in the upper black branch, ${|-\pi /2\rangle }_8^{2-}{|·\rangle }_8^2{|\pi /2\rangle }_8^{2-}$ (solid line), ${|-\pi /2\rangle }_8^{2-}|4_{+}{\rangle }_8^2{|\pi /2\rangle }_8^{2-}$ (dashed line), and ${|-\pi /2\rangle }_8^{2-}{|\pi /2\rangle }_8^{2+}{|\pi /2\rangle }_8^{2-}$ (dot-dashed line); in the lower blue branch, ${|\pi \rangle }_8^{2-}{|·\rangle }_8^2{|0\rangle }_8^{2-}$ (solid line), ${|\pi \rangle }_8^{2-}|4_{+}{\rangle }_8^2{|0\rangle }_8^{2-}$ (dashed line), and ${|\pi \rangle }_8^{2-}{|\pi /2\rangle }_8^{2+}{|0\rangle }_8^{2-}$ (dot-dashed line). (b) Initial states of the cluster are as follows: $|4_{-}{\rangle }_8^2{|·\rangle }_8^2{|4_{-}\rangle }_8^2$ (solid line), $|4_{-}{\rangle }_8^2|4_{+}{\rangle }_8^2{|4_{-}\rangle }_8^2$ (dashed line), and $|4_{-}{\rangle }_8^2{|\pi /2\rangle }_8^{2+}{|4_{-}\rangle }_8^2$ (dot-dashed line). The interaction strength is $U=100J$. Numerical parameters are the same as in Figs. 4 and 5, and the curves terminate when the accumulated cutoff error equals ${10}^{-2}$. The gray lines are obtained from the equivalent effective model, with the time step increased to $1/10J$.

Figure 7

Total population (${\sum }_{j=1}^{63}\langle {\widehat{a}}_j^{†}{\widehat{a}}_j\rangle +{\sum }_{j=98}^{160}\langle {\widehat{a}}_j^{†}{\widehat{a}}_j\rangle$) of monomers outside the dimer cluster of 32 sites surrounded by empty lattice, ${|·\rangle }_{64}^0$, on both sides. (a) Initial states of the cluster are as follows: in the upper black branch, ${|-\pi /2\rangle }_8^{2-}{|·\rangle }_8^2{|\pi /2\rangle }_8^{2-}{|-\pi /2\rangle }_8^{2-}$ (solid line), ${|-\pi /2\rangle }_8^{2-}|4_{+}{\rangle }_8^2{|\pi /2\rangle }_8^{2-}{|-\pi /2\rangle }_8^{2-}$ (dashed line), and ${|-\pi /2\rangle }_8^{2-}{|\pi /2\rangle }_8^{2+}{|\pi /2\rangle }_8^{2-}{|-\pi /2\rangle }_8^{2-}$ (dot-dashed line); in the lower blue branch, ${|\pi \rangle }_8^{2-}{|·\rangle }_8^2{|0\rangle }_8^{2-}{|\pi \rangle }_8^{2-}$ (solid line), ${|\pi \rangle }_8^{2-}|4_{+}{\rangle }_8^2{|0\rangle }_8^{2-}{|\pi \rangle }_8^{2-}$ (dashed line), and ${|\pi \rangle }_8^{2-}{|\pi /2\rangle }_8^{2+}{|0\rangle }_8^{2-}{|\pi \rangle }_8^{2-}$ (dot-dashed line). (b) Initial states of the cluster are as follows: $|4_{-}{\rangle }_8^2{|·\rangle }_8^2|4_{-}{\rangle }_8^2{|4_{-}\rangle }_8^2$ (solid line), $|4_{-}{\rangle }_8^2|4_{+}{\rangle }_8^2|4_{-}{\rangle }_8^2{|4_{-}\rangle }_8^2$ (dashed line), and $|4_{-}{\rangle }_8^2{|\pi /2\rangle }_8^{2+}|4_{-}{\rangle }_8^2{|4_{-}\rangle }_8^2$ (dot-dashed line). (c) Initial states of the cluster are as follows: ${|\pi \rangle }_8^{2-}{|·\rangle }_8^2{|0\rangle }_8^{2-}{|4_{-}\rangle }_8^2$ (solid line), ${|\pi \rangle }_8^{2-}|4_{+}{\rangle }_8^2{|0\rangle }_8^{2-}{|4_{-}\rangle }_8^2$ (dashed line), and ${|\pi \rangle }_8^{2-}{|\pi /2\rangle }_8^{2+}|4_{-}{\rangle }_8^2{|4_{-}\rangle }_8^2$ (dot-dashed line). Simulations were performed with the effective model. Bond dimensions $\chi =300$ are used, and the time-step size is $1/10J$. The curves terminate when the accumulated cutoff error equals ${10}^{-1}$.

Figure 8

Density of unpaired particles b, or a holes, in the cluster (left column), and particles a (right column), in the lattice with an MI cluster of a-b dimers spanning 24 sites surrounded by empty lattice, ${|·\rangle }_{32}^0$, on both sides. The initial state of the cluster ${|-\pi /2\rangle }_8^{2-\mathrm{a}}{|·\rangle }_8^2{|\pi /4\rangle }_8^{2-\mathrm{a}}$ corresponds to two a-hole defects moving to the left with velocity $2J_{\mathrm{a}}$, and to the right with velocity $\sqrt{2}{J}_{\mathrm{a}}$, respectively, while all particles a are dimerized with particles b (no b-hole defects). The parameters are $U_{\mathrm{a}}=U_{\mathrm{b}}=60J_{\mathrm{b}}$, $U_{\mathrm{ab}}=40J_{\mathrm{b}}$, and $J_{\mathrm{a}}=J_{\mathrm{b}}$ (top panel), $J_{\mathrm{a}}=2J_{\mathrm{b}}$ (central panel), and $J_{\mathrm{a}}=\frac{1}{2}{J}_{\mathrm{b}}$ (bottom panel). A TEBD algorithm with bond dimension $\chi =100$ is used for the time evolution with a fourth-order Trotter decomposition and time-step size $1/50J_{\mathrm{b}}$, with the particle number conservation for each species explicitly included in the MPS.

Figure 9

Dynamics of an initially localized unpaired particle b, or a hole, in the cluster (left column), and an unpaired particle a, b hole, in the cluster (right column), for the initial cluster state $|4_{-\mathrm{a}}{\rangle }_8^2{|·\rangle }_8^2{|4_{-\mathrm{b}}\rangle }_8^2$. All parameters are as in Fig. 8, and the bond dimension of the TEBD is $\chi =200$.

Figure 10

Comparison of the full Bose-Hubbard dynamics [Eq. (1)] with $U=100J$ (a) and (b), and with the effective model [Eq. (9)] (c). The initial state contains an MI cluster of dimers on sites $j\ge 1$ and localized monomers at sites $j=-1$ and $j=2$. The density of dimers is shown in (a), and the density of monomers in (b)–(d). Note that the cluster boundary is shifted upon particle crossing, which manifests in (a) as a smoothing of step in the dimer density. The effective Hamiltonian without moving boundaries, $\widehat{H}={\sum }_{j=1}^{L-1}{\widehat{H}}_j^{\left[\Theta \right(j\left)\right]}$, yields the dynamics of (d).