Effects of interactions on periodically driven dynamically localized systems

It is known that there are lattice models in which noninteracting particles get dynamically localized when periodic $\delta$-function kicks are applied with a particular strength. We use both numerical and analytical methods to study the effects of interactions in three different models in one dimension. The systems we have considered include spinless fermions with interactions between nearest-neighbor sites, the Hubbard model of spin-1/2 fermions, and the Bose-Hubbard model with on-site interactions. We derive effective Floquet Hamiltonians up to second order in the time period of kicking. Using these we show that interactions can give rise to a variety of interesting results such as two-body bound states in all three models and dispersionless few-particle bound states with more than two particles for spinless fermions and bosons. We substantiate these results by exact diagonalization and stroboscopic time evolution of systems with a few particles. We derive a pseudo-spin-1/2 limit of the Bose-Hubbard system in the thermodynamic limit and show that a special case of this has an exponentially large number of degenerate eigenstates of the effective Hamiltonian. Finally, we study the effect of changing the strength of the $\delta$-function kicks slightly away from perfect dynamical localization; we find that a single particle remains dynamically localized for a long time after which it moves ballistically.

Figure 1

Numerically obtained eigenvalues of the Floquet operator as compared with the analytical expressions in Eqs. (44) and (45), for $V=1,T=0.5$, and $\gamma =1$. All other eigenvalues are zero. We have two particles on 20 sites.

Figure 2

Time evolution of a two-particle state for four cases: (i) $V=0$, no kicking, (ii) $V=1$, no kicking, (iii) $V=0$, with kicking, and (iv) $V=1$, with kicking. In all cases $\gamma =1$ and $T=0.5$. There are two particles on 20 sites, and they are initially located at two adjacent sites. The color shows the expectation value of the particle number at different sites.

Figure 3

Time evolution of a state with two particles on 20 sites in the presence of kicking, for four cases. In (i) the two particles are initially on adjacent sites and there is no interaction ($V=0$). The state is dynamically localized due to kicking. In (ii)–(iv), the initial distance between the particles is progressively increased from two to four lattice spacings, and interactions are present with $V=1$. The color shows the expectation value of the particle number at different sites. Note that with increasing initial spacing the overlap with the two-particle bound states gets reduced, and the states get more localized.

Figure 4

Numerically obtained eigenvalues of the Floquet operator as compared with the analytical expression in Eq. (46), for $V=1,T=0.5$ and $\gamma =1$, for $n=3,4,5$ particles on 12 sites. Note that for each $n$, we have $N=12$ eigenvalues which are nondispersing.

Figure 5

Time evolution of a four-particle state: (i) $V=1$, no kicking, and (ii) $V=1$, with kicking. In both cases, $\gamma =1$ and $T=0.5$. There are four particles on 12 sites, and they are initially located on four adjacent sites. The second row shows that the particles do not move even in the presence of interactions.

Figure 6

Numerically obtained eigenvalues of the effective Hamiltonian as compared with the analytical expressions in Eqs. (55) and (59), for $U=1,T=0.25$, and $\gamma =1$. All other eigenvalues are zero. We have one $\uparrow$ and one $\downarrow$ particle on 20 sites.

Figure 7

Time evolution of a two-particle state for four cases: (i) $U=0$, no kicking, (ii) $U=1$, no kicking, (iii) $U=0$, with kicking, and (iv) $U=1$, with kicking. In all cases $\gamma =1$ and $T=0.25$. There are two particles, with spins $\uparrow$ and $\downarrow$, on 20 sites, and they are initially located at the same site. The colors of the dots on the outer (inner) ring show the expectation values of the number of up (down) spin particles at different sites.

Figure 8

Plot of $v_{\max }$ versus $\alpha$ for $T=0.5$ and $\gamma =1$. The solid red line shows the analytical result obtained from Eq. (96), while the black squares show the result obtained numerically from a study of the propagation of a particle as discussed in the text.

Figure 9

Plot of $m_2$ versus $t$ for $\alpha =3.12$, $T=0.5$, and $\gamma =1$. The particle is initially at a site in the middle of a system with 2000 sites. At short times, $m_2$ alternates between two values depending on whether $t/T$ is an odd or even integer. At long times, a power law fit between $m_2$ and $t$ shows that $m_2$ increases as $t^{2.0}$, implying that the particle is moving ballistically.