Effective Hamiltonians for Rapidly Driven Many-Body Lattice Systems: Induced Exchange Interactions and Density-Dependent Hoppings

We consider 1D lattices described by Hubbard or Bose-Hubbard models, in the presence of periodic high-frequency perturbations, such as uniform ac force or modulation of hopping coefficients. Effective Hamiltonians for interacting particles are derived using an averaging method resembling classical canonical perturbation theory. As is known, a high-frequency force may renormalize hopping coefficients, causing interesting phenomena such as coherent destruction of tunneling and creation of artificial gauge fields. We find explicitly additional corrections to the effective Hamiltonians due to interactions, corresponding to nontrivial processes such as single-particle density-dependent tunneling, correlated pair hoppings, nearest neighbor interactions, etc. Some of these processes arise also in multiband lattice models, and are capable of giving rise to a rich variety of quantum phases. The apparent contradiction with other methods, e.g., Floquet-Magnus expansion, is explained. The results may be useful for designing effective Hamiltonian models in experiments with ultracold atoms, as well as in the field of ultrafast nonequilibrium magnetism. An example of manipulating exchange interaction in a Mott-Hubbard insulator is considered, where our corrections play an essential role.

Figure 1

Exchange interaction $J_{\mathrm{ex}}$ in the driven fermionic two-site model as a function of the driving strength ${\mathcal{E}}_0$. Parameters: $U=10$, “bare” tunneling $J=-1$, $\omega =16$. Harmonic driving changes the effective tunneling constant $\mathbf{(}J\to J_e=J{J}_0({\mathcal{E}}_0)\mathbf{)}$ affecting the exchange interaction in the zeroth order in $1/\omega$: $J_{\mathrm{ex}}^{(0)}=[2J_e^2({\mathcal{E}}_0)/U]$. Solid curves, from up to down: bare exchange interaction $J_{\mathrm{ex}}^{(0)}$; theoretical prediction $J_{\mathrm{ex}}^{(2)}$ with corrections up to the second order in $1/\omega$ taken into account; $J_{\mathrm{ex}}^{(4)}$ with corrections up to the fourth order in $1/\omega$ taken into account; $J_{\mathrm{ex}}^{(\infty )}$ with infinite order of terms in $1/\omega$ expansion taken into account (with leading order in $U$ contribution in each term) (the lowest solid curve). Dashed curve (nearly indiscernible from the lowest solid curve): numerical value of the exchange interaction (kindly provided by J. Mentink). In the regions where $J_e$ is strongly suppressed by driving (around zeros of $J_{\mathrm{ex}}^{(0)}$), the second order corrections become insufficient, and one needs to include fourth and higher orders in $1/\omega$. $J_{\mathrm{ex}}^{(2),(4),\dots }$ are defined as the half-distance between the two lowest levels of the effective Hamiltonian of the 2-site model [36].