Boltzmann relaxation dynamics of strongly interacting spinless fermions on a lattice

Motivated by the recent interest in nonequilibrium phenomena in quantum many-body systems, we study strongly interacting fermions on a lattice by deriving and numerically solving quantum Boltzmann equations that describe their relaxation to thermodynamic equilibrium. The derivation is carried out by inspecting the hierarchy of correlations within the framework of the $1/Z$ expansion. Applying the Markov approximation, we obtain the dynamic equations for the distribution functions. Interestingly, we find that in the strong-coupling limit, collisions between particles and holes dominate over particle-particle and hole-hole collisions—in stark contrast to weakly interacting systems. As a consequence, our numerical simulations show that the relaxation timescales strongly depend on the type of excitations (particles or holes or both) that are initially present.

Figure 1

Sketch of a square lattice with a checker-board pattern as an example for a charge-density wave state (left) with a quasiparticle (middle) and quasihole (right) excitation. By definition of the model, the original fermions (blue dots) can only move to the nearest neighboring lattice sites, i.e., one step in horizontal or vertical direction, but not along the diagonal. Due to the strong repulsion $V$, a quasiparticle (middle) and quasihole (right) can only move to next-to-nearest neighboring lattice sites, which involves second-order tunneling processes such as co-tunneling of two fermions (middle) or sequential tunneling of one fermion (right). We also see that neither two quasiparticles (middle) nor two quasiholes (right) can occupy nearest neighboring lattice sites.

Figure 2

Band structure for a two-dimensional square lattice for $J/V={10}^{-3}$. We can see the Brillouin zone and parts of the adjacent ones. The gap is shrunk by six orders of magnitude to show the $\mathbf{k}$ dependence of both bands in one plot.

Figure 3

Time series (in units of $J^2/V^3)$ of the evolution of a low-energy excitation with symmetric initial conditions. For both bands at $t=0$, a set of states that is close to but not at the gap is perturbed.

Figure 4

Probability distribution at $t={10}^2{J}^2/V^3$ for the case of the symmetric initial conditions. Note that the occupation probabilities for the quasiholes are plotted as $1-f^{-}$.

Figure 5

Times series (in units of $J^2/V^3)$ of the evolution of a high energy excitation. The upper (quasiparticle) band is initially perturbed in the diamond shaped minimum, the lower (quasihole) band in the zone center.

Figure 6

Probability distribution at $t={10}^2{J}^2/V^3$ for the case of the asymmetric initial conditions. Note that the occupation probabilities for the quasiholes are plotted as $1-f^{-}$.