Quench dynamics in lattices above one dimension: the free fermionic case

We begin a systematic investigation of quench dynamics in higher-dimensional lattice systems considering the case of non-interacting fermions with conserved particle number. We prepare the system in a translational-invariant non-equilibrium initial state -- the simplest example being a classical configuration with fermions at fixed positions on the lattice -- and let it to evolve in time. We characterise the system's dynamics by measuring the entanglement between a finite connected region and its complement. We observe the transmutation of entanglement entropy into thermodynamic entropy and investigate how this process depends on the shape and orientation of the region with respect to the underlying lattice. Interestingly, we find that irregular regions display a distinctive multi-slope entanglement growth, while the dependence on the orientation angle is generically fairly weak. This is particularly true for regions with a large (discrete) rotational symmetry group. The main tool of our analysis is the celebrated quasiparticle picture of Calabrese and Cardy, which we generalise to describe the case at hand. Specifically, we show that for generic initial configurations (even when restricting to classical ones) one has to allow for the production of multiplets involving ${n>2}$ quasiparticles and carrying non-diagonal correlations. We obtain quantitatively accurate predictions -- tested against exact numerics -- and propose an efficient Monte Carlo-based scheme to evaluate them for arbitrary connected regions of generic higher dimensional lattices.