Equilibration via Gaussification in Fermionic Lattice Systems

In this Letter, we present a result on the nonequilibrium dynamics causing equilibration and Gaussification of quadratic noninteracting fermionic Hamiltonians. Specifically, based on two basic assumptions—clustering of correlations in the initial state and the Hamiltonian exhibiting delocalizing transport—we prove that non-Gaussian initial states become locally indistinguishable from fermionic Gaussian states after a short and well controlled time. This relaxation dynamics is governed by a power-law independent of the system size. Our argument is general enough to allow for pure and mixed initial states, including thermal and ground states of interacting Hamiltonians on large classes of lattices as well as certain spin systems. The argument gives rise to rigorously proven instances of a convergence to a generalized Gibbs ensemble. Our results allow us to develop an intuition of equilibration that is expected to be more generally valid and relates to current experiments of cold atoms in optical lattices.

Figure 1

The right panel shows a numerical study of the spreading of the support of a fermionic annihilation operator described in Eq. (2) under the evolution of the quadratic hopping Hamiltonian $H=-\sum _j(f_j^{†}{f}_{j+1}+f_{j+1}^{†}{f}_j)$ on a one-dimensional chain of 150 sites with periodic boundary conditions. The support expands ballistically, creating the Lieb-Robinson cone. The left panel shows the suppression of different elements of the propagator in time. The plot at the top shows the evolution of the maximum taken over the full lattice ${\max }_k|W_{75,k}(t)|$, where in the lower plot the maximum inside the inner region of the Lieb-Robinson cone (between the red dashed lines) is plotted. The maximum taken over the full lattice is reached for $k$ in the wave front (indicated by the blue curve in the right panel) and the suppression goes as $t^{-1/3}$, while in the bulk of the cone, the suppression is proportional to $t^{-1/2}$. The suppression stabilizes, once the wave fronts collide.

Figure 2

Numerical study of the evolution of a nearest-neighbor density-density correlator for system sizes $V=32$, 64, 128 under the quadratic hopping Hamiltonian $H=-\sum _j^V(f_j^{†}{f}_{j+1}+f_{j+1}^{†}{f}_j)$. The initial states are the ground states of the interacting spinless Fermi-Hubbard model $H_{\mathrm{FH}}=H+U\sum _j^V{n}_j{n}_{j+1}+\sum _j{\omega }_j{n}_j$ with $U=2$, and weak on site disorder $w_j$ drawn independently from a Gaussian distribution with variance $1/4$. For consistency, we have initially drawn 128 random numbers $w_j$ and used the first $V$ of them for different system sizes. All calculations were performed with periodic boundary conditions at half filling. The difference between the expectation values in the state $\rho$ and its Gaussified version ${\rho }_G$ of the density-density correlator between sites 27 and 28 as a function of time is suppressed approximately like $t^{-1}$, as indicated by the black line. At late times, due to the finite size of the system, recurrences occur, leading to an increase of the difference. Increasing the system size only shifts the recurrence time $t_{\mathrm{Rec}}$ but leaves the decay behavior unchanged. The visible oscillations depend on details of the model and initial state. The time evolution was performed by using Eq. (2).