Relaxation to Gaussian and generalized Gibbs states in systems of particles with quadratic Hamiltonians

We present an elementary, general, and semiquantitative description of relaxation to Gaussian and generalized Gibbs states in lattice models of fermions or bosons with quadratic Hamiltonians. Our arguments apply to arbitrary initial states that satisfy a mild condition on clustering of correlations. We also show that similar arguments can be used to understand relaxation (or its absence) in systems with time-dependent quadratic Hamiltonians and provide a semiquantitative description of relaxation in quadratic periodically driven (Floquet) systems.

Figure 1

Magnitude of the single-particle propagator $|G_{xy}\left(t\right)|=|J_{x-y}\left(t\right)|$ for the model described by Eq. (11).

Figure 2

Schematic showing spreading of operators in the model of Eq. (11) and how this leads to the decay of connected correlation functions as time $t$ increases. The points $x_j$ are the locations of the operators on the left side of Eq. (55); the points $y_j$ are for a representative term in the sum on the right side of this equation. Each $y_j$ must lie inside the backward light cone of $x_j$ in order for the propagator $G_{x_j{y}_j}\left(t\right)$ to be nonzero. Configurations of the $y$'s in which the distance between any pair is much greater than the correlation length $\xi$ (as is the case in the figure) give negligible contributions due to clustering of correlations in the initial state. This effectively restricts the sum over $y_1\cdots y_4$ to a region of size $\sim {\xi }^{3}t$ [Eq. (56)].

Figure 3

Schematic showing how the spreading of operators in $d=2$ dimensions causes the connected 4-point function $\langle \langle {\widehat{\psi }}_{x_1}^{a_1}\left(t\right){\widehat{\psi }}_{x_2}^{a_2}\left(t\right){\widehat{\psi }}_{x_3}^{a_3}\left(t\right){\widehat{\psi }}_{x_4}^{a_4}\left(t\right)\rangle \rangle$ to decay as time $t$ increases. As in Eq. (104), this function is expressed as a weighted sum of connected 4-point functions at time zero, $\langle \langle {\widehat{\psi }}_{y_1}^{b_1}{\widehat{\psi }}_{y_2}^{b_2}{\widehat{\psi }}_{y_3}^{b_3}{\widehat{\psi }}_{y_4}^{b_4}\rangle \rangle$. Cluster decomposition ensures that only configurations of the $y$'s of the form depicted in the left panel contribute to the sum; configurations like that shown in the right panel do not, because the connected function $\langle \langle {\widehat{\psi }}_{y_1}^{b_1}{\widehat{\psi }}_{y_2}^{b_2}{\widehat{\psi }}_{y_3}^{b_3}{\widehat{\psi }}_{y_4}^{b_4}\rangle \rangle$ is negligible. This restriction in allowed phase space is ultimately responsible for the power-law decay of all connected 3- and higher-point functions, as explained in the text.

Figure 4

Paradigmatic spreading behaviors of 1-particle propagators. This list is certainly not exhaustive, but the spreading behaviors shown may be regarded as typical.