Time Evolution of Correlation Functions in Quantum Many-Body Systems

We give rigorous analytical results on the temporal behavior of two-point correlation functions—also known as dynamical response functions or Green’s functions—in closed many-body quantum systems. We show that in a large class of translation-invariant models the correlation functions factorize at late times $⟨A(t)B{⟩}_{\beta }\to ⟨A{⟩}_{\beta }⟨B{⟩}_{\beta }$, thus proving that dissipation emerges out of the unitary dynamics of the system. We also show that for systems with a generic spectrum the fluctuations around this late-time value are bounded by the purity of the thermal ensemble, which generally decays exponentially with system size. For autocorrelation functions we provide an upper bound on the timescale at which they reach the factorized late time value. Remarkably, this bound is only a function of local expectation values and does not increase with system size. We give numerical examples that show that this bound is a good estimate in nonintegrable models, and argue that the timescale that appears can be understood in terms of an emergent fluctuation-dissipation theorem. Our study extends to further classes of two point functions such as the symmetrized ones and the Kubo function that appears in linear response theory, for which we give analogous results.

Figure 1

Plots of $\delta (\epsilon )$ (top) and $a(\epsilon )$ (bottom) for distribution $v_{\alpha }\equiv [{\rho }_{jj}|A_{jk}{|}^2/C^A(0)]$ in Theorem 3, obtained by exact diagonalization and a Monte Carlo approximation. The plots were generated with 10 000 sampled frequency intervals. Small values of $\delta$ imply equilibration occurs for long enough times, while the value of $a$ controls the prefactor in the equilibration timescale $T_{eq}\equiv [3\pi a(\epsilon )/{\sigma }_G]$ derived from Eq. (11). For small $\epsilon$ one can satisfy both $\delta \ll 1$ and $a\sim \mathcal{O}(1)$, and this becomes increasingly so for larger system sizes.

Figure 2

Comparison of the upper bound in Eq. (11) (rhs) with the simulated evolution of the time-averaged correlation function (lhs) as a function of time, for increasing number of spins $L$. The evolution obtained from the upper bound approaches the exact dynamics of the system for larger system size.