Time evolution of correlation functions in quantum many-body systems

Álvaro M. Alhambra, Jonathon Riddell, Luis Pedro García-Pintos

Published 2019-06-26Version 1

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 models the correlation functions factorize at late times $\langle A(t) B\rangle_\beta \rightarrow \langle A \rangle_\beta \langle B \rangle_\beta$, thus proving that dissipation emerges out of the unitary dynamics of the system. We also show that the fluctuations around this late-time value are bounded by the purity of the thermal ensemble, which generally decays exponentially with system size. For auto-correlation 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 non-integrable 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.