# Time evolution of correlation functions in quantum many-body systems

## TL;DR

This paper rigorously analyzes the time evolution of two-point correlation functions in quantum many-body systems, showing how they factorize at late times and providing bounds on the timescales involved, with implications for understanding thermalization.

## Contribution

It provides the first rigorous bounds on the timescales for correlation functions to reach equilibrium in quantum many-body systems, extending to various types of two-point functions.

## Key findings

- Correlation functions factorize at late times in translation-invariant models.
- Fluctuations around late-time values are bounded by the purity of the thermal state.
- The timescale for auto-correlation functions to reach equilibrium is size-independent and bounded by local expectation values.

## Abstract

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 $\langle A(t) B\rangle_\beta \rightarrow \langle A \rangle_\beta \langle B \rangle_\beta$, thus proving that dissipation emerges out of the unitary dynamics of the system. We also show that 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 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.