Analysis of the buildup of spatiotemporal correlations and their bounds outside of the light cone

TL;DR

This paper derives bounds on the growth of correlations outside the light cone in non-relativistic quantum systems, linking initial decay properties to their dynamical evolution, and verifies results in the Luttinger model.

Contribution

It establishes a bound for correlation functions outside the light cone based on initial power-law decay and confirms the theoretical predictions with an exactly solvable model.

Findings

Correlation bounds match initial decay exponents

Verification in the Luttinger model confirms theoretical predictions

Correlation tails outside the light cone decay with the same power-law as initial state

Abstract

In non-relativistic quantum theories the Lieb-Robinson bound defines an effective light cone with exponentially small tails outside of it. In this work we use it to derive a bound for the time evolution of the correlation function of two local disjoint observables if the initial state has a power-law decay. We show that the exponent of the power-law of the bound is identical to the initial (equilibrium) decay. We explicitly verify this result by studying the full dynamics of the susceptibilities and correlations in the exactly solvable Luttinger model after a sudden quench from the non-interacting to the interacting model.