Smooth and fast versus instantaneous quenches in quantum field theory

TL;DR

This paper investigates the relationship between smooth fast and instantaneous quantum quenches in free scalar field theory, showing how correlators and energy scaling behave at different times and distances, with implications for interacting theories.

Contribution

It provides a detailed analysis of the differences and similarities between smooth and instantaneous quenches, especially in the UV behavior and late-time correlators, extending previous work.

Findings

Smooth quenches approach instantaneous quenches at late times and large distances.

Correlators scale universally with quench time at early times and small distances.

Excess energy density diverges for high dimensions in the zero quench time limit.

Abstract

We examine in detail the relationship between smooth fast quantum quenches, characterized by a time scale $\delta t$, and {\em instantaneous quenches}, within the framework of exactly solvable mass quenches in free scalar field theory. Our earlier studies \cite{dgm1,dgm2} highlighted that the two protocols remain distinct in the limit $\delta t \rightarrow 0$ because of the relation of the quench rate to the UV cut-off, i.e., $1/\delta t\ll\Lambda$ always holds in the fast smooth quenches while $1/\delta t\sim\Lambda$ for instantaneous quenches. Here we study UV finite quantities like correlators at finite spatial distances and the excess energy produced above the final ground state energy. We show that at late times and large distances (compared to the quench time scale) the smooth quench correlator approaches that for the instantaneous quench. At early times, we find that for small spatial separation and small $\delta t$, the correlator scales universally with $\delta t$, exactly as in the scaling of renormalized one point functions found in earlier work. At larger separation, the dependence on $\delta t$ drops out. The excess energy density is finite (for finite $m\delta t$) and scales in a universal fashion for all $d$. However, the scaling behaviour produces a divergent result in the limit $m\delta t \rightarrow 0$ for $d\ge4$, just as in an instantaneous quench, where it is UV divergent for $d \geq 4$. We argue that similar results hold for arbitrary interacting theories: the excess energy density produced is expected to diverge for scaling dimensions $\Delta > d/2$.