Dynamical scaling symmetry and asymptotic quantum correlations for time-dependent scalar fields

TL;DR

This paper extends the concept of scaling symmetry to time-dependent quantum systems, revealing invariant measures of quantum correlations and analyzing their late-time behavior, including instabilities, zero-modes, and implications for cosmology and black holes.

Contribution

It generalizes dynamical scaling symmetry to time-dependent quantum fields, linking quantum correlations and instabilities, and explores their implications in cosmological and black-hole contexts.

Findings

Quantum correlations are invariant under dynamical scaling symmetry.

Late-time behavior shows relations between correlations, instabilities, and zero-modes.

Entropy oscillates between area-law and volume-law in scalar fields.

Abstract

In time-independent quantum systems, entanglement entropy possesses an inherent scaling symmetry that the energy of the system does not have. The symmetry also assures that entropy divergence can be associated with the zero modes. We generalize this symmetry to time-dependent systems all the way from a coupled harmonic oscillator with a time-dependent frequency, to quantum scalar fields with time-dependent mass. We show that such systems have dynamical scaling symmetry that leaves the evolution of various measures of quantum correlations invariant -- entanglement entropy, GS fidelity, Loschmidt echo, and circuit complexity. Using this symmetry, we show that several quantum correlations are related at late-times when the system develops instabilities. We then quantify such instabilities in terms of scrambling time and Lyapunov exponents. The delayed onset of exponential decay of the Loschmidt echo is found to be determined by the largest inverted mode in the system. On the other hand, a zero-mode retains information about the system for a considerably longer time, finally resulting in a power-law decay of the Loschmidt echo. We extend the analysis to time-dependent massive scalar fields in $(1 + 1)-$dimensions and discuss the implications of zero-modes and inverted modes occurring in the system at late-times. We explicitly show the entropy scaling oscillates between the \emph{area-law} and \emph{volume-law} for a scalar field with stable modes or zero-modes. We then provide a qualitative discussion of the above effects for scalar fields in cosmological and black-hole space-times.