Evolution of complexity following a quantum quench in free field theory

TL;DR

This paper studies the time evolution of circuit complexity in a free scalar quantum field after a smooth mass quench, using the $F_{2}$ cost from Jefferson–Myers. By organizing the problem into momentum‑space SU$(1,1)$ gates, it reduces to independent geodesics for each mode and derives a closed expression for the complexity in terms of Bogoliubov data and quench parameters. The main finding is a two‑phase evolution: an early near‑linear regime whose slope tracks $δm/δt$ and a later saturation with oscillations around a mean value; the saturation time is set by the quench scale $δt$, and the growth can be either positive or negative depending on whether the mass increases or decreases. This behavior mirrors holographic intuition in chaotic systems while highlighting distinct time‑scales in a free theory, and it motivates extending the analysis to interacting theories and exploring different cost functionals for potential closer holographic ties.

Abstract

Using a recent proposal of circuit complexity in quantum field theories introduced by Jefferson and Myers, we compute the time evolution of the complexity following a smooth mass quench characterized by a time scale $δt$ in a free scalar field theory. We show that the dynamics has two distinct phases, namely an early regime of approximately linear evolution followed by a saturation phase characterized by oscillations around a mean value. The behavior is similar to previous conjectures for the complexity growth in chaotic and holographic systems, although here we have found that the complexity may grow or decrease depending on whether the quench increases or decreases the mass, and also that the time scale for saturation of the complexity is of order $δt$ (not parametrically larger).

Figures (1)

• Figure 1: Time dependence of the complexity for many combinations of the mass difference $\delta m=m_\text{out}-m_\text{in}$, the quench rate $\delta t$, and the scale $M$ characterizing the reference state. All the plots were made with specific numerical values of the parameters chosen for presentation purposes, but the qualitative behavior above was checked to hold generically. Figure (a) has $\delta t=1,M=2,m_\text{in}=1$ and $m_\text{out}$ is changed as indicated by the inset; (b) has $\delta t=1,M=2,m_\text{in}=6$ and varying $m_\text{out}$; (c) has $m_\text{in}=1,m_\text{out}=2,M=2$ and varying $\delta t$; and (d) was made with $m_\text{in}=1,m_\text{out}=2,\delta t=1$ while $M$ was changed.