Circuit complexity for free Fermion with a mass quench

TL;DR

This paper calculates the complexity of free fermionic states undergoing a mass quench, revealing saturation behavior and proportionality to mass difference, with implications for holographic complexity in strongly coupled systems.

Contribution

It adapts Hackl et al.'s counting method to all compact groups and analyzes the time evolution of complexity in fermionic systems after a mass quench.

Findings

Complexity saturates at late times for rotational invariant states.

Complexity growth is proportional to mass difference.

Features of excited states and non-rotational references are identified.

Abstract

By using a recent approach proposed by Hackl $et\, al.$ to evaluate the complexity of the free fermionic Gaussian state, we compute the complexity of the Dirac vacuum state as well as the excited state of the Fermi system with a mass quench. First of all, we review the counting method given by Hackl $et\, al.$, and demonstrate that the result can be adapted to all of the compact transformation group $G$. Then, we utilize this result to study the time evolution of the complexity of these states. We show that, for the rotational invariant reference state, the total complexity of the incoming vacuum state will saturate the value of the instantaneous vacuum state at the late time, with a typical timescale to achieve the final stable state. Moreover, we find that the complexity growth under the sudden quench is directly proportional to the mass difference, which shares similar behaviors with the holograph complexity growth rate in an AdS-Vaidya black hole with a shock wave, even though the dual boundary CFT is strongly coupled. Finally, we obtain some features of the excited state and the non-rotational reference state.