We expand on the investigation of the universal scaling properties in the early time behaviour of fast but smooth quantum quenches in a general d-dimensional conformal field theory deformed by a relevant operator of dimension Δ with a time-dependent coupling. The quench consists of changing the coupling from an initial constant value λ 1 by an amount of the order of δλ to some other final value λ 2, over a time scale δt. In the fast quench limit where δt is smaller than all other length scales in the problem, δt ≪ λ 1 1/(Δ − d) , λ 2 1/(Δ − d) , δλ 1/(Δ − d), the energy (density) injected into the system scales as δℰ ∼ (δλ)2(δt) d− 2Δ . Similarly, the change in the expectation value of the quenched operator at times earlier than the endpoint of the quench scales as ⟨OΔ⟩∼δλ(δt)d−2Δ, with further logarithmic enhancements in certain cases. While these results were first found in holographic studies, we recently demonstrated that precisely the same scaling appears in fast mass quenches of free scalar and free fermionic field theories. As we describe in detail, the universal scaling refers to renormalized quantities, in which the UV divergent pieces are consistently renormalized away by subtracting counterterms derived with an adiabatic expansion. We argue that this scaling law is a property of the conformal field theory at the UV fixed point, valid for arbitrary relevant deformations and insensitive to the details of the quench protocol. Our results highlight the difference between smooth fast quenches and instantaneous quenches where the Hamiltonian abruptly changes at some time.

Document Type


Publication Date


Notes/Citation Information

Published in Journal of High Energy Physics, v. 2015, no. 2, article 167, p. 1-68.

© The Authors.

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Digital Object Identifier (DOI)


Funding Information

Article funded by SCOAP3.