Universality in fast quantum quenches

TL;DR

This paper investigates universal scaling laws in the early time behavior of fast, smooth quantum quenches in conformal field theories, revealing that certain energy and operator expectation values follow predictable power-law scaling.

Contribution

It demonstrates that the universal scaling laws observed in holographic models also apply to free scalar and fermionic theories, emphasizing their fundamental nature in conformal field theories.

Findings

Energy density scales as $( ext{delta lambda})^2 ( ext{delta t})^{d-2 riangle}$

Operator expectation value scales as $ ext{delta lambda} ( ext{delta t})^{d-2 riangle}$

Scaling laws are independent of quench protocol details

Abstract

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 $\Delta$ with a time-dependent coupling. The quench consists of changing the coupling from an initial constant value $\lambda_1$ by an amount of the order of $\delta \lambda$ to some other final value $\lambda_2$, over a time scale $\delta t$. In the fast quench limit where $\delta t$ is smaller than all other length scales in the problem, $ \delta t \ll \lambda_1^{1/(\Delta-d)}, \lambda_2^{1/(\Delta-d)}, \delta \lambda^{1/(\Delta-d)}$, the energy (density) injected into the system scales as $\delta{\cal E} \sim (\delta \lambda)^2 (\delta t)^{d-2\Delta}$. Similarly, the change in the expectation value of the quenched operator at times earlier than the endpoint of the quench scales as $<{\cal O}_\Delta> \sim \delta \lambda (\delta t)^{d-2\Delta}$, 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.