Universality in fast quantum quenches

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 << lambda(1)(1)/((Delta-d)), lambda(1)(2)/((Delta-d)), delta lambda(1)/((Delta-d)), the energy (density) injected into the system scales as delta epsilon similar to (delta lambda)(2)(delta t)(2-2 Delta). Similarly, the change in the expectation value of the quenched operator at times earlier than the endpoint of the quench scales as (O-Delta) similar to 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.