Thermalization and Revivals after a Quantum Quench in Conformal Field Theory

We consider a quantum quench in a finite system of length L described by a $1+1$-dimensional conformal field theory (CFT), of central charge c, from a state with finite energy density corresponding to an inverse temperature βL. For times t such that /2<t<(L-)/2 the reduced density matrix of a subsystem of length is exponentially close to a thermal density matrix. We compute exactly the overlap F of the state at time t with the initial state and show that in general it is exponentially suppressed at large L/β. However, for minimal models with $c<1$ (more generally, rational CFTs), at times which are integer multiples of $L/2$ (for periodic boundary conditions, L for open boundary conditions) there are (in general, partial) revivals at which F is $O(1)$, leading to an eventual complete revival with F=1. There is also interesting structure at all rational values of $t/L$, related to properties of the CFT under modular transformations. At early times t(Lβ)1/2 there is a universal decay F~exp(-(πc/3)Lt²/β(β²+4t²)). The effect of an irrelevant nonintegrable perturbation of the CFT is to progressively broaden each revival at $t=nL/2$ by an amount O(n1/2).