Thermalization and Revivals after a Quantum Quench in Conformal Field Theory

Abstract: We consider a quantum quench in a finite system of length $L$ described by a 1+1-dimensional CFT, of central charge $c$, from a state with finite energy density corresponding to an inverse temperature $\beta\ll L$. For times $t$ such that $\ell/2<t<(L-\ell)/2$ the reduced density matrix of a subsystem of length $\ell$ is exponentially close to a thermal density matrix. We compute exactly the overlap $\cal F$ of the state at time $t$ with the initial state and show that in general it is exponentially suppressed at large $L/\beta$. 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 $\cal F$ is $O(1)$, leading to an eventual complete revival with ${\cal F}=1$. There is also interesting structure at all rational values of $t/L$, related to properties of the CFT under modular transformations. At early times $t!\ll!(L\beta)^{1/2}$ there is a universal decay ${\cal F}\sim\exp\big(!-!(\pi c/3)Lt^2/\beta(\beta^2+4t^2)\big)$. The effect of an irrelevant non-integrable perturbation of the CFT is to progressively broaden each revival at $t=nL/2$ by an amount $O(n^{1/2})$.