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 conformal field theory (CFT), of central charge c, from a state with finite energy density corresponding to an inverse temperature β≪L. For times t such that ℓ/2<t<(L-ℓ)/2 the reduced density matrix of a subsystem of length ℓ 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)Lt^{2}/β(β^{2}+4t^{2})). The effect of an irrelevant nonintegrable perturbation of the CFT is to progressively broaden each revival at t=nL/2 by an amount O(n^{1/2}).