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 $\ensuremath{\beta}\ensuremath{\ll}L$. For times $t$ such that $\ensuremath{\ell}/2<t<(L\ensuremath{-}\ensuremath{\ell})/2$ the reduced density matrix of a subsystem of length $\ensuremath{\ell}$ is exponentially close to a thermal density matrix. We compute exactly the overlap $\mathcal{F}$ of the state at time $t$ with the initial state and show that in general it is exponentially suppressed at large $L/\ensuremath{\beta}$. 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 $\mathcal{F}$ is $O(1)$, leading to an eventual complete revival with $\mathcal{F}=1$. There is also interesting structure at all rational values of $t/L$, related to properties of the CFT under modular transformations. At early times $t\ensuremath{\ll}(L\ensuremath{\beta}{)}^{1/2}$ there is a universal decay $\mathcal{F}\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}(\ensuremath{\pi}c/3)L{t}^{2}/\ensuremath{\beta}({\ensuremath{\beta}}^{2}+4{t}^{2}))$. The effect of an irrelevant nonintegrable perturbation of the CFT is to progressively broaden each revival at $t=nL/2$ by an amount $O({n}^{1/2})$.

Перевод пока недоступен