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 $\beta \ll L$. 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 $\mathcal{F}$ 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 $\mathcal{F}$ 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 $t\ll (L\beta {)}^{1/2}$ there is a universal decay $\mathcal{F}\sim \exp (-(\pi c/3)L{t}^2/\beta ({\beta }^2+4t^2))$. The effect of an irrelevant nonintegrable perturbation of the CFT is to progressively broaden each revival at $t=nL/2$ by an amount $O(n^{1/2})$.

Figure 1

Quasiparticle configuration leading to the feature in the return amplitude at $t=L/4$ for periodic boundary conditions. The pairs emitted a distance $L/2$ apart must be correlated, leading to an exponential suppression.

Figure 2

Log of the return amplitude for the Ising CFT starting from a disordered state for $0<2t/L<2$, with $\pi \beta /L=0.1$. The vertical axis has been shifted so as to expose the mean plateau behavior. This shows the initial Gaussian decay and revival at $t=L$. The negative peak at $t=L/2$ is due to destructive interference between two kinds of quasiparticles. Smaller Gaussian peaks are seen at rational values with small denominators. The positive peaks are mapped by the modular group to the initial peak, and the negative ones are mapped to the feature at $2t/L=1$.

Figure 3

Same as above with $\pi \beta /L=0.01$. Now there is structure at more rational values, and we see the predicted $1/m^2$ dependence of the heights of nearby peaks with denominators $m$.