Thermalization and Revivals after a Quantum Quench in Conformal Field Theory

John Cardy

doi:10.1103/PhysRevLett.112.220401

paper

Thermalization and Revivals after a Quantum Quench in Conformal Field Theory

arXiv:1403.3040 · doi:10.1103/PhysRevLett.112.220401

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

arXiv comment5 pages, 3 figures

Thermalization and Revivals after a Quantum Quench in Conformal Field Theory

Abstract

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