Out-of-equilibrium protocol for R\'enyi entropies via the Jarzynski equality

In recent years entanglement measures, such as the von Neumann and the R\'enyi entropies, provided a unique opportunity to access elusive feature of quantum many-body systems. However, extracting entanglement properties analytically, experimentally, or in numerical simulations can be a formidable task. Here, by combining the replica trick and the Jarzynski equality we devise a new effective out-of-equilibrium protocol for measuring the equilibrium R\'enyi entropies. The key idea is to perform a quench in the geometry of the replicas. The R\'enyi entropies are obtained as the exponential average of the work performed during the quench. We illustrate an application of the method in classical Monte Carlo simulations, although it could be useful in different contexts, such as in Quantum Monte Carlo, or experimentally in cold-atom systems. The method is most effective in the quasi-static regime, i.e., for a slow quench, where it allows to obtain the R\'enyi entropies in a single realization of the protocol. As a benchmark, we present results for the R\'enyi entropies in the Ising universality class in $1$$+$$1$ dimensions, which are found in perfect agreement with the well-known Conformal Field Theory (CFT) predictions.