Revivals imply quantum many-body scars
Álvaro M. Alhambra, Anurag Anshu, Henrik Wilming
Abstract
We derive general rigorous results relating revivals in the dynamics of quantum many-body systems to the entanglement properties of energy eigenstates. For a $D$-dimensional lattice system of $N$ sites initialized in a low-entangled and short-range correlated state, our results show that a perfect revival of the state after a time of at most $O(\mathrm{poly}(N))$ implies the existence of at least $O(\sqrt{N}/{log}^{2D}(N))$ ``quantum many-body scars'': energy eigenstates with energies placed in an equally spaced ladder and with R\'enyi entanglement entropy of at most $O(log(N))+O(|\ensuremath{\partial}A|)$ for any region $A$ of the lattice. This shows that quantum many-body scars are a necessary consequence of revivals, independent of particularities of the Hamiltonian leading to them. We also present results for approximate revivals and for revivals of expectation values of observables and prove that the duration of revivals of states has to become vanishingly short with increasing system size.