Equilibration of Small and Large Subsystems in Field Theories and Matrix Models

Abstract

It has been recently shown that small subsystems of finite quantum systems generically equilibrate. We extend these results to infinite-dimensional Hilbert spaces of field theories and matrix models. We consider a quench setup, where initial states are chosen from a microcanonical ensemble of finite energy in free theory, and then evolve with an arbitrary non-perturbative Hamiltonian. Given a dynamical assumption on the expectation value of particle number density, we prove that small subsystems reach equilibrium at the level of quantum wave-function, and with respect to all observables. The picture that emerges is that at higher energies, larger subsystems can reach equilibrium. For bosonic fields on a lattice, in the limit of large number of bosons per site, all subsystem smaller than half equilibrate. In the Hermitian matrix model, by contrast, this occurs in the limit of large energy per matrix element, emphasizing the importance of the $O(N^2)$ energy scale for the fast scrambling conjecture. Applying our techniques to continuum field theories on compact spaces, we show that the density matrix of small momentum-space observables equilibrate. Finally, we discuss the connection with scrambling, and provide a sufficient condition for a time-independent Hamiltonian to be a scrambler in terms of the entanglement entropy of its energy eigenstates.