Nonperturbative dynamical effects in nearly-scale-invariant systems: The action of breaking scale invariance

In this work, we develop a general formalism that categorizes the action of broken scale invariance on the nonequilibrium dynamics of nonrelativistic quantum systems. This approach is equally applicable to both strongly and weakly interacting systems. We show that any small deviation from the strongly interacting fixed point, in three spatial dimensions, leads to nonpertubative effects in the long-time dynamics, dramatically altering the dynamics observed at the scale-invariant fixed point. As a concrete example, we apply this approach to the nonequilibrium dynamics for the interacting two-body problem and for a noninteracting quantum gas in the presence of an impurity, both in three spatial dimensions. Slightly away from the resonantly interacting scale invariant fixed point, we show that the dynamics are altered by a nonperturbative log-periodic beat. The presence of the beat depends only on deviating from the resonant fixed point, while the frequency depends on the microscopic parameters of the system.