New insights into four-boson renormalization group limit cycles

Using machine learning techniques, we verify that the emergence of renormalization group limit cycles beyond the unitary limit is transferred from the three-boson subsystems to the whole four-boson system. Focussing on four identical bosons, we first generate populations of synthetic singular potentials within the latent space of a boosted ensemble of variational autoencoders. After introducing the limit cycle loss for measuring the deviation of a given renormalization group flow from limit cycle behavior, we minimize it by applying an elitist genetic algorithm to the generated populations. The fittest potentials are observed to accumulate around the inverse-square potential, which we prove to generate limit cycles for four bosons and which is already known to produce limit cycles in the three-boson system. This also indicates that a four-body term does not enter low-energy observables at leading order, since we do not observe any additional scale to emerge.