Researchers have developed a quantitative theory describing the emergence of chaos in quantum many-body systems as their integrability is broken. Using a circuit model of free fermions, to which an adjustable density of integrability-breaking gates is added, the study reveals the microscopic mechanisms underlying the transition from integrable behavior at early times to a chaotic state at late times.

The study focuses on out-of-time-ordered correlators (OTOCs), key tools for characterizing quantum chaos. The integrability-breaking gates act as local spacetime hotspots, locally amplifying the OTOCs. The accumulation of these hotspots eventually leads to the full development of chaos in the system. This approach allows for a detailed understanding of how information and complexity propagate within the system as non-integrability is introduced.

The results explicitly identify the characteristic time and length scales governing this crossover. Furthermore, the research details how the properties of chaotic OTOCs, such as the butterfly velocity (which measures information spreading) and front broadening, depend on the parameter quantifying the integrability breaking. This work provides a robust framework for understanding and predicting the behavior of complex quantum systems at the boundary between integrability and chaos.