Uncomputably complex renormalisation group flows

Renormalisation group methods are among the most important techniques for analysing the physics of many-body systems: by iterating a renormalisation group map, which coarse-grains the description of a system and generates a flow in the parameter space, physical properties of interest can be extracted. However, recent work has shown that important physical features, such as the spectral gap and phase diagram, may be impossible to determine, even in principle. Following these insights, we construct a rigorous renormalisation group map for the original undecidable many-body system that appeared in the literature, which reveals a renormalisation group flow so complex that it cannot be predicted. We prove that each step of this map is computable, and that it converges to the correct fixed points, yet the resulting flow is uncomputable. This extreme form of unpredictability for renormalisation group flows had not been shown before and goes beyond the chaotic behaviour seen previously.