  Renormalisation group (RG) methods provide one of the most important
techniques for analysing the physics of many-body systems, both analytically
and numerically. By iterating an RG map, which "course-grains" the description
of a many-body system and generates a flow in the parameter space, physical
properties of interest can be extracted even for complex models. RG analysis
also provides an explanation of physical phenomena such as universality. Many
systems exhibit simple RG flows, but more complicated -- even chaotic --
behaviour is also known. Nonetheless, the structure of such RG flows can still
be analysed, elucidating the physics of the system, even if specific
trajectories may be highly sensitive to the initial point. In contrast, recent
work has shown that important physical properties of quantum many-body systems,
such as its spectral gap and phase diagram, can be uncomputable.
  In this work, we show that such undecidable systems exhibit a novel type of
RG flow, revealing a qualitatively different and more extreme form of
unpredictability than chaotic RG flows. In contrast to chaotic RG flows in
which initially close points can diverge exponentially, trajectories under
these novel uncomputable RG flows can remain arbitrarily close together for an
uncomputable number of iterations, before abruptly diverging to different fixed
points that are in separate phases. The structure of such uncomputable RG flows
is so complex that it cannot be computed or approximated, even in principle. We
give a mathematically rigorous construction of the block-renormalisation-group
map for the original undecidable many-body system that appeared in the
literature (Cubitt, P\'erez-Garcia, Wolf, Nature 528, 207-211 (2015)). We prove
that each step of this RG map is computable, and that it converges to the
correct fixed points, yet the resulting RG flow is uncomputable.
