  We apply a functional renormalisation group to systems of four bosonic atoms
close to the unitary limit. We work with a local effective action that includes
a dynamical trimer field and we use this field to eliminate structures that do
not correspond to the Faddeev-Yakubovsky equations. In the physical limit, we
find three four-body bound states below the shallowest three-body state. The
values of the scattering lengths at which two of these states become bound are
in good agreement with exact solutions of the four-body equations and
experimental observations. The third state is extremely shallow. During the
evolution we find an infinite number of four-body states based on each
three-body state which follow a double-exponential pattern in the running
scale. None of the four-body states shows any evidence of dependence on a
four-body parameter.
