We use the functional renormalisation group to study the spectrum of three- and four-body states in bosonic systems around the unitary limit. Our effective action includes all energy-independent contact interactions in the four-atom sector and we int
roduce a running trimer field to eliminate couplings that involve the atom-atom-dimer channel. The results show qualitatively similar behaviour to those from exact approaches. The truncated action we use leads to overbinding of the two four-body states seen in those treatments. It also generates a third state, although only for a very narrow range of two-body scattering lengths.
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.
