  Soft condensed matter structures often challenge us with complex many-body
phenomena governed by collective modes spanning wide spatial and temporal
domains. In order to successfully tackle such problems mesoscopic
coarse-grained (CG) statistical models are being developed, providing a
dramatic reduction in computational complexity. CG models provide an
intermediate step in the complex statistical framework of linking the
thermodynamics of condensed phases with the properties of their constituent
atoms and molecules. These allow us to offload part of the problem to the CG
model itself and reformulate the remainder in terms of reduced CG phase space.
However, such exchange of pawns to chess pieces, or ``Hamiltonian
renormalization'', is a radical step and the thermodynamics of the primary
atomic and CG models could be markedly different. Here, we present a
comprehensive study of the phase diagram including binodal and interfacial
properties of a novel soft CG model, which includes finite-range attraction and
supports liquid phases. Although the model is rooted in similar arguments to
the Lennard-Jones (LJ) atomic pair potential, its phase behaviour is
qualitatively different from that of LJ and features several anomalies such as
an unusually broad liquid range, change in concavity of the liquid coexistence
branch with variation of the model parameters, volume contraction on fusion,
temperature of maximum density in the liquid phase and negative thermal
expansion in the solid phase. These results provide new insight into the
connection between simple potential models and complex emergent condensed
matter phenomena.
