  Ca$_{10}$Cr$_7$O$_{28}$ is a novel spin-$1/2$ magnet exhibiting spin liquid
behaviour which sets it apart from any previously studied model or material.
However, understanding Ca$_{10}$Cr$_7$O$_{28}$ presents a significant
challenge, because the low symmetry of the crystal structure leads to very
complex interactions, with six inequivalent magnetic sites in the unit cell.
Here we explore the origin of the spin-liquid behaviour in
Ca$_{10}$Cr$_7$O$_{28}$, starting from the simplest microscopic model
consistent with experiment - a Heisenberg model on a single bilayer of the
breathing-kagome (BBK) lattice. We use a combination of classical Monte Carlo
(MC) simulation and (semi-)classical Molecular Dynamics (MD) simulation to
explore the thermodynamic and dynamic properties of this model, and compare
these with experimental results for Ca$_{10}$Cr$_7$O$_{28}$. We uncover
qualitatively different behaviours on different timescales, and argue that the
ground state of Ca$_{10}$Cr$_7$O$_{28}$ is born out of a slowly-fluctuating
"spiral spin liquid", while faster fluctuations echo the U(1) spin liquid found
in the kagome antiferromagnet. We also identify key differences between
longitudinal and transverse spin excitations in applied magnetic field, and
argue that these are a distinguishing feature of the spin liquid in the BBK
model.
