  K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ is a highly one-dimensional spin-1/2
inequilateral diamond-chain antiferromagnet. Spinon continuum and spin-singlet
dimer excitations are observed in the inelastic neutron scattering spectra,
which is in excellent agreement with a theoretical prediction: a dimer-monomer
composite structure, where the dimer is caused by strong antiferromagnetic
(AFM) coupling and the monomer forms an almost isolated quantum AFM chain
controlling low-energy excitations. Moreover, muon spin rotation/relaxation
spectroscopy shows no long-range ordering down to 90~mK, which is roughly three
orders of magnitude lower than the exchange interaction of the quantum AFM
chain. K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ is, thus, regarded as a compound that
exhibits a Tomonaga-Luttinger spin liquid behavior at low temperatures close to
the ground state.
