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

Tomonaga-Luttinger spin liquid in the spin-1/2 inequilateral diamond-chain compound K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$