Quantum gas-liquid condensation in an attractive Bose gas

Gas-liquid condensation (GLC) in an attractive Bose gas is studied on the basis of statistical mechanics. Using some results in combinatorial mathematics, the following are derived. (1) With decreasing temperature, the Bose-statistical coherence grows in the many-body wave function, which gives rise to the divergence of the grand partition function prior to Bose-Einstein condensation. It is a quantum-mechanical analogue to the GLC in a classical gas (quantum GLC). (2) This GLC is triggered by the bosons with zero momentum. Compared with the classical GLC, an incomparably weaker attractive force creates it. For the system showing the quantum GLC, we discuss a cold helium 4 gas at sufficiently low pressure.