  We demonstrate that the thermodynamics of one-dimensional Lieb-Liniger bosons
can be accurately calculated in analytic fashion using the polylog function in
the framework of the thermodynamic Bethe ansatz. The approach does away with
the need to numerically solve the thermodynamic Bethe ansatz (Yang-Yang)
equation. The expression for the equation of state allows the exploration of
Tomonaga-Luttinger liquid physics and quantum criticality in an archetypical
quantum system. In particular, the low-temperature phase diagram is obtained,
along with the scaling functions for the density and compressibility. It has
been shown recently by Guan and Ho (arXiv:1010.1301) that such scaling can be
used to map out the criticality of ultracold fermionic atoms in experiments. We
show here how to map out quantum criticality for Lieb-Liniger bosons. More
generally the polylog function formalism can be applied to a wide range of
Bethe ansatz integrable quantum many-body systems which are currently of
theoretical and experimental interest, such as strongly interacting
multi-component fermions, spinor bosons and mixtures of bosons and fermions.
