  The one-dimensional interacting Bose-Fermi mixtures, exhibiting quantum phase
transitions at zero temperature, are particularly valuable for the study of
quantum critical phenomena. In the present paper, we analytically study quantum
phase diagram, equation of state and quantum criticality of the Bose-Fermi
mixture using the thermodynamic Bethe ansatz equations. We show that
thermodynamical properties display universal scaling behaviour at quantum
criticality. Furthermore, quantum criticality of the Bose-Fermi mixture in an
harmonic trap is also studied within the local density approximation. We thus
demonstrate that the phase diagram and critical properties of the bulk system
provide insights into understanding universal features of many-body critical
phenomena.
