We study the quantum criticality of the Lifshitz ϕ 4 -theory below the upper critical dimension. Two fixed points, one Gaussian and the other non-Gaussian, are identified with zero and finite interaction strengths, respectively. At zero temperature the particle density exhibits different powerlaw dependences on the chemical potential in the weak and strong interaction regions. At finite temperatures, critical behaviors in the quantum disordered region are mainly controlled by the chemical potential. In contrast, in the quantum critical region critical scalings are determined by temperature. The scaling ansatz remains valid in the strong interaction limit for the chemical potential, correlation length, and particle density, while it breaks down in the weak interaction one. As approaching the upper critical dimension, physical quantities develop logarithmic dependence on dimensionality in the strong interaction region. These results are applied to spin-orbit coupled bosonic systems, leading to predictions testable by future experiments. Introduction.-Quantum phase transitions, uniquely driven by quantum fluctuations, appear when the ground state energy encounters non-analyticity via tuning a nonthermal parameter. Physical properties around quantum critical points (QCPs) are of extensive interests because the interplay between quantum and thermal critical fluctuations strongly influence the dynamical and thermodynamic quantities, giving rise to rich quantum critical properties beyond the classical picture [1,2]. Quantum critical fluctuations are believed to be responsible for various emergent phenomena, including the non-Fermi liquid behaviors in heavy fermion systems, unconventional superconductivity, and novel spin dynamics in onedimensional quantum magnets [3][4][5][6].