Quantum criticality of bosonic systems with the Lifshitz dispersion

We study a novel type of quantum criticality of the Lifshitz ${\ensuremath{\varphi}}⁴$ theory below the upper critical dimension ${d}_u=z+{d}_c=8$, where the dynamic critical exponent $z=4$ and the spatial upper critical dimension ${d}_c=4$. 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 power-law 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. 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.