Lifshitz Effects on Vector Condensate Induced by a Magnetic Field

By numerical and analytical methods, we study in detail the effects of the Lifshitz dynamical exponent $z$ on the vector condensate induced by an applied magnetic field in the probe limit. Concretely, in the presence of the magnetic field, we obtain the Landau level independent of $z$, and also find the critical value by coupling a Maxwell complex vector field and SU(2) field into a (3+1)-dimensional Lifshitz black hole, respectively. The research results show that for both two models with the lowest Landau level, the increasing $z$ improves the response of the critical temperature to the applied magnetic field even without the charge density, and the analytical results uphold the numerical results. In addition, we find even in the Lifshitz black hole, the Maxwell complex vector model is still a generalization of the SU(2) Yang-Mills model. Furthermore, we construct the square vortex lattice and discuss the implications of these results.