  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.
