Phase transition in anisotropic holographic superfluids with arbitrary $z$ and $\alpha$

Einstein-dilaton-$U(2)$ gauge field theory is considered in a spacetime characterised by $\alpha$ and $z$, which are the hyperscaling violation factor and the dynamical critical exponent respectively. We obtain the critical values of chemical potential $\mu_c$ that is defined on its boundary dual fluid and derives phase transition from spatially isotropic to anisotropic phase for the various values of the $\alpha$ and $z$. To do so, we first apply Sturm-Liouville theory and estimate the upper bounds of the critical values of the chemical potential. We also employ a numerical method in the ranges of $1 \leq z \leq 4$ and $0 \leq \alpha \leq 4$ to check if the Sturm-Liouville method correctly estimates the critical values of the chemical po10 percent error ranges. Finally, we compute free energy density of the dual fluid by using its gravity dual and check if the system shows phase transition at the critical values of the chemical potential $\mu_c$ for the given parameter region of $\alpha$ and $z$. Interestingly, it is observed that the anisotropic phase is more favoured than the isotropic phase for small values of $z$ and $\alpha$. However, for large values of $z$ and $\alpha$, the anisotropic phase is not favoured.tential. It turns out that the two methods are agreed within