Phase transition in anisotropic holographic superfluids with arbitrary dynamical critical exponent &ITz&IT and hyperscaling violation factor alpha

Abstract

Einstein-scalar-U(2) gauge field theory is considered in a spacetime characterized by alpha and z, which are the hyperscaling violation factor and the dynamical critical exponent, respectively. We consider a dual fluid system of such a gravity theory characterized by temperature T and chemical potential mu It turns out that there is a superfluid phase transition where a vector order parameter appears which breaks SO(3) global rotation symmetry of the dual fluid system when the chemical potential becomes a certain critical value. To study this system for arbitrary z and alpha, 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 <= z <= 4 and 0 <= alpha <= 4 to check if the Sturm-Liouville method correctly estimates the critical values of the chemical potential. It turns out that the two methods are agreed within 10 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 favored than the isotropic phase for relatively small values of z and alpha. However, for large values of z and alpha, the anisotropic phase is not favored.