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 le z le 4 and 0 le alpha le 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 _mathrm{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.
Highlights
The precise map of fluid/gravity duality is given in [20]
An interesting research topic along these directions is the thermodynamic phase transition from normal fluids/conductor to superfluids/conductor. This holographic superfluid/superconductor phase transition is accompanied by a condensation such as the appearance of Cooper pairs in the superconducting phase
The holographic superconductor model was established in [2], which shows a complex scalar field condensation resulting from a spontaneous symmetry breaking of global U(1) and it corresponds to an order parameter in the second order phase transition via a holographic interpretation
Summary
Where M and N are 5-dimensional (5-D) spacetime indices, running from 0 to 4, gM N is the spacetime metric, V0, γ , λU and λY M are real constants and κ5 is the 5-D gravity constant. We start with a solution having a generic hyperscaling violating factor α and a temporal anisotropy factor z Such solutions already appeared in [13], an Einstein-dilaton theory with two different U (1) gauge fields. When there is no anisotropy, i.e. ω(r ) = 0, the background geometry becomes that of 5-D charged black brane solutions, which are given by f (r ). We would like to explore a spatial anisotropy in this background by turning on Bx11 (= ω(r )) without considering its back reactions to the background geometry.. We would like to explore a spatial anisotropy in this background by turning on Bx11 (= ω(r )) without considering its back reactions to the background geometry.5 This limit can be obtained by demanding that the Yang–Mill coupling is taken to be infinity, i.e.
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