Fermi-like Liquid From Einstein-DBI-Dilaton System

We have obtained an expression of the entropy density depending on the scale transformation of the spatial directions in the field theory. It takes the following form in $d+1$ dimensional bulk spacetime: $s\sim T^{\f{\delta(d-1)-\theta}{z}}_H$,where$z$and$T_H$ are the dynamical exponent and temperature in the field theory, respectively. $\theta$ is related to the scaling violation exponent, whereas $\delta$ gives us the information about the scaling behavior of the spatial field theoretic direction. This we demonstrate by finding solutions to the Einstein-DBI-dilaton system in generic spacetime dimensions. Upon restricting to $d=3$, we show the linear temperature dependence of the specific heat and inverse quadratic temperature dependence of the resistivity for $z=2,\theta =0$ and $\delta =1$, which resembles that of the Fermi-like liquid. Whereas for $z=2,\theta =-2$ and $\delta =0$ gives us a solution that is conformal to $Ad{S}_2\otimes R^2$, which resembles with the non-Fermi-like liquid. Moreover, it shows the logarithmic violation of the entanglement entropy when the entangling region is of the strip type.