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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: $s\sim T^{\f{δ(d-1)-θ}{z}}_H$, where $z$ and $T_H$ are the dynamical exponent and temperature in the field theory, respectively. $θ$ is related to the scaling violation exponent, whereas $δ$ 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 $z=2,θ=0$ and $δ=1$, which resembles that of the Fermi-like liquid. Whereas for $z=2, θ=-2$ and $δ=0$ gives us a solution that is conformal to $AdS_2\otimes R^2$, 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.