  Applying recursive renormalization group transformations to a scalar field
theory, we obtain an effective quantum gravity theory with an emergent extra
dimension, described by a dual holographic Einstein-Klein-Gordon type action.
Here, the dynamics of both the dual order-parameter field and the metric tensor
field originate from density-density and energy-momentum tensor-tensor
effective interactions, respectively, in the recursive renormalization group
transformation, performed approximately in the Gaussian level. This linear
approximation in the recursive renormalization group transformation for the
gravity sector gives rise to a linearized quantum Einstein-scalar theory along
the $z-$directional emergent space. In the large $N$ limit, where $N$ is the
flavor number of the original scalar fields, quantum fluctuations of both
dynamical metric and dual scalar fields are suppressed, leading to a classical
field theory of the Einstein-scalar type in $(D+1)$-spacetime dimensions. We
show that this emergent background gravity describes the renormalization group
flows of coupling functions in the UV quantum field theory through the extra
dimension. More precisely, the IR boundary conditions of the gravity equations
correspond to the renormalization group $\beta$-functions of the quantum field
theory, where the infinitesimal distance in the extra-dimensional space is
identified with an energy scale for the renormalization group transformation.
Finally, we also show that this dual holographic formulation describes quantum
entanglement in a geometrical way, encoding the transfer of quantum
entanglement from quantum matter to classical gravity in the large $N$ limit.
We claim that this entanglement transfer serves as a microscopic foundation for
the emergent holographic duality description.
