We show that renormalization group(RG) flow can be viewed as a gradual wave
function collapse, where a quantum state associated with the action of field
theory evolves toward a final state that describes an IR fixed point. The
process of collapse is described by the radial evolution in the dual
holographic theory. If the theory is in the same phase as the assumed IR fixed
point, the initial state is smoothly projected to the final state. If in a
different phase, the initial state undergoes a phase transition which in turn
gives rise to a horizon in the bulk geometry. We demonstrate the connection
between critical behavior and horizon in an example, by deriving the bulk
metrics that emerge in various phases of the U(N) vector model in the large N
limit based on the holographic dual constructed from quantum RG. The gapped
phase exhibits a geometry that smoothly ends at a finite proper distance in the
radial direction. The geometric distance in the radial direction measures a
complexity : the depth of RG transformation that is needed to project the
generally entangled UV state to a direct product state in the IR. For gapless
states, entanglement persistently spreads out to larger length scales, and the
initial state can not be projected to the direct product state. The obstruction
to smooth projection at charge neutral point manifests itself as the long
throat in the anti-de Sitter space. The Poincare horizon at infinity marks the
critical point which exhibits a divergent length scale in the spread of
entanglement. For the gapless states with non-zero chemical potential, the bulk
space becomes the Lifshitz geometry with the dynamical critical exponent two.
The identification of horizon as critical point may provide an explanation for
the universality of horizon. We also discuss the structure of the bulk tensor
network that emerges from the quantum RG.