  We introduce a novel momentum space entanglement renormalization group (MERG)
scheme for the topologically ordered (T.O.) ground state of the 2D Hubbard
model on a square lattice (\cite{anirbanmotti,anirbanmott2}) using a unitary
quantum circuit comprised of non-local unitary gates. At each MERG step, the
unitary quantum circuit disentangles a set of electronic states, thereby
transforming the tensor network representation of the many-particle state. By
representing the non-local unitary gate as a product of two-qubit disentangler
gates, we provide an entanglement holographic mapping (EHM) representation for
MERG. Using entanglement based measures from quantum information theory and
complex network theory, we study the emergence of topological order in the bulk
of the EHM. The MERG reveals distinct holographic entanglement features for the
normal metallic, topologically ordered insulating quantum liquid and Ne\'{e}l
antiferromagnetic symmetry-broken ground states of the 2D Hubbard model at
half-filling found in Ref.\cite{anirbanmotti}. An MERG analysis of the quantum
critical point of the hole-doped 2D Hubbard model found in
Ref.\cite{anirbanmott2} reveals the evolution of the many-particle entanglement
of the quantum liquid ground state with hole-doping, as well as how the
collapse of Mottness is responsible for the emergence of d-wave
superconductivity. We perform an information theoretic analysis of the EHM
network, demonstrating that the information bottleneck principle is responsible
for the distillation of entanglement features in the heirarchical structure of
the EHM network. As a result, we construct a deep neural network (DNN)
architecture based on our EHM network, and employ it for predicting the onset
of topological order. We also demonstrate that the DNN is capable of
distinguishing between the topologically ordered and gapless normal metallic
phases.
