  We perform a unitary renormalization group (URG) study of the 1D fermionic
Hubbard model. The formalism generates a family of effective Hamiltonians and
many-body eigenstates arranged holographically across the tensor network from
UV to IR. The URG is realized as a quantum circuit, leading to the entanglement
holographic mapping (EHM) tensor network description. A topological
$\Theta$-term of the projected Hilbert space of the degrees of freedom at the
Fermi surface are shown to govern the nature of RG flow towards either the
gapless Tomonaga-Luttinger liquid or gapped quantum liquid phases. This results
in a nonperturbative version of the Berezenskii-Kosterlitz-Thouless (BKT) RG
phase diagram, revealing a line of intermediate coupling stable fixed points,
while the nature of RG flow around the critical point is identical to that
obtained from the weak-coupling RG analysis. This coincides with a phase
transition in the many-particle entanglement, as the entanglement entropy RG
flow shows distinct features for the critical and gapped phases depending on
the value of the topological $\Theta$-term. We demonstrate the Ryu-Takyanagi
entropy bound for the many-body eigenstates comprising the EHM network,
concretizing the relation to the holographic duality principle. The scaling of
the entropy bound also distinguishes the gapped and gapless phases, implying
the generation of very different holographic spacetimes across the critical
point. Finally, we treat the Fermi surface as a quantum impurity coupled to the
high energy electronic states. A thought-experiment is devised in order to
study entanglement entropy generated by isolating the impurity, and propose
ways by which to measure it by studying the quantum noise and higher order
cumulants of the full counting statistics.
