  We present a unified framework for the renormalisation of the Hamiltonian and
eigenbasis of a system of correlated electrons, unveiling thereby the interplay
between electronic correlations and many-particle entanglement. For this, we
extend substantially the unitary renormalization group (URG) scheme introduced
in Refs.\cite{anirbanmotti,anirbanmott2,mukherjee2020}. We recast the RG as a
discrete flow of the Hamiltonian tensor network, i.e., the collection of
various $2n$-point scattering vertex tensors comprising the Hamiltonian. The
renormalisation progresses via unitary transformations that block diagonalizes
the Hamiltonian iteratively via the disentanglement of single-particle
eigenstates. This procedure incorporates naturally the role of quantum
fluctuations. The RG flow equations possess a non-trivial structure, displaying
a feedback mechanism through frequency-dependent dynamical self-energies and
correlation energies. The interplay between various UV energy scales enables
the coupled RG equations to flow towards a stable fixed point in the IR. The
effective Hamiltonian at the IR fixed point generically has a reduced parameter
space, as well as number of degrees of freedom, compared to the microscopic
Hamiltonian. Importantly, the vertex tensor network is observed to govern the
RG flow of the tensor network that denotes the coefficients of the
many-particle eigenstates. The RG evolution of various many-particle
entanglement features of the eigenbasis are, in turn, quantified through the
coefficient tensor network. In this way, we show that the URG framework
provides a microscopic understanding of holographic renormalisation: the RG
flow of the vertex tensor network generates a eigenstate coefficient tensor
network possessing a many-particle entanglement metric. We find that the
eigenstate tensor network accommodates sign factors arising from fermion
exchange.
