  An outstanding challenge involves understanding the many-particle
entanglement of liquid states of quantum matter that arise in systems of
interacting electrons. The Fermi liquid (FL) in $D$ spatial dimensions shows a
violation of the area-law in real-space entanglement entropy of a subsystem (of
length $L$), $S_{EE} \sim L^{D-1}\ln L$, widely believed to be a hallmark
signature of the ground state of a gapless quantum critical system of
interacting fermions. In this work, we apply a $T=0$ renormalisation group
approach to a prototype of the FL in momentum (or, $k$)-space, unveiling
thereby the RG relevant quantum fluctuations (due to forward and tangential
scattering) from which long-range entanglement arises. A similar analysis of
non-Fermi liquids such as the 2D marginal Fermi liquid (MFL) and the 1D
Tomonaga-Luttinger liquid (TLL) reveals a universal logarithmic violation of
the area-law in gapless electronic liquids for a subsystem defined within a
$k$-space window (of size $\Lambda$) proximate to the Fermi surface, with a
proportionality constant that depends on the nature of the underlying Fermi
surface. We extend this analysis to the gapped quantum liquids emergent from
the destabilisation of the Fermi surface by quantum fluctuations arising from
backscattering processes. Indeed, we find that the $k$-space entanglement
signatures of gapped quantum liquids appear to be governed by the nature of the
Fermi surface (e.g., nested or not) from which they emerge, as well as the
nature of their parent gapless metallic liquid (e.g., FL, MFL etc.). This is
confirmed by our finding an enhanced entanglement entropy for the nodal MFL
present at the quantum critical point recently discovered in the 2D Hubbard
model at optimal hole-doping. Our work thus paves the way for an entanglement
based classification of quantum liquids emergent from the criticality of
interacting fermionic matter.
