, ,

# Universal entanglement signatures of quantum liquids as a guide to fermionic
criticality

Siddhartha Patra1,∗, Anirban Mukherjee1 and Siddhartha Lal1
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Department of
Physical Sciences, Indian Institute of Science Education and Research-Kolkata,
W.B. 741246, India  ∗Author to whom any correspondence should be addressed.

###### Abstract

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) shows a violation of the area-law in real-
space entanglement entropy of a subsystem, believed to be a signature of the
ground state of a gapless quantum critical system of interacting fermions.
Here, we apply a $T=0$ renormalisation group approach to the FL, unveiling the
long-wavelength quantum fluctuations 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, with a
proportionality constant that depends on the nature of the underlying Fermi
surface. We extend this analysis to classify the gapped quantum liquids
emergent from the destabilisation of the Fermi surface by RG relevant quantum
fluctuations arising from backscattering processes.

##  1 Introduction

There has been a surge recently in applying quantum information-theoretic
tools towards understanding the nature of entanglement and quantum correlation
encoded within the ground states and low-lying excitations of strongly
correlated systems [1, 2, 3]. For a system with local interactions among its
constituents, the entanglement of a subsystem is expected to follow an area-
law (i.e., scale with the size of its boundary) [2]. There are, however, some
notable departures. First, a gapped topologically ordered system possesses, in
addition to the area-law term, a sub-dominant piece that arises from its
topological properties and encodes true long-range entanglement [4, 5, 6].
Another exception involves the finding of signatures of volume-law
entanglement in some quantum critical systems arising from the existence of
gapless quantum fluctuations at all lengthscales [7, 8, 9, 10, 11]. A final
exception is found in gapless quantum liquids of interacting electrons in
spatial dimensions $d>1$, and forms the focus of our work. The best understood
gapless electronic liquids are metals belonging to the Fermi liquid paradigm
[12, 13, 14]. These quantum critical systems are known to possess long-range
entanglement in the form a modified area-law: $S_{EE}\sim L^{d-1}\ln L$ for
$d$-dimensional subsystem of spatial length $L$ [15, 16, 17, 18, 19, 20, 21].
The extra $\ln L$ is conjectured to arise from the presence of gapless long-
wavelength quasiparticle excitations lying proximate to a Fermi surface [22,
15, 23, 16, 24]. Indeed, this result is believed to be the higher dimensional
analogue of the entanglement scaling observed in quantum critical phases of
interacting quantum spin and electron systems in 1D: $S_{EE}^{1D}\sim\ln L$
[25, 26, 27, 28]. A similar violation of the area-law is also expected in
other quantities of such gapless liquids, such as the fluctuation in the
number of particles within a subsystem [29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44].

In addition to the FL, we will also focus here on a particular variant of the
2D non-Fermi liquid known as the marginal Fermi liquid (MFL). Proposed on
phenomenological grounds as the parent metal of high-temperature
superconductivity in the cuprates [45, 46, 47], a first-principles derivation
of the effective theory for the MFL (e.g., low-energy Hamiltonian, nature of
gapless excitations etc.) was obtained only recently from a detailed
renormalisation group study of the 2D Hubbard model [48, 49, 50].
Interestingly, the modified area-law discussed above for the real-space
entanglement entropy of the FL is also proposed to hold for non-Fermi liquids
such as the MFL [51], hinting at a possible universality connecting various
gapless electronic quantum liquids despite the obvious differences in some of
their key properties (e.g., nature of their low-energy excitations etc.).

Gapless electronic liquids are different. A deeper understanding of the
entanglement properties of gapless quantum liquids such as the FL and MFL
necessitates an approach that lies beyond the well-established Ginzburg-
Landau-Wilson (GLW) paradigm. This is can be argued as follows: the effective
theories for phases falling within the GLW paradigm describe the quantised
excitations of classical scalar (or bosonic) field degrees of freedom that lie
above symmetry broken ground states that are short-range entangled at best.
They are characterised by order parameters that correspond to the ground state
expectation values of real-space local operators, and correlation functions
that satisfy the cluster decomposition property. On the other hand, the
excitations of a system of interacting fermions carry sign factors that arise
from the exchange of particles and do not have a classical origin. Further, a
filled Fermi volume at $T=0$ is the ground state for gapless fermionic quantum
liquids, owes its origin to the Pauli exclusion principle, and is
characterised by a topological quantum number called the Luttinger volume that
is robust against the inclusion certain kinds of inter-particle interactions
[52, 53, 54, 55, 56, 57, 58]. Indeed, these liquids are described by effective
theories that contain only the physics of the low-energy long-wavelength
degrees of freedom proximate to the Fermi surface [59, 60]. The remarkable
simplicity of these theories is that they are comprised purely of terms that
are number-diagonal in the momentum (or, $k$)-space single-electron Fock state
occupation number operators $\hat{n}_{\vec{k}}$ (see equations eq.(7) and
eq.(8) below). This indicates that the $k$-space many-particle wavefunctions
of these quantum liquids are direct product in form, i.e., they are separable
in terms of the single-electron Fock states and do not encode any entanglement
among them. Instead, as mentioned above, these states of electronic quantum
matter display long-range entanglement in real-space. From a renormalisation
group (RG) perspective, these low-energy (IR) theories correspond to universal
scale-invariant quantum critical fixed points obtained from the coarse-
graining of bare (UV) theories of interacting electrons possessing translation
invariance.

Setting out the goals. We first seek the quantum fluctuations in the UV from
which the real-space long-range entanglement of these universal IR theories is
emergent. As the low-energy degrees of freedom in the IR are effectively
decoupled from their UV counterparts, meeting this challenge necessitates an
investigation of the forward and tangential scattering related RG relevant
quantum fluctuations that are resolved under the flow from UV to IR. In
keeping with the Wilsonian approach to critical phenomena, this involves
understanding the signatures of universality that are likely encoded within
the many-particle entanglement wrought from such quantum fluctuations. The
answer to this question holds the potential to offer crucial insight on
whether certain aspects of the physics of candidate non-Fermi liquids (e.g.,
the marginal Fermi liquid) show commonalities with metals belonging to the
Fermi liquid paradigm, even if some other aspects (e.g., the nature of the
low-lying excitations) are qualitatively different. To the best our knowledge,
such a study of the RG evolution of the $k$-space entanglement has been
attempted only for the case of scalar (or bosonic) field theories [61].

Systems of interacting fermions, on the other hand, involve constraints on the
inter-particle scattering mechanisms arising from the Pauli exclusion
principle as well as phase space constraints due to the existence of a well-
defined Fermi volume and a bounding Fermi surface [59, 60]. We recall that the
quantum fluctuations in such systems thus contain fermion-exchange related
signs, rendering their study difficult. Recent studies indicate that the sign
structure of the many-particle wavefunction significantly affect the nature of
the entanglement encoded in them [62, 63]. This is of particular significance
to the study of non-Fermi liquid gapless states that are emergent at quantum
critical points related to the collapse of a Fermi liquid metal: a recent
numerical study based on an ansatz wavefunction incorporating correlations
from long-ranged backflow effects of the interacting fermions revealed a
crossover from a volume-law scaling of the subsystem entanglement entropy to
the modified area-law familiar for the Fermi liquid effective theory [64]. The
crossover takes place across a distance scale related to the physics of
critical backflow interactions. In addition, it is pertinent to enquire on
whether there exist similar imprints of universality hidden within the
backscattering related RG relevant quantum fluctuations that destabilise the
Fermi surface of these gapless metallic states and lead to the emergence of
topologically ordered gapped liquid states of quantum matter (e.g., the 2D
Mott liquid found in the 2D Hubbard model on the square lattice at
$1/2$-filling [48] and the Cooper pair insulator for the reduced BCS model
with a circular Fermi surface [65]). This amounts to an attempt at a
systematic classification of the states of quantum matter based on their
entanglement content, and looks beyond the short range entangled states that
belong to the Ginzburg-Landau-Wilson paradigm of broken symmetries and real-
space local order parameters.

The strategy. In order to meet these goals, we adopt a RG approach that we
have formulated recently for the investigation of criticality observed in
systems of interacting fermions [48, 49, 66, 57, 50, 58, 65, 67, 68].
Formulated using only many-particle unitary transformations to the bare
Hamiltonian, the procedure reveals IR fixed point theories and their low-
energy wavefunctions. An iterative application of the unitaries to the IR
wavefunctions then generates a family of states spanning towards the UV. This
facilitates a quantitative study of the RG evolution of many-body
correlations, as well as several many-particle entanglement features, from the
ground state and lowest lying excited states [58, 65, 67, 69, 68]. Thus, we
first employ this strategy to obtain the UV wavefunctions of several gapless
and gapped quantum liquid states that are unitarily connected to their IR
counterparts. We then compute the scaling of the entanglement entropy of a
block of states in $k$-space (lying proximate to the IR Fermi energy) with the
block size ($\Lambda$).

We note that there is a small, but growing, body of work on adopting a field
theoretic approach towards understanding the dependence of many-particle
entanglement on scale in systems with inter-particle interactions. While most
of these works apply the Wilsonian renormalisation group towards obtaining
entanglement measures from the effective action of interacting scalar theories
in real-space [70, 71, 72, 73, 74] and in momentum-space [61, 75, 76, 77, 78,
79, 80, 81, 82, 83, 84, 85], some recent works along these lines have also
been devoted to the study of interacting fermions at finite densities in
momentum-space [86, 87]. Our work, on the other hand, takes the Hamiltonian
renormalisation route in momentum-space towards the same goal. As discussed in
the concluding section, our methods corresponds to a tensor network that
readily admits an exact holographic mapping.

##  2 Methods

We now present the main methods employed by us. As these methods have been
employed by us in several recent works [48, 49, 66, 57, 50, 58, 65, 67, 69,
68], we present only an overview here and refer the reader to A as well as our
earlier works for further details.

###  2.1 Unitary renormalization group (URG) method

The URG method iteratively decouples quantum fluctuations in higher energy
Fock states by applying a many-body unitary operator, generating thereby a
low-energy effective Hamiltonian. For a general electronic system with
Hamiltonian $\mathcal{H}_{(N)}$ with total $N$ Fock states, Mutually non-
commuting terms in the Hamiltonian are the source of quantum fluctuation in
the occupation of the electronic Fock states. We first label the Fock states
according to the eigenvalues of the diagonal part of the bare Hamiltonian,
$\epsilon_{N}>\epsilon_{N-1}>\cdots$. The unitary operator $\mathcal{U}_{N}$
then decoupled the $N^{th}$ Fock state from all other states, i.e., it removes
quantum fluctuation present in this state. The low-energy effective
Hamiltonian after the first step of URG is

$\displaystyle\mathcal{H}_{(N-1)}=\mathcal{U}_{N}\mathcal{H}_{(N)}\mathcal{U}_{N}^{\dagger}~{}.$
(1)

Subsequently, the unitary operator $\mathcal{U}_{N-1}$ is used to remove the
quantum fluctuations in the Fock state $N-1$, and so on. The general form of
the unitary operator for $j^{th}$ step of URG is given as

$\displaystyle\mathcal{U}_{j}$ $\displaystyle=$
$\displaystyle(1+\eta_{j}-\eta_{j}^{\dagger})/\sqrt{2}\quad,\quad\eta_{j}^{\dagger}=\frac{1}{\omega-\textrm{Tr}(\mathcal{H}_{(j)}\hat{n}_{j})}c_{j}^{\dagger}\textrm{Tr}(\mathcal{H}_{(j)}c_{j})~{},$
(2)

where $\omega$ characterises the energyscale for quantum fluctuations, and an
equivalence of $\omega$ has been established with that for thermal
fluctuations at a finite-temperature [48, 49, 57, 50]. Iterative application
of the unitaries generates a series of effective Hamiltonians with
progressively lower RG energy scale,
$\mathcal{H}_{(N)},\mathcal{H}_{(N-1)},\cdots$. The fixed point effective
theory is reached at the effective Hamiltonian $\mathcal{H}_{(j^{*})}$, when
further decoupling is not possible due to the vanishing of the denominator of
$\eta_{j^{*}}$.

At any step of the URG [66, 48, 49, 57, 50, 58, 69, 65], the Hamiltonian in
the rotated basis is given by

$\displaystyle U_{k\sigma}\hat{H}U_{k\sigma}^{\dagger}$
$\displaystyle=\frac{1}{2}Tr_{k\sigma}(\hat{H})+\tau_{k\sigma}Tr_{k\sigma}(H\tau_{k\sigma})+\tau_{k\sigma}\\{c_{k\sigma}^{\dagger}Tr_{k\sigma}(\hat{H}c_{k\sigma}),\hat{\eta}_{k\sigma}\\}~{}.$
(3)

It is important to note that while
$\hat{n}_{k\sigma}\hat{H}(1-\hat{n}_{k\sigma})\neq 0$ (i.e., there exist non-
trivial quantum fluctuations in the occupation of single-particle Fock state
given by $n_{\sigma}$) prior to the application of the unitary operator, the
unitary operation removes such quantum fluctuation.

$\displaystyle\hat{n}_{k\sigma}U_{k\sigma}\hat{H}U_{k\sigma}^{\dagger}(1-\hat{n}_{k\sigma})=0~{}\Rightarrow~{}[\hat{n}_{k\sigma},U_{k\sigma}\hat{H}U_{k\sigma}^{\dagger}]=0~{},~{}~{}$
(4)

upon the application of $U$. The degree of freedom $n_{k\sigma}$ is thus
rendered an integral of motion (IOM) of the RG flow. The RG equations can then
be obtained from the condition eq.(4). The diagonal part of the Hamiltonian
$H^{D}_{(k\sigma)}$ with respect to the Fock state ($k\sigma$) has the
property $[H^{D}_{(k\sigma)},\hat{n}_{k\sigma}]=0$. Then, the operators
$\eta_{k\sigma}^{\dagger}$ and $\eta_{k\sigma}$ are given as

$\displaystyle\eta_{k\sigma}^{\dagger}$ $\displaystyle=$
$\displaystyle\frac{1}{\hat{\omega}_{k\sigma}-Tr_{N}(\hat{n}_{k\sigma}H^{D}_{(k\sigma)})\hat{n}_{k\sigma}}\mathcal{T}_{k\sigma}^{\dagger}\quad,\quad\eta_{k\sigma}=\frac{1}{\hat{\omega}_{k\sigma}-Tr_{k\sigma}((1-\hat{n}_{k\sigma})H^{D}_{(k\sigma)})(1-\hat{n}_{k\sigma})}\mathcal{T}_{k\sigma}~{},$
(5)

where
$\mathcal{T}_{k\sigma}^{\dagger}=Tr_{k\sigma}(H_{(k\sigma)}c_{k\sigma})c_{k\sigma}^{\dagger}$,
and
$\mathcal{T}_{k\sigma}=c_{k\sigma}Tr_{k\sigma}(c_{k\sigma}^{\dagger}H_{(k\sigma)})$.
Such decoupling leads to the emergence of a lower energy effective Hamiltonian
$U_{k\sigma}\hat{H}U^{\dagger}_{k\sigma}$. Further iterative application of
unitary operators on effective Hamiltonians leads to the decoupling of further
higher energy (UV) Fock states and the emergence of lower energy effective
Hamiltonians. Importantly, an equivalence of the quantum fluctuation scale
$\omega$ with an energyscale for finite-temperature thermal fluctuations has
also been established in Refs.[48, 49, 57, 50]. RG flow equations for various
couplings are derived from the series of effective Hamiltonians. The RG flow
attains a stable effective Hamiltonian at the IR fixed point when further
decoupling is not possible. The URG has been applied in deriving analytic
expressions for the low-energy effective theories of several gapless (e.g.,
Fermi liquid, Marginal Fermi liquid, Tomonaga-Luttinger liquid) as well as
gapped quantum liquid states of matter discussed in the introduction (e.g.,
Mott liquid, Cooper Pair Insulator, etc.) [48, 49, 66, 57, 50, 58, 65]. We
will next see that the URG motivates a renormalisation group scheme for the
study of many-particle entanglement in these quantum liquids.

###  2.2 Momentum-space entanglement renormalisation group (MERG)

Having set out the URG flow to the IR fixed point Hamiltonian, we can now
sketch the MERG algorithm by which to study the renormalisation group flow of
entanglement in momentum-space. First, we solve for the wavefunctions of the
ground state and low-lying excited states of the fixed point Hamiltonian. Now,
by acting with the unitary operators of the URG in reverse on, say, the ground
state $|\Psi_{0}\rangle$, we systematically re-entangle the low-energy degrees
of freedom lying proximate to the Fermi surface with the hitherto decoupled
Fock states (i.e., describing the IOMs of the URG flow). In this way, we
generate under the MERG flow a family of many-body wavefunctions that are
unitarily connected to $|\Psi_{0}\rangle$ by quantum fluctuations in momentum-
space [58, 65, 67, 69]. The explicit form of the wavefunction obtained after
the first step of the MERG flow is given by

$\displaystyle|\Psi_{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\bigg{[}1+Q_{k\sigma}c_{k\sigma}Tr_{k\sigma}(c_{k\sigma}^{\dagger}H_{(k\sigma)})-P_{k\sigma}Tr_{k\sigma}(H_{(k\sigma)}c_{k\sigma})c_{k\sigma}^{\dagger}\bigg{]}|\Psi_{0}\rangle~{},$
$\displaystyle\hat{Q}_{k\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{\hat{\omega}_{k\sigma}-Tr_{k\sigma}((1-\hat{n}_{k\sigma})H^{D}_{(k\sigma)})(1-\hat{n}_{k\sigma})}\quad,\quad\hat{P}_{k\sigma}=\frac{1}{\hat{\omega}_{k\sigma}-Tr_{k\sigma}(\hat{n}_{k\sigma}H^{D}_{(k\sigma)})\hat{n}_{k\sigma}}~{}.$
(6)

Having obtained a series of many-body wavefunctions from the MERG, we compute
from them the RG evolution of several measures of many-particle entanglement
and many-body correlations.

##  3 Models of quantum matter

We introduce the various states of quantum matter whose entanglement has been
studied using the methods outlined above. In order to focus the discussion, we
present all technical details (e.g., the derivation of URG flow equations that
lead to various IR fixed point theories) in A.

###  3.1 Gapless quantum liquids

We investigate the momentum-space entanglement features of a variety of
gapless states of fermionic quantum matter in this work, each of which is
described briefly below.

1\. Fermi liquid and Marginal Fermi liquid for the square lattice:
The IR fixed point Hamiltonian describing the excitations of the FL phase is
[12, 13, 14]

$\displaystyle H^{\hat{s}}_{FL}$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{k,\sigma}\epsilon_{k}n_{k\sigma}+\displaystyle\sum_{\begin{subarray}{c}kk^{\prime}\in\\{\hat{s},\hat{s}^{\prime}\\}\\\
\sigma,\sigma^{\prime}\end{subarray}}V_{\begin{subarray}{c}kk^{\prime}\sigma\sigma^{\prime}\end{subarray}}~{}n_{k,\sigma}n_{k^{\prime},\sigma^{\prime}}~{},$
(7)

where $\hat{s}$ and $\hat{s}^{\prime}$ correspond to excitations in directions
normal to the Fermi surface, and an analytic expression for the RG evolution
of the interaction coupling
$V_{\begin{subarray}{c}kk^{\prime}\sigma\sigma^{\prime}\end{subarray}}$ has
been obtained from a unitary renormalisation group (URG) study of the 2D
Hubbard model on the square lattice at large hole doping [49, 50]. Further,
the IR fixed point effective Hamiltonian for the MFL phase is obtained from
the URG analysis of the 2D Hubbard model at half-filling, as well as optimal
doping, as as [48, 49, 50]

$\displaystyle H^{\hat{s}}_{MFL}$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{k,\sigma}\epsilon_{k}n_{k\sigma}+\displaystyle\sum_{\begin{subarray}{c}\tilde{K}=(kk^{\prime}k^{\prime\prime}\sigma)\\\
kk^{\prime}k^{\prime\prime}\in\hat{s}\\\
\sigma\end{subarray}}\mathcal{R}^{\hat{s}}_{\tilde{K}}n_{k,\sigma}n_{k^{\prime},-\sigma}\bigg{(}1-n_{k^{\prime\prime}\sigma}\bigg{)}~{},$
(8)

where the RG evolution for the interaction coupling
$\mathcal{R}^{\hat{s}}_{\tilde{K}}$ shows the emergence of an excitation
different from the Fermi liquid kind. We can see from eq.(8) that the low-
lying excitations of the MFL are described by three-particle composite objects
comprised of two electrons and a hole. Further, various features conjectured
for the MFL phenomenology (e.g., the vanishing of the Landau quasiparticle
residue at energies proximate to the Fermi surface [45, 46, 47]) have been
established from the MFL effective Hamiltonian eq.(8) [48, 49, 50].
Importantly, all backscattering related quantum fluctuations in $k$-space are
rendered irrelevant under the RG flows to the FL and MFL fixed point theories.
On the other hand, quantum fluctuations arising from RG relevant forward and
tangential scattering processes of interacting electrons are of interest in
understanding the nature of the entanglement. Finally, in unveiling the
entanglement features of the gapped quantum liquids (e.g., Mott liquid, Cooper
pair insulator) obtained by destabilising the FL and MFL fixed points, we will
consider the quantum fluctuations arising from RG relevant backscattering
processes.

Figure 1:  Momentum-space construction of reduced system studied under MERG.
Four directions
($\hat{s}_{0},\hat{s}_{1},\mathcal{\hat{T}}\hat{s}_{0},\mathcal{\hat{T}}\hat{s}_{1}$)
are chosen normal to the Fermi surface in the 2D Brillouin zone; two
($\hat{s}_{0},\mathcal{\hat{T}}\hat{s}_{0}$, lines of red circles) are
labelled as the nodal directions, and two
($\hat{s}_{1},\mathcal{\hat{T}}\hat{s}_{1}$, lines of blue circles) as the
near-anti-nodal. Each $\hat{s}$-direction is comprised of 28 electronic Fock
states (red and blue circles). Pairs of electronic Fock states with opposite
momentum and spin ($\vec{k}\uparrow,-\vec{k}\downarrow$) form Anderson
pseudospins. The bold blue and red lines show the two opposite momentum
present at two ends, participating in the pseudospin formation.

2\. Fermi liquid with circular Fermi surface in 2D:
In contrast to the electronic differentiation in the electronic dispersion
that is present for the case of the FL and MFL phases on the square lattice,
the FL phase for the case of a circular Fermi surface is isotropic in the
angular direction. Thus, in order to understand the dependence of various
entanglement and scattering signatures of the FL phase on the electronic
differentiation, we have also studied the momentum space entanglement scaling
of an FL phase proximate to a circular Fermi surface [65]. Though the nature
of the forward scattering for both cases is similar, the difference in the
nature of tangential scatterings makes them different. Further, the symmetry
of the circular Fermi surface allows one to focus on the momentum space
scaling for any one radial direction; this renders possible the simulation of
a larger number of steps of the MERG, and thereby obtain a larger family of
wavefunctions ranging between the IR and UV.

3\. Tomonaga-Luttinger Liquid in 1D:
In order to understand the dependence of the nature of the many-particle
entanglement on the dimensionality of the underlying electronic system, we
have studied the gapless quantum liquid phase in one spatial dimension known
as the Tomonaga-Luttinger Liquid (TLL) [58]. While this phase corresponds to a
non-Fermi liquid, the logarithmic entanglement scaling of this phase in real-
space [28, 43] hints at an underlying universality across dimensions. Due to
the simplicity of the physics being confined to one spatial dimension, we can
once again simulate a larger family of wavefunctions from the MERG flow.

4\. Quantum fluctuations proximate to a quantum critical point:
An important class of gapless quantum matter are those found at quantum
critical points. Here, we study the entanglement features of the gapless
quantum liquid obtained at the quantum critical point (QCP) obtained recently
by some of us at optimal hole-doping in the 2D Hubbard model [57, 50]. The
four nodal directions of the Fermi surface were observed to be gapless at this
critical point, whereas other directions perpendicular to the Fermi surface
were found to be gapped (and comprised of the 2D Mott liquid). We expect a
volume law behaviour for the entanglement entropy arising from the strong
quantum fluctuations present at such a QCP.

###  3.2 Gapped quantum liquids

In order to contrast the entanglement features of various gapless quantum
liquids with their gapped counterparts, we have performed MERG calculations of
two examples of the latter as well.

1\. 2D Cooper Pair Insulator:
The Cooper pair insulator is an insulating phase emergent from a RG relevant
pairing instability of the FL phase. The momentum space symmetry of the
emergent CPI phase ($C_{4}$ or $U(1)$) [49, 50, 65] corresponds to the
symmetry of the Fermi surface of the parent metallic phase (optimally hole-
doped Fermi surface of the square lattice and circular Fermi surface
respectively).

2\. 2D Mott liquid:
The 2D Mott liquid phase is an insulating phase emergent from an instability
of the Marginal Fermi liquid phase of the 2D Hubbard model on the square
lattice at $1/2$-filling. It arises from RG relevant backscattering of charge
pseudospin degrees of freedom [48, 50, 69], and possesses the $C_{4}$ symmetry
of the square Fermi surface of a half-filled tight-binding problem on the
square lattice.

##  4 Results

Forming a reduced system. In order to study the $T=0$ FL and MFL phases
obtained recently for the Hubbard model [57, 50], we consider the square Fermi
volume of the $1/2$-filled tight-binding problem on the square lattice and
divide the momentum-space window of low-momenta around the square Fermi
surface into several directions normal to it (see Fig.1). We note that
electrons among all the directions perpendicular to the Fermi surface interact
with one other, and will therefore contribute to the renormalization group
computation. We consider scattering processes among all such electronic states
in implementing the URG and MERG formalisms.

Our investigations of entanglement involve simulating the RG evolution of
many-body wavefunctions. For this, we apply iteratively the inverse many-
particle unitary transformations of the URG method to the IR ground state
wavefunction of, say, the FL (see Method). This generates quantum fluctuations
by coupling the electronic Fock states near the Fermi surface to those farther
away via forward and tangential scattering processes. Note that any two-
particle scattering process can always be decomposed in forward, backward, and
tangential scattering. In this way, we obtain a family of wavefunctions
ranging from IR to UV, and involving a systematic growth of the size of the
effective Hilbert space of interacting electrons. This $k$-space entanglement
renormalisation group (MERG) process is, however, rendered very challenging
due to the fact that the many-particle Hilbert space scales exponentially in
the system size. In order to make the MERG analysis tractable, we construct a
reduced subspace of the FL and MFL systems in $k$-space for an underlying 2D
square lattice (Fig.1). We consider only the quantum fluctuations generated by
scattering processes along only four such directions
($\hat{s}_{0},\hat{s}_{1},\mathcal{\hat{T}}\hat{s}_{0},\mathcal{\hat{T}}\hat{s}_{1}$).
The direction $\hat{s}_{0}$ and its inverse ($\mathcal{\hat{T}}\hat{s}_{0}$)
are along the nodal direction of the square Brillouin zone, while
$\hat{s}_{1}$ and its inversion ($\mathcal{\hat{T}}\hat{s}_{1}$) are near the
anti-nodal direction. The Fock states residing on the four
$\hat{s}-$directions considered form a minimal set that captures all the
possible (forward, backward, and tangential) scattering processes. We have
used this construction to capture difference and universality among various
gapless and gapped quantum liquids.

Figure 2: Progression of the MERG flow. The black circles represent up-
pseudospins (pairs of occupied electronic Fock states), and the white circles
represent down-pseudospins (pairs of unoccupied electronic Fock states). The
red vertical line represents the Fermi surface. The red shaded box represents
the $k$-space window ($\Lambda_{0}$) of the IR fixed point theory, while the
light purple and light blue shaded boxes represent the windows ($\Lambda_{1}$
and $\Lambda_{2}$) of electronic Fock states coupled by quantum fluctuations
after the first and second steps of the MERG respectively.

As a study of the 112 fermionic states (i.e., a Hilbert space dimension of
$\sim 10^{33}$) is intractable, we focus the study on the quantum fluctuations
of the holon (pairs of unoccupied electronic Fock states) and doublon (pairs
of occupied electronic Fock states) degrees of freedom in the hole-doped 2D
Hubbard model. As shown in Fig.1, this corresponds to a smaller sub-space
defined in terms $56$ pseudospin degrees of freedom (defined in terms of
electronic Fock states with opposite momenta) and a reduced Hilbert space size
$\sim 10^{16}$ [88, 48, 49, 69]. A further drastic simplification is made by
ignoring the RG irrelevant backscattering related quantum fluctuations: this
allows us to study a reduced system of 28 pseudospins residing on only one
side of the Brillouin zone, say, the $\hat{s}_{0}$ and $\hat{s}_{1}$
directions. In each of these directions, we have chosen 14 pseudospin states,
seven outside and seven inside the Fermi surface. We have labelled all 28
pseudo-spins using integers in Fig.2. Specifically, Fock states residing
outside or inside the Fermi surface along a particular $\hat{s}-$direction are
labelled such that states nearest/farthest to the Fermi surface are associated
with highest/lowest integer respectively. Further, along a particular
$\hat{s}-$direction, states inside the Fermi surface are labelled by a higher
integer value than those outside the Fermi surface.

The MERG proceeds via the iterative application of a set of unitaries
($U_{(j)}^{\dagger}~{},~{}j\in\\{1,2,\cdots,6\\}$) to the IR fixed point
ground state and lowest-lying excited state wavefunctions of the gapless FL
and MFL liquids that systematically couple the $4$ electronic Fock states
($6,13,20,27$) lying within the window proximate to the Fermi surface to the
$24$ Fock states lying outside.

Figure 3: Construction of ground and lowest-lying excited states of FL and
MFL. The black circles represent up-pseudospins (pairs of occupied electronic
Fock states), and the white circles represent down-pseudospins (pairs of
unoccupied electronic Fock states). The red color vertical line in the middle
represents the Fermi surface, while the grey and white boxes represent the
occupied and unoccupied states below and above the Fermi surface respectively.
The blue dashed box represents the window of the IR gapless quantum liquid.
(a) FL ground state configuration: all states occupied inside the Fermi
surface, and unoccupied outside. (d) MFL ground state configuration (identical
to that of the FL). We have chosen two different excited states for both the
FL and MFL phases. (b) FL excited state, with excitation in the near-anti-
nodal direction. (c) FL excited state, with excitation in the nodal direction.
(e) MFL excited state, with excitation in the near-anti-nodal direction. (f)
MFL excited state, with excitation in the nodal direction.

The IR wavefunctions for the ground state and lowest-lying excited states of
the reduced system taken for the FL and MFL phases in shown in Fig.3. The IR
fixed-point ground state involving the four Fock states ($6,13,20,27$) are
identical for both the FL and MFL (see Fig.3(a, d)): all state below the Fermi
surface are occupied, and those above it are unoccupied. This is a consequence
of the fact that both these gapless liquids have been established as
satisfying Luttinger’s theorem [52, 53, 54, 55, 56, 57, 58]. The lowest lying
excited states of the FL and MFL involve configurations with holes below the
Fermi surface and occupied states above it. We consider two such possible
excited states in the reduced model: the near-anti-node (Fig.3(b) for the FL
and (e) for the MFL) and nodal (Fig.3(c) for the FL and (f) for the MFL)
excited states. The IR ground state of the FL is written in terms of the
pseudospin degrees of freedom as
$|\Psi^{FL}_{0}\rangle=|\phi^{FL}_{E}\rangle\otimes|\Psi_{IOM}\rangle_{(0)}$,
where
$|\phi^{FL}_{E}\rangle=|\uparrow_{20}\uparrow_{27}\rangle\otimes|\downarrow_{6}\downarrow_{13}\rangle$
represents the window of Fock states ($6,13,20,27$) proximate to the Fermi
surface and the configuration of the Fock states lying outside these windows
(whose occupation numbers correspond to integrals of motion (IOMs) of the URG
method) is given by

$\displaystyle|\Psi_{IOM}\rangle_{(0)}=\displaystyle\prod_{i=0}^{5}|\downarrow_{i}\rangle~{}\otimes\displaystyle\prod_{i=7}^{12}|\downarrow_{i}\rangle~{}\otimes\displaystyle\prod_{i=14}^{19}|\uparrow_{i}\rangle\otimes\displaystyle\prod_{i=21}^{26}|\uparrow_{i}\rangle~{}.~{}~{}$
(9)

The first step of the MERG then couples the four Fock states ($5,12,19,26$)
with $(6,13,20,27)$ via the application of the unitary operator
$\mathcal{U}_{(6)}^{\dagger}$:
$|\Psi^{FL}_{1}\rangle=\mathcal{U}_{(6)}^{\dagger}|\Psi^{FL}_{0}\rangle$ , and
so onwards till we obtain a UV wavefunction that is unitarily connected to the
IR state we started from.

We then employ the UV wavefunction to study the scaling of the von-Neumann
entanglement entropy of a subsystem in $k$-space with its size. This is done
by first selecting a block in $k$-space of size $(2\Lambda)$, chosen
symmetrically about the Fermi surface and along the $\hat{s}_{0}$ and
$\hat{s}_{1}$ directions. Small $k$-space blocks contain degrees of freedom
corresponding to the largest real-space lengthscales and vice versa. By
tracing over all degrees of freedom except those within the chosen block, a
reduced density matrix is then obtained for the block ($\rho_{\Lambda}$). The
von-Neumann entanglement entropy
($S_{EE}=-{\textrm{T}r}\left[\rho_{\Lambda}\ln\rho_{\Lambda}\right]$) is
computed for the $k$-space block. Understanding the entanglement signatures in
the UV wavefunctions of FL and MFL $(|\Psi_{j}\rangle,~{}j>0)$ is the primary
goal of this work.

Figure 4: Scaling of entanglement entropy ($S_{EE}$) and number fluctuations
($\Delta N$) within a window in $k$-space with window size ($\Lambda$) for
gapless liquids. (a) $S_{EE}$ and (b) $\Delta N$ for FL ground state along the
nodal direction $\hat{s}_{0}$. (c) $S_{EE}$ and (d) $\Delta N$ for MFL ground
state along the nodal direction $\hat{s}_{0}$. (e) and (f) show scaling of
$S_{EE}$ for excited states of FL along nodal ($\hat{s}_{0}$) and antinodal
($\hat{s}_{1}$) directions respectively. Similarly, (g) and (h) show scaling
of $S_{EE}$ for excited states of MFL along nodal ($\hat{s}_{0}$) and
antinodal ($\hat{s}_{1}$) directions respectively. While $four$ MERG steps
where performed for case (a), $five$ were performed in all other cases
(b,e,f,c,d,g,h).

Scaling of $S_{EE}$ for gapless quantum liquids : Fig.4(a) and Fig.4(c) show
the scaling of $S_{EE}$ $k$-space block size ($\Lambda$) along the nodal
direction of the UV ground states of the FL and MFL obtained from the
optimally doped and overdoped 2D Hubbard model respectively. Both gapless
quantum liquids show logarithmic scaling of block entanglement in $k$-space
with decrease in block size $\Lambda$, i.e., upon focusing on quantum
fluctuations lying proximate to the Fermi surface: $S_{EE}\sim\ln\Lambda$. The
logarithmic scaling is clear for a smaller $\Lambda$ window near the Fermi
surface, and deviations away from the logarithmic scaling are observed as
$\Lambda$ is increased to large values. The deviations are, however, different
for various types of phases. The momentum space entanglement signature
suggests a corresponding logarithmic scaling in real-space as well (in large
system size limit), with deviations for for small and intermediate real-space
scales [89, 90].

Further, the number fluctuations ($\Delta N=\sqrt{<N^{2}>-<N>^{2}}$) within
the block $\Lambda$ also show similar logarithmic scaling for both the FL and
MFL: $\Delta N\sim\ln\Lambda$, as shown in Fig.4(b) and Fig.4(d) respectively.
These results correspond to a modified area-law for these gapless liquids
(upon accounting for all directions normal to the Fermi surface):
$S_{EE}^{gapless}\sim\Lambda\ln\Lambda$. Further, this confirms the existence
of long-range entanglement in $k$-space encoded within the UV wavefunctions of
both these gapless quantum liquids, arising from long-wavelength quantum
fluctuations at low energies.

The results for the scaling of $S_{EE}$ with $\Lambda$ for the two excited
state configurations (i.e., along the nodal ($\hat{s}_{0}$) and near-anti-
nodal ($\hat{s}_{1}$) directions normal to the Fermi surface) of the FL and
MFL phases are shown in Figs.4(e), 4(f) and Figs.4(g), 4(h) respectively.
While all results show a logarithmic scaling for $S_{EE}$ with $\Lambda$,
$S_{EE}\sim(1/\alpha)\ln\Lambda$, the values of the coefficient $\alpha_{FL}$
are similar for the two excited states of the FL ($\alpha_{FL}\sim 2$). This
likely arises from the presence of strong forward scattering normal to the
Fermi surface, as well as tangential scattering between the two $\hat{s}$
directions. On the other hand, the values of $\alpha_{MFL}$ are quite
different for the two excited states of the MFL, and
$\alpha_{MFL}>>\alpha_{FL}$. This displays the strong electronic
differentiation among the $\hat{s}$ directions in the MFL phase, and the fact
that only strong forward scattering processes determine the entanglement of
this gapless liquid. Further, we have also found that both the 2D Fermi liquid
with circular Fermi surface and the 1D Tomonaga-Luttinger liquid show
logarithmic scaling behaviour of $S_{EE}$ with $\Lambda$. In Fig.5(a) and (b),
we show that the coefficient $\alpha\to 2$ for both these gapless quantum
liquids upon computing the $S_{EE}$ from wavefunctions after a sufficiently
large number of MERG steps. These results suggest that while the Fermi liquid
and several non-Fermi liquids show the same modified area-law for the scaling
of $S_{EE}$ with $\Lambda$, the value of the proportionality constant
$1/\alpha$ is sensitive to the nature of the underlying Fermi surface.

Figure 5: Variation of the inverse slope ($\alpha$) of the scaling of $S_{EE}$
with $\ln\Lambda$ with number of MERG steps. The coefficient $\alpha$ is
observed to saturate to $\alpha\sim 2$ for the Fermi liquid in two dimensions
with a circular Fermi surface (blue circles), and the Tomonaga-Luttinger
liquid in one dimension (black diamonds).

Scaling of $S_{EE}$ for gapped quantum liquids : In order to understand
whether $k$-space entanglement is a good diagnostic of fermionic criticality,
we need to study RG relevant quantum fluctuations of various kinds. Having
accounted above for the forward and tangential scattering related quantum
fluctuations that lead to gapless quantum liquids, we now turn to the gapped
quantum liquids that arise from backscattering. The ground state wavefunctions
for the states studied below have been obtained from Refs.[48, 49, 58, 65],
and adopted to the reduced system shown in Figs.1 and 2 similarly to that
discussed earlier for gapless liquids. Fig.6(a) shows the linear scaling of
$S_{EE}$ with $\Lambda$ for the gapped 2D Mott liquid along the nodal
direction $\hat{s}_{0}$. As we now argue, this can be understood as an area-
law scaling in momentum space. The nature of the effective Hamiltonian for the
2D Mott liquid [48] and 2D Cooper pair insulator [49] obtained from
instabilities of a strongly nested Fermi surface involves a strong coupling
between all the electronic degrees of freedom within the gapped low-energy
window lying proximate to the Fermi surface. As a result, the ground state
wavefunction of these gapped quantum liquid phases cannot be decomposed into a
direct product of states along each s-hat direction normal to the Fermi
surface [48, 91], leading to the area-law momentum space scaling of
$S_{\textrm{EE}}\sim\Lambda$ for the 2D Mott liquid observed in Fig.6(a).

On the other hand, the momentum space scaling of
$S_{\textrm{EE}}\sim\ln\Lambda$ (with a proportionality coefficient
$\alpha\sim 2$) for a gapped 2D Cooper pair insulator phase along a given
direction normal to a circular Fermi surface observed in Fig.6(b) likely
arises from it’s non-nested nature [65], while the area law scaling (similar
to Fig.6(a)) is expected for the Cooper pair insulator arising from the
strongly nested 2D Fermi surface [49]. Further, the nature of the parent metal
(from which these gapped phases are emergent) also appears to be an important
factor: the MFL is the parent metal for the 2D Mott liquid [48, 50], while the
FL is the parent meal for the CPI studied in Fig.6(b) [65]. The Mott liquid
and Cooper pair insulating phases are special types of gapped quantum liquids
comprised of pseudo-spin degrees of freedom.

Figure 6: Scaling of entanglement entropy ($S_{EE}$) with window size
($\Lambda$) for gapped liquids. (a) 2D Mott liquid along nodal ($\hat{s}_{0}$)
direction of a 2D nested Fermi surface. (b) 2D Cooper pair insulator (CPI)
along a radius of the circular Fermi surface. (c) and (d): Scaling of $S_{EE}$
with $\Lambda$ for gapless MFL along nodal direction ($\hat{s}_{0}$) and 2D
CPI along near-anti-nodal direction ($\hat{s}_{1}$) at the quantum critical
point of the optimally hole-doped 2D Hubbard model [49] respectively. While
$four$ MERG steps where performed for case (b), $five$ were performed in all
other cases (a,c,d).

In order to understand the dependence of these scaling laws of $S_{EE}$ on the
nature of the underlying Fermi surface, we studied the case of the quantum
critical point obtained recently in the optimally-doped 2D Hubbard model [49]:
here, the four nodal directions contain gapless MFL liquids while the rest of
the Fermi surface is composed of a gapped 2D Cooper pair insulator liquid.
Thus, in Figs.6(c) and (d), we present the scaling of $S_{EE}$ with $\Lambda$
for the nodal MFL (along $\hat{s}_{0}$) and near-anti-nodal CPI liquid (along
$\hat{s}_{1}$) respectively. While the CPI again shows the logarithmic scaling
behaviour (albeit with a coefficient $\alpha<1$), the MFL shows linear scaling
of $S_{EE}$ with $\Lambda$ and reflects on an enhanced long-range entanglement
along the nodal directions due to an underlying quantum critical Fermi
surface.

##  5 Discussions

A modified area-law in $k$-space. By studying a reduced subspace containing
the relevant $k$-space scattering processes (i.e., along and between certain
directions normal to the Fermi surface), our results reveal a modified area-
law scaling in both the block entanglement ($S_{EE}$) and number fluctuations
($\Delta N$) for the UV ground states and lowest lying excited states
unitarily connected to the Fermi liquid (FL), marginal Fermi liquid (MFL) and
Tomonaga-Luttinger liquid (TLL) gapless quantum liquids (Figs.4 and 5) :
$S_{EE}\propto\log(\Lambda/\Lambda_{F})$ and $\Delta
N\propto\log(\Lambda/\Lambda_{F})$ (where $\Lambda_{F}$ is the Fermi momentum,
and which we set to unity hereafter). This is in agreement with findings for
the real-space block entanglement entropy and number fluctuations computed
from the IR state of the FL, and reveals that the forward and tangential
scattering related RG relevant quantum fluctuations near the Fermi surface
encode the long-range $k$-space entanglement of the modified area-law even at
UV scales. This is consistent with the conjectured duality mapping between the
real-space and $k$-space subsystem entanglement entropies of a system of non-
interacting fermions in $d$-spatial dimensions [92]. Further, it shows that
the $k$-space quantum fluctuations of the UV theory are systematically
converted into their real-space counterparts along the URG flow.

Universality of Gapless Liquids. Importantly, the similarity of the results
obtained for both FL, MFL and TLL offer striking evidence on the universality
of the entanglement properties of various gapless quantum liquids that arise
from systems of interacting electrons. This suggests a unification of the
Fermi liquid paradigm with non-Fermi liquid metals that are qualitatively
different in terms of their low-lying excitations. Interestingly, we find that
the proportionality constant $\alpha$ in the relation
$S_{EE}\propto(1/\alpha)\ln\Lambda$ is very different for the different
metallic systems: $\alpha\sim 2$ for the FL (whether obtained from a 2D Fermi
volume with a $U(1)$ symmetric circular Fermi surface (Fig.5(a)), or from the
strongly rounded $C(4)$ symmetric Fermi surface of the FL obtained in the 2D
Hubbard at large hole-doping [49] (Fig.4(a), (b), (e) and (f))) and the TLL
(with a 2 point Fermi surface, Fig.5(b)). On the other hand, we find
$\alpha>>1$ for the MFL obtained in the 2D Hubbard at optimal hole-doping [49]
(Fig.4(c), (d), (g) and (h))). We believe that this difference arises from the
fact that the MFL is governed by strongly RG relevant forward scattering in
directions normal to the Fermi surface, and that MFL states on different
normals show variations arising from the differentiation in the electronic
dispersion everywhere along the $C(4)$ symmetric Fermi surface of the 2D
square lattice [48, 49]. By contrast, the presence of subdominant RG relevant
tangential scattering processes in the FL between different directions normal
to the Fermi surface remove the electronic differentiation inherent in the MFL
[49].

Classification of Gapped Quantum Liquids. We also analyse the $k$-space block
entanglement entropy ($S_{EE}$) of the UV states that are unitarily connected
to the gapped topologically ordered quantum liquids emergent from the
destabilisation of the Fermi surface of the FL and MFL gapless metals [48, 49,
50, 65]. As shown in Figs.6(a) and (b), we find that while the Cooper pair
insulating (CPI) state for the 2D FL with circular Fermi surface [65] shows
logarithmic scaling of $S_{EE}$ with $\Lambda$ (with the inverse coefficient
$\alpha\sim 2$), the Mott liquid state in the 2D Hubbard model at
$1/2$-filling [48, 50] shows an area-law scaling with $\Lambda$. We believe
that this striking difference arises from the fact that the 2D Mott liquid is
emergent from the backscattering related instability of a strongly nested
singular Fermi surface of the underlying 2D tight-binding model (i.e,
containing van Hove singularities), while the 2D CPI state emergent from the
minimal nesting of a circular Fermi surface (i.e., that of diametrically
opposite Fermi points). Further, the parent metal of the former is the 2D MFL
[48], while that of the latter is the 2D FL [65]. Thus, we conclude that the
entanglement features of a gapped quantum liquid appear to be determined by
the nature of the Fermi surface and parent metal from which it emerges.
Further evidence of this dependence is seen by considering the scaling
behaviour of the $k$-space block entanglement entropy ($S_{EE}$) of the
quantum critical state of matter found recently in the 2D Hubbard model on the
square lattice at optimal doping [49]. At this novel quantum critical point
(QCP), only the four nodal directions of the 2D Brillouin zone for the square
lattice are found to be comprised of gapless metallic MFLs, while there exists
a gapped 2D CPI quantum liquid in the antinodal regions of the rounded $C(4)$
symmetric hole-doped Fermi surface of the underlying 2D tight-binding model.
We find that $S_{EE}$ scales logarithmically with $\Lambda$ for this 2D CPI
state (Fig.6(d)), while the nodal MFLs (about four point-like singular quantum
critical Fermi surfaces) show a volume-law scaling (Fig.6(c)). Indeed, this
appears to be consistent with the findings of Ref.[64] for $S_{EE}$ of a
gapless metal at a quantum critical Fermi surface.

Holographic evolution of entanglement. Finally, the unitary renormalisation
group (URG) method has also been shown to provide an explicit demonstration of
the holographic principle [93, 94]. By this, we mean that the evolution of the
Hamiltonian from UV to IR via the unitaries corresponds to a tensor network
that admits an exact holographic mapping, i.e., it provides a precise
relationship between the coupled degrees of freedom within one layer of the
network with those that are emergent in the next [95, 96]. Further, this
generates a eigenstate coefficient tensor network possessing an entanglement
metric [57]. The renormalisation of the entanglement corresponds to the
evolution of the many-particle Hilbert space geometry. By employing this
method, we unveil several distinct holographic signatures of Fermi and non-
Fermi liquids in measures of entanglement (see B) and many-body correlations
(see C). An important challenge for the future will be to characterise the
entanglement properties of gapless liquids using multipartite quantum
information measures, as has been achieved recently for topologically ordered
gapped liquids [97]. Other future studies include analysing the deviations
from logarithmic scaling for any subtle signatures of universality, and
investigating the microscopic origin of long-ranged entanglement in quantum
critical liquid studied above as well as elsewhere [98, 99].

The authors thank S. Pal, Abhirup Mukherjee, R. K. Singh, A. Dasgupta, A.
Ghosh, S. Sinha and A. Taraphder for several discussions and feedback. S. P.
and A. M. thanks the CSIR, Govt. of India and IISER Kolkata for funding
through research fellowships. S. Lal thanks the SERB, Govt. of India for
funding through MATRICS grant MTR/2021/000141 and Core Research Grant
CRG/2021/000852. We also thank two anonymous referees for their helpful
comments and questions, which greatly improved our presentation.

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##  Appendix A Details of the URG flow equations

We obtain the effective Hamiltonian for the FL and MFL IR fixed point theories
for the case of the half-filled square lattice by performing a URG analysis of
the 2D Hubbard model [48, 49]. The Hamiltonian RG flow equation is

$\displaystyle\Delta
H_{(j)}=\displaystyle\sum_{l}\textrm{Tr}_{j,l}(c_{j,l}^{\dagger})c_{j,l}~{}G_{(j),l}~{}c_{j,l}^{\dagger}\textrm{Tr}_{j,l}(H_{(j)}c_{j,l})~{},$
(10)

where
$G_{(j),l}=[\omega-\hat{n}_{j,l}\textrm{Tr}_{j,l}(H^{D}_{(j)}\hat{n}_{j,l})]^{-1}$
and $H^{D}_{(j)}$ is the diagonal part of the Hamiltonian at the RG step $j$.
This change in the Hamiltonian is then decomposed in different scattering
channels: namely, forward scattering $\Delta H^{F}_{(j)}$, backscattering
$\Delta H^{B}_{(j)}$, tangential scattering $\Delta H^{T}_{(j)}$, and the
3-particle interaction term $\Delta H^{3}_{(j)}$. This leads to the following
RG equations

$\displaystyle\Delta H^{F}_{(j)}$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{k,k^{\prime},l^{\prime}}c_{k^{\prime},l}^{\dagger}c_{k^{\prime},l^{\prime}}^{\dagger}\frac{4(V(\delta)_{l}^{(j)})^{2}\tau_{j,l}\tau_{j,l^{\prime}}}{G_{j,l}^{-1}-V(\delta)_{l}^{(j)}\tau_{j,l}\tau_{j,l^{\prime}}}c_{k,l^{\prime}}c_{k,l}~{},~{}~{}~{}~{}~{}~{}~{}\textrm{where
,}~{}\tau_{j,l}=(\hat{n}_{j,l}-\frac{1}{2})$ (11) $\displaystyle\Delta
H^{B}_{(j)}$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{k,k^{\prime},l^{\prime}}c_{k^{\prime},l}^{\dagger}c_{k^{\prime},l^{\prime}}^{\dagger}\frac{4V(\delta)_{l}^{(j)}K(\delta)_{l}^{(j)}\tau_{j,l}\tau_{j,l^{\prime}}}{G_{j,l}^{-1}-V(\delta)_{l}^{(j)}\tau_{j,l}\tau_{j,l^{\prime}}}c_{k,l^{\prime}}c_{k,l}~{},~{}~{}~{}$
(12) $\displaystyle\Delta H^{T}_{(j)}$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{k,k^{\prime},m,n}c_{k,m}^{\dagger}c_{k,m^{\prime}}^{\dagger}\frac{(\Gamma^{(j)})^{2}(L_{j}^{2}-L_{j}^{z2}-L_{j}^{z})}{\omega-\tilde{\epsilon}^{c}_{j,avg}L_{j}^{z}-\Gamma^{(j)}L_{j}^{z2}}c_{k^{\prime},n^{\prime}}c_{k^{\prime},n}~{},~{}~{}~{}$
(13) $\displaystyle\Delta H^{3}_{(j)}$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{k^{\prime\prime},k^{\prime},l^{\prime},l^{\prime\prime}}c_{k^{\prime},l}^{\dagger}c_{k^{\prime},l^{\prime}}^{\dagger}c_{j,l}\frac{V_{l}^{(j)}(\delta)V_{l}^{(j)}(\delta^{\prime})\tau_{j,l}}{\omega-\tilde{\epsilon}_{j,l}\tau_{j,l}}c_{j,l^{\prime\prime}}^{\dagger}c_{k^{\prime\prime},l}c_{k^{\prime\prime},l^{\prime}}$
(14)
$\displaystyle+\displaystyle\sum_{k^{\prime\prime},p^{\prime},l^{\prime},l^{\prime\prime}}c_{p^{\prime},l}^{\dagger}c_{p^{\prime},l^{\prime}}^{\dagger}c_{j,l^{\prime}}\frac{K_{l}^{(j)}(\delta)K_{l}^{(j)}(\delta^{\prime})\tau_{j,l}}{\omega-\tilde{\epsilon}_{j,l}\tau_{j,l}}c_{j,l^{\prime\prime}}^{\dagger}c_{k^{\prime\prime},l}c_{k^{\prime\prime},l^{\prime}}$
$\displaystyle+\displaystyle\sum_{\Lambda^{\prime}<\Lambda_{j},p^{\prime},k^{\prime\prime}}c_{p^{\prime},l}^{\dagger}c_{p^{\prime},l^{\prime}}^{\dagger}c_{j^{\prime},l^{\prime}}\frac{8R^{(j)}_{l,\delta\delta^{\prime\prime(j)}}R^{(j)}_{l,\delta^{\prime}\delta^{\prime\prime(j)}}\tau_{i_{1}}\tau_{i_{2}}\tau_{i_{3}}}{G_{j,l,3}^{-1}-R^{(j)}_{l,\delta^{\prime\prime},\delta^{\prime\prime}}\tau_{i_{1}}\tau_{i_{2}}\tau_{i_{3}}}c_{j,l^{\prime\prime}}^{\dagger}c_{k^{\prime\prime},l}c_{k^{\prime\prime},l^{\prime}}~{}.$

In the above equations, $(j,l)$ represents the state
$|\vec{k}_{\Lambda_{j},\hat{s}}\sigma\rangle$,
$(j,l^{\prime})\equiv|\vec{k}_{-\Lambda_{j}+\delta,T\hat{s}\sigma}\rangle$,
and $(j^{\prime},l)\equiv|\vec{k}_{\Lambda_{j},\hat{s}\sigma}\rangle$.
Further, $V(\delta)^{(j)}_{l},K(\delta)^{(j)}_{l}$ and $\Gamma^{(j)}_{l}$ are
the forward scattering density-density interaction, backscattering and
tangential scattering couplings respectively, while $R_{\delta\delta}^{(j)}$
is the interaction coupling for the 3-particle (i.e., 2-electron-1-hole)
composites objects. $\delta$ represents the off-resonant scattering momenta.
The components of the pseudospin $\vec{L}$,
$L^{+}=\sum_{m}c_{j,m}^{\dagger}c_{j,m^{\prime}}^{\dagger}$,
$L_{j}^{z}=(1/2)\sum_{m}(\hat{n}_{j,m}+\hat{n}_{j,m^{\prime}}-1)$ etc. follow
the standard spin-algebra. Further, the electronic Fock states are labelled as
$i_{1}:(\vec{k}_{\Lambda_{j},\hat{s},\sigma}),i_{2}:(\vec{k}_{-\Lambda_{j}+\delta^{{}^{\prime\prime}},T\hat{s},-\sigma}),i_{3}:(\vec{k}_{\Lambda^{\prime}\hat{s},\sigma})$.
$G_{j,l,3}^{-1}=(\epsilon_{j,l}+\epsilon_{j,l^{\prime}})/2-\Delta\mu_{eff}-\omega$,
where $\omega$ is the quantum fluctuation scale. From eq.(10), we extract the
coupling RG equations as follows

$\displaystyle\Delta V_{l,\hat{s}}^{(j)}$ $\displaystyle=$
$\displaystyle-\frac{(V_{l,\hat{s}}^{(j)})^{2}}{\frac{1}{2}E_{j,\hat{s}}-\omega+\frac{1}{4}V_{l,\hat{s}}^{(j)}}~{},$
(15) $\displaystyle\Delta K_{l,\hat{s}}^{(j)}$ $\displaystyle=$
$\displaystyle-\frac{(K_{l,\hat{s}}^{(j)})^{2}}{\omega-\frac{1}{2}E_{j,\hat{s}}+\frac{1}{4}K_{l,\hat{s}}^{(j)}}~{},$
(16) $\displaystyle\Delta\Gamma_{\hat{s}}^{(j)}$ $\displaystyle=$
$\displaystyle-\frac{N_{F}^{2}(\Gamma_{\hat{s}}^{(j)})^{2}}{\frac{1}{N^{\prime}}\sum_{s}E_{j,\hat{s}}-\omega+\frac{1}{4}\Gamma_{\hat{s}}^{(j)}}~{},$
(17) $\displaystyle\Delta R^{(j)}_{l,\hat{s}}$ $\displaystyle=$
$\displaystyle\frac{(V_{l,\hat{s}}^{(j)})^{2}}{\omega-\epsilon_{j,\hat{s}}}+\frac{(K_{l,\hat{s}}^{(j)})^{2}}{\omega-\epsilon_{j,\hat{s}}}+\frac{(R_{l,\hat{s}}^{(j)})^{2}}{\frac{1}{2}E_{j,\hat{s}}-\omega+\frac{1}{8}R_{l,\hat{s}}}~{},$
(18)

where
$E_{j,\hat{s}}=(\epsilon_{\Lambda_{j,\hat{s}}}+\epsilon_{\Lambda_{j,-\hat{s}}}-2\Delta\mu_{eff})$,
and $\Delta\mu_{eff}=-\Delta U/2$, $N_{F}$ is the number of electronic states
on the Fermi surface, and $\epsilon_{\Lambda_{j,\hat{s}}}$ is the kinetic
energy of an electron lying on the direction $\hat{s}$ normal to the Fermi
surface and at a momentum-space distance $\Lambda_{j}$ from the Fermi energy.
It was shown in Refs.[48, 49] that the gapless FL is obtained in the overdoped
2D Hubbard model with the couplings $V$ and $\Gamma$ dominant under RG, while
the MFL is obtained for filling ranging from zero (i.e., a $1/2$-filled
system) till optimal hole-doping at sufficiently high quantum fluctuation
scale $\omega$ due to the coupling $R$ being dominant under the RG flow.
Similarly, at small quantum fluctuations scales proximate to the Fermi energy,
the gapped 2D Mott quantum liquid is obtained due to the backscattering
coupling $K$ being dominant under RG for hole-doping ranging from zero till
optimal doping.

##  Appendix B RG evolution of various measures of Entanglement

###  B.1 Many-particle Block Entanglement Entropy (MBE)

Figure 7: Three choices of momentum-space blocks to compute block entanglement
entropy from each member of the family of wavefunctions obtained from the MERG
. Block-1 includes all the nodes on the direction $\hat{s}_{1}$. Block-2
includes the nodes outside the Fermi surface on $\hat{s}_{1}$ and inside the
Fermi surface on $\hat{s}_{0}$. Block-3 includes all the nodes residing
outside the Fermi surface.

As discussed in the main manuscript, we study a reduced system involving 28
pseudospin degrees of freedom formed from 56 electronic Fock states. While
there is no entanglement among the low-energy states that comprise the IR
fixed point theory in momentum (or, $k$)-space, nontrivial entanglement is
emergent from the MERG method. This is due to the introduction of RG-
irrelevant quantum fluctuations that iteratively couple the low-energy states
within the low-energy window (namely, the states $6,13,20$ and $27$) with the
24 states that lie outside it and farther away from the Fermi surface as the
MERG flow proceeds towards the UV. We measure the variation of the
entanglement entropy of subsystems of blocks of states in $k$-space in the
many-body state with the MERG steps.

As shown in the Fig.7, we have chosen three blocks of states in $k$-space
blocks comprising a total of $14$ states. Subsystem choice $1$ consists of
states along the near-anti-nodal direction $\hat{s}_{1}$ normal to the Fermi
surface. Thus, the entanglement entropy measured here involves the
entanglement between the states in $\hat{s}_{1}$ and those in $\hat{s}_{0})$
(i.e., the nodal direction normal to the Fermi surface), thus offering a
measure of the entanglement driven by quantum fluctuations generated via
tangential scatterings between the two $\hat{s}$ directions. Subsystem choice
2, on the other hand, consists of states outside the Fermi surface in
$\hat{s}_{1}$ but it in $\hat{s}_{0}$; the entanglement entropy measures here
the entanglement arising from quantum fluctuations involving both tangential
and forward scatterings. Another subsystem choice is that shown in block 3,
involving states outside the Fermi surface in both the $\hat{s}$ directions.
The entanglement entropy here measures the $k$-space entanglement driven due
to excitations that lead from within states the Fermi surface to those outside
it, arising from quantum fluctuations involving both forward and tangential
scattering events.

Figure 8: Variation of the block entanglement entropy for the choice of Block
1 of Fig.7 with MERG step number for FL and MFL ground and nodal excited
states. Left: Red curve represents the entanglement scaling for the FL ground
state, and the blue one represents the FL nodal excited state. Right: Red
curve represents the entanglement scaling for the MFL ground state, and the
blue curve represents MFL nodal excited state. Figure 9: Variation of the
block entanglement entropy for the choice of Block 2 of Fig.7 with MERG step
number for FL and MFL ground and nodal excited states. Left: Red curve
represents the entanglement scaling for the FL ground state, and the blue one
represents the FL nodal excited state. Right: Red curve represents the
entanglement scaling for the MFL ground state, and the blue curve represents
MFL nodal excited state. Figure 10: Variation of the block entanglement
entropy for the choice of Block 3 of Fig.7 with the MERG step number for FL
and MFL ground and nodal excited states. Left: Red curve represents the
entanglement scaling for the FL ground state, and the blue one represents the
FL nodal excited state. Right: Red curve represents the entanglement scaling
for the MFL ground state, and the blue curve represents MFL nodal excited
state.

The results presented in Figs.8, 9 and 10) of the variation of the many-
particle block entanglement (MBE) with the number of MERG steps show
distinctions between the FL and MFL states. As block 1 is very sensitive to
entanglement arising from quantum fluctuations due to tangential scattering
events, the stark difference between the numbers achieved by the MBE in the UV
for the case of block 1 in the FL and MFL reveals the dominance of tangential
scattering events in the former over the latter. On the other hand, the
similar values of the MBE achieved in the UV for the blocks 2 and 3 shows the
importance of forward scattering events in both the FL and MFL phases. Taken
together, these results show that while the IR ground state of the FL is
obtained from the decoupling of quantum fluctuations that arise from both
forward and tangential scattering, the MFL is obtained from the decoupling of
quantum fluctuations that arise almost completely from forward scattering.

###  B.2 Ryu-Takayanagi upper bound for Entanglement

Figure 11: Top: Variation of entanglement entropy of blocks of momentum-space
states of varying sizes with MERG step number for the FL ground state.
Different coloured curves correspond to different block sizes (shown in
inset). Bottom: Scaling of entanglement entropy of blocks of momentum-space
states of varying sizes with respect to MERG step number with MERG step number
for the MFL ground state.

Given that the URG method (from which the unitaries are obtained for the MERG)
is known to correspond to an exact holographic mapping [95, 57], we will now
check whether the entanglement entropies of various sizes of subsystems taken
from the family of many-body wavefunctions generated by the MERG satisfy the
Ryu-Takayanagi upper bound [100]. This bound is defined as follows:

$\displaystyle S^{max}_{n}\leq nS^{max}_{1}~{},$ (19)

where $S^{max}_{n}$ corresponds to the largest entanglement entropy of a
subsystem containing $n$ constituents (and $n\geq 1,n\in\mathcal{Z}$). For
this, we first compute the entanglement entropy of various block sizes across
the MERG energy scales ranging from the UV to the IR fixed point, as shown in
the Figure.11. The blocks have been chosen to lie along the near-anti-nodal
direction $\hat{s}_{1}$ and outside the Fermi surface. The smallest block size
contains one pseudospin state ($6$), while the largest contains all $7$ states
residing on the $\hat{s}_{1}$ direction outside the Fermi surface (see legends
in inset). Fig.11 shows that the block entanglement entropies increase from
the IR and increase linearly in the UV for the FL (i.e., for the largest block
size ($6,5,4,3,2,1,0$)), while they saturate for the MFL. This increase in
entanglement in the FL phase is likely due to the strong growth of tangential
scattering in addition to forward scattering. Fig.12 shows a comparison of
$S(6)^{max}$ (associated with the largest block size $(6,5,4,3,2,1)$) for both
the FL and MFL with $6\times S(1)^{max}$, and reveals that the Ryu-Takayanagi
upper bound is indeed obeyed for both these gapless quantum liquids. The
growth of the upper bound in the UV energy scale for the FL phase (as compared
to the saturation witnessed for the MFL phase) is again likely due to the
presence of strong tangential scattering processes.

Figure 12: MERG evolution of the Ryu-Takayanagi upper bound. The blue plus
symbols represent the entanglement entropy of the largest block of block size
six, and the red pentagons represent the upper bound by $nS(1)^{max}$, where
$n=6$. The top and bottom figures show that the scaling of the largest block
entanglement entropy is satisfied for the FL and MFL phase respectively.

###  B.3 Mutual Information

Figure 13: Reduced $k$-space system of 28 pseudospins is divided into four
blocks $A,B,C$ and $D$. The blocks $A$ and $B$ reside outside the Fermi
surface, while the blocks $C$ and $D$ reside inside the Fermi surface. $E_{F}$
represents the Fermi energy. $\hat{s}_{0}$ and $\hat{s}_{1}$ represents the
nodal and near-anti-nodal directions perpendicular to the Fermi surface.

Mutual information between two subsystems measures the total classical as well
as quantum correlations between them. Definition of mutual information between
two subsystems $A$ and $B$ is given as

$\displaystyle I_{2}(A:B)=S(A)+S(B)-S(AB)\geq 0~{},$ (20)

where $S(A_{i})$ is the von-Neumann entanglement entropy of the subset $A_{i}$
with the rest etc. Here, we compute the mutual information between two
momentum-space blocks. For this, we divide the entire $28$ pseudospin system
into four blocks ($A,B,C,D$), as shown in Fig.13, and compute the mutual
information between some of them for each member of the family of
wavefunctions generated by the MERG. The scaling of some of these mutual
informations is presented in Fig.14.

Figure 14: Variation of Mutual information ($I_{2}(A;B)$) between blocks $A$
and $B$ (left figure). and Mutual Information ($I_{2}(A;D)$) between blocks
$A$ and $D$ (right figure) with MERG step number.

We are mainly interested in the mutual informations $I_{2}(A,B)$ and
$I_{2}(A,D)$: which capture the correlation arising from tangential and
forward scattering. Note that $I_{2}(A,B)$ captures the correlations between
the blocks $A$ and $B$ connected via tangential and forward scattering,
whereas $I_{2}(A,B)$ captures that between $A$ and $D$ connected only through
forward scattering. Fig.14(left) shows that $I_{2}(A,B)$ for the ground and
excited states of the FL increases with the number of MERG steps, in keeping
with the steady growth in tangential scattering events as the MERG proceeds
towards the UV. This is in contrast with our finding for ground and excited
states of the MFL case, which remain zero throughout the MERG; this is
consistent with RG irrelevant tangential scattering processes for the MFL.
Further, the $I_{2}(A,D)$ among the blocks $A,D$ shown in Fig.14(right) shows
the importance of forward scattering is far greater in the UV for the MFL than
it is for the FL.

###  B.4 Tripartite Information

Tripartite information measures the joint entropy among three subsystems
$(A,B,C)$ [5, 101, 97], and is defined as

$\displaystyle I_{3}(A,B,C)$ $\displaystyle=$ $\displaystyle
S(A)+S(B)+S(C)-S(A\cup B)-S(B\cup C)-S(C\cup A)+S(A\cup B\cup C)~{},$ (21)
$\displaystyle=$ $\displaystyle I_{2}(A,B)+S(C)-S(B\cup C)-S(C\cup A)+S(A\cup
B\cup C)~{}.$

Figure 15: Variation of the tripartite information $I_{3}(20,27,6)$ among the
pseudospin states ($20,27,6$) with MERG step number for the FL (large red
diamond) and MFL (small blue diamond) ground states.

We are mainly interested in the tripartite information among the three
pseudospin states ($20,27,6$) proximate to the Fermi surface: the states
$6,27$ and $20,27$ are connected via forward and tangential scattering
respectively. The MERG scaling of $I_{3}(6,20,27)$ for the FL and MFL ground
states is shown in Figure.15: while $I_{3}(6,20,27)$ is found to grow
monotonically for the FL as the MERG proceeds to the UV, it is found to remain
strictly zero throughout for the MFL. As seen above for the mutual
informations, the growth of $I_{3}(6,20,27)$ for the FL is clearly driven by
the growth in tangential scattering processes in the UV. Interestingly, the
strict vanishing of $I_{3}(6,20,27)$ for the MFL ground state appears to show
a constraint among various entanglement measures

$\displaystyle
I_{3}{(6,20,27)}=S(6)-S(20,6)-S(27,6)+S(20,27,6)=0~{},~{}~{}~{}$ (22)

where we have used the fact that $I_{2}(20,27)=0$ in the MFL ground state. We
note, however, that the individual von-Neumann entanglement entropies in
eq.(22) are nonzero. Thus, the relations $I_{2}(20,27)=0$ and
$I_{3}{(6,20,27)}=0$ appear to be invariants of the MERG flow, capturing the
decoupling between various $\hat{s}$ directions normal to the Fermi surface
for the MFL.

##  Appendix C RG evolution of various many-body correlations

###  C.1 Correlations generated due to pseudospin-flip scattering

The low-energy window is formed by the states $(6,13,2,27)$ proximate to the
Fermi surface at the IR fixed point, and increases as the MERG progresses
towards the UV. We define a correlation measure $C^{(j),\hat{s}_{1}}_{sf}$ in
order to capture the pseudospin-flip correlations among the pseudospin states
present in the low-energy window near the near-anti-nodal direction
$\hat{s}_{1}$

$\displaystyle C^{(j),\hat{s}_{1}}_{sf}$ $\displaystyle=$
$\displaystyle\bigg{(}\langle\Psi^{(j)}|\displaystyle\sum^{\mathcal{U}^{\hat{s}_{1}}_{em}}_{i<j}(S_{i}^{+}S_{j}^{-}+S_{j}^{-}S_{j}^{+})|\Psi^{(j)}\rangle\bigg{)}/\bigg{(}\displaystyle\sum^{\mathcal{U}^{\hat{s}_{1}}_{em}}_{i<j}1\bigg{)}~{},~{}~{}~{}$
(23)

where $|\Psi^{(j)}\rangle$ is the wave function at the $j^{th}$ MERG step
(i.e., $j=0$ represents the IR fixed point and increases towards the UV),
$\mathcal{U}_{em}$ is the set of pseudospin states lying in the direction
$\hat{s}_{1}$ within the low-energy window at the $j^{th}$ MERG step. (Note
that $\mathcal{U}_{em}$ does not include the decoupled electronic Fock states
present outside the emergent window.)

Figure 16: Variation of the correlations generated by pseudospin-flip
scattering processes ($\langle S^{+}S^{-}+\textrm{h.c.}\rangle$) within the
low-energy window with MERG step number. Left: Red and blue curves show
variation of the correlation for the FL ground and excited state respectively.
Right: Red and blue curves show variation of the correlation scaling for the
MFL ground and excited states respectively.

As shown in Fig.16(left), the MERG scaling of the correlation
$C^{(j),\hat{s}_{1}}_{sf}$ for the FL ground and excited states increases and
finally saturates in the UV scale. This is due to the fact that the phase
space for tangential scattering grows monotonically with the MERG. On the
other hand, $C^{(j),\hat{s}_{1}}_{sf}$ correlation shows a non-monotonic
behaviour for the MFL case (Fig.16(right)), likely arising due to an initial
increase in the phase space for forward scattering at the beginning of the
MERG flow and its eventual decrease as the flow reaches the UV (and highlights
the absence of tangential scattering in the MFL).

###  C.2 MERG scaling of the occupation of pseudospin states

Figure 17: Variation of the expectation value of the occupation of the states
$20$ and $27$ with MERG step number for the FL and MFL ground states.

Fig.17 shows the MERG flow of the average number occupation of the pseudospin
states $20$ and $27$ (just below the Fermi surface) for both the FL and MFL
ground states shown. We see that both $\langle\hat{n}_{20}\rangle$ (blue
triangle) and $\langle\hat{n}_{27}\rangle$ decay much faster for the FL
(curves with blue triangles and red circles respectively) than for the MFL
(curves with blue diamonds and red squares respectively). The similar nature
of decays for the FL shows the weak electronic differentiation between the
directions $\hat{s}_{0}$ and $\hat{s}_{1}$ due to the strong tangential
scattering. On the other hand, $\langle\hat{n}_{20}\rangle$ does not decay at
all under the MERG flow for the MFL ground state while
$\langle\hat{n}_{20}\rangle$ decays slowly, displaying the electronic
differentiation inherent in the MFL.

###  C.3 Non-local strings

Figure 18: Two choices of the non-local string parity operators. Left: String
$AD$ containing states in only one direction perpendicular to the Fermi
surface ($\hat{s}_{1}$), and stretching across the Fermi surface. Right:
String $AB$ lying outside the Fermi surface but spread over both $\hat{s}$
directions $\hat{s}_{0}$ and $\hat{s}_{1}$.

Another way by which to probe how tangential and forward scattering shape the
FL and MFL states can be learnt by computing the MERG evolution of the
expectation value of the $k$-space non-local operator
$Z=\prod_{k}\sigma^{z}_{k}$, $\langle Z\rangle$ for appropriately chosen
strings of pseudospin states in $k$-space. As shown in Fig.18, we have chosen
two strings of length 14 nodes each for our analysis. The string $AD$ passes
across the Fermi surface but along only the near-anti-nodal direction
$\hat{s}_{1}$, whereas the string $AB$ lies completely outside the Fermi
surface but across the two $\hat{s}$ directions. The operator $Z$ corresponds
to a parity operator defined for the pseudospin states in $k$-space, i.e.,
sign of $\langle Z\rangle$ depends on the number of down pseudospins: negative
for an odd number of down pseudospins (odd parity) and positive for an even
number of down pseudospins (even parity).

Figure 19: Variation of the expectation value of the non-local string operator
$\langle\hat{\mathcal{Z}}\rangle$ for two different string choices AD (left
plot) and AB (right plot) (shown in Fig.18) with MERG step number. The blue
curve shows variation for the MFL ground state, and the red curve for the FL
ground state.

First, we find from Fig.19(left) that $\langle Z_{AD}\rangle$ undergoes a
crossover for the FL ground state under the MERG flow from a state with a
well-defined parity in the IR ($\langle Z_{AD}\rangle=-1$) to one that does
not have a well-defined parity ($\langle Z_{AD}\rangle=0$) in the UV likely
due to the increased tangential scattering. On the other hand, $\langle
Z_{AD}\rangle=-1$ for the MFL throughout the MERG flow, i.e., the MFL ground
state maintains its parity under forward scattering. Secondly, $\langle
Z_{AB}\rangle$ undergoes a crossover under the MERG flow for both the FL and
the MFL from states with a well-defined parity ($\langle Z_{AB}\rangle=+1$) to
one that does not have a well-defined parity ($\langle Z_{AB}\rangle=0$). In
the MFL, this arises from the absence of tangential scattering between
subsystems $A$ and $B$. On the other hand, the vanishing of $\langle
Z_{AB}\rangle$ for the FL likely arise from the tangential scattering events
that connect states inside the Fermi surface to those outside ($D$ with $B$,
and $C$ with $A$).

###  C.4 Fidelity in RG scale

Fidelity is a measure of overlap between two wavefunctions defined as
$F(\alpha,\beta)=|\langle\psi_{\alpha}|\psi_{\beta}\rangle|^{2}$. We now study
the fidelity between the wavefunctions of the FL and MFL ground and excited
states as the MERG proceeds from IR to UV. We define
$|\Psi^{FL}_{G/E,0}\rangle$ and $|\Psi^{MFL}_{G/E,0}\rangle$ as the IR fixed-
point ground ($G$) and excited ($E$) state wavefunctions for the FL and MFL
phases respectively. Similarly,
$|\Psi^{FL}_{G/E,i}\rangle,|\Psi^{MFL}_{G/E,i}\rangle$ are the ground ($G$)
and excited ($E$) state wavefunctions for the FL and MFL at the $i^{th}$ MERG
step. Then, we define the four possible fidelities between the ground and
excited state wavefunctions of the FL and MFL phases at the $0^{th}$ and
$i^{th}$ MERG steps as given by
$F(\alpha\beta,i)=\langle\Psi^{FL}_{\alpha,0}|\Psi^{MFL}_{\beta,i}\rangle$,
where the indices $\alpha$ and $\beta$ can either be $G$ (ground state) or $E$
(excited state). The MERG flow of the four different fidelities
$F(\alpha\beta,i)=\langle\Psi^{FL}_{\alpha,0}|\Psi^{MFL}_{\beta,i}\rangle$ is
shown in Fig.20.

Figure 20: Left: Variation of four fidelities
$F(\alpha\beta,i)=\langle\Psi^{FL}_{\alpha,0}|\Psi^{MFL}_{\beta,i}\rangle$,
constructed using the overlaps of the FL and MFL ground and excited states
(i.e., the indices $\alpha$ and $\beta$ can either be $G$ (ground state) or
$E$ (excited state)), with MERG step number. Right: This shows the square of
the absolute value of the four fidelities
$F(\alpha\beta,i)=\langle\Psi^{FL}_{\alpha,0}|\Psi^{MFL}_{\beta,i}\rangle$.

We can see from Fig.20(left) that the fidelity between the FL and MFL ground
states is perfect at the IR fixed point, $F_{GG,i=0}=1$. This is simply
because both these gapless quantum liquids satisfy Luttinger’s theorem, and
have the same ground state. However, $F_{GG}$ falls monotonically to zero. On
the other hand, the IR excited states of the FL and MFL are orthogonal to the
common IR ground state and to one another, as indicated by the near vanishing
of the other three fidelities at the zeroth MERG step. Further, this
orthogonality is maintained under the MERG flow to the UV, as the unitary
operations of the MERG maintain the orthogonality structure of these states.
Fig.20(right) shows that the MERG evolution of the square of the real part of
the fidelities $F(\alpha\beta,i)$ follows the same trend as $F(\alpha\beta,i)$
itself, reinforcing the fact that the phases of the four fidelities do not
play an important role.