# The Entanglement Entropy between Short Range Correlations and the Fermi Sea
in Nuclear Structure

Ehoud Pazy  Massachusetts Institute of Technology, Cambridge, MA, 02139 USA
111On leave from Department of Physics, NRCN, P.O.B. 9001, Beer-Sheva 84190,
Israel

###### Abstract

We calculate the nuclear structure orbital entanglement entropy of short range
correlations (SRC) based on the nuclear scale separation. Specifically, the
entanglement between the SRC orbitals and the rest of the system. It should be
stressed that this is a single nucleon not a pair entanglement entropy between
the proton and neutron. The entanglement arises from the probability for a
nucleon to occupy a momentum state above the Fermi momentum. We separate the
momentum space of the nucleus into two parts such that nucleons can occupy the
meanfield part of the wave function, i.e. Fermi sea (FS) and separately the
high-momentum SRC part. The orbital entropy we obtain is between these two
parts where we essentially define two momentum subspaces, one containing all
the low momentum FS states and the other the high-momentum part as a SRC
”orbital” state. For the calculation we employ the decoupling of low and high-
momenta which was established by the similarity normalization group the SRC is
viewed as a further ”orbital” which can be multiply occupied. Since the
probability of the occupation of a single SRC is given by the nuclear contact
we are able to obtain a simple general expression of the orbital entanglement
entropy for SRC by employing the generalized contact formalism. This general
formula for the SRC orbital entanglement entropy of a nuclear structure in
terms of the nuclear contact, allows us to obtain the scaling of the entropy
in terms the mass number, $A$. We find that, unlike the entanglement entropy
of many quantum systems which scales with the surface area, the orbital
entanglement entropy associated with the SRC in large nuclei is linearly
dependent on $A$, i.e., it is shown to be extensive.

##  I Introduction

Entanglement entropy, is an entropy arising not from a lack of knowledge, due
to thermal fluctuations, of the micro-state the system is in, but is rather an
entropy arising due to the entanglement between sub-parts of the system. It
arises when one chooses to integrate over part of the system. As such, this
entropy can also exists for the ground state. The entropy of the reduced state
of a sub-region tends to grow like the boundary area of that sub-region, often
with a small, typically logarithmic correction, rather than extensively with
the volume Plenio10 . The scaling of the entanglement with the area of a
region is referred to as an area law and holds insights with regard to the
distribution of quantum correlations in many-body systems and thus offers a
measure of the complexity of the system. Since entanglement is an important
resource for quantum computation, sensing, as well as for communication, there
is a growing interest in calculating the entanglement entropy of different
many-body systems. One obtains area laws for one and higher-dimensional
lattices, for fermionic or spin degrees of freedom Amico08 . The entanglement
entropy has been calculated for electronic orbitals Rissler06  and area laws
have been obtained for condensed matter systems Laflorencie16 . The
entanglement entropy is also a good measure for identifying correlations in
many-body systems and as such it was also used as a tool for identifying
topological order Jiang12 . In addition to the theoretical interest in trying
to understand the quantum structure of the nucleus, calculating the
entanglement entropy of nuclear structure has practical implications because
it can play a role as an organizational principle. Entanglement entropy
measures the distribution of wave-function coefficients, and thus it acts as a
measure for the viability of truncation of a computational model space. As
such it has many implications regarding the reduction of the exponentially
large computational basis Gorton19  where knowledge of the entanglement
entropy allows one to require less computational resources while retaining
similar accuracy. More specifically, Ref. Legeza15  demonstrates how employing
a quantum entropy based optimization procedure can effectively reduce the
number of block states needed to provide an accurate calculation for the
ground and first excited state of 28Si. Recently it has also been shown how
nuclear structure can be investigated on a quantum computer where the
evaluation of the entanglement entropy plays an important role in assessing
the required quantum resources in the number of qubits and circuit depths
Stetcu21 .

The calculation of the entanglement entropy for nuclear structures has
commenced with an initial study of the single-orbital entropy and two-orbital
mutual information performed for 28Si, 56Ni and 64Ge employing the density
matrix renormalization group (DMRG) Legeza15 . It continued, rather recently,
with the calculation for the state of two interacting particles Kruppa21  and
shortly after with a detailed numerical calculation for the entanglement
entropy of 4He and 6He Robin21 . Mode entanglement has also been investigated
under the framework of the Lipkin model Faba21 . In this work we modify the
calculation of the mode (orbital) entanglement by considering the entanglement
entropy for a subspace of the system. The subspace for which we calculate the
mode entanglement entropy consists of the high-momentum states above the Fermi
momentum for the SRC states. It should be stressed that we are not calculating
the entropy between the nucleons in the SRC but rather consider the SRC as a
further single state which the nucleons can multiply occupy. Due to the
universality of the SRC state we are able to obtain a general formula for the
entanglement entropy in terms of the high-momentum states rather then having
to perform specific calculations for different nuclei. A similar approach of
treating the entropy of a Fermi system with a mean-field (MF) part and a
correlated part was recently employed in Ref. Bulgac22 . There the orbital
entanglement entropy is given simply by the zero temperature limit of the one-
body entropy of a Fermi system. Whereas typically entanglement is used as a
diagnostic tool for defining quantum correlations Klco22 , here this role is
reversed and we use the correlations as a basis to evaluate entropy
entanglement. This sort of approach allows us also to look at the spatial
scaling of the entanglement entropy identifying possible area laws.

Calculating the nuclear structure is highly complex. Due to the repulsive-core
of nucleon-nucleon interactions, combined with the large numbers of many-body
states, it was initially considered to be unachievable in a reliable way. It
should be noted that it is primarily the entanglement of the many-body nuclear
wave function that complicates the calculation since the system can not be
well approximated by separable states. The status of nuclear structure
calculations has greatly advanced since the 1970’s through the use of modern
field theoretic techniques. Major tools were developed in enabling the study
of nuclear structure, e.g. quantum Monte Carlo Epelbaum11 ; Carlson15 ,
Coupled Cluster Hagen10 , no core Shell Model Barrett11  and Similarity
Renormalization Group (SRG). In the current context we employ the SRG
transformation which results in scale seperation. This formalism also helps
resolve the following apparent contradiction Bogner12 ; Tropiano21 . On the
one hand, short range correlations (SRC) have been experimentally observed
Egiyan03 ; Egiyan06 ; Subedi08 ; Fomin12 ; Hen17 ; Torres20 ; Torres21 ;
Piasetzky06  in which one observes components in the high resolution nuclear
wave function with relative pair momenta greater than the Fermi momentum
Korover14 ; HenSci14 ; Korover21 ; Duer18 ; Duer19 ; Schmookler19 ; Schmidt21
. On the other hand, the shell model for the nucleus, which has been highly
successful for calculating nuclear properties deShalit63  seems to include no
such short range structure thus seemingly contradicting the SRC description.
Through SRG it was recently demonstrated Bogner12 ; Tropiano21  that these two
viable seemingly contradicting descriptions for the nucleus, the FS like
independent particle model and SRC can in fact be connected. This sort of
connection has also been previously studied and demonstrated for the 4He case
Neff15 . Employing a SRG transformation specifically enables one to obtain a
factorization of the long from short-distance physics, providing a clean scale
separation. The SRC are manifested when considering high RG resolution.
Whereas in the low-RG resolution the features of SRC phenomenology are clearly
identified using simple two-body operators and local-density approximations
with uncorrelated wave functions reconciling the contrasting pictures. One can
thus naively view SRC as the two-body extension to the single independent
particle form of the shell model. In this naive view the physics of the
nucleus is encapsulated in one plus correlated two-body model which are scale
separated. This separation of scales is extremely effective for the
calculation of the entanglement entropy of the nucleus. In the present work we
exemplify how the mode entanglement entropy is given in terms of SRC instead
of considering single particle states above the Fermi surface we consider the
single particles to be occupying a universal SRC state. For simplicity, we
only consider the SRC composed of neutron-proton pairs which are in a $s$-wave
configuration. The extension to other isospin SRC is more elborate, however,
neutron-proton pairs have been found experimentally to constitute the vast
majority of SRC. In only considering such pairs we in fact neglect the
contribution of neutron-neutron and proton-proton pairs to the entanglement
entropy. However this contribution has been shown to be small. In fact in the
electron scattering experiments kinematically designed to measure SRC a
typical ratio of proton-neutron SRC pairs to proton-proton SRC pairs was found
to be around 20 Subedi08 ; Hen17 ; Piasetzky06 ; Korover14 ; HenSci14 ;
Korover21 ; Duer18 ; Duer19 .

We start by a brief reminder of the orbital entanglement entropy given in Sec.
II. In Sec. III the generalized contact formalism (GCF) is succinctly
introduced. We then extend the definition of the orbital entanglement entropy
to include an entanglement entropy for a subspace of the momentum space which
we define as the ”orbital” entanglement entropy of the SRC. The orbital
enatnglement entropy of the SRC is then analytically calculated in terms of
the nuclear contact. In Sec. IV we compare our results with previously
obtained results from the literature for the entanglement entropy of 4He
Robin21  and those for the (one-body) entropy of a Fermi system Bulgac22  .
The results for the entanglement entropy for the SRC of the nuclear structure
are summarized and future research directions are discussed in Sec. V.

##  II Orbital entanglement entropy

The entanglement entropy measures the correlation between two parts of a
bipartite system Plenio07 ; Horodecki09 . In the case of distinguishable
particles, the notion of the entanglement is based on the structure of the
tensor product for the Hilbert spaces of the subsystems. However, trying to
define a similar concept for identical particles is not straightforward, since
the decomposition into particle subsystems does not correspond to a tensor
product structure of the Fock space. In order to address this issue, the
notion of the mode (orbital) entanglement was introduced Zanardi02 ;
Gittings02 ; Shi03 .

One can define the eigenstates of the nucleus, $|\Psi\rangle$, by choosing a
single particle basis defined by the quantum numbers
$\\{i\\}=\\{n_{i},J_{i},l_{i},m_{i},s_{i},\tau_{i}\\}$, which correspond to
the principal quantum number, the total angular momentum, the orbital angular
momentum, and the spin, isospin projections. The nucleus eigenstates can then
be written as a linear combination of Slater determinants $|\phi\rangle$ for
the nucleon wave functions

$|\Psi\rangle=\sum_{\eta}\mathcal{A}_{\eta}|\phi_{\eta}\rangle,$ (1)

where the Slater determinant is given in terms of applying creation operators
on the true particle vacuum $|0\rangle$

$|\phi_{\eta}\rangle=\prod_{i\in\eta}^{A}a_{i}^{\dagger}|0\rangle,$ (2)

where $A$ is the number of nucleons in the nucleus.

Employing this notation, the single orbital entanglement entropy $S_{i}^{(1)}$
is given in terms of the one-orbital density matrix Robin21

$\rho^{(i)}=\begin{pmatrix}1-\gamma_{ii}&0\\\ 0&\gamma_{ii}\end{pmatrix},$ (3)

where the occupation of orbital, $i$, is given by
$\gamma_{ii}=\langle\Psi|a_{i}^{\dagger}a_{i}|\Psi\rangle$.

To obtain the entanglement entropy essentially one needs to perform a
calculation of the von Neumann entropy of the partial density matrix, Eq. (3)

$S_{i}^{(1)}=-Tr[\rho^{(i)}\ln\rho^{(i)}]=-\sum_{k=1}^{2}\omega_{k}^{(i)}\ln\omega_{k}^{(i)},$
(4)

where $\omega_{k}$ are the eigenvalues of $\rho^{(i)}$

For the sake of completeness we also present the two-orbital entanglement
entropy which is similarly given by

$S_{ij}^{(2)}=-Tr[\rho^{(ij)}\ln\rho^{(ij)}]=-\sum_{k=1}^{4}\eta_{k}^{(ij)}\ln\eta_{k}^{(ij)},$
(5)

where $\rho^{(ij)}$ is the following $4\times 4$ matrix with eigenvalues
$\eta_{k}^{(ij)}$ Robin21

$\rho^{(ij)}=\begin{pmatrix}&1-\gamma_{ii}-\gamma_{ij}+\gamma_{ijij}&0&0&0\\\
&0&\gamma_{jj}-\gamma_{ijij}&\gamma_{ij}&0\\\
&0&\gamma_{ij}&\gamma_{ii}-\gamma_{ijij}&0\\\
&0&0&0&\gamma_{ijij}\end{pmatrix},$

defined in terms of $\gamma_{ij}=\langle\Psi|a_{j}^{\dagger}a_{i}|\Psi\rangle$
and
$\gamma_{ijij}=\langle\Psi|a_{i}^{\dagger}a_{j}^{\dagger}a_{j}a_{i}|\Psi\rangle$.

##  III Entanglement entropy in terms of the nuclear contact

Having introduced the concept of orbital entanglement entropy, we modify it by
considering the entanglement entropy for the subspcae of high-momentum states.
We consider this subspace as aseparate ”orbital” , SRC state, which can be
multiply occupied. In this format the mode entropy for high-momentum states is
given in terms of SRC instead of single particle states. We show specifically
that the splitting of the wave function into two parts, one part is a MF part
and the other part is an SRC part which is defined by a single two-body wave
function in space with undefined center of mass cordinate. This separation
which is the basis of GCF, provides a very convenient framework for this
calculation. One does not need to know the specific wave function for the SRC
pair just to realize it is universal and as such it is its occupation that
defines its entanglement entropy.

The GCF fundamentally complements the shell model enabling one to incorporate
SRC into the model. Though the shell model is essentially a MF theory, it
provides detailed information about the nuclear shell structure. One of the
major success of the shell model is explaining the stability of a certain
number of nucleons in nuclei called the magic numbers. However, it does not
capture the complete picture of the nucleus structure. The shell model fails
to predict the occupancy of the shells since in particular it does not
describe the strong short range nucleon-nucleon part of the potential which is
responsible for the SRC. The SRC are manifested in a high-momentum tail of the
nucleon momentum distribution with momenta exceeding the Fermi momentum.
Experimental evidence for the existence of SRC was obtained by performing
electron scattering experiments. SRC related physics was observed by choosing
the right kinematics in these experiments, specifically inclusive scattering
at Bjorken $x_{B}>1$ Egiyan03 ; Egiyan06 ; Fomin12  and also by performing
exclusive experiments in which the two-body currents could be identified and
distinguished from final state interactions Piasetzky06 ; Korover14 ; HenSci14
.

Starting off with the ground state of the nucleus with hard interactions,
i.e., low-RG resolution, there are both high-momentum and low-momentum
contributions to the wave function. By employing a SRG transformation to high-
RG resolution it was demonstrated that operator expectation values exhibit
factorization in the two-nucleon system Bogner12 ; Tropiano21 . Through this
factorization a model can be obtained in which SRC, which are identified as
components of the nuclear wave function in which nucleon pairs with relative
momenta well above the Fermi momentum, naturally emerge. This factorization is
the basis of the GCF WeissPRC ; Weiss15 ; Weiss17 ; Weiss16 , which is a model
introduced to provide a simple framework for describing the SRC. The GCF is a
very convenient theoretical tool for analyzing experiments designed to probe
SRC in the nucleus HenSci14 ; Alvioli13 ; ArrHigRos12 ; Wiringa14 . It has
been benchmarked against ab-initio many-body calculations, proving its
validity, and was successfully applied to a wide range of topics, most notably
to the analysis of electron scattering measurements Weiss18 ; Weiss19 ; Duer19
; Pybus20 ; Weiss21 . The contact which is at the heart of the GCF will be
shown to be an instrumental tool for expressing the SRC entanglement entropy.
Currently the GCF does not treat three-body effects. However, the three-body
contributions are expected to be less important than those of the leading two-
body.

###  III.1 The nuclear contact

The GCF is based on the factorization ansatz of the many-body wave function
into a two-body problem of a nucleon pair close in space (correlated), which
is a universal part, common to all nuclei, multiplied by a particular part,
that depends on the specific state. The specific state part describes the
remaining, $A-2$, nucleons in the nucleus WeissPRC  and also depends on the
center of mass coordinate for the nucleons pair

$\Psi\xrightarrow[r_{ab}\rightarrow
0]{}\sum_{\alpha}\varphi_{ab}^{\alpha}(\bm{r}_{ab})A_{ab}^{\alpha}\left(\bm{R}_{ab},\\{\bm{r}_{c}\\}_{c\not=a,b}\right).$
(6)

Here $\varphi_{ab}^{\alpha}$ are the two-body universal functions defining the
SRC, $A_{ab}^{\alpha}$ are the so called regular part of the many-body wave
function, the index $\alpha$ defines the quantum numbers for the two-body
states and the indices $ab$ define the SRC in terms of isospin, $pp$, $pn$ and
$nn$ pairs as well as the specific particle indices in $\bm{r}_{ab}$ and
$\bm{R}_{ab}$. This sort of approximation is viable when the wave function is
viewed on a short length scale, important for calculating short range
observables. For simplicity and since most experimental data on SRC is in the
kinematical region where neutron-proton pairs dominate Subedi08 ; Hen17 ;
Piasetzky06 ; Korover14 ; HenSci14 ; Korover21 ; Duer18 ; Duer19 , we consider
in this work only neutron-proton pairs in the deuteron like quantum state
(quasi-deuteron). These pairs are defined by
$\alpha\equiv(S=1,\pi=+1,J=1,M=0\pm 1)$, where $S$ is the total spin of the
pair, $\pi$ is their parity, and $J$ and $M$ are the quantum numbers defining
the SRC, which are the total angular momentum of the pair and its projection
respectively. This state has been shown to match experimental data to a high
precision Subedi08 ; Hen17 ; Piasetzky06 ; Korover14 ; HenSci14 ; Korover21 ;
Duer18 ; Duer19 . Since we will be considering only this proton neutron
quantum state for the SRC pair we will for simplicity eliminate $pn$ index
aswell as the quantum state index $\alpha$. Under this approximation the
nuclear contact in GCF is defined as

$C=N(A,Z)\langle A|A\rangle,$ (7)

where $N(A,Z)$ is the number of pairs one can produce from $Z$ protons and,
$A-Z$, neutrons. In this work we will only consider symmetric nuclei such that
$Z=A/2$.

The function $\varphi_{ab}(\bm{r}_{ab})$ in Eq. (6) is a function of solely
the distance between the SRC proton and neutron, $\bm{r}_{ab}$, and not their
center of mass coordinate $\bm{R}_{ab}$ which appears in
$A_{ab}\left(\bm{R}_{ab},\\{\bm{r}_{c}\\}_{c\not=a,b}\right)$. This function
is obtained by solving the two-body, zero energy, Schrödinger equation with
the full many-body potential.

Switching to momentum states the one nucleon momentum distribution

$n({\bf q})=\langle\Psi|a_{\bf q}^{\dagger}a_{\bf q}|\Psi\rangle.$ (8)

can be expressed by employing the contact Eq. (7) for large momentum limit
$|{\bf q}|\ll k_{F}$. It is well approximated by WeissPRC

$n({\bf q})=C|\phi({\bf q})|^{2},$ (9)

where $\phi({\bf q})$ is the Fourier transform of $\varphi_{ab}(\bm{r}_{ab})$
in Eq. (6) and the normalization for, $n({\bf q})$ for a $N=Z$ nucleus is
given such that, $n({\bf q})$, gives the fraction of the one body momentum
above $k_{F}$ Weiss18

${\int_{k_{F}}^{\infty}n({\bf q})d{\bf q}\over\int_{0}^{\infty}n({\bf q})d{\bf
q}}={C\over(A/2)}$ (10)

Since the contact is obtained by tracing out the degrees of freedom exterior
to the SRC pair, it is clear why it plays an important part in determining the
entanglement entropy of a SRC pair with respect to the rest of the nucleus
structure.

###  III.2 Orbital entanglement entropy of SRC in terms of the contact

To calculate the entanglement entropy between the SRC and the rest of the
system we define the momentum states with $q>k_{F}$ as an SRC state which is a
relatively good approximation even though excitations near the Fermi level are
due to long range correlations Weiss19 . Thus we divide the momentum space
into two distinct regions, the high-momentum region defining the SRC and the
mean field region defined by, $q<k_{F}$. We will further treat the high-
momentum states as a single SRC ”orbital” which can be multiply occupied,
since we have defined it by all high-momentum states.

In calculating the entanglement entropy of a particle occupying a momentum
state above the Fermi energy, we essentially extend the notion of orbital
entanglement entropy to that of a subspace entanglement entropy. We divided
the momentum space into two separate sub spaces, $\Phi^{i}$ which includes all
single particle states in which $q<k_{F}$, i.e. FS, and $\Phi^{i}_{\perp}$
which includes all single particle states in which $q>k_{F}$ which is
essentially what we refer to as the SRC ”orbital”. Defining a projection
operator $\hat{P}$ which projects onto states in $\Phi^{i}_{\perp}$ we define
the analog of $\gamma_{ii}$ in Eq. (3) to be
$\gamma_{{}_{SRC}}=\langle\Psi|\hat{P}|\Psi\rangle$.

To calculate the SRC orbital entanglement entropy one first has to calculate
the occupancy probability for a single SRC $\gamma_{{}_{SRC}}$ which is
defined through Eq. (10)

$\gamma_{{}_{SRC}}={C\over(A/2)}\equiv c.$ (11)

Where we have defined for convenience a normalized contact $c$ which to a good
approximation does not depend on $A$.

Employing the SRC occupancy, Eq. (11), a one ”orbital” SRC density matrix,
$\rho^{(SRC)}$ as in Eq. (3) can be obtained with the following eigenvalues:
$\omega_{k}^{(SRC)}=\\{1-c,c\\}$. The orbital entanglement entropy for a
single SRC is directly obtained through Eq. (4)

$S^{SRC}=-\left[c\ln\left({c\over 1-c}\right)+\ln{(1-c)}\right].$ (12)

It should be noted that under the normalization, Eq. (10) the probability of a
particle being in FS is given by $\gamma_{{}_{FS}}=1-\gamma_{{}_{SRC}}$. Thus,
as should be expected, the eigenvalues for the FS are given by
$\omega_{k}^{(FS)}=\\{c,1-c\\}$. So that the orbital entanglement of the whole
FS subspace is also given by Eq. (12).

To obtain the total SRC orbital entanglement entropy one has to multiply the
single SRC entanglement, Eq. (12) entropy by plugging the expression for the
number of nucleons a certain nucleon can be paired with $N(A,Z)=A/2$ as
expressed in Eq. (7). Thus the total SRC entanglement entropy is obtained,

$S_{A}^{SRC}=-{A\over 2}\left[c\ln\left({c\over
1-c}\right)+\ln{(1-c)}\right].$ (13)

This simple general analytic expression for the orbital entanglement of the
SRC part of the orbital entanglement entropy in terms of the reduced contact
is viable to any nucleus under the approximation that SRC are defined by the
subspace of states with momentum above the Fermi momentum.

###  III.3 Scaling of the SRC entanglement entropy

The above result, Eq. (13), shows an explicit linear dependence of the SRC
entanglement entropy on $A$. This is a rather surprising result since the SRC
entanglement primarily scales like the nuclei volume $A$, making the
entanglement entropy extensive which means that it does not obey an area law
as could be expected. This extensive scaling of the entanglement entropy can
be traced back to the number of available nucleons a nucleon can pair with to
form a SRC, $N(A,Z)\propto A$. There might be a further dependence on $A$
through the normalized contact, $c$. However, the normalized contact is
expected to be $A$ independent for nuclei with a large $A$ for which the Fermi
gas approximation is expected to be valid. The linear dependence of the
contact on $A$ and the independence of the reduced contact in the Fermi gas
approximation can be inferred from Ref. Weiss19b . There the contact was
calculated by taking the limit when interparticle distance $r$ goes to zero of
the two-body Fermi gas correlation function $g(r)$. The obtained result for
the Fermi gas model was shown to be linearly dependent on the number of
nucleons in the nucleus and the slope of the linear dependence on $A$ was then
estimated by comparison to experimental results as well as to calculations
based on the AV18 potential Weiss19b . The obtained result for the reduced
contact for a symmetric nuclei was given as $c_{pn}^{FG}=0.146\pm 0.002$. Thus
under the Fermi gas approximation the SRC entanglement entropy, Eq. (13), is
simply linear in $A$.

The reduced contact was also calculated using a variational Monte Carlo (VMC)
method and calculations were compared to experimental results and found to be
in agreement. Through the VMC results one can also justify numerically the
neglect of other channels differing form the proton-neutron channel
$(S=1,\pi=+1,J=1,M=0\pm 1)$. The proton-proton and neutron-neutron calculated
contacts were shown to be an order of magnitude smaller then the proton-
neutron channel and the $S=0$ proton-neutron channel is even smaller. The VMC
calculated values in momentum space, for the reduced contact in Ref. Weiss18 ,
range from $c_{pn}\approx 0.125$ for 4He, 8Be, 9Be, 10B to $c_{pn}=0.106$ for
6Li, 7Li and a higher value $c_{pn}=0.168$ for 12C. The experimental values
obtained for 4He were slightly higher than the VMC calculation
$c_{pn}^{exp}\approx 0.149$ as was the experimental evaluated contact for 12C
which was $c_{pn}^{exp}\approx 0.168$. It is difficult to determine if there
is a slight mass number dependence of the reduced contact due to the limited
range of nuclei calculated. As a rough approximation one can take the value
for the contact as $c_{pn}\approx 0.15$ and as independent of $A$. However, it
seems that there could be some corrections to the simple linear (extensive)
dependence of the entanglement entropy on $A$.

A word of caution should be given when estimating the value for the reduced
contact by calibrating the result through experimental data. Whereas the
relevant scale for momentum separation is well defined theoretically by SRG it
is not always clear that this was the relevant scale in the experiments. The
obtained experimental results depends on physical scale through the specific
kinematics. These two scales, the theoretical and experimental, need not
necessarily coincide. This issue is elaborated in detail in Tropiano21 . The
scale mismatch if occurring can cause errors in estimating the reduced contact
and thus also effect the entanglement entropy estimation through Eq. (13).

##  IV Comparison to previous results

Calculating the entanglement entropy for a nuclei can involve considerable
computational efforts when considering single particle energy levels. As such
it can only be preformed for few-body systems, due to the extensive
computational effort which scales rapidly due to combinatorics. In the
following section we will compare our results to such a few-body calculation
Robin21 . It should however be noted that we have calculated the ”orbital”
entanglement entropy of a subspace of high-momentum states rather then a
single orbital, moreover one should also note that the orbital entropy
calculation is basis dependent and as such the comparison to the few-body
results should be taken with a grain of salt. In addition below we present a
comparison with respect to a calculation of the one body entropy which was
calculated for a Fermi system with a MF part and a correlated part Bulgac22 ,
this calculation used the same basis we have however in a different method. We
demonstrate how taking the zero temperature limit for the one body entropy one
and summing over the high-momentum states one obtains the same result.

The sole source for an entanglement entropy calculation involving single
particle energy levels which we found easy to compare to quantitatively was
the calculation performed for 4He and 6He Robin21 . The calculation of the
entanglement entropy was performed by a full diagonalization on a 7 shell
model space. It should be noted that the calculation that was performed with
only 6 major active shells, containing the first 114 single-particle states,
was not large enough to give satisfactory results. On the other side it seems
rather simple using our SRC entanglement expression Eq. (13) to calculate the
SRC entanglement entropy once the relevant reduced contact is known. Though
one should keep in mind that we have only considered a subspace SRC ”orbital”
considering the orbital entanglement entropy resulting from the distinction
between FS or the SRC states. It is thus of interest to try to evaluate how
much of the entanglement entropy can be related to this seperation. In this
section we compare the results for the entanglement entropy of 4He from Ref.
Robin21  to the SRC entanglement entropy from our analytical expression with
the appropriate reduced contact and mass number for 4He.

In Ref. Robin21  a comparison was made between different basis states to find
the appropriate base in which to work in. It was found that the two most
accurate basis sets were NAT, a term for natural, and VNAT, a term for, for
variational natural which gave the most accurate results. The total sum of the
single orbital entanglement entropy for the calculations in these basis gave a
result of $S_{tot}^{(1)}(^{4}He)=1.006$. In calculating the single-orbital
entanglement entropy for 4He solely from Eq. (13) we obtain a value of
$S^{SRC}(^{4}He)=0.74$ for $c_{pn}=0.12$ and $S^{SRC}(^{4}He)=0.86$ for the
value for the reduced contact obtained experimentally $c_{pn}^{exp}=0.15$
Weiss18 . Surprisingly, the results are relatively similar to those of Ref.
Robin21  especially keeping in mind the simple SRC ”orbital” like formalism.
Moreover, contributions from proton-proton and neutron-neutron SRC pairs were
not considered as well as two-orbital entanglement entropy involving pairs of
nucleons. Taking this comparison literally one can claim that probably much of
the nuclear entanglement entropy is due to the SRC separation from the FS,
i.e., that the main source for the entanglement entropy is obtained by the
possibility for nucleons to excited to high-momentum states. However, due to
the use of a different basis for the calculations, one can only view the
relative agreement between them as a trend. One should keep in mind that the
calculation of the entropy entanglement in Robin21  was performed for the low-
RG case, whereas the results obtained in this work, Eq. (13) are for the high-
RG case. In Ref. Tropiano21  a SRG connection was established showing that
calculations performed in the high-RG resolution involving SRC pairs in the
GCF can be mapped via SRG to the low-RG resolution were he nucleons are best
described by a FS. It still needs to be shown how the SRG connects these two
methods for calculating the entanglement entropy and how it continuously
transform from one to the other. An indication for this connection can be
found in the claim made in Ref. Robin21  that the most important couplings are
$1s$-$1s$, $1s$-$1p_{1/2}$ and $1p_{1/2}$-$1p_{1/2}$ since they are related to
deuteron-type correlations. A further qualitative comparison can also be made
to Ref. Legeza15  in which the single orbital and two-orbital mutual
information was calculated employing the DMRG for 28Si, 56Ni and 64Ge. The
total entanglement entropy for these calculations was not presented thus
allowing only a qualitative comparison. The authors of of Ref. Legeza15  state
that they could see significant entanglement between proton-neutron maximally
aligned states for the $p$3/2 and $f$ 5/2 orbits. However, it is also claimed
that the fact that the two orbital mutual information is approximately equal
for the $p$1/2, $p$3/2, and $f$ 5/2 orbits, and independent of their $M$
projections, is an indication for the presence of a strong $T=1$ proton-proton
and neutron-neutron pairing coherence. Whereas in our calculations we have
neglected such SRC pairs. Future work should establish how the entanglement
entropy is effected by the SRG transformation.

A calculation for the nuclear orbital entanglement entropy involving SRC was
recently preformed in Ref. Bulgac22 . The starting point for that calculation
was the one-body Fermi system Boltzmann entropy defined in terms of the
fermion momentum distribution $n(k)$

$S=-g\int{dk\over{(2\pi)}^{3}}n(k)\ln
n(k)-g\int{dk\over{(2\pi)}^{3}}[1-n(k)]\ln[1-n(k)],$ (14)

where in the nuclear case $g=4$ is the spin-isospin degeneracy factor. The
one-body entropy was calculated depending on the temperature from which one
can obtain the orbital entanglement entropy by taking the zero temperature
limit. The SRC were incorporated into the calculation by considering the
following nucleon momentum distribution

$n(k)=\eta(k_{0})\begin{cases}n_{mf}(k)&\text{if}\ k\leq k_{0}\\\ {C_{pn}\over
k^{4}}&\text{if}\ k_{0}<k<\Lambda,\end{cases}$ (15)

where $\Lambda$ is a cut-off and will be assumed $\Lambda=\infty$. The result
for the SRC orbital entanglement entropy can be obtained from Eqs. (14, 15) by
integrating over all momentum states above the Fermi momentum. This result is
matched to the SRC entanglement entropy as defined by Eq. (13) the details are
described in the Appendix.

##  V Summary and Discussion

Basing our work on previously obtained theoretical results demonstrating that
through applying an SRG transformation on the nuclear structure one obtains
scale separation Tropiano21 , we have calculated the nuclear structure orbital
entanglement entropy for SRC. It should be stressed that the SRC orbital
entanglement entropy we have calculated is a single nucleon entropy related to
the probability of finding a nucleon in the FS or as part of an SRC pair. This
is a rather simplified model in which the SRC is identified with the high-
momentum subspace and considered as a single ”orbital”. Thus a nucleon can
occupy either one of the FS orbitals or an SRC pair. Essentially, based on the
scale separation we decoupled the Hilbert space such that

$\mathcal{H}=\mathcal{H}_{FS}\otimes\mathcal{H}_{SRC}.$ (16)

It is a product of a FS Hilbert subspace, $\mathcal{H}_{FS}$ and the SRC
Hilbert subspace $\mathcal{H}_{SRC}$. Based on this decomposition, Eq. (16) we
calculated the reduced density matrix for the SRC momentum subspace

$\rho_{{}_{SRC}}=\text{Tr}_{{}_{FS}}\rho$ (17)

where $\rho=|\Psi\rangle\langle\Psi|$. Employing the GCF formalism allowed us
to obtain a simple analytic expression for the reduced density matrix, Eq.
(17), in terms of the reduced nuclear contact, Eq. (11) and through it to
obtain a simple analytical expression for the entanglement entropy, Eq. (13).
Through this general formula we have been able to demonstrate that the orbital
entanglement entropy for the high-momentum states, i.e., the SRC orbital part
of the nuclear structure, scales extensively and not according to an area law.

More specifically, we utilized in our calculations the fact that in the high-
RG resolution the nuclear structure can be viewed as a bipartite system where
the high-momentum short length scale properties of the system are given in
terms of SRC pairs. Based on this we have performed our calculation for the
SRC entanglement entropy in the framework of the orbital entanglement entropy,
in which the entanglement is given by the probability an ”orbital” is
occupied. In the calculation we considered the high-momentum states as the SRC
part of the many-body wave function, which can be occupied by many pairs. The
reduced nuclear contact for a SRC pair which is obtained by integrating out
the long length scales in the nucleus corresponding to the low-momentum
degrees of freedom and then normalizing the result, naturally appears as the
probability measure for finding a single SRC pair. Using this connection, one
can define a single orbital SRC density matrix , Eq. (17) in terms of the
reduced nuclear contact, Eq. (11). From the SRC density matrix one can obtain
the SRC entanglement entropy through the von Neumann entropy, Eq. (12).
Summing over the entanglement entropy for all SRC pairs we obtained that the
nuclear orbital entanglement entropy for the high-momentum, i.e., SRC part,
Eq. (13). One can notice that it obeys an extensive entropy law, since it
scales linearly with the mass number $A$ and since the nuclear structure
density is essentially constant this means it scales like the volume of the
nucleus.

Even though in obtaining our results quite a few contributions were neglected,
e.g., neutron-neutron and proton-proton SRC still we find them to be
relatively close to the more elaborate work performed in calculating the
entanglement entropy for 4He Robin21 . This can be considered as a
coincidental agreement between the results since in considering the SRC as a
single orbital we summed over all high-momentum states. However, it might be
seen as an indication that most of the nuclear entanglement entropy is related
to the ability to excite nucleons to high momentum states as part of SRC and
thus our general Eq. (13) is a good measure for the entanglement entropy of
all nuclei.

The work presented here is essentially preliminary. It introduced the idea
that the SRG scale separation is an invaluable tool for performing the
calculations for the nuclear structure orbital entanglement entropy and
demonstrating the scaling of the SRC orbital entanglement entropy. Further
research has yet to incorporate proton-proton and neutron-neutron SRC and
establish how the orbital entanglement entropy is modified by the SRG
transformation.

## Acknowledgments

I wish to thank C. W. Johnson and O. Gorton for sharing their results, C.
Robin for discussions. R. Weiss ,G. A. Miller and A. Bulgac for their valuable
advice and comments. Special thanks to O. Hen and N. Barnea for insightful
discussions, invaluable comments and Or for his generous hospitality at the
LNS which made this work possible. This work was supported by the PAZY
Foundation.

## Appendix

A calculation of the one-body entropy was preformed in Ref. Bulgac22  for
which the zero temperature case corresponds to the orbital entanglement
entropy. The SRC were incorporated into that calculation by considering the
nucleon momentum distribution described in Eq. (15) choosing as we did here
that the distinction in momentum between SRC and the bulk FS is given by the
Fermi momentum $k_{0}=k_{F}$. For this choice one obtains $\eta(k_{F})=0.25$.
The contact in this case is given by Bulgac22

$C(k_{F})=\eta(k_{F})n_{mf}(k_{F})k_{F}^{4}.$ (18)

The SRC entanglement entropy is obtained by considering all momentum states
above the Fermi momentum as the SRC ”orbital”. Thus Inserting the expression
for $n(k>k_{F})$ from Eq. (15) into the equation for the entanglement entropy,
Eq. (14) and approximating the integrals one obtains

$S\approx-{g\over
2\pi^{2}}\left[{C\over{k_{F}}}\ln\left({C\over{k_{F}}^{4}}\over{1-{C\over{k_{F}}^{4}}}\right)+{k_{F}}^{3}\ln\left(1-{C\over{k_{F}}^{4}}\right)\right].$
(19)

Associating the reduced contact with $c_{pn}={C/{k_{F}}^{4}}$ one obtains

$S\approx-3A\left[c_{pn}\ln\left(c_{pn}\over{1-c_{pn}}\right)+\ln\left(1-{c_{pn}}\right)\right]$
(20)

where the different prefactor with regards to Eq. (13) is due to the
difference in defining the fraction of which in Eq. (10) was simply given by
$A/2$ whereas in Bulgac22  it is given by

${n(k>k_{F})\over n_{0}}={3C(k_{F})\over{k_{F}}^{4}}.$ (21)

This difference can be related to the number of SRC channels, e.g., proton-
proton, neutron-proton, considered.

## References

  * (1) J. Eisert, M. Cramer, and M.B. Plenio, Rev. Mod. Phys. 82, 277 (2010).
  * (2) L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008).
  * (3) J. Rissler, R. M. Noack, and S. R. White, Chem. Phys. 323, 519 (2006).
  * (4) N. Laflorencie, Phys. Rep. 646, 1 (2016).
  * (5) HC. Jiang, Z. Wang, and L. Balents, Nature Physics 8, 902 (2012).
  * (6) O. C. Gorton, Efficient modeling of nuclei through coupling of proton and neutron wavefunctions, Master’s thesis, San Diego State University, 2018 (unpublished).
  * (7) O. Legeza, L. Veis, A. Poves, and J. Dukelsky, Phys. Rev. C 92, 051303(R) (2015).
  * (8) I. Stetcu, A. Baroni, and J. Carlson, arXiv:2110.06098 [nucl-th].
  * (9) A. T. Kruppa, J. Kovacs, P. Salamon, and O. Legeza, J. Phys. G: Nucl. Part. Phys. 48, 025107 (2021).
  * (10) C. Robin ,M. J. Savage, and N. Pillet, Phys. Rev. C 103, 034325 (2021).
  * (11) J. Faba, V. Martin, and L. Robledo, Phys. Rev. A 104 032428 (2021);ibid. arXiv:2203.09400.
  * (12) A. Bulgac, arXiv:2203.12079
  * (13) N. Klco, A. Roggero and M. J. Savage, Rep. Prog. Phys. in press (2022). arXiv:2107.04769
  * (14) E. Epelbaum, H. Krebs, D. Lee, UG. Meissner, Phys. Rev. Lett. 106192501 (2011).
  * (15) J. Carlson, et al. Rev. Mod. Phys. 87, 1067 (2015).
  * (16) G. Hagen, T. Papenbrock, D. J. Dean, and M. Hjorth-Jensen, Phys. Rev. C 82, 034330 (2010).
  * (17) B. R. Barrett, P. Navrátil, J. P. Vary, Prog. Part. Nucl. Phys. 69, 131 (2013).
  * (18) S. K. Bogner, and D. Roscher, Phys. Rev. C 86, 064304 (2012).
  * (19) A. J. Tropiano, S. K. Bogner, and R. J. Furnstahl, Phys. Rev. C 104, 034311 (2021).
  * (20) K. Egiyan, et. al., Phys. Rev. C 68, 014313 (2003).
  * (21) K. Egiyan, et. al., Phys. Rev. Lett. 96, 082501 (2006).
  * (22) R. Subedi et al., Science 320, 1476 (2008).
  * (23) N. Fomin, et.al., 2012, Phys. Rev. Lett. 108, 092502 (2012).
  * (24) O. Hen, G. A. Miller, E. Piasetzky, and L. B. Weinstein, Rev. Mod. Phys. 89, 045002 (2017).
  * (25) R. Cruz-Torres, et. al., Phys. Rev. Lett. 124, 212501 (2020).
  * (26) R. Cruz-Torres, et. al., Nature Physics 17, 306 (2021).
  * (27) E. Piasetzky, M. Sargsian, L. Frankfurt, M. Strikman, and J.W. Watson, Phys. Rev. Lett. 97, 162504 (2006).
  * (28) I. Korover, et al. , Phys. Rev. Lett. 113, 022501 (2014).
  * (29) O. Hen, et al. , Science 346, 614 (2014).
  * (30) I. Korover, et al. ,Phys. Lett. B 820, 136523 (2021).
  * (31) M. Duer, et al. , Nature (London) 560, 617 (2018).
  * (32) M. Duer, et. al., Phys. Rev. Lett. 122, 172502 (2019).
  * (33) B. Schmookler et al., Nature (London) 566, 354 (2019).
  * (34) A. Schmidt, et. al., Nature 578, 540 (2021).
  * (35) A. de-Shalit, and I. Talmi, Nuclear Shell Theory (Academic Press, New York 1963).
  * (36) T. Neff, H. Feldmeier and W. Horiuchi, Phys. Rev. C 92, 024003 (2015).
  * (37) M. B. Plenio, and S. Virmani, Quantum Inf. Comput. 7, 1 (2007).
  * (38) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Med. Phys. 81, 865 (2009).
  * (39) P. Zanardi Phys. Rev. A 65, 042101 (2002).
  * (40) J. R. Gittings, A. J. Fisher, Phys. Rev. A 66, 032305 (2002).
  * (41) Y. Shi, Phys. Rev. A 67, 024301 (2003).
  * (42) R. Weiss, B. Bazak, and N. Barnea, Phys. Rev. Lett. 114, 012501 (2015).
  * (43) R. Weiss and N. Barnea, Phys. Rev. C 96, 041303(R) (2017).
  * (44) R. Weiss, B. Bazak, and N. Barnea, Eurp. Phys. J. A, 52, 92 (2016).
  * (45) R. Weiss, B. Bazak, and N. Barnea, Phys. Rev. C 92, 054311 (2015).
  * (46) M. Alvioli, C. Ciofi degli Atti, L. P. Kaptari, C. B. Mezzetti, and H. Morita, Phys. Rev. C 87, 034603 (2013); M. Alvioli, C. Ciofi degli Atti and H. Morita, Phys. Rev. Lett. 100, 162503 (2008).
  * (47) J. Arrington, D. Higinbotham, G. Rosner, M. Sargsian, Prog. Part. Nucl. Phys. 67, 898 (2012).
  * (48) R. B. Wiringa, R. Schiavilla, Steven C. Pieper, and J. Carlson, Phys. Rev. C 89, 024305 (2014).
  * (49) R. Weiss, R. Cruz-Torres, N. Barnea, E. Piasetzky and O. Hen, Phys. Lett. B 780, 211 (2018).
  * (50) R. Weiss, I. Korover, E. Piasetzky, O. Hen and N. Barnea, Phys. Lett. B 791, 242 (2019).
  * (51) J. R. Pybus, et. al., Phys. Lett. B 805, 135429 (2020).
  * (52) R. Weiss, A. W. Denniston, J. R. Pybus, O. Hen, E. Piasetzky, A. Schmidt, L. B. Weinstein, and N. Barnea, Phys. Rev. C 103, L031301 (2021).
  * (53) R. Weiss, A. Scmidt, G. A. Miller and N. Barnea, Phys. Lett. B 790, 484 (2019).