# Deciphering Effects of Relativistic Kinematics, Dimensionality, Interactions
and Short-Range Correlations on the Ratio of Quartic over Quadratic Nuclear
Symmetry Energies

Bao-Jun Cai<EMAIL_ADDRESS>Quantum Machine Learning Laboratory, Shadow
Creator Inc., Shanghai 201208, China  Bao-An Li<EMAIL_ADDRESS>Department of Physics and Astronomy, Texas A$\&$M University-Commerce,
Commerce, TX 75429-3011, USA

###### Abstract

While ample evidence for the so-called empirical parabolic law of the Equation
of State (EOS) of isospin asymmetric nuclear matter (ANM) has been obtained in
many studies within both non-relativistic and relativistic nuclear many-body
theories using various interactions, it has been unclear if there is any
fundamental physics reason for the small quartic symmetry energy compared to
the quadratic one even as the ANM approaches pure neutron matter. Within both
relativistic and non-relativistic Free Fermi Gas (FFG) models in coordinate
spaces of arbitrary dimension $d$ with and without considering Short-Range
Correlations (SRC) as well as non-linear Relativistic Mean Field (RMF) models,
we study effects of relativistic kinematics, dimensionality, interactions and
SRC on the ratio $\Psi(\rho)$ of quartic over quadratic symmetry energies in
ANM EOSs. We found that the ratio $\Psi(\rho)$ in the FFG model depends
strongly on the dimension $d$. While it is very small already in the normal 3D
space, it could be even smaller in spaces with reduced dimensions for sub-
systems of particles in heavy-ion reactions and/or whole neutron stars due to
constraints, collectivities and/or symmetries. We also found that the ratio
$\Psi(\rho)$ could theoretically become very large only at the ultra-
relativistic limit far above the density reachable in neutron stars. On the
other hand, nuclear interaction directly and/or indirectly through SRC-induced
high-momentum nucleons affect significantly the density dependence of
$\Psi(\rho)$ compared to the relativistic FFG model prediction. The SRC
affects significantly not only the kinetic energy of symmetric nuclear matter
but also the ratio $\Psi(\rho)$ while the relativistic corrections are found
negligible. Although we found no fundamental physics reason for the
$\Psi(\rho)$ to be very small especially at high densities, the results may
help better understand the EOS of dense neutron-rich matter.

###### pacs:

21.65.-f, 21.30.Fe, 24.10.Jv

##  I Introduction and conclusions

The energy per nucleon $E(\rho,\delta)$ in cold neutron-rich nucleonic matter
at density $\rho$ and isospin asymmetry
$\delta=(\rho_{\textrm{n}}-\rho_{\textrm{p}})/\rho$ with $\rho_{\textrm{n}}$
and $\rho_{\textrm{p}}$ being the neutron and proton densities, respectively,
is a basic input for calculating the Equation of State (EOS) for various
applications in both nuclear physics and astrophysics. It is usually expanded
in even powers of $\delta$ as

$E(\rho,\delta)\approx
E_{0}(\rho)+E_{\textrm{sym}}(\rho)\delta^{2}+E_{\textrm{sym,4}}(\rho)\delta^{4}+\mathcal{O}(\delta^{6})$
(1)

in terms of the energy per nucleon $E_{0}(\rho)\equiv E(\rho,0)$ in symmetric
nuclear matter (SNM), the symmetry energy (quadratic term)
$E_{\textrm{sym}}(\rho)\equiv
2^{-1}\partial^{2}E(\rho,\delta)/\partial\delta^{2}|_{\delta=0}$, the fourth-
order symmetry energy (quatic term)
$E_{\textrm{sym,4}}(\rho)\equiv{24}^{-1}\partial^{4}E(\rho,\delta)/\partial\delta^{4}|_{\delta=0}$,
etc. The first three terms at the nuclear saturation density $\rho_{0}\approx
0.16\,\textrm{fm}^{-3}$, namely $E_{0}(\rho_{0})$,
$E_{\textrm{sym}}(\rho_{0})$ and $E_{\textrm{sym,4}}(\rho_{0})$ are
empirically constrained to be about
$E_{0}(\rho_{0})\approx-16\,\textrm{MeV},E_{\textrm{sym}}(\rho_{0})\approx
32\,\textrm{MeV}$ LiBA13 ; LiBA19 ; LiBA21  and
$E_{\textrm{sym,4}}(\rho_{0})\lesssim 2\,\textrm{MeV}$ Bom91 ; Lee98 ; Ste06 ;
Cai12 ; Gon17 ; PuJ17 , indicating that the small-quantity expansion in terms
of $\delta^{2}$ for the $E(\rho,\delta)$ in asymmetric nuclear matter (ANM) is
very effective and sufficient for many physics purposes. The small ratio of
the quartic over the quadratic symmetry energy terms was confirmed using both
the relativistic and non-relativistic phenomenological models as well as
several more microscopic many-body theories Lee98 ; Bom91 ; Ste06 ; Cai12 ;
Gon17 ; PuJ17 . Thus, there are many empirical evidences for the so-called
empirical parabolic law of the EOS of neutron-rich matter Bom91 , i.e, safely
truncating the expansion of Eq. (1) at $\delta^{2}$. Interestingly, however,
certain quantities/processes in neutron stars/astrophysics were found to be
significantly affected by even a very small quartic and/or even higher order
terms in the expanded $E(\rho,\delta)$, see, e.g., Ref. Cai12  and references
therein.

Considering the aforementioned findings and empirical evidences, some
interesting questions emerge naturally. For instance, why is the isospin
quartic term so small compared with its preceding term? Does there exist any
deep physics reason for the seemingly quick convergence in expanding the
$E(\rho,\delta)$ while the $\delta$ may be not small (in fact, it may approach
1 not only theoretically but also actually in some regions of neutron stars)?
Within most of the energy density functional formalisms in the literature, the
kinetic and potential contributions to the $E(\rho,\delta)$ are clearly
separated. Generally, the $E(\rho,\delta)$ is written as the sum of nucleon
kinetic energies in a Free Fermi Gas (FFG) and their interaction energies
evaluated using various nuclear many-body theories. The first indication that
the (kinetic) quartic symmetry energy may be comparable with the (kinetic)
quadratic symmetry energy is from studying effects of the strong isospin
dependence of nucleon-nucleon short range correlations (SRCs) Cai16 . The
latter leads to a higher fraction of protons compared to neutrons in the high
momentum tail (HMT) in the single-nucleon momentum distribution functions in
neutron-rich matter as evidenced by findings of several recent SRC experiments
Hen14 ; Due18 ; Sch19 ; Sch20 . Incorporating approximately experimental
findings regrading the SRC-induced HMTs in a non-relativistic Fermi gas model,
the (kinetic) quadratic symmetry energy was found to decrease Xulili ; Hen15 ;
LiBA15  while the quartic one increases significantly compared to the FFG
model predictions without considering the SRC effects Cai16 . More
quantitatively, the kinetic quartic symmetry energy at $\rho_{0}$ was found to
be as large as about 7 MeV Cai16 . For a recent review, see, e.g., Ref. LCCX18
. It is well known that two-body SRCs in cold nuclear matter push nucleons
from below to above the Fermi surface up to about two times the nucleon Fermi
momentum. Since the latter is proportional to $\rho^{1/3}$, at high densities
relativistic effects naturally become important. Thus, both the SRC and
relativistic kinematics are expected to be important for nucleons in the HMT.
However, it is still unclear how the relativistic kinematics and SRC effects
may be intertwined in determining the quartic relative to the quadratic
symmetry energy, i.e., $\Psi(\rho)\equiv
E_{\textrm{sym,4}}(\rho)/E_{\textrm{sym}}(\rho)$ (this ratio naturally becomes
the ratio of the respective kinetic energies in FFG models). Moreover, nuclear
interactions are naturally expected to affect the ratio $\Psi(\rho)$. It is
thus also interesting to know the relative kinetic and potential contributions
to the various high-order symmetry energies.

In this work, to help shed some new light on the issues mentioned above, we
investigate the following topics:

  1. (1)

The ratio $\Psi$ in a non-relativistic FFG model in coordinate spaces of
general dimension $d$=1 to 4. We also discuss briefly some possible situations
where a sub-system of particles during heavy-ion reactions or all particles in
whole neutron stars can be considered approximately as moving in 1D or 2D due
to constraints, collectivities and/or symmetries.

  2. (2)

The ratio $\Psi$ in a 3D ($d$=3) relativistic FFG model.

  3. (3)

Nuclear interaction effects on the ratio $\Psi$ within a nonlinear
relativistic mean field (RMF) model with respect to the relativistic FFG model
predictions.

  4. (4)

Relativistic corrections to the first four terms in the kinetic EOS of
neutron-rich nucleonic matter and the ratio $\Psi$ in the presence of SRC-
induced HMT in the single-nucleon momentum distribution.

Our main findings can be summarized as follows:

  1. (1)

The ratio $\Psi$ in the FFG model depends strongly on the dimension $d$ of the
coordinate space in which nucleons move around. While in the normal 3D space,
the ratio $\Psi$ is very small, it can be even smaller or much larger in
spaces with reduced or extended dimensions. The smallness of $\Psi$ in the
conventional 3D space is thus nothing special. Moreover, relativistic
corrections to the kinetic energy of SNM as well as the quadratic and quartic
symmetry energies all depend strongly on the dimensionality of the system
considered.

  2. (2)

The ratio $\Psi$ could become very large if either the density is large enough
(in the ultra-relativistic limit) or nucleon-nucleon interactions are taken
into account (either indirectly via the SRC-induced HMT or directly through
the effective nuclear interactions).

  3. (3)

In the presence of the SRC-induced HMT, relativistic corrections to both the
quadratic and quartic kinetic symmetry energies are small mainly due to the
small $k_{\textrm{F}}/M$ ratio with $k_{\textrm{F}}$ and $M$ being the Fermi
momentum and nucleon mass, respectively.

##  II
$E_{\textmd{sym},4}^{\textmd{kin}}(\rho)/E^{\textmd{kin}}_{\textmd{sym}}(\rho)$
in Free Fermi Gas models

###  II.1 Non-relativistic FFG Model predictions in 3D

As a useful reminder of the basic formalisms and notations used in the present
work, we briefly recall here predictions of the non-relativistic FFG model in
a 3D coordinate space. For nucleons in ANM, the FFG single-nucleon momentum
distributions are step functions, i.e.,
$n_{\textbf{k}}^{J}(\rho,\delta)=\Theta(k_{\textrm{F}}^{J}-|\textbf{k}|)$
where $J=\textrm{n}$ and p. The nucleon density $\rho_{J}$ and the
corresponding Fermi momentum $k_{\textrm{F}}^{J}$ are related by
$k_{\textrm{F}}^{J}=k_{\textrm{F}}(1+\tau_{3}^{J}\delta)^{1/3}$, where
$\tau_{3}^{\textrm{n}}=+1$ and $\tau_{3}^{\textrm{p}}=-1$ are the third
components of the isospin vector of neutrons and protons, respectively. The
average kinetic energy per nucleon in ANM is then straightforwardly obtained
as

$\displaystyle E^{\textrm{kin}}(\rho,\delta)=$
$\displaystyle\left.\left[\int_{0}^{k_{\textrm{F}}^{\textrm{n}}}\frac{\textbf{k}^{2}}{2M}\mathrm{d}\textbf{k}+\int_{0}^{k_{\textrm{F}}^{\textrm{p}}}\frac{\textbf{k}^{2}}{2M}\mathrm{d}\textbf{k}\right]\right/2\int_{0}^{k_{\textrm{F}}}\mathrm{d}\textbf{k}$
$\displaystyle=$
$\displaystyle\frac{3}{5}\frac{k_{\textrm{F}}^{2}}{2M}\frac{1}{2}\left[(1+\delta)^{5/3}+(1-\delta)^{5/3}\right]$
$\displaystyle\approx$
$\displaystyle\frac{3k_{\textrm{F}}^{2}}{10M}+\frac{k_{\textrm{F}}^{2}}{6M}\delta^{2}+\frac{k_{\textrm{F}}^{2}}{162M}\delta^{4}+\frac{7k_{\textrm{F}}^{2}}{4374M}\delta^{6}.$
(2)

One can then find that $E_{0}^{\textrm{kin}}(\rho)=3k_{\textrm{F}}^{2}/10M$,
$E_{\textrm{sym}}^{\textrm{kin}}(\rho)={k_{\textrm{F}}^{2}}/{6M}$,
$E_{\textrm{sym,4}}^{\textrm{kin}}(\rho)={k_{\textrm{F}}^{2}}/{162M}$, and
$E_{\textrm{sym,6}}^{\textrm{kin}}(\rho)=7k_{\textrm{F}}^{2}/4374M$. The ratio
$\Psi\equiv
E^{\textrm{kin}}_{\textrm{sym,4}}(\rho)/E^{\textrm{kin}}_{\textrm{sym}}(\rho)$
is 1/27. By using the Fermi momentum
$k_{\textrm{F}}=(3\pi^{2}\rho/2)^{1/3}\approx 263\,\textrm{MeV}$ corresponding
to the saturation density $\rho_{0}\approx 0.16\,\textrm{fm}^{-3}$, we obtain
$E_{0}^{\textrm{kin}}(\rho_{0})\approx 22.2\,\textrm{MeV}$,
$E_{\textrm{sym}}^{\textrm{kin}}(\rho_{0})\approx 12.3\,\textrm{MeV}$,
$E_{\textrm{sym,4}}^{\textrm{kin}}(\rho_{0})\approx 0.45\,\textrm{MeV}$, and
$E_{\textrm{sym},6}^{\textrm{kin}}(\rho)\approx 0.12\,\textrm{MeV}$,
respectively.

The above well known predictions by the non-relativistic FFG model serve as
useful references in evaluating effects of the relativistic kinematics and SRC
as well as nuclear interactions missing in this model. For example, since
$E_{0}^{\textrm{kin}}(\rho_{0})\approx 22.2\,\textrm{MeV}$, to reproduce the
empirical binding energy of about $-16\,\textrm{MeV}$ at $\rho_{0}$, an
interaction contribution of about $-38.2$ MeV is required. Similarly, since
the kinetic symmetry energy is about 12.3 MeV, an interaction contribution of
about 20 MeV would be needed to reproduce the total empirical nuclear symmetry
energy of about 32 MeV at $\rho_{0}$ LiBA13 . While this model predicts a
fourth-order kinetic symmetry energy of about 0.45 MeV and several microscopic
theories and phenomenological models predict a total
$E_{\textrm{sym,4}}(\rho_{0})$ less than about 2 MeV Lee98 ; Bom91 ; Ste06 ;
Cai12 ; Gon17 ; PuJ17 , there is currently no community census on the
empirical value of $E_{\textrm{sym,4}}(\rho_{0})$. Needless to say, the
situation is worse for the sixth-order symmetry
$E_{\textrm{sym},6}^{\textrm{kin}}(\rho_{0})$ (both the kinetic term and the
total). We thus can not judge whether the relative kinetic contribution to the
total symmetry energy is decreasing or not going from the $\delta^{2}$ to
higher-order terms.

###  II.2 FFG Model predictions for $\langle k^{\sigma}\rangle$ with
$\sigma\geq 2$ and
$E_{\textmd{sym},4}^{\textmd{kin}}(\rho)/E^{\textmd{kin}}_{\textmd{sym}}(\rho)$
in $d$-Dimensions

As shown above, the FFG model predicts that the ratio
$E^{\textrm{kin}}_{\textrm{sym,4}}(\rho)/E^{\textrm{kin}}_{\textrm{sym}}(\rho)=1/27$.
To put this result in appropriate perspective, noticing that the relativistic
kinetic energy $T(\textbf{k})=\sqrt{\textbf{k}^{2}+M^{2}}-M$ of a nucleon can
be expanded as

$\displaystyle
T(\textbf{k})\approx\frac{\textbf{k}^{2}}{2M}\left(1-\frac{\textbf{k}^{2}}{4M^{2}}+\frac{\textbf{k}^{4}}{8M^{4}}-\frac{5\textbf{k}^{6}}{64M^{6}}+\frac{7\textbf{k}^{8}}{128M^{8}}\right),$
(3)

to order $\textbf{k}^{10}$, here we first consider mathematically ANM in a
coordinate space of arbitrary dimension $d$ and calculate the average $\langle
k^{\sigma}\rangle$ of $k^{\sigma}$ with $\sigma\geq 2$ being an integer. The
$\langle k^{\sigma}\rangle$ is a function of density and isospin asymmetry,
i.e.,

$\displaystyle\langle k^{\sigma}(\rho,\delta)\rangle\equiv$
$\displaystyle\left.\left[\int_{0}^{k_{\textrm{F}}^{\textrm{n}}}k^{\sigma}\mathrm{d}^{d}\textbf{k}+\int_{0}^{k_{\textrm{F}}^{\textrm{p}}}k^{\sigma}\mathrm{d}^{d}\textbf{k}\right]\right/2\int_{0}^{k_{\textrm{F}}}\mathrm{d}^{d}\textbf{k}$
$\displaystyle=$
$\displaystyle\Upsilon\left[(1+\delta)^{1+\sigma/d}+(1-\delta)^{1+\sigma/d}\right],$
(4)

where $\sigma\in\textrm{Z}$ and $\Upsilon$ a coefficient independent of
$\delta$. We emphasize that the power $\sigma$ and dimension $d$ appear in the
combination of $\sigma/d$. We also notice that the case $\sigma=2$ corresponds
to the conventional non-relativistic kinetic energy, and $\sigma=4$ (with the
energy scales as $k^{4}/M^{3}$) induces the first relativistic correction and
$\sigma=6$ induces the second relativistic correction, etc, as indicated by
the expansion in Eq. (3). In addition,
$k_{\textrm{F}}^{J}=k_{\textrm{F}}(1+\tau_{3}^{J}\delta)^{1/d}$ is the nucleon
Fermi momentum with the Fermi momentum in symmetric matter given by
$k_{\textrm{F}}(\rho)=[\rho
2^{d-2}\pi^{d/2}\Gamma(d/2+1)]^{1/d}\sim\rho^{1/d}$.

The quantity $\langle k^{\sigma}(\rho,\delta)\rangle$ can be expanded around
$\delta=0$ as

$\displaystyle\langle k^{\sigma}(\rho,\delta)\rangle$ $\displaystyle=$
$\displaystyle\sum_{j=0}k_{\textrm{sym},2j}^{\sigma}(\rho)\delta^{2j}$ (5)
$\displaystyle\approx$ $\displaystyle
k_{0}^{\sigma}(\rho)+k_{\textrm{sym}}^{\sigma}(\rho)\delta^{2}+k^{\sigma}_{\textrm{sym,4}}(\rho)\delta^{4}+\mathcal{O}(\delta^{6})\cdots,$

where $k_{0}^{\sigma}(\rho)\equiv k_{\textrm{sym},0}^{\sigma}(\rho)$ and
$k_{\textrm{sym}}^{\sigma}(\rho)\equiv k_{\textrm{sym,2}}^{\sigma}(\rho)$.
Then from Eq. (II.2), it is straightforward to obtain the ratio

$\displaystyle\Psi^{\sigma}_{d}(j=2)\equiv\left.{k^{\sigma}_{\textrm{sym,4}}(\rho)}\right/{k^{\sigma}_{\textrm{sym}}(\rho)}=\frac{1}{12}\left(\frac{\sigma}{d}-2\right)\left(\frac{\sigma}{d}-1\right).$
(6)

Depending on the value of $\sigma/d$, this ratio is not necessarily as small
as the
$E^{\textrm{kin}}_{\textrm{sym,4}}(\rho)/E^{\textrm{kin}}_{\textrm{sym}}(\rho)=1/27$
in the FFG model. For example, if one takes $\sigma=4$ and $d=1$, i.e., the
average of the momentum to the fourth power (corresponding to the first
relativistic correction) in dimension 1, one obtains $\Psi^{4}_{1}(j=2)=1/2$.
Moreover, from the above relation, one has

$\lim_{d\to\infty}\frac{k^{\sigma}_{\textrm{sym,4}}(\rho)}{k^{\sigma}_{\textrm{sym}}(\rho)}=\frac{1}{6},$
(7)

i.e., in an imagined space of infinite dimensions the ratio $\Psi$ takes the
value of $1/6\approx 16.7$%, which is independent of the power $\sigma$ and is
about five times the FFG model prediction in 3D.

Fig. 1: (Color Online). The ratio
$\Psi_{d}^{\sigma}(j)=k^{\sigma}_{\textrm{sym},2j+2}/k^{\sigma}_{\textrm{sym},2j}$
as a function of $j$ with different $d$ by fixing $\sigma=2$ (first line from
(a) to (d)), $\sigma=4$ (second line from (e) to (h)) and $\sigma=6$ (third
line from (i) to (l)). The vertical dashed black line corresponds to $j=1$.

More generally, one can obtain the ratio between any two adjacent terms in
expanding the $\langle k^{\sigma}(\rho,\delta)\rangle$ as a function of
$\delta^{2j}$. Specifically, the ratio of the coefficient of the $(2j+2)$-term
and the $2j$-term, i.e.,
${k^{\sigma}_{\textrm{sym},2j+2}}/{k^{\sigma}_{\textrm{sym},2j}}\equiv\Psi_{d}^{\sigma}(j)$
(thus the $\Psi_{d}^{\sigma}(j=2)$ defined in (6) is a special case) is

$\displaystyle\Psi_{d}^{\sigma}(j)=\frac{1}{(2j+1)(2j+2)}\left(\frac{\sigma}{d}-2j+1\right)\left(\frac{\sigma}{d}-2j\right).$
(8)

In Fig. 1 the ratio $\Psi_{d}^{\sigma}(j)$ as a function of $j$ with different
$d$ (from 1 to 4) is shown by fixing $\sigma=2$ (first row from (a) to (d),
conventional kinetic energy), $\sigma=4$ (second row from (e) to (h), first
relativistic correction) and $\sigma=6$ (third line from (i) to (l), second
relativistic correction). From the first row of Fig. 1 (panels (a) to (d)),
one can see that the ratio of the kinetic quartic over quadratic symmetry
energies (characterized by $j=1$) is a very “special” point (at the bottom of
the curves) irrespective of the dimension $d$. In particular, for the special
case of $\sigma=2$ and $d=3$ (kinetic energy or the square of the momentum in
dimensions 3), the ratio $\Psi_{3}^{2}(j)$ takes the form
${k^{\sigma}_{\textrm{sym},2j+2}}/{k^{\sigma}_{\textrm{sym},2j}}=({18j^{2}-21j+5})/({18j^{2}+27j+9})$,
while if $\sigma=d=2$ (i.e., the square of the momentum in dimensions 2), then
${k^{\sigma}_{\textrm{sym},2j+2}}/{k^{\sigma}_{\textrm{sym},2j}}=(2j^{2}-3j+1)/(2j^{2}+3j+1)$.

Similarly, the corresponding ratio from the first relativistic correction to
the kinetic EOS (the average of $k^{4}$) is shown in the second row of Fig. 1
with different $d$ by fixing $\sigma$ at four. Except the case
$(\sigma,d)=(4,1)$ (panel (e)), all the other three sets (panels (f) to (h))
predict that the minimum of $\Psi_{d}^{4}$ occurs at $j=1$. The second
relativistic correction effect is shown in the third row of Fig. 1, here for
$d\leq 2$ the minimum of $\Psi_{d}^{6}$ occurs at $j\neq 1$. From these
graphs, one finds that $j=1$ is in fact a special point with respect to the
variation of the dimension $d$ (around 3) and power $\sigma$ (around 2). Since
the $\sigma$ and dimension $d$ appear in the combination of $\sigma/d$ in
$\Psi_{d}^{\sigma}(j)$, the panels (a), (f) and (k) of Fig. 1 are identical.

Generally, we have from Eq. (8) that

$\lim_{j\to\infty}{k^{\sigma}_{\textrm{sym},2j+2}}/{k^{\sigma}_{\textrm{sym},2j}}=1.$
(9)

It indicates that the radius of convergence of the expansion of the kinetic
EOS in terms of $\delta$ is simply $R_{\delta}=1$ Wel16 , irrespective of the
integer $\sigma$. For $\sigma=0$, the expression (8) gives a nonzero value,
i.e., $\lim_{\sigma\to 0}\Psi_{d}^{\sigma}(j)=(2j-1)(2j)/(2j+1)(2j+2)$,
although in this case both the $k_{\textrm{sym},2j+2}^{\sigma}$ and
$k_{\textrm{sym},2j}^{\sigma}$ are approaching zero (but their ratio is not
necessarily zero).

To this end, it is worth noting that all up-to-today calculations of ANM EOS
are performed in dimensions 3. Then, why is it physically meaningful and
possibly useful practically besides mathematically doable as we have shown
above to consider the kinetic EOS in spaces with dimensions other than 3?
First of all, we notice that constraints, symmetries and/or collectivities can
all reduce the minimum number of degrees of freedom or the dimensions of the
coordinate space necessary to fully describe a many-particle system. In fact,
many exciting novel features/breakthroughs have been established in systems
with reduced dimensions, see, e.g., Refs. And82 ; Sar11 ; Pit16  for reviews.
New experimental techniques have been developed to generate not only pure 1D
or 2D systems but also two-component many-particle systems with mixed
dimensions. For instance, the dimensionality of space for cold atoms can be
changed by means of strong optical lattices Blo08 ; Lew07 ; Lew12 ; Gro17 .
Interesting new physics has been revealed from studying 1D and 2D systems dD1
; dD2 ; dD3 ; dD4 ; Pet16 ; Pet00 ; Mor15 ; Ber11 , such as, a two-species
Fermi gas with mixed dimensions (one component in 1D or 2D while the other one
in 3D) dD5 , a three-dimensional resonant Bose-Fermi mixture at zero
temperature Ber13  as well as the two-dimensional Fermi-Bose dimers with a
stable $p$-wave resonant interaction Baz18 . Moreover, new physics insights
regarding the relationship between the two-body energy spectrum and the
scattering phase shifts can be obtained by studying two particle scatterings
in a $d$-dimensional space Tan . It was also shown that a multiple-body
problem could be mapped to a two-body problem in a higher dimension Guo .
Thus, explorations of many-body systems in spaces with different dimensions
may provide important insights into some physics problems.

While we are not aware of any study about the EOS in spaces other than 3D in
nuclear physics and/or astrophysics, we notice that the production of particle
jets or their squeeze-out from the participant region in a specific direction
may effectively create a 1D sub-system while the collective flow in the
reaction plane may create approximately a 2D sub-system in intermediate-
relativistic energy heavy-ion reactions. In addition, non-rotating neutron
stars are often assumed to be spherically symmetric and their structures are
uniquely determined by the 1D (in the radial direction only) Tolman-
Oppenheimer-Volkov (TOV) equations. In such models, neutron stars are
essentially 1D systems. Moreover, neutron star mergers may create bursts of
$\gamma$-rays and/or other particles/waves preferentially in certain
directions similar to the collective phenomena in relativistic heavy-ion
collisions. It is also well known that the dimensionality plays an important
role in simulating supernova explosions. As indicated by Eq. (6), the ratio
$E^{\textrm{kin}}_{\textrm{sym,4}}(\rho)/E^{\textrm{kin}}_{\textrm{sym}}(\rho)$
is zero in both 1D and 2D (in fact the quartic terms in 1D and 2D are
identically zero), while it is $1/27$ in 3D as normally used in preparing the
EOS for modeling neutron stars and various studies of heavy-ion reactions.
Future studies of nuclear EOSs in 1D and 2D and their observational effects in
collectively moving sub-systems in heavy-ion reactions and/or neutron stars
might be interesting.

###  II.3 Relativistic FFG Model Predictions in 3D

The relativistic kinetic energy per nucleon in ANM (in 3D) is obtained
straightforwardly from

$\displaystyle E^{\textrm{kin}}(\rho,\delta)=$
$\displaystyle\left.\left[\sum_{J=\textrm{n,p}}\int_{0}^{k_{\textrm{F}}^{J}}\mathrm{d}\textbf{k}\left[\sqrt{\textbf{k}^{2}+M^{2}}-M\right]\right]\right/2\int_{0}^{k_{\textrm{F}}}\mathrm{d}\textbf{k},$
(10)

by generalizing the nucleon’s dispersion relation from $\textbf{k}^{2}/2M$ to
$\sqrt{\textbf{k}^{2}+M^{2}}-M$. We recall that for small $\delta x$, one has
mathematically

$\displaystyle\sum_{\Delta=\pm 1}\textrm{arcsinh}\,(x+\Delta\delta x)\approx
2\textrm{arcsinh}\,x-\frac{x}{(x^{2}+1)^{3/2}}\delta x^{2}$
$\displaystyle-\frac{1}{4}\frac{x(2x^{2}-3)}{(x^{2}+1)^{7/2}}\delta
x^{4}-\frac{1}{24}\frac{x(8x^{4}-40x^{2}+15)}{(x^{2}+1)^{11/2}}\delta x^{6},$
(11)

and $\sqrt{1+\delta x}\approx 1+2^{-1}\delta x-8^{-1}\delta
x^{2}+16^{-1}\delta x^{3}-(5/128)\delta x^{4}+(7/256)\delta
x^{5}-(21/1024)\delta x^{6}$. Using these approximations, one can obtain the
following analytical expressions for the components of the
$E^{\textrm{kin}}(\rho,\delta)$ when it is expanded as an even-power series of
$\delta$

$\displaystyle E_{0}^{\textrm{kin}}(\rho)=$
$\displaystyle\frac{3M}{8\nu^{2}}\left[\left(1+2\nu^{2}\right)\left({1+\nu^{2}}\right)^{1/2}-\frac{\textrm{arcsinh}\,\nu}{\nu}\right]-M,$
(12) $\displaystyle E_{\textrm{sym}}^{\textrm{kin}}(\rho)=$
$\displaystyle\frac{M}{6}\frac{\nu^{2}}{\sqrt{1+\nu^{2}}},$ (13)
$\displaystyle E_{\textrm{sym,4}}^{\textrm{kin}}(\rho)=$
$\displaystyle\frac{M}{648}\frac{\nu^{2}(10\nu^{4}+11\nu^{2}+4)}{(1+\nu^{2})^{5/2}},$
(14) $\displaystyle E_{\textrm{sym,6}}^{\textrm{kin}}(\rho)=$
$\displaystyle\frac{M}{34992}\frac{\nu^{2}}{(1+\nu^{2})^{9/2}}$
$\displaystyle\times\left(176\nu^{8}+428\nu^{6}+477\nu^{4}+260\nu^{2}+56\right),$
(15)

where the dimensionless quantity $\nu$ is defined as
$\nu={k_{\textrm{F}}}/{M}.$

At the non-relativistic limit with $\nu\ll 1$, the above expressions are
reduced to

$\displaystyle E_{0}^{\textrm{kin}}(\rho)\approx$
$\displaystyle\frac{3k_{\textrm{F}}^{2}}{10M}\left(1-\frac{5}{28}\nu^{2}+\frac{5}{72}\nu^{4}-\frac{25}{704}\nu^{6}\right),$
(16) $\displaystyle E^{\textrm{kin}}_{\textrm{sym}}(\rho)\approx$
$\displaystyle\frac{k_{\textrm{F}}^{2}}{6M}\left(1-\frac{1}{2}\nu^{2}+\frac{3}{8}\nu^{4}-\frac{5}{16}\nu^{6}\right),$
(17) $\displaystyle E^{\textrm{kin}}_{\textrm{sym,4}}(\rho)\approx$
$\displaystyle\frac{k_{\textrm{F}}^{2}}{162M}\left(1+\frac{1}{4}\nu^{2}-\frac{25}{32}\nu^{6}\right),$
(18) $\displaystyle E^{\textrm{kin}}_{\textrm{sym,6}}(\rho)\approx$
$\displaystyle\frac{7k_{\textrm{F}}^{2}}{4374M}\left(1+\frac{1}{7}\nu^{2}-\frac{5}{112}\nu^{6}\right),$
(19)

where the in-front coefficient of each expression is the non-relativistic
correspondence. It is necessary to point out that the first-order relativistic
correction to the kinetic symmetry energy, i.e.,
$-2^{-1}\nu^{2}\cdot(k_{\textrm{F}}^{2}/6M)=-12^{-1}k_{\textrm{F}}^{4}/M^{3}$,
was first given in Ref. Fri05 , and its value at $\rho\approx
0.16\,\textrm{fm}^{-3}$ is found to be about $-$0.48 MeV.

On the other hand, at the ultra-relativistic limit with $\nu\gg 1$, we can
similarly obtain ($\mu=\nu^{-1}$) the following

$\displaystyle E^{\textrm{kin}}_{0}(\rho)\approx$
$\displaystyle\frac{3k_{\textrm{F}}}{4}\left(1-\frac{4\mu}{3}+\mu^{2}+\frac{1-4\ln(2/\mu)}{8}\mu^{4}\right),$
(20) $\displaystyle E^{\textrm{kin}}_{\textrm{sym}}(\rho)\approx$
$\displaystyle\frac{k_{\textrm{F}}}{6}\left(1-\frac{1}{2}\mu^{2}+\frac{3}{8}\mu^{4}\right),$
(21) $\displaystyle E^{\textrm{kin}}_{\textrm{sym,4}}(\rho)\approx$
$\displaystyle\frac{5k_{\textrm{F}}}{324}\left(1-\frac{7}{5}\mu^{2}+\frac{81}{40}\mu^{4}\right),$
(22) $\displaystyle E^{\textrm{kin}}_{\textrm{sym,6}}(\rho)\approx$
$\displaystyle\frac{11k_{\textrm{F}}}{2187}\left(1-\frac{91}{44}\mu^{2}+\frac{729}{176}\mu^{4}\right),$
(23)

where each in-front coefficient is the corresponding expression in the ultra-
relativistic case.

The density dependence of the above quantities, i.e., (16) to (23), is rather
different for small and large densities. Take the symmetry energy as an
example, we find in the non-relativistic limit that
$E^{\textrm{kin}}_{\textrm{sym}}(\rho)\approx k_{\textrm{F}}^{2}/6M\sim
k_{\textrm{F}}\nu\sim\rho^{2/3}\ll k_{\textrm{F}}$ since $\nu\ll 1$. While in
the ultra-relativistic limit, $E_{\textrm{sym}}^{\textrm{kin}}(\rho)\approx
k_{\textrm{F}}/6\sim M\nu\sim\rho^{1/3}\gg M$ since now $\nu\gg 1$. All these
quantities can take any positive values, while their ratios are bounded.

Now considering the ratio between the fourth-order and second-order terms, we
immediately obtain

$\Psi=\frac{E_{\textrm{sym,4}}^{\textrm{kin}}(\rho)}{E_{\textrm{sym}}^{\textrm{kin}}(\rho)}=\frac{1}{108}\frac{4+11\nu^{2}+10\nu^{4}}{1+2\nu^{2}+\nu^{4}}.$
(24)

In the non-relativistic limit, the above expression is reduced to $\Psi\approx
1/27+\nu^{2}/36-\nu^{6}/36+\mathcal{O}(\nu^{8})\approx\Psi_{\textrm{NR}}=1/27\approx
3.7\%$, which is the familiar result obtained earlier (in subsection II.1). On
the other hand, we have in the ultra-relativistic limit that
$\Psi\to\Psi_{\textrm{UR}}=5/54\approx 9.3$% by taking $\nu\to\infty$ in (24),
since now $\Psi\approx
5/54-1/12\nu^{2}+1/9\nu^{4}-5/36\nu^{6}+\mathcal{O}(\nu^{-8})$. Thus, between
the two limits we have $1/27\leq\Psi\leq 5/54$. The enhancement from
$\Psi_{\textrm{NR}}$ to $\Psi_{\textrm{UR}}$ is $1/18$ which is $150$% of
$\Psi_{\textrm{NR}}$, although the enhancement itself is still a small
quantity. It is also interesting to point out that the Fermi momentum
$k_{\textrm{F}}$ is the only scale in the ultra-relativistic situation, since
here the static mass $M\ll k_{\textrm{F}}$, and thus could be neglected
safely. In particular, we have $E_{\textrm{sym}}^{\textrm{kin}}(\rho)\approx
k_{\textrm{F}}/6$ and $E_{\textrm{sym,4}}^{\textrm{kin}}(\rho)\approx
5k_{\textrm{F}}/324$. In this sense, the relativistic effects enhance the
quartic contribution (compared with the conventional quadratic kinetic
symmetry energy). Moreover, if one takes $\nu=k_{\textrm{F}}/M\approx 1$
roughly as the boundary between the non-relativistic and the relativistic
regions, then $\rho\approx 45\rho_{0}$ is found. It indicates that for
conventional nuclear physics even for neutron stars ($\rho\lesssim
10\rho_{0}$), the smallness of the quartic kinetic symmetry energy could be
partially explained as due to the fact that the problems encountered in these
studies are effectively non-relativistic.

Fig. 2: (Color Online). Panel (a): The $\nu$ (equivalently $\rho$) dependence
of the ratio $\Psi$ in the relativistic FFG model (red) in comparison with the
nonlinear RMF model prediction (blue lines). Panel (b): results with a general
RMF construction different from the FSUGold set, see detailed descriptions in
the text.

##  III $E_{\textmd{sym},4}(\rho)/E_{\textmd{sym}}(\rho)$ in Nonlinear
Relativistic Mean Field Models: Effects of Nuclear Interactions

In order to investigate effects of nuclear effective interactions on the ratio
$\Psi$, we give an example in this section adopting the nonlinear relativistic
mean field (RMF) model. In particular, the $\Psi$ obtained from the RMF using
the FSUGold parameter set Tod05  is shown in the left panel of Fig. 2 with the
blue solid line, while the dashed magenta line is the relativistic FFG model
prediction according to Eq. (24). It is clearly shown that the interaction
changes significantly the FFG model prediction for $\Psi$ (indicated by the
red arrows in panel (a)), at both low and high densities. The total quadratic
and quartic symmetry energy in the nonlinear RMF model at very high densities
(ultra-relativistic limit) are,

$\displaystyle E_{\textrm{sym}}(\rho)\approx$
$\displaystyle\frac{k_{\textrm{F}}}{6}+\frac{c_{\omega}^{2/3}\rho^{1/3}}{2\Lambda_{\textrm{V}}},$
(25) $\displaystyle E_{\textrm{sym,4}}(\rho)\approx$
$\displaystyle\frac{5k_{\textrm{F}}}{324}+\frac{c_{\omega}^{5/3}\rho^{1/3}}{6\Lambda_{\textrm{V}}^{2}},$
(26)

respectively, and they both scale as $\rho^{1/3}$ Cai12 . Here the coupling
parameters $c_{\omega}$ and $\Lambda_{\textrm{V}}$ characterize the self-
interaction among the four $\omega$ mesons as well as the coupling between the
$\omega$ meson and the $\rho$ meson, respectively. They are introduced into
the nonlinear RMF Lagrangian $\mathcal{L}_{\textrm{non-RMF}}$ through the
$4^{-1}c_{\omega}g_{\omega}^{4}(\omega_{\mu}\omega^{\mu})^{2}$ and
$2^{-1}\Lambda_{\textrm{V}}g_{\rho}^{2}g_{\omega}^{2}\omega_{\mu}\omega^{\mu}\vec{\rho}_{\nu}\cdot\vec{\rho}^{\nu}$
terms, where $g_{\omega}$ and $g_{\rho}$ are two coupling constants between
nucleons and the vector mesons Cai12 , respectively. Consequently, we obtain
the ratio of the quartic over the quadratic symmetry energy in the ultra-
relativistic limit as

$\Psi=\Psi^{\textrm{RMF}}_{\textrm{UR}}\equiv\left.\left(\frac{5a}{324}+\frac{c_{\omega}^{5/3}}{6\Lambda_{\textrm{V}}^{2}}\right)\right/\left(\frac{a}{6}+\frac{c_{\omega}^{2/3}}{2\Lambda_{\textrm{V}}}\right),$
(27)

where $a=\sqrt[3]{3\pi^{2}/2}$, see, e.g., Ref. Cai12  for more details on
these analytical expressions. In the FSUGold parameter set, $c_{\omega}=0.01$
and $\Lambda_{\textrm{V}}=0.24$, and consequently
$\Psi_{\textrm{UR}}^{\textrm{RMF}}\approx 7.8$%. As indicated in the panel
(a), this limit is reached quickly as soon as $\nu=k_{F}/M$ becomes larger
than about 1. We notice that in both the numerator and denumerator of the
above expression, the first term is the kinetic while the second one is the
potential contribution. The potential/kinetic ratio is
$(c_{\omega}^{2/3}/2\Lambda_{\textrm{V}})/(a/6)\approx 0.24$ and
$(c_{\omega}^{5/3}/6\Lambda_{\textrm{V}}^{2})/(5a/324)\approx 0.04$,
respectively, indicating that the kinetic part is dominant in the FSUGold
parameter set for both the $E_{\textrm{sym}}(\rho)$ and
$E_{\textrm{sym,4}}(\rho)$ at the ultra-relativistic limit. Moreover, if the
coupling constant $c_{\omega}$ or $1/\Lambda_{\textrm{V}}$ is very small (near
zero), then the high-density limit of $\Psi$ naturally approaches $5/54$ as
predicted by the relativistic FFG model, see expression (24).

To further explore effects of nuclear interactions on the ratio $\Psi$, we now
go beyond the FSUGold parameter set and check purely theoretically what will
happen if the potential term $c_{\omega}^{5/3}/6\Lambda_{\textrm{V}}^{2}$ is
far larger than the kinetic term $5a/324$ and in the meanwhile
$c_{\omega}^{2/3}/2\Lambda_{\textrm{V}}$ is much larger than $a/6$. Obviously
according to Eq. (27), the $\Psi$ then approaches another constant
$\xi=c_{\omega}/3\Lambda_{\textrm{V}}$ independent of the density $\rho$ by
neglecting the kinetic contributions. In this case, the fourth-order symmetry
energy is not necessarily smaller than the quadratic one, i.e., the
$E_{\textrm{sym},4}(\rho)$ could be comparable or even be (much) larger than
the $E_{\textrm{sym}}(\rho)$ if the density is large enough. In other words,
nucleon-nucleon interaction could change significantly the relative strength
of the quadratic and quartic symmetry energies at large densities.
Numerically, the ratio $\Psi$ could even approach infinity if the coupling
constant $\Lambda_{\textrm{V}}$ is selected to be very small (but not zero).
Of course, this may only happen at extremely high densities where there is no
experimental and/or theoretical constraints and may not be reachable anywhere.
Thus, this exercise may be only for satisfying our intellectual curiosity.

In order to check the above expectations, we show in the right panel of Fig. 2
an example of a relevant RMF construction for $\Psi$ (blue solid line), the
$\Psi_{\textrm{UR}}^{\textrm{RMF}}$ factor (black dotted line) by setting
artifically the constant $\xi=c_{\omega}/3\Lambda_{\textrm{V}}=4.8$. Here the
coupling constant $c_{\omega}$ is still fixed at 0.01 as in the FSUGold, but
the coupling constant $\Lambda_{\textrm{V}}$ is adjusted according to
$\Lambda_{\textrm{V}}=c_{\omega}/3\xi\approx 7\times 10^{-4}$ (which is much
smaller than the one used in the FSUGold set). Thus, overall, the newly
constructed RMF parameter set is different from the FSUGold one. It is seen
that as the dimensionless quantity $\nu$ becomes large, the RMF prediction
naturally approaches $\Psi_{\textrm{UR}}^{\textrm{RMF}}$. It also approaches
the constant factor $\xi$ since the interaction parts in both the quadratic
and quartic symmetry energies are dominant over their kinetic parts. More
quantitatively, under these conditions for $c_{\omega}$ and
$\Lambda_{\textrm{V}}$ we have $c_{\omega}^{2/3}/2\Lambda_{\textrm{V}}\approx
33.42$, which is far larger than the kinetic term $a/6\approx 0.41$, while
similarly $c_{\omega}^{5/3}/6\Lambda_{\textrm{V}}^{2}\approx 160.41$ which is
also far larger than the kinetic contribution $5a/324\approx 0.04$. It is
necessary to point out that a factor $\nu$ about 40 corresponds to a density
$\rho$ about $2.9\times 10^{6}\rho_{0}$, which is thus only theoretically
meaningful. Since the large-$\rho$ limit is also the relativistic limit, we
find that the combination of the nucleon-nucleon interactions and the
relativistic corrections (here come into play in the potential part)
essentially modify the prediction on the ratio $\Psi$, although the
relativistic corrections alone can not effectively affect the value of $\Psi$.
In this sense, the nucleon-nucleon interactions (here through the effective
potentials in the EOS of ANM) are very important for the ratio $\Psi$ as one
generally expects.

On the opposite side, i.e., in the ultra-low density limit, the kinetic
symmetry energy from the nonlinear RMF models could be expanded as
$E^{\textrm{kin}}_{\textrm{sym}}(\rho)\approx(k_{\textrm{F}}^{2}/6M)\cdot[1-2^{-1}(k_{\textrm{F}}/M)^{2}+M^{-1}(g_{\sigma}/m_{\sigma})^{2}\rho]+\cdots\sim
b\nu^{2}+\cdots$, while the potential part is similarly approximated as
$E_{\textrm{sym}}^{\textrm{pot}}(\rho)\approx
2^{-1}(g_{\rho}/m_{\rho})^{2}\rho\cdot[1-\Lambda_{\textrm{V}}(g_{\rho}/m_{\rho})^{2}(g_{\omega}/m_{\omega})^{4}\rho^{2}]+\cdots\sim
b^{\prime}\nu^{3}+\cdots$, where $b$ and $b^{\prime}$ are two constants,
$g_{\sigma}$ is the coupling constant between the nucleon and the $\sigma$
meson, and $m_{\sigma},m_{\rho}$ and $m_{\omega}$ are the static masses of the
$\sigma,\rho$ and the $\omega$ mesons, respectively. Thus the ratio
$E_{\textrm{sym}}^{\textrm{pot}}(\rho)/E_{\textrm{sym}}^{\textrm{kin}}(\rho)\approx(b^{\prime}/b)\nu$
approaches zero as $\rho\to 0$. One can find similarly that at ultra-low
densities, the fourth-order symmetry energy is dominated by its kinetic part,
actually according to the non-relativistic FFG prediction. More
quantitatively, we have $E_{\textrm{sym},4}^{\textrm{kin}}(\rho)\approx
k_{\textrm{F}}^{2}/162M+\cdots\sim c\nu^{2}+\cdots$, and similarly
$E_{\textrm{sym},4}^{\textrm{pot}}(\rho)\approx-72^{-1}(g_{\sigma}/m_{\sigma})^{2}\rho\nu^{4}+\cdots\sim
c^{\prime}\nu^{7}+\cdots$ by carefully considering the exact expression for
the fourth-order symmetry energy Cai12 , where $c$ and $c^{\prime}$ are
another two constants. Consequently, the ratio
$E_{\textrm{sym},4}^{\textrm{pot}}(\rho)/E_{\textrm{sym},4}^{\textrm{kin}}(\rho)\approx(c^{\prime}/c)\nu^{5}$
which is much smaller than 1 since $\nu\ll 1$ for low densities. From these
low-density expansions, we can conclude that although the nucleon-nucleon
interactions could modify the ratio $\Psi$, it is still mainly determined by
the kinetic contributions at low densities.

##  IV Relativistic Corrections to Kinetic Symmetry Energies in the Presence
of Short-Range Correlations

###  IV.1 Nucleon Momentum Distribution with SRC-induced High-Momentum
Nucleons

In this subsection, we briefly recall the single nucleon momentum distribution
function encapsulating a SRC-induced HMT and the relevant parameters. Based on
predictions of microscopic nuclear many-body theories and relevant
experimental findings Hen14 ; Due18 ; Sch19 ; Sch20 , the single-nucleon
momentum distribution in ANM can be parametrized as Cai16 ,

$n^{J}_{\textbf{k}}(\rho,\delta)=\left\\{\begin{array}[]{ll}\Delta_{J},{}{}&0<|\textbf{k}|<k_{\textrm{F}}^{J},\\\
\displaystyle{C}_{J}\left({k_{\textrm{F}}^{J}}/{|\textbf{k}|}\right)^{4},{}{}&k_{\textrm{F}}^{J}<|\textbf{k}|<\phi_{J}k_{\textrm{F}}^{J}.\end{array}\right.$
(28)

Here, $\Delta_{J}$ is the depletion of the Fermi sphere with respect to the
step function in the FFG model. The three parameters $\Delta_{J}$, ${C}_{J}$
and $\phi_{J}$ are constrained by the fraction of nucleons in the HMT

$x_{J}^{\textrm{HMT}}=\left.\int_{k_{\textrm{F}}^{J}}^{\phi_{J}k_{\textrm{F}}^{J}}n_{\textbf{k}}^{J}\mathrm{d}\textbf{k}\right/{\displaystyle\int_{0}^{\phi_{J}k_{\textrm{F}}^{J}}n_{\textbf{k}}^{J}\mathrm{d}\textbf{k}}=3C_{{J}}\left(1-\frac{1}{\phi_{{J}}}\right),$
(29)

and the normalization condition

$\frac{2}{(2\pi)^{3}}\int_{0}^{\infty}n^{J}_{\textbf{k}}(\rho,\delta)\mathrm{d}\textbf{k}=\rho_{J}=(k_{\textrm{F}}^{J})^{3}/3\pi^{2}.$
(30)

Only two of the three parameters $\Delta_{J}$, $C_{J}$ and $\phi_{J}$ are
independent. Using the last two as independent parameters and assuming they
have the same isospin dependence, i.e,
$Y_{J}=Y_{0}(1+Y_{1}\tau_{3}^{J}\delta)$ Cai16 , the associated parameters
were then constrained to be about $C_{0}\approx 0.161\pm
0.015,C_{1}\approx-0.25\pm 0.07,\phi_{0}\approx 2.38\pm 0.56$ and
$\phi_{1}\approx-0.56\pm 0.10$ using information from the SRC experiments
Hen14 ; Hen16x , see, e.g., Ref. LCCX18  for a recent review.

Fig. 3: (Color Online). Kinetic parts of the EOS of ANM with its relativistic
corrections for
$E_{0}^{\textrm{kin}}(\rho_{0}),E_{\textrm{sym}}^{\textrm{kin}}(\rho_{0}),E_{\textrm{sym,4}}^{\textrm{kin}}(\rho_{0})$
and $E_{\textrm{sym,6}}^{\textrm{kin}}(\rho_{0})$. Here the saturation density
$\rho_{0}\approx 0.16\pm 0.01\,\textrm{fm}^{-3}$, $\sigma=2$ is the non-
relativistic term while $\sigma=4,6,8$ and $10$ correspond to the first four
relativistic corrections to the kinetic parts of the EOS originated from Eq.
(3).

The isospin dependence of the SRC-induced HMT was found to affect especially
the kinetic part of nuclear symmetry energy Xulili ; Hen15 ; Car12 ; Vid11 ;
Lov11 . It has also been found to affect significantly properties of neutron
stars, such as the mass-radius relation, tidal polarizability, cooling rate
and crust-core transition density Cai16a ; Sou20-a ; Sou20-b ; HLu22 .
Interestingly, the enhanced pressure due to the SRC-induced high momentum
nucleons can also help balance the reduction of the maximum mass caused by the
possible existence of dark matter particles in neutron stars Lou22 . There is
also a long history of studying effects of SRC on nuclear reactions Ant93 .
More recently, the SRC-induced HMT has been found to affect particularly the
dynamics and emissions of energetic nucleons, hard photons and/or the
production of deeply sub-threshold particles in nuclear reactions LiBA15 ;
Yong17-a ; Yong17-b ; Wang17 ; Guo21 . Interestingly, a very recently analysis
of the energy spectra of protons emitted in reactions of 47 MeV/u projectiles
with Sn and Au targets provides new evidence for the $1/k^{4}$ shape of the
HMTs in the intrinsic momenta spectra of the projectiles Chris .

###  IV.2 First Four Relativistic Corrections to the Kinetic EOS of ANM
Considering the SRC-induced HMT

As shown in Eq. (3), the first four relativistic corrections to the nucleon
kinetic energy are given by “$-\textbf{k}^{2}/4M^{2}$”,
“$\textbf{k}^{4}/8M^{4}$”, “$-5\textbf{k}^{6}/64M^{6}$” and
“$7\textbf{k}^{8}/128M^{8}$”, respectively. In calculating the average energy
per nucleon in ANM, after integrating over the momentum $k$, the dependence of
the kinetic energy on momentum $k$ is transformed into the corresponding
dependence on the Fermi momentum $k_{\textrm{F}}\sim\rho^{1/3}$. Specifically,
for a general $\sigma$ the term $\textbf{k}^{\sigma}/M^{\sigma-1}$ in the
kinetic energy leads to the $3k_{\textrm{F}}^{\sigma}/(\sigma+3)M^{\sigma-1}$
term in the kinetic EOS if the step function for the momentum distribution
$n_{\textbf{k}}^{J}$ is adopted. Similarly, the corresponding term for the
kinetic symmetry energy is $\sigma k_{\textrm{F}}^{\sigma}/6M^{\sigma-1}$. For
example, for the first-order relativistic correction to the kinetic energy,
one has the relevant corrections for the kinetic EOS of SNM and for the
kinetic symmetry energy as $-3k_{\textrm{F}}^{4}/56M^{3}$ and
$-k_{\textrm{F}}^{4}/12M^{3}$ Fri05 , respectively, see also the expressions
(16) and (17).

However, when the SRC-induced HMT is considered, the analytical expression for
$\langle\textbf{k}^{\sigma}\rangle$ becomes non-trivial. The
$\langle\textbf{k}^{\sigma}\rangle$ with a general power $\sigma$ at order
$\delta^{0}$ (corresponding to SNM), $\delta^{2}$ (corresponding to the
quadratic symmetry energy) and $\delta^{4}$ (corresponding to the quartic
symmetry energy) are given in the appendix in details. For our interest here,
terms in the expansion (3) will be investigated numerically below using
$\sigma=2,4,6,8$ and $10$. In addition, a similar expression for the
$\langle\textbf{k}^{\sigma}\rangle$ at order $\delta^{6}$, which is relevant
for the derivation of the sixth-order symmetry energy, could also be obtained
similarly. While we do not give its explicit expression in the appendix due to
its very complicated form, numerical results for the sixth-order symmetry
energy will also be presented below.

Using the parameters $\phi_{0},\phi_{1},C_{0}$ and $C_{1}$ given in the last
subsection, we can obtain the magnitudes of all terms in the kinetic EOS as
well as their relativistic corrections considering the SRC effects. In
particular, we have found that
$E_{0,\textbf{k}^{2}}^{\textrm{kin}}(\rho_{0})\approx 40.47\pm
9.23\,\textrm{MeV}$ adopting $\rho_{0}\approx 0.16\pm 0.01\,\textrm{fm}^{-3}$,
here the subscript “$0,\textbf{k}^{2}$” reminds us that this term originates
from the non-relativistic kinetic energy $\textbf{k}^{2}/2M$, thus it is not a
relativistic correction. To the next order in $\textbf{k}^{\sigma}$, we then
have $E_{0,\textbf{k}^{4}}^{\textrm{kin}}(\rho_{0})\approx-1.68\pm
1.11\,\textrm{MeV}$, which originates from the first relativistic correction
“$-\textbf{k}^{4}/8M^{3}$” (dividing the zeroth order term $\textbf{k}^{2}/2M$
gives “$-\textbf{k}^{2}/4M^{2}$” in Eq. (3)). Similarly, contributions from
the next three terms of the relativistic corrections, i.e.,
“$\textbf{k}^{6}/16M^{5}$”, “$-5\textbf{k}^{8}/128M^{7}$”, and
“$7\textbf{k}^{10}/256M^{9}$”, are found to be about $0.21\pm
0.25\,\textrm{MeV},-0.04\pm 0.07\,\textrm{MeV}$ and $0.01\pm
0.02\,\textrm{MeV}$, respectively. It is obvious that the fourth-order
relativistic correction with its value about $0.01\pm 0.02\,\textrm{MeV}$ is
much smaller than the (leading) non-relativistic contribution, indicating that
considering the first four relativistic corrections is enough for estimating
the overall relativistic effects.

The magnitudes of the kinetic EOS of SNM together with its first four
relativistic corrections are shown in the panel (a) of Fig. 3, where the non-
relativistic FFG prediction about $22.10\pm 0.92\,\textrm{MeV}$ is also shown
for a comparison (marked as “FFG”). The enhancement of the kinetic EOS of SNM
considering the SRC-induced HMT is known for some time, see, e.g., Refs. Hen14
; Xulili ; Cai16a  and references therein. By adding the non-relativistic
kinetic EOS of SNM and its first four relativistic corrections, one then
obtains,

$E_{0}^{\textrm{kin}}(\rho_{0})\approx\sum_{\sigma=2\ell}^{\ell=1\sim
5}E_{0,\textbf{k}^{\sigma}}^{\textrm{kin}}(\rho_{0})\approx 38.98\pm
8.32\,\textrm{MeV}.$ (31)

Compared with 40.47 MeV, a reduction about 1.49 MeV on the (non-relativistic)
$E_{0}^{\textrm{kin}}(\rho_{0})$ is generated due to the relativistic effects,
and the relative correction is about 4%.

Similarly, we can obtain the first four relativistic corrections to the
quadratic kinetic symmetry energy as $2.11\pm 1.87\,\textrm{MeV},-0.46\pm
0.58\,\textrm{MeV},0.12\pm 0.20\,\textrm{MeV}$ and $-0.03\pm
0.07\,\textrm{MeV}$, respectively. The quadratic kinetic symmetry energy
itself considering the SRC-induced HMT is found to be about $-13.80\pm
9.57\,\textrm{MeV}$. Based on these results, we have

$E_{\textrm{sym}}^{\textrm{kin}}(\rho_{0})\approx\sum_{\sigma=2\ell}^{\ell=1\sim
5}E_{\textrm{sym},\textbf{k}^{\sigma}}^{\textrm{kin}}(\rho_{0})\approx-12.01\pm
8.23\,\textrm{MeV}.$ (32)

Compared with the original quadratic kinetic symmetry energy encapsulating the
HMT but no relativistic correction, i.e., $-13.80\pm 9.57\,\textrm{MeV}$, the
relativistic corrections generate an enhancement about 1.79 MeV, corresponding
to a relative correction about 13%, see the panel (b) of Fig. 3. The value
either $-13.80\,\textrm{MeV}$ (without) or $-12.01\,\textrm{MeV}$ (with)
relativistic corrections for $E_{\textrm{sym}}^{\textrm{kin}}(\rho_{0})$ is
consistent with the value of $-16.94\,\textrm{MeV}$ obtained directly from a
nonlinear RMF model including the HMT Cai16a .

It is important to emphasize that the SRC-induced reduction of the quadratic
kinetic symmetry energy with respect to the FFG prediction of about $12.28\pm
0.51\,\textrm{MeV}$ at saturation density is also known for some time when the
SRC effects are considered in microscopic nuclear many-body theories or simple
phenomenlogical models, see, e.g., Ref. Carb14 ; LCCX18  for reviews. These
studies include the Brueckner-Hartree-Fock (BHF) approach using the Av18
potential plus the Urbana IX three-body force Vid11 , the self-consistent
Green’s function (SCGF) theories adopting different microscopic interactions
(N3LO, Av18, Nij1, and CD Bonn) Car12 , the Fermi hypernetted chain (FHNC)
method Lov11 , and phenomenological models Xulili ; Hen15 . See the Fig. 41 in
Ref. LCCX18  for a summary of the results from some of these studies in
comparison with the values extracted from analyzing the SRC data within a
neutron-proton dominance model Hen15 . We also note that the quadratic kinetic
symmetry energy was obtained approximately in some of the earlier studies by
using the difference between the EOSs of pure neutron matter (PNM) and SNM
adopting the parabolic approximation of the ANM EOS.

Moreover, the kinetic fourth- and sixth-order symmetry energies considering
the first four relativistic corrections in the presence of SRC are about
$6.06\pm 1.77\,\textrm{MeV}$ and $1.07\pm 0.54\,\textrm{MeV}$, respectively,
i.e.,

$\displaystyle
E_{\textrm{sym},4}^{\textrm{kin}}(\rho_{0})\approx\sum_{\sigma=2\ell}^{\ell=1\sim
5}E_{\textrm{sym,4},\textbf{k}^{\sigma}}^{\textrm{kin}}(\rho_{0})\approx
6.06\pm 1.77\,\textrm{MeV},$ (33) $\displaystyle
E_{\textrm{sym,6}}^{\textrm{kin}}(\rho_{0})\approx\sum_{\sigma=2\ell}^{\ell=1\sim
5}E_{\textrm{sym,6},\textbf{k}^{\sigma}}^{\textrm{kin}}(\rho_{0})\approx
1.07\pm 0.54\,\textrm{MeV}.$ (34)

More specifically, the contributing terms for the fourth-order symmetry energy
$E_{\textrm{sym},4}^{\textrm{kin}}(\rho_{0})$ are $7.12\pm 2.30\,\textrm{MeV}$
(non-relativistic contribution), $-1.36\pm 0.99\,\textrm{MeV}$, $0.37\pm
0.44\,\textrm{MeV}$, $-0.10\pm 0.17\,\textrm{MeV}$ and $0.03\pm
0.06\,\textrm{MeV}$, respectively. Similarly, those for the sixth-order
symmetry energy $E_{\textrm{sym,6}}^{\textrm{kin}}(\rho_{0})$ are $1.01\pm
0.51\,\textrm{MeV}$ (non-relativistic contribution), $0.09\pm
0.08\,\textrm{MeV}$, $-0.06\pm 0.07\,\textrm{MeV}$, $-0.007\pm
0.05\,\textrm{MeV}$ and $0.01\pm 0.05\,\textrm{MeV}$, respectively. See the
panels (c) and (d) of Fig. 3, respectively, for illustrations. From these
results, one finds that the relativistic corrections on the kinetic fourth-
and sixth-order symmetry energies are about $-1.06\,\textrm{MeV}$ and
$0.06\,\textrm{MeV}$, with the relative effects about 15% and 6%,
respectively. It is thus obvious that the absolute change on the sixth-order
symmetry energy due to the relativistic corrections could be safely neglected
compared with its non-relativistic value.

Finally, it is interesting to compare the ratio $\Psi$ between the kinetic
quartic and quadratic symmetry energies with and without the relativistic
corrections in the presence of SRC-induced high momentum nucleons. Based on
the results presented above, the $\Psi$ changes from
$\Psi\approx-7.12/13.80\approx-51.6\%$ (without) to
$\Psi\approx-6.06/12.01\approx-50.4\%$ (with) the relativistic corrections.
Moreover, considering the extreme case of PNM, the overall relativistic
corrections can be evaluated by adding $E_{0}^{\textrm{kin}}(\rho_{0})$,
$E_{\textrm{sym}}^{\textrm{kin}}(\rho_{0})$,
$E_{\textrm{sym},4}^{\textrm{kin}}(\rho_{0})$ and
$E_{\textrm{sym},6}^{\textrm{kin}}(\rho_{0})$ alltogether. This sum is 34.8
MeV for the non-relativistic FFG model and 34.1 MeV by including the (first
four) relativistic corrections, both are close to the non-relativistic FFG
model prediction about $3k_{\textrm{n}}^{2}/10M\approx 35.1\,\textrm{MeV}$
where $k_{\textrm{n}}=(3\pi^{2}\rho)^{1/3}=2^{1/3}k_{\textrm{F}}$ is the
neutron Fermi momentum in the PNM. Thus, although the relativistic corrections
to each kinetic energy term may be large or small, the overall effects are
rather small (the relative effect is about 2%), since some changes are
positive while others are negative. Particularly, for the four kinetic
energies, only the (quadratic) symmetry energy is reduced considering the SRC-
induced HMT, while the other three higher order symmetry energies are all
enhanced, compared with their FFG predictions (i.e., without HMT), see Fig. 3.
These results indicate clearly that the SRC-induced HMT is more fundamental
and has much larger impacts on the ANM EOS than the relativistic corrections.
Although the SRC effects are introduced through the HMT in the single nucleon
momentum distribution function $n_{\textbf{k}}^{J}$ and affect apparently only
the kinetic parts of ANM EOS in the models considered here, the SRC is
fundamentally due to the tensor force in the neutron-proton isosinglet
interaction channel Hen14 ; Hen16x . Thus, in this sense, the finding of this
section is consistent with that in section III, i.e., the nucleon-nucleon
interactions could change the value of $\Psi$ significantly.

##  V Summary

In summary, to pin down the EOS of dense neutron-rich matter has long been a
major science driver in both astrophysics and nuclear physics. Most studies
within both non-relativistic and relativistic nuclear many-body theories and
phenomenological models with and/or without considering the isospin dependence
of SRC indicate that the $E(\rho,\delta)$ converges quickly when it is
expanded in terms of even powers of $\delta$. Thus, the so-called empirical
parabolic law of ANM EOS seems to be valid even as $\delta\rightarrow 1$.
While there are many empirical evidences available in the literature using
various many-body approaches and interactions, it has been unclear why the
isospin quartic symmetry energy is so small compared with the quadratic one
and if there is any deep physics reason for the seemingly quick convergence in
expanding the $E(\rho,\delta)$.

In this work, we tried to decipher effects of relativistic kinematics,
dimensionality, interactions and SRC on the ratio $\Psi$ of isospin quartic
over quadratic symmetry energies as transparent as possible. Within both
relativistic and non-relativistic FFG models in coordinate spaces of arbitrary
dimension $d$ with and without considering SRC effects as well as the 3D RMF
models at both low and high density limits, we learned a few things that might
be useful for the community to further understand the EOS of dense neutron-
rich nuclear matter. In particular, we found that the ratio $\Psi$ in the FFG
model depends strongly on the dimension $d$. While the ratio $\Psi$ is very
small already in the normal 3D space, it could be even smaller in spaces with
reduced dimensions. Based on this finding and stimulated by the many
interesting new physics found in two-component cold atoms moving in either the
same or mixed 1D or 2D spaces, we pointed out a few situations where the EOS
in spaces with reduced dimensions may be useful for heavy-ion reactions and/or
neutron stars. We also found that the ratio $\Psi$ could theoretically become
very large only at the ultra-relativistic limit far above the density
reachable in neutron stars. On the other hand, the non-linear RMF mode using
the FSUGold parameter set predicts a $\Psi$ value around $2-8\%$ and it has
significantly different density dependence compared to the relativistic FFG
model prediction. In the relativistic FFG model incorporating the SRC-induced
HMT, the SRC affects significantly not only the kinetic energy of SNM but also
the ratio $\Psi$ while the relativistic corrections are negligible. In
conclusion, while the relativistic kinematics and dimensionality may affect
appreciably the ratio $\Psi$, it is the nuclear interactions and the
associated SRC dominate the ratio $\Psi$. We also found no fundamental physics
reason for the $\Psi$ to be very small especially at high densities.

## Acknowledgement

This work is supported in part by the U.S. Department of Energy, Office of
Science, under Award Number DE-SC0013702, the CUSTIPEN (China-U.S. Theory
Institute for Physics with Exotic Nuclei) under the US Department of Energy
Grant No. DE-SC0009971.

##  Appendix A Analytical Expressions for $\langle\textbf{k}^{\sigma}\rangle$
in Presence of SRC-induced HMT

According to the definition of the average of $\textbf{k}^{\sigma}$, namely

$\langle|\textbf{k}|^{\sigma}\rangle=\langle\textbf{k}^{\sigma}\rangle=\left.\langle\textbf{k}^{\sigma}(\rho,\delta)\rangle=\sum_{J=\textrm{n,p}}\int_{0}^{\phi_{J}k_{\textrm{F}}^{J}}\textbf{k}^{\sigma}n_{\textbf{k}}^{J}(\rho,\delta)\mathrm{d}\textbf{k}\right/2\int_{0}^{k_{\textrm{F}}}\mathrm{d}\textbf{k},$
(a35)

one obtains by noticing that
$2\int_{0}^{k_{\textrm{F}}}\mathrm{d}\textbf{k}=\rho[2/(2\pi)^{3}]^{-1}$,

$\displaystyle\langle\textbf{k}^{\sigma}\rangle\left(\mbox{at order
}\delta^{0}\right)\equiv\left(\frac{1}{\rho}\frac{2}{(2\pi)^{3}}\sum_{J=\textrm{n,p}}\int_{0}^{\phi_{J}k_{\textrm{F}}^{J}}\textbf{k}^{\sigma}n_{\textbf{k}}^{J}(\rho,\delta)\mathrm{d}\textbf{k}\right)\left(\mbox{at
order
}\delta^{0}\right)=\frac{2[(1-3C_{0})\phi_{0}+3C_{0}]k_{\textrm{F}}^{\sigma+1}}{(\sigma+1)\phi_{0}}+\frac{2C_{0}k_{\textrm{F}}^{\sigma+1}(\phi_{0}^{\sigma-3}-1)}{\sigma-3},$
(a36)

and similarly the quadratic and the quartic contributions for
$\textbf{k}^{\sigma}$,

$\displaystyle\langle\textbf{k}^{\sigma}\rangle\left(\mbox{at order
}\delta^{2}\right)\equiv\left(\frac{1}{\rho}\frac{2}{(2\pi)^{3}}\sum_{J=\textrm{n,p}}\int_{0}^{\phi_{J}k_{\textrm{F}}^{J}}\textbf{k}^{\sigma}n_{\textbf{k}}^{J}(\rho,\delta)\mathrm{d}\textbf{k}\right)\left(\mbox{at
order }\delta^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{3k_{\textrm{F}}^{\sigma}}{\sigma+3}\Bigg{[}\frac{3C_{0}\phi_{1}}{\phi_{0}}(\phi_{1}-C_{1})+(\sigma+3)C_{0}C_{1}\phi_{1}e^{(\sigma-1)\ln\phi_{0}}+\frac{1}{2}\frac{\sigma+3}{\sigma-1}C_{0}\phi_{1}^{2}(\sigma-1)(\sigma-2)e^{(\sigma-1)\ln\phi_{0}}+(\sigma+3)C_{0}C_{1}\phi_{1}e^{(\sigma-1)\ln\phi_{0}}$
$\displaystyle\hskip
28.45274pt-\left(1+\frac{\sigma}{3}\right)\Bigg{[}\frac{3C_{0}\phi_{1}}{\phi_{0}}+3C_{0}C_{1}\left(1-\frac{1}{\phi_{0}}\right)-(\sigma+3)C_{0}\phi_{1}e^{(\sigma-1)\ln\phi_{0}}-\frac{\sigma+3}{\sigma-1}C_{0}C_{1}\left[e^{(\sigma-1)\ln\phi_{0}}-1\right]\Bigg{]}$
$\displaystyle\hskip
28.45274pt+\frac{\sigma(\sigma+3)}{18}\Bigg{[}1-3C_{0}\left(1-\frac{1}{\phi_{0}}\right)+\frac{\sigma+3}{\sigma-1}C_{0}\left[e^{(\sigma-1)\ln\phi_{0}}-1\right]\Bigg{]}\Bigg{]},$
(a37) $\displaystyle\langle\textbf{k}^{\sigma}\rangle\left(\mbox{at order
}\delta^{4}\right)\equiv\left(\frac{1}{\rho}\frac{2}{(2\pi)^{3}}\sum_{J=\textrm{n,p}}\int_{0}^{\phi_{J}k_{\textrm{F}}^{J}}\textbf{k}^{\sigma}n_{\textbf{k}}^{J}(\rho,\delta)\mathrm{d}\textbf{k}\right)\left(\mbox{at
order }\delta^{4}\right)$ $\displaystyle=$
$\displaystyle\frac{3k_{\textrm{F}}^{\sigma}}{\sigma+3}\Bigg{[}\frac{3C_{0}\phi_{1}^{3}}{\phi_{0}}(\phi_{1}-C_{1})+\frac{\sigma+3}{24}C_{0}\phi_{1}^{4}(\sigma-2)(\sigma-3)(\sigma-4)e^{(\sigma-1)\ln\phi_{0}}+\frac{\sigma+3}{6}C_{0}C_{1}\phi_{1}^{3}(\sigma-2)(\sigma-3)e^{(\sigma-1)\ln\phi_{0}}$
$\displaystyle\hskip
28.45274pt+\left(1+\frac{\sigma}{3}\right)\Bigg{[}\frac{3C_{0}\phi_{1}^{2}}{\phi_{0}}(C_{1}-\phi_{1})+\frac{\sigma+3}{6}C_{0}\phi_{1}^{3}(\sigma-2)(\sigma-3)e^{(\sigma-1)\ln\phi_{0}}+\frac{\sigma+3}{2}C_{0}C_{1}\phi_{1}^{2}(\sigma-2)e^{(\sigma-1)\ln\phi_{0}}\Bigg{]}$
$\displaystyle\hskip
28.45274pt+\frac{\sigma(\sigma+3)}{18}\Bigg{[}\frac{3C_{0}\phi_{1}}{\phi_{0}}(\phi_{1}-C_{1})+\frac{\sigma+3}{2}C_{0}\phi_{1}^{2}(\sigma-2)e^{(\sigma-1)\ln\phi_{0}}+(\sigma+3)C_{0}C_{1}\phi_{1}e^{(\sigma-1)\ln\phi_{0}}\Bigg{]}$
$\displaystyle\hskip
28.45274pt+\left(\frac{\sigma}{18}-\frac{\sigma^{3}}{162}\right)\Bigg{[}3C_{0}\left[\frac{\phi_{1}}{\phi_{0}}+C_{1}\left(1-\frac{1}{\phi_{0}}\right)\right]-(\sigma+3)C_{0}\phi_{1}e^{(\sigma-1)\ln\phi_{0}}-\frac{\sigma+3}{\sigma-1}C_{0}C_{1}\left[e^{(\sigma-1)\ln\phi_{0}}-1\right]\Bigg{]}$
$\displaystyle\hskip
28.45274pt+\frac{1}{1944}\sigma(\sigma-6)(\sigma-3)(\sigma+3)\left[1-3C_{0}\left(1-\frac{1}{\phi_{0}}\right)+\frac{\sigma+3}{\sigma-1}C_{0}\left[e^{(\sigma-1)\ln\phi_{0}}-1\right]\right]\Bigg{]}.$
(a38)

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# Large Upper Critical Field of Superconductivity in the Single Crystal U6Co

Dai Aoki1,2 E-mail<EMAIL_ADDRESS>Ai Nakamura1 Fuminori Honda1 DeXin
Li1 and Yoshiya Homma1 1IMR1IMR Tohoku University Tohoku University Oarai
Oarai Ibaraki Ibaraki 311-1313 311-1313 Japan
2INAC/PHELIQS Japan
2INAC/PHELIQS CEA-Grenoble CEA-Grenoble 38054 Grenoble 38054 Grenoble France
France

###### Abstract

We grew single crystals of U6Co by the self-flux method and measured the
magnetic susceptibility, resistivity, and specific heat. The magnetic
susceptibility shows very small anisotropy and weak temperature dependence,
indicating small spin-susceptibility. Superconductivity was clearly observed
in the resistivity, susceptibility, and specific heat at $T_{\rm c}\sim
2.3\,{\rm K}$. The upper critical field was remarkably large, $7.9$ and
$6.6\,{\rm T}$ for $H\parallel[001]$ and $[110]$, respectively, in the
tetragonal structure, indicating that the ellipsoidal Fermi surface is
slightly suppressed along the $[001]$ direction according to the effective
mass model. The specific heat shows a large jump at $T_{\rm c}$ with $\Delta
C/\gamma T_{\rm c}=1.58$, and the field dependence of the specific heat at low
temperatures shows an almost linear increase. These experimental results are
well explained by the BCS model in the dirty limit condition. U6Co is most
likely a conventional $s$-wave superconductor with a full superconducting gap.

Superconductivity is one of the most fascinating topics in condensed matter
physics. In particular, there are many unconventional superconductors
correlated with magnetism in heavy fermion systems, A simple criterion for the
conventional superconductivity is the so-called Hill limit [1], which
indicates the distance between $f$-electron atoms and the direct overlap of
the $f$-wave functions. If the distance is lower than the Hill limit
($d<3.5\,{\rm\AA}$ for uranium compounds), the wave function of $f$-electrons
forms a narrow band of itinerant electrons and the high density of states at
the Fermi level, which is favorable for conventional superconductivity with
phonon interactions. On the other hand, a large distance with
$d>3.5\,{\rm\AA}$ creates a magnetic order, which is unfavorable for
conventional superconductivity.

U6X (X=Mn, Fe, Co, Ni) is an interesting system that shows superconductivity
with a relatively high $T_{\rm c}$ compared with other uranium-based
superconductors. [2, 3] The values of $T_{\rm c}$ of these materials are
$2.31\,{\rm K}$ (U6Mn), $3.78\,{\rm K}$ (U6Fe), $2.33\,{\rm K}$ (U6Co), and
$0.33\,{\rm K}$ (U6Ni). Indeed, U6Co has the second highest $T_{\rm c}$ among
all the known uranium-based superconductors. By alloying the compounds from Mn
to Ni, which tunes the number of valence electrons of the transition-element,
$T_{\rm c}$ changes smoothly and takes a maximum at U6Fe. This tendency had
been discussed from the viewpoint of the correlation between magnetism and
superconductivity because of the analogy of Slater–Pauling curves. [3] Note
that U6X was suggested to be close to the boundary between superconducting and
ferromagnetic ground states, [4] which is now extensively studied for
ferromagnetic superconductors in heavy fermion systems [5]. The smooth
variation of $T_{\rm c}$ also indicates that the superconductivity in this
system is robust against disorder and impurities, suggesting conventional
superconductivity. The upper critical field $H_{\rm c2}$ for $T\to 0$ is,
however, markedly large in U6Fe and U6Co. In U6Fe, for example, $H_{\rm
c2}(0)$ is $13.1$ and $10.4\,{\rm T}$ for $H\parallel[001]$ and $[110]$,
respectively, in the tetragonal structure. [6]. There are no reports on
$H_{\rm c2}(0)$ in U6Co down to low temperatures, but a large $H_{\rm c2}(0)$
is inferred from the large initial slope of $H_{\rm c2}$. [7] The penetration
depth [8] and NMR [9] experiments suggest BCS-type conventional
superconductivity in U6Co. All the previous experiments in U6Co were done with
polycrystalline samples. Although conventional superconductivity is most
likely realized in U6Co, it is important to clarify the superconducting
properties using a single crystal, especially focusing on the large $H_{\rm
c2}$. Therefore, we grew single crystals of U6Co and measured the resistivity,
magnetic susceptibility, and specific heat down to low temperatures.

U6Co forms the U6Mn-type tetragonal crystal structure with the space group
$I4/mcm$ (#140, $D_{4h}^{18}$), as shown in Fig. 1. The unit cell containing
four molecules is relatively large with the lattice parameters of
$a=10.36\,{\rm\AA}$ and $c=5.21\,{\rm\AA}$. The two sites of the U atom, U1
and U2 are located in the $c$-plane, forming square lattices for each U site.
These U-planes are stacked along the $c$-axis, and the square lattice in the
adjacent U-plane is slightly rotated in the $c$-plane. The Co atom is located
at $z=0.25$ between the U-planes. The distance between U atoms is quite small.
The first nearest neighbor is $d1=2.68\,{\rm\AA}$ between U1 sites, and the
second nearest neighbor is $d2=2.72\,{\rm\AA}$ between U2 sites. These values
are far below the Hill limit, implying the direct overlap of 5$f$-wave
functions and the itinerant nature of 5$f$-electrons.

Figure 1: (Color online) Tetragonal crystal structure of U6Co. The U atom has
two sites as denoted by U1 and U2. The distances for the first and second
nearest neighbors of the U atom are denoted by d1 and d2, respectively.

Single crystals of U6Co were grown by the self-flux method. Figure 2(a) shows
the binary phase diagram of U-Co [10]. The conventional Czochralski pulling
method with stoichiometric amounts is not applicable. Thus, we attempted to
grow U6Co single crystals by the self-flux method. The starting materials of U
and Co with off-stoichiometric amounts, U:Co = 76:24 (atomic %), shown in Fig.
2(a), were put in an alumina crucible, which was sealed in a quartz ampoule.
The quartz ampoule was then heated to $830\,^{\circ}{\rm C}$ in an electrical
furnace. The temperature was maintained for $300\,{\rm h}$ and then decreased
to $740\,^{\circ}{\rm C}$ with a slow cooling rate of $0.4\,^{\circ}{\rm
C/h}$. The excess Co was removed by spinning off in the centrifuge at this
temperature. Many single crystals of U6Co with a bar shape elongated along the
$c$-axis were obtained, as shown in Fig. 2(b). The single crystals were
oriented using the Laue photograph, and were cut with a spark cutter.

Figure 2: (Color online) (a) U-Co alloy phase diagram. The data are taken from
Ref. Mas90. The arrow indicates the starting composition for the self-flux
method. (b) Photograph of a U6Co single crystal grown by the self-flux method.

The magnetic susceptibility was measured using a commercial SQUID magnetometer
at temperatures down to $2\,{\rm K}$. The specific heat was measured at
temperatures down to $0.36\,{\rm K}$ and at fields up to $9\,{\rm T}$ using a
relaxation technique in a 3He cryostat. The resistivity was measured by the
four-probe AC method at temperatures down to $0.03\,{\rm K}$ and at fields up
$14\,{\rm T}$ in a 3He cryostat and in a dilution refrigerator.

Figure 3 shows the temperature dependence of magnetic susceptibility for
$H\parallel[001]$ and $[110]$. The susceptibility shows quite small anisotropy
and almost temperature-independent behavior down to $2.3\,{\rm K}$, indicating
Pauli paramagnetism. Below $2.3\,{\rm K}$, the susceptibility starts to
decrease, revealing a diamagnetic signal due to the superconducting
transition, as shown in the inset of Fig. 3.

Figure 3: (Color online) Temperature dependence of the magnetic susceptibility
at $H=1\,{\rm T}$ for $H\parallel[110]$ and $[001]$ in U6Co. The inset shows
the temperature dependence of magnetization at low temperatures measured below
$0.001\,{\rm T}$.

Figure 4(a) shows the temperature dependence of resistivity for the electrical
current along the $[001]$ direction. The resistivity decreases with decreasing
temperature with a convex curvature and shows a superconducting transition.
The resistivity starts to drop around $2.8\,{\rm K}$ and becomes zero at
$2.33\,{\rm K}$. The midpoint of the resistivity drop, which we define as the
superconducting transition temperature hereafter, is $T_{\rm c}=2.36\,{\rm
K}$. The resistivity follows $T^{2}$ dependence below $7\,{\rm K}$, that is,
$\rho=\rho_{0}+AT^{2}$, with the coefficient
$A=0.012\,\mu\Omega\\!\cdot\\!{\rm cm\,K^{-2}}$ and the residual resistivity
$\rho_{0}=54\,\mu\Omega\\!\cdot\\!{\rm cm}$. Despite the large residual
resistivity, the superconducting transition is rather sharp, indicating that
the superconductivity of U6Co is not very sensitive to the sample quality.
This also implies conventional superconductivity in U6Co.

Figure 4: (Color online) (a) Temperature dependence of the resistivity for
$J\parallel[001]$ in U6Co. The inset shows the low-temperature part. (b) Field
dependence of the longitudinal magnetoresistance for $J\parallel[001]$ and
$H\parallel[001]$ at different constant temperatures from $2.3$ to $0.4\,{\rm
K}$ with $0.1\,{\rm K}$ steps.

In order to determine the superconducting phase diagram, we measured the
magnetoresistance for $H\parallel[001]$ and $[110]$. Figure 4(b) shows the
magnetoresistance at different temperatures for $H\parallel[001]$. The
superconducting transition is sharp even at high fields, thus, we could
determine the temperature dependence of the upper critical field $H_{\rm c2}$,
as shown in Fig. 5. Here, we define $H_{\rm c2}$ at the midpoint of the
resistivity drop. The values of $H_{\rm c2}$ are remarkably large for both
$H\parallel[001]$ and $[110]$. At the lowest temperature, $0.03\,{\rm K}$,
$H_{\rm c2}$ is $7.85\,{\rm T}$ for $H\parallel[001]$ and $6.56\,{\rm T}$ for
$H\parallel[110]$. The initial slope of $H_{\rm c2}$ is $-dH_{\rm
c2}/dT=4.3\,{\rm T/K}$ for $H\parallel[001]$ and $3.4\,{\rm T/K}$ for
$H\parallel[110]$. In general, the Pauli limiting field is given by $H_{\rm
P}=\sqrt{2}\Delta/(g\mu_{\rm B})=1.86T_{\rm c}$, assuming the $g$-factor $g=2$
and the superconducting gap $\Delta$, with $2\Delta/(k_{\rm B}T_{\rm c})=3.53$
based on the weak coupling BCS model. In this case, $H_{\rm P}$ in U6Co is
estimated to be $4.3\,{\rm T}$, which is much smaller than the measured
$H_{\rm c2}$ for both directions. Thus, strong-coupling superconductivity or a
small $g$-factor is required if we assume the spin-singlet state. As shown
later, the specific heat jump at $T_{\rm c}$, namely, $\Delta C/\gamma T_{\rm
c}$ is only $10{\%}$ larger than that for the weak coupling BCS model.
Therefore, a small $g$-factor is expected. In fact, the weakly temperature
dependent susceptibility for $[001]$ and $[110]$ indicates very small spin-
susceptibility, which is consistent with a small $g$-value. For $H_{\rm
c2}\leq H_{\rm P}$, one can expect $g<1.2$.

Figure 5: (Color online) Temperature dependence of the upper critical field
$H_{\rm c2}$ for $H\parallel[001]$ and $[110]$ determined by the field and
temperature dependences of resistivity in U6Co. The solid lines are guides to
the eyes.

We show in Fig. 6 the angular dependence of $H_{\rm c2}$ determined by the
magnetoresistance at $30\,{\rm mK}$. By rotating the field angle, $H_{\rm c2}$
smoothly increases from $6.56\,{\rm T}$ for $[110]$ to $7.85\,{\rm T}$ for
$[001]$. The angular dependence of $H_{\rm c2}$ can be fitted by the so-called
effective mass model, where an ellipsoidal Fermi surface with anisotropic
effective mass is assumed. The solid line in the main panel of Fig. 6 is the
result of fitting by the equation $H_{\rm c2}(\theta)=H_{\rm
c2}(90^{\circ})/\sqrt{\sin^{2}\theta+(m_{c}/m_{a})\cos^{2}\theta}$. We obtain
$m_{c}/m_{a}=0.699$ and $k_{c}/k_{a}=0.84$. The effective Fermi surface is
almost spherical with slight suppression along the $[001]$ direction.

Figure 6: (Color online) Angular dependence of the upper critical field
$H_{\rm c2}$ at $30\,{\rm mK}$ from $H\parallel[001]$ to $[110]$. The solid
line shows the results of fitting by the effective mass model. The inset shows
the field dependence of magnetoresistance for different field directions.
$H_{\rm c2}$ was defined as the midpoint of the resistivity drop.

Figure 7(a) shows the temperature dependence of the specific heat. The inset
of Fig. 7(a) shows the total specific heat in the form of $C/T$ vs $T$. The
specific heat jump indicates bulk superconductivity. In the normal state below
$4\,{\rm K}$, the data is well fitted by $C/T=\gamma+\beta T^{2}$. By
extrapolating the fitting to $0\,{\rm K}$, we subtract the phonon contribution
and extract the electronic specific heat $C_{\rm el}$ from the total specific
heat, as shown in the main panel of Fig. 7, which also satisfies the entropy
balance. The Sommerfeld coefficient is $\gamma\simeq 20\,{\rm
mJ\,K^{-1}U\mbox{-}mol^{-1}}$, indicating moderately enhanced heavy electrons.
The value of $A/\gamma^{2}$ is approximately consistent with the so-called
Kadowaki–Woods ratio. $T_{\rm c}$ determined by the entropy balance is
$2.17\,{\rm K}$, which is slightly lower than the value of $2.33\,{\rm K}$
from the resistivity measurements, but is not very far. The jump of the
specific heat is $\Delta C/\gamma T_{\rm c}=1.58$, which is slightly larger
but close to the weak coupling BCS value of $1.43$.

The low-temperature data from $0.35$ to $0.71\,{\rm K}$ were well fitted by
the BCS asymptotic formula [11], $C_{\rm
el}/T=\gamma_{0}+3.15(\Delta/1.76k_{\rm B}T)^{5/2}\gamma\exp(-\Delta/k_{\rm
B}T)$, as shown in the main panel in Fig. 7(a). We obtain the gap energy,
$\Delta=3.97\,{\rm K}$, and $2\Delta/k_{\rm B}T_{\rm c}=3.7$, which is also
close to the weak coupling BCS value.

The thermodynamic critical field $H_{\rm c}$ can be calculated from the
difference in free energy between the superconducting and normal states, that
is, $\Delta F(T)=\mu_{0}H_{\rm c}^{2}(T)/2=\int_{T_{\rm c}}^{T}[S_{\rm
s}(T)-S_{\rm n}(T)]dT$, where $S_{\rm s}$ and $S_{\rm n}$ are the entropies in
the superconducting and normal states, respectively, which are calculated as
$S=\int C_{\rm el}/TdT$. We obtain $H_{\rm c}(0)$ as $0.065\,{\rm T}$, which
roughly agrees with the value reported previously, $0.072\,{\rm T}$ [12].

In order to investigate the superconducting gap structure, we measured the
field dependence of the specific heat at a low temperature, $0.41\,{\rm K}$,
as shown in Fig. 7(b). The temperature is approximately equal to $0.19T_{\rm
c}$. $C/T$ increases almost linearly with a slight convex curvature, implying
full-gap superconductivity at the first approximation, although measurements
at lower temperatures and in different field directions are required to obtain
a conclusion.

Figure 7: (Color online) (a) Temperature dependence of the electronic specific
heat at zero field in the form of $C_{\rm el}/T$ vs $T$ in U6Co. The red solid
line below $0.71\,{\rm K}$ shows the results of fitting by the BCS full gap
model. The inset shows the total specific heat in the form of $C/T$ vs $T$.
The red solid line in the inset shows the specific heat in the normal state
extrapolated to $0\,{\rm K}$ as a function of $C/T=\gamma+\beta T^{2}$. (b)
Field dependence of the specific heat in the form of $C/T$ vs $H$ for
$H\parallel[110]$ at $0.41\,{\rm K}$, which is approximately equal to
$0.19\,T_{\rm c}$.

On the basis of the BCS theory assuming a spherical Fermi surface, we
calculate some parameters of the superconductivity. [13] From the starting
parameters of $\gamma=20\,{\rm mJK^{-2}U\mbox{-}mol^{-1}}$,
$\rho_{0}=54\,\mu\Omega\\!\cdot\\!{\rm cm}$, $T_{\rm c}=2.3\,{\rm K}$, and the
lattice parameters, we obtain the initial slope of $H_{\rm c2}$ as $-dH_{\rm
c2}/dT=3.5\,{\rm T/K}$ in the case of the dirty limit. This is close to the
values obtained in the experiments, namely, $3.36$ and $4.30\,{\rm T/K}$ for
$H\parallel[110]$ and $[001]$, respectively. Taking the average of the
experimental initial slope, we obtain the effective Fermi surface as
$1.6\times 10^{21}\,{\rm m^{-2}}$, ($4.3\times 10^{20}\,{\rm m^{-2}}$) for the
dirty (clean) limit. The mean free path is $l=67\,{\rm\AA}$, ($18\,{\rm\AA}$).
The BCS coherence length for $T\to 0$ is $\xi_{0}=0.18\hbar v_{\rm F}/k_{\rm
B}T_{\rm c}=84\,{\rm\AA}$, ($38\,{\rm\AA}$). The mean free path is smaller
than the coherence length, revealing that the present sample is in the dirty
limit condition. This suggests that U6Co is a conventional BCS superconductor.
The coherence lengths estimated from $H_{\rm c2}=\phi_{0}/2\pi\xi^{2}$ are
$\xi=65$ and $71\,{\rm\AA}$ for $H\parallel[001]$ and $[110]$, respectively.
These values are close to the BCS coherence length $\xi_{0}$ estimated in the
dirty limit.

Note that similar $H_{\rm c2}$ curves have also been reported for U6Fe. [6]
Since U6Fe has a slightly larger $T_{\rm c}$ ($\sim 4\,{\rm K}$) than U6Co,
$H_{\rm c2}$ is also large, $13.1\,{\rm T}$ for $H\parallel[001]$ and
$10.4\,{\rm T}$ for $[110]$. These values can be scaled by a factor of $1.7$
from those of U6Co. The angular dependence of $H_{\rm c2}$ with the effective
mass anisotropy $m_{c}/m_{a}=0.64$ in U6Fe is also similar to that of U6Co.

In summary, we succeeded in growing single crystals of U6Co by the self-flux
method. The magnetic susceptibility is almost temperature-independent and is
quite isotropic between $H\parallel[001]$ and $[110]$, indicating small spin-
susceptibility compared with orbital susceptibility, which is consistent with
the small $g$-factor estimated from the specific heat and the Pauli limit. The
upper critical field $H_{\rm c2}$ and its initial slope are relatively large,
indicating a moderately enhanced heavy fermion system. The anisotropy of
$H_{\rm c2}$ can be well explained by the effective mass model, which reveals
the almost spherical Fermi surface suppressed slightly along the $[001]$
direction. The superconducting parameters obtained using the BCS model in the
dirty limit condition agree reasonably well with the experiments, indicating
an $s$-wave superconductor with most likely a full gap of U6Co.

## Acknowledgements

We thank J. P. Brison, Y. Ōnuki, K. Ishida, and Y. Shimizu for fruitful
discussion. This work was supported by ERC starting grant (NewHeavyFermion),
KAKENHI (25247055, 15H05884, 15H05882, 15K21732, 16H04006), ICC-IMR, and
REIMEI.

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