$E_{\mathrm{sym}}(\rho_{0})\equiv S$ and $L(\rho_{0})\equiv{L}$ to determine
the parameters $\beta_{\mathrm{IS}}$ and $\lambda_{\mathrm{IS}}$.
Consequently, the isospin-related parameters are found to be

$\displaystyle\beta_{\mathrm{IS}}=$ $\displaystyle
6[5E_{\mathrm{sym}}(\rho_{0})-{L}-3E^{\mathrm{kin}}_{\mathrm{sym}}(\rho_{0})-5X+Y],$
(5.25) $\displaystyle\lambda_{\mathrm{IS}}=$ $\displaystyle
3[2E^{\mathrm{kin}}_{\mathrm{sym}}(\rho_{0})+4X-Y-4E_{\mathrm{sym}}(\rho_{0})+{L}],$
(5.26)

where

$\displaystyle E_{\mathrm{sym}}^{\mathrm{kin}}(\rho_{0})=$
$\displaystyle{k_{\mathrm{F}}^{2}(\rho_{0})}/{6M},~{}~{}$ (5.27)
$\displaystyle X=$
$\displaystyle-\frac{1}{2}\mu\theta\left(\frac{k_{\mathrm{F}}(\rho_{0})}{\Lambda}\right)^{3}\left/\left[1+\theta\left(\frac{k_{\mathrm{F}}(\rho_{0})}{\Lambda}\right)^{3}\right]^{2}\right.,~{}~{}$
(5.28) $\displaystyle Y=$
$\displaystyle-3\mu\theta\left(\frac{k_{\mathrm{F}}(\rho_{0})}{\Lambda}\right)^{3}\left/\left[1+\theta\left(\frac{k_{\mathrm{F}}(\rho_{0})}{\Lambda}\right)^{3}\right]^{3}\right..$
(5.29)

In Fig. 6 the symmetry energy functions with different $L$ values and a fixed
$E_{\mathrm{sym}}(\rho_{0})=32\,\mathrm{MeV}$ are shown. For a comparison, the
constraint on the symmetry energy from the isobaric analog states (IAS) [130]
is also shown. It is obvious that at large densities the symmetry energy may
increase or decrease depending on the value of $L$. For example, with
$L=50\,\mathrm{MeV}$, the symmetry energy starts to decrease at about 1.6
$\rho_{0}$. It is necessary to point out that the density dependence of the
symmetry energy given here is so simple that when generalized to large
densities one should be more cautious. All the other higher order coefficients
could be obtained from the expressions given above, e.g., the curvature of the
symmetry energy,

$\displaystyle K_{\mathrm{sym}}=$ $\displaystyle
6E^{\mathrm{kin}}_{\mathrm{sym}}(\rho_{0})+6{L}-20E_{\mathrm{sym}}(\rho_{0})$
$\displaystyle-\mu\left[10\left[\theta\left(\frac{k_{\mathrm{F}}(\rho_{0})}{\Lambda}\right)^{3}\right]^{3}-16\left[\theta\left(\frac{k_{\mathrm{F}}(\rho_{0})}{\Lambda}\right)^{3}\right]^{2}+\theta\left(\frac{k_{\mathrm{F}}(\rho_{0})}{\Lambda}\right)^{3}\right]\left/\left[1+\theta\left(\frac{k_{\mathrm{F}}(\rho_{0})}{\Lambda}\right)^{3}\right]^{4}\right..$
(5.30)

If one takes $\rho_{0}\approx 0.16\pm
0.02\,\mathrm{fm}^{-3},E_{\mathrm{sym}}(\rho_{0})\approx 30\pm
4\,\mathrm{MeV},L\approx 60\pm 30\,\mathrm{MeV}$, then
$K_{\mathrm{sym}}\approx-166\pm 197\,\mathrm{MeV}$, which is also reasonable
with respect to the results currently available in the literature [131, 132,
133, 134, 135, 136, 137, 138, 139, 140].

Fig. 7: Fourth-order symmetry energy with different $L$ and fixed
$E_{\mathrm{sym}}(\rho_{0})=32\,\mathrm{MeV}$ (left) and the ratio
$E_{\mathrm{sym,4}}(\rho)/E_{\mathrm{sym}}(\rho)$ (right).

In addition, the fourth-order symmetry energy could also be obtained via the
HVH theorem as,

$\displaystyle{E_{\mathrm{sym},4}(\rho)=\frac{k_{\mathrm{F}}^{2}}{162M}+\frac{1}{648}\beta_{\mathrm{IS}}\left(\frac{\rho}{\rho_{0}}\right)^{4/3}+\frac{1}{81}\lambda_{\mathrm{IS}}\left(\frac{\rho}{\rho_{0}}\right)^{5/3}-\frac{1}{4}\mu\left(\frac{\rho}{\rho_{0}}\right)\left[\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}\right]^{3}\left/\left[1+\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}\right]^{4}\right.}.$
(5.31)

In this toy model, the fourth-order symmetry energy is totally determined by
the symmetry energy and the EOS of SNM via the parameters
$\beta_{\mathrm{IS}},~{}\lambda_{\mathrm{IS}}$ and the parameter $\mu$. See
Fig. 7 for the density dependence of the $E_{\mathrm{sym},4}(\rho)$ (left) and
that of the ratio $E_{\mathrm{sym,4}}(\rho)/E_{\mathrm{sym}}(\rho)$ (right),
adopting different values of $L$ while fixing the $E_{\mathrm{sym}}(\rho_{0})$
at 32 MeV. We have found that the fourth-order symmetry energy
$E_{\mathrm{sym,4}}(\rho_{0})$ at saturation density is generally $\lesssim
2\,\mathrm{MeV}$ (indicated by the red “+”). This value is also consistent
with available empirical findings [79, 141, 142, 143].

Finally, the EOS of pure neutron matter (PNM) obtained in the parabolic
approximation and the one including the fourth-order symmetry energy can then
be obtained, i.e., $E_{\mathrm{n}}(\rho)\approx
E_{0}(\rho)+E_{\mathrm{sym}}(\rho)$ and $E_{\mathrm{n}}(\rho)\approx
E_{0}(\rho)+E_{\mathrm{sym}}(\rho)+E_{\mathrm{sym,4}}(\rho)$, respectively.
The expression for the latter is given by

$\displaystyle E_{\mathrm{n}}(\rho)\approx$
$\displaystyle\frac{383}{810}\frac{k_{\mathrm{F}}^{2}}{M}+\frac{\alpha}{2}\left(\frac{\rho}{\rho_{0}}\right)+\left(\frac{3\beta}{7}+\frac{109\beta_{\mathrm{IS}}}{648}\right)\left(\frac{\rho}{\rho_{0}}\right)^{4/3}+\left(\frac{3\lambda}{8}+\frac{28\lambda_{\mathrm{IS}}}{81}\right)\left(\frac{\rho}{\rho_{0}}\right)^{5/3}$
$\displaystyle+\frac{\phi}{3}\left(\frac{\rho}{\rho_{0}}\right)^{2}+\mu\left(\frac{\rho}{\rho_{0}}\right)\left[\frac{x-\ln(1+x)}{x^{2}}-\frac{1}{2}\frac{x}{(1+x)^{2}}-\frac{1}{4}\frac{x^{3}}{(1+x)^{4}}\right],$
(5.32)

with $x=\theta(k_{\mathrm{F}}/\Lambda)^{3}$. For small values of $x$, the
above $E_{\mathrm{n}}(\rho)$ becomes,

$\displaystyle E_{\mathrm{n}}(\rho)\approx$
$\displaystyle\frac{383}{810}\frac{k_{\mathrm{F}}^{2}}{M}+\frac{\alpha}{2}\left(\frac{\rho}{\rho_{0}}\right)+\left(\frac{3\beta}{7}+\frac{109\beta_{\mathrm{IS}}}{648}\right)\left(\frac{\rho}{\rho_{0}}\right)^{4/3}+\left(\frac{3\lambda}{8}+\frac{28\lambda_{\mathrm{IS}}}{81}\right)\left(\frac{\rho}{\rho_{0}}\right)^{5/3}$
$\displaystyle+\frac{\phi}{3}\left(\frac{\rho}{\rho_{0}}\right)^{2}+\frac{1}{2}\mu\left(\frac{\rho}{\rho_{0}}\right)\left[1-\frac{5}{3}\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}+\frac{5}{2}\left[\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}\right]^{2}-\frac{39}{10}\left[\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}\right]^{3}+\frac{19}{3}\left[\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}\right]^{4}\right].$
(5.33)

Numerically, we found that the fourth-order symmetry energy induces a
contribution $\lesssim 0.5\,\mathrm{MeV}$ to the PNM EOS at densities
$\rho\approx 2\rho_{0}$, see Fig. 8. For a comparison, the constraint on the
PNM EOS from the chiral perturbation theory ($\chi\mathrm{PT}$) [144] is also
shown. It is seen that the constructed toy model satisfies the constraint from
the chiral perturbation theory.

Fig. 8: Toy model EOS for PNM obtained in the parabolic approximation and the
one including the fourth-order symmetry energy in comparison with the chiral
perturbation theory prediction.

So far we have constructed a phenomenological single-nucleon potential
$U_{J}(\rho,\delta,\mathbf{k})$ in the form of (5.1). The resulting EOS of ANM
can safely fulfills the empirical constraints (a)-(f) mentioned at the
beginning of this subsection. In particular, the EOS of SNM is consistent with
the collective flow constraint on the pressure [128] and the optical potential
constraint on the momentum dependence of the isoscalar potential in SNM [94].
In addition, the EOS of PNM from our phenomenological potential is consistent
with the microscopic calculations using chiral effective interactions [144].
Moreover, the symmetry energy around $\rho_{0}$ is also consistent with
existing constraints from analyzing various terrestrial experiments (e.g, the
analysis of isobaric analog states [130]) and neutron star observations.
Besides, the potential (5.1) thus designed is also consistent with the HVH
theorem, i.e., the isovector symmetry potential and the second-order symmetry
potential should be related as in Eq. (2.52). Of course, here we have no
desire to construct a celebrated effective model for the EOS of ANM. Indeed,
advanced nuclear many-body theories either phenomenological or microscopic can
predict numerically more accurately the EOS of neutron-rich matter at certain
densities and/or isospin asymmetries. However, analytical expressions for the
$U_{J}(\rho,\delta,\mathbf{k})$ and all the characteristics of the
corresponding EOS with only 9 parameters that can describe satisfactorily
essentially all known constraints on the EOS are invaluable for many purposes.
As we shall show next, serving as a reliable basis in the $\epsilon$-expansion
approach these expressions facilitate the study of nuclear EOSs in spaces with
dimensions different from 3.

### 5.2. Perturbing the EOS of spatial dimension $d_{\mathrm{f}}=3$

We now study the EOS of ANM in spaces with dimensions near the reference
dimension $d_{\mathrm{f}}=3$ based on the $\epsilon$-expansion formalism with
the 3D toy model EOS developed in the last subsection. The EOS of SNM in
dimension $d+\epsilon$ with $\epsilon$ being the perturbative dimension is
given by the formula (4.15), where the term originated from the momentum-
dependence of the single-nucleon potential $U_{0}$ (see (5.2)) could be
written out as,

$\frac{\sigma}{\rho}\int_{0}^{\rho}\left(\left.\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right|_{|\mathbf{k}|=\overline{k}^{f}_{\mathrm{F}}}\cdot\overline{k}^{f}_{\mathrm{F}}\right)\mathrm{d}f=-\frac{\sigma}{\rho}\frac{3\mu\nu}{\rho_{0}}\int_{0}^{\rho}\mathrm{d}f\frac{f^{2}}{(1+\nu
f)^{2}},$ (5.34)

where $\nu=3\pi^{2}\theta/2\Lambda^{3}$. The integration in the above equation
can be carried out analytically, leading to

$\int_{0}^{\rho}\mathrm{d}f\frac{f^{2}}{(1+\nu
f)^{2}}=\frac{1}{\nu^{3}}\left[\frac{x(x+2)}{x+1}-\ln(x+1)\right],$ (5.35)

where $x=\theta({k_{\mathrm{F}}}/{\Lambda})^{3}$ as introduced earlier is a
function of density through $k_{\mathrm{F}}$. Consequently,

$\displaystyle\frac{\sigma}{\rho}\int_{0}^{\rho}\left(\left.\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right|_{|\mathbf{k}|=\overline{k}^{f}_{\mathrm{F}}}\cdot\overline{k}^{f}_{\mathrm{F}}\right)\mathrm{d}f=$
$\displaystyle-3\sigma\mu\left(\frac{\rho}{\rho_{0}}\right)\frac{1}{x^{2}}\left[\frac{x^{2}+2x}{x+1}-2\ln(x+1)\right]$
(5.36) $\displaystyle\approx$
$\displaystyle-3\sigma\mu\left(\frac{\rho}{\rho_{0}}\right)\left(\frac{1}{3}x-\frac{1}{2}x^{2}+\frac{3}{5}x^{3}\right),~{}~{}\mbox{for
small $x$},$ (5.37)

where the second line (approximation) holds for small values of $x$
(corresponding to low densities). In fact the limit $x\approx 1$ is equivalent
to a density about $\rho\approx 2.85\,\mathrm{fm}^{-3}\approx 18\rho_{0}$
using $\theta=1.3$ and $\Lambda=750\,\mathrm{MeV}$ (see Tab. 1). Moreover, the
saturation density $\rho_{0}$ corresponds to $x$ about $x\approx 0.056$. Thus,
the above small-$x$ approximation is very reasonable/effective for our
discussions of situations with densities up to several times of $\rho_{0}$.
Another interesting feature indicated by (5.37) is that the correction is
positive since $\mu$ is negative (see Tab. 2). This is consistent with the
general analysis given in the paragraph after formula (4.21), i.e., the
momentum-dependence of the nuclear potential tends to reduce (enhance) the EOS
of SNM in lower (higher) dimensions compared with the 3D case. In fact, it is
a direct consequence of the well-constrained momentum dependence of the
$U_{0}$ (e.g., see the right panel of Fig. 2).

Combining (5.37) with other relevant terms gives the EOS of SNM in
perturbative dimension $\epsilon$ as,

$\displaystyle E_{0}(\rho)=$
$\displaystyle\frac{3}{10M}\left(\frac{3\pi^{2}}{2}\right)^{2/3}\rho^{2/3}+\frac{\alpha}{2}\left(\frac{\rho}{\rho_{0}}\right)+\frac{3\beta}{7}\left(\frac{\rho}{\rho_{0}}\right)^{4/3}+\frac{3\lambda}{8}\left(\frac{\rho}{\rho_{0}}\right)^{5/3}+\frac{\phi}{3}\left(\frac{\rho}{\rho_{0}}\right)^{2}$
$\displaystyle+\mu\left(\frac{\rho}{\rho_{0}}\right)\left[\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}-\ln\left(1+\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}\right)\right]\left/\theta^{2}\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{6}\right.$
$\displaystyle+\epsilon\left[\frac{3\overline{k}_{\mathrm{F}}^{2}}{5M}\left(\sigma+\frac{1}{15}\right)-3\sigma\mu\left(\frac{\rho}{\rho_{0}}\right)\frac{1}{x^{2}}\left[\frac{x^{2}+2x}{x+1}-2\ln(x+1)\right]\right].$
(5.38)

Similarly, the pressure could be worked out as,

$\displaystyle P_{0}(\rho)=$
$\displaystyle\frac{1}{5M}\left(\frac{3\pi^{2}}{2}\right)^{2/3}\rho^{5/3}+\rho\left[\frac{\alpha}{2}\left(\frac{\rho}{\rho_{0}}\right)+\frac{4\beta}{7}\left(\frac{\rho}{\rho_{0}}\right)^{4/3}+\frac{5\lambda}{8}\left(\frac{\rho}{\rho_{0}}\right)^{5/3}+\frac{2\phi}{3}\left(\frac{\rho}{\rho_{0}}\right)^{2}\right]$
$\displaystyle+\mu\rho\left(\frac{\rho}{\rho_{0}}\right)\left[1\left/\left[1+\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}\right]\right.-1\left/\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}\right.+\left[\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}\right]^{-2}\ln\left[1+\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}\right]\right]$
$\displaystyle+\epsilon\rho\left[\frac{\overline{k}_{\mathrm{F}}^{2}}{10M}\left(2\sigma-\frac{1}{5}\right)+3\sigma\mu\left(\frac{\rho}{\rho_{0}}\right)\left[\frac{1}{x^{2}}\left[\frac{x^{2}+2x}{x+1}-2\ln(x+1)\right]-\frac{x}{(x+1)^{2}}\right]\right],$
(5.39)

where the relevant momentum-dependent term in the pressure is,

$\displaystyle-\frac{\sigma}{\rho}\int_{0}^{\rho}\left(\left.\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right|_{|\mathbf{k}|=\overline{k}^{f}_{\mathrm{F}}}\cdot\overline{k}^{f}_{\mathrm{F}}\right)\mathrm{d}f+\left.\sigma\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right|_{|\mathbf{k}|=\overline{k}_{\mathrm{F}}}\cdot\overline{k}_{\mathrm{F}}$
$\displaystyle=$ $\displaystyle
3\sigma\mu\left(\frac{\rho}{\rho_{0}}\right)\left[\frac{1}{x^{2}}\left[\frac{x^{2}+2x}{x+1}-2\ln(x+1)\right]-\frac{x}{(x+1)^{2}}\right]$
(5.40) $\displaystyle\approx$ $\displaystyle
3\sigma\mu\left(\frac{\rho}{\rho_{0}}\right)\left(-\frac{2}{3}x+\frac{3}{2}x^{2}-\frac{12}{5}x^{3}\right),~{}~{}\mbox{for
small $x$}.$ (5.41)

Fig. 9: EOS of SNM and the corresponding pressure adopting different values of
$\epsilon$.

For small $x$ and $\epsilon=-1$ (in 2D), one has

$\displaystyle E_{0}^{(\mathrm{2D})}(\rho)\approx$
$\displaystyle\frac{3\overline{k}_{\mathrm{F}}^{2}}{10M}\left(\frac{13}{15}-2\sigma\right)+\frac{1}{2}\mu\left(\frac{\rho}{\rho_{0}}\right)\left(1+2x\sigma\right)$
$\displaystyle+\frac{\alpha}{2}\left(\frac{\rho}{\rho_{0}}\right)+\frac{3\beta}{7}\left(\frac{\rho}{\rho_{0}}\right)^{4/3}+\frac{3\lambda}{8}\left(\frac{\rho}{\rho_{0}}\right)^{5/3}+\frac{\phi}{3}\left(\frac{\rho}{\rho_{0}}\right)^{2},$
(5.42)

from which it is clear that the correction to the EOS of SNM from the
momentum-dependent interaction term (characterized by $2x\sigma$) is generally
smaller than that from the kinetic part (characterized by $2\sigma+2/15$), due
to the smallness of $x\approx 0.056$ near the saturation density. If one
introduces an effective coupling “constant”

$\mu_{\mathrm{eff}}(\rho)=\mu(1+2x\sigma)$ (5.43)

and an effective mass related to the $\sigma$ as,

$M_{\sigma}^{\mathrm{eff}}=M\left(\frac{13}{15}-2\sigma\right)^{-1},$ (5.44)

then the EOS of SNM in 2D has the same form as that in 3D, i.e.,

$E_{0}^{(\mathrm{2D})}(\rho)=\frac{3\overline{k}_{\mathrm{F}}^{2}}{10M_{\sigma}^{\mathrm{eff}}}+\frac{1}{2}\mu_{\mathrm{eff}}(\rho)\left(\frac{\rho}{\rho_{0}}\right)+\frac{\alpha}{2}\left(\frac{\rho}{\rho_{0}}\right)+\frac{3\beta}{7}\left(\frac{\rho}{\rho_{0}}\right)^{4/3}+\frac{3\lambda}{8}\left(\frac{\rho}{\rho_{0}}\right)^{5/3}+\frac{\phi}{3}\left(\frac{\rho}{\rho_{0}}\right)^{2},$
(5.45)

see Eq. (5.3) and the few equations following it. The effective mass
$M_{\sigma}^{\mathrm{eff}}\approx 2.6M$ in 2D is much larger than the bare
mass in 3D, indicating the nucleon is now becoming too heavy to move easily
(i.e., it is more bounded). However, the coupling “constant”
$\mu_{\mathrm{eff}}$ is density dependent, i.e., the density dependence of the
SNM EOS in 2D (or generally in other dimensions) is modified due to the
$\mu_{\mathrm{eff}}$-term.

In Fig. 9 we show the nucleon specific energy $E_{0}(\rho)$ (left) and
pressure $P_{0}(\rho)$ (right) in SNM adopting 5 perturbative dimensions,
i.e., $\epsilon=-2,-1,0,1$ and $2$ (corresponding to $d$ from 1 to 5). It is
seen that as the dimension decreases, the EOS of SNM becomes much more bounded
and as $d$ increases, the $E_{0}(\rho)$ becomes less negative. In particular,
if the $d$ is larger than some critical dimension, the $E_{0}(\rho)$ becomes
totally positive, indicating the system is unbounded and there would be no
saturation density in these dimensions. Noticing that our results given here
are from perturbating the EOS in 3D, if one adopts the exact scheme, the
qualitative conclusion will not change, i.e., the nucleon specific energy in
SNM eventually increases as the $d$ increases and there would be no saturation
density above certain critical dimension. This phenomenon actually can be
traced back to a deep principle behind the high-dimensional geometry:
According to the relation between the Fermi momentum $k_{\mathrm{F}}$ and the
density $\rho$, i.e., (2.18), one can equivalently write the density in terms
of $k_{\mathrm{F}}$ as
$\rho=k_{\mathrm{F}}^{d}/[{2^{d-2}\pi^{d/2}\Gamma(d/2+1)}]$. The volume of a
high-dimensional sphere with radius $k_{\mathrm{F}}$ is $(2\pi)^{d}\rho/4$.
For a fixed Fermi momentum $k_{\mathrm{F}}$, the volume of the sphere
approaches zero as $d\to\infty$ [54]. Inversely, if one fixes the density
$\rho$, then the Fermi momentum $k_{\mathrm{F}}$ should approach infinity as
$d\to\infty$, indicating that the kinetic EOS of SNM namely
$dE_{\mathrm{F}}/(d+2)\sim E_{\mathrm{F}}=k_{\mathrm{F}}^{2}/2M$ (as
$d\to\infty$, see (2.18)) will always denominate over the potential part,
irrespective of the form of the latter.

Fig. 10: Saturation density $\rho_{\mathrm{sat},\epsilon}$ as a function of
$\epsilon$.

Another interesting feature of Fig. 9 is that perturbatively the saturation
density inferred from $E_{0}(\rho)$ (as the minimum of the curves in the left
panel) does not necessarily correspond to the vanishing point of $P_{0}(\rho)$
(the right panel). The reason is simple: As one does perturbative
calculations, the $d$-dependences of the $E_{0}(\rho)$ and $P_{0}(\rho)$ are
different. More specifically, we have
$E_{0}^{\mathrm{kin}}(\rho)=dk_{\mathrm{F}}^{2}/2(d+2)M$ while
$3P^{\mathrm{kin}}_{0}(\rho)/\rho=L^{\mathrm{kin}}_{0}(\rho)=k_{\mathrm{F}}^{2}/(d+2)M$
(see formula (2.49)). When the dimension $d$ is perturbed around
$d_{\mathrm{f}}$, the self-consistent relation
$P_{0}^{\mathrm{kin}}(\rho)=\rho^{2}\mathrm{d}E_{0}^{\mathrm{kin}}(\rho)/\mathrm{d}\rho$
may be lost. Similar phenomenon also occurs for the isospin-dependent part of
the EOS of ANM. If one takes the exact calculations in $d$D instead of the
perturbation on $\epsilon$, then the self-consistency will naturally be
maintained (e.g., it could be straightforwardly shown using the single-nucleon
potential
$U_{J}(\rho,\delta)=\sum_{i}[\alpha_{i}+\alpha_{i}^{\mathrm{IS}}\exp((\theta_{i}-1)\tau_{3}^{J}\delta)](\rho/\rho_{0})^{\vartheta_{i}}$
with the exact solution of the constraining equations for certain empirical
characteristics). On the other hand, the (perturbative) potential of the EOS
of SNM fulfills the self-consistent relation, i.e.,

$\rho^{2}\frac{\mathrm{d}}{\mathrm{d}\rho}\left[\frac{\sigma}{\rho}\int_{0}^{\rho}\left(\left.\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right|_{|\mathbf{k}|=\overline{k}^{f}_{\mathrm{F}}}\cdot\overline{k}^{f}_{\mathrm{F}}\right)\mathrm{d}f\right]=\rho\sigma\left.\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right|_{|\mathbf{k}|=\overline{k}_{\mathrm{F}}}\cdot\overline{k}_{\mathrm{F}}-\sigma\int_{0}^{\rho}\left(\left.\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right|_{|\mathbf{k}|=\overline{k}^{f}_{\mathrm{F}}}\cdot\overline{k}^{f}_{\mathrm{F}}\right)\mathrm{d}f,$
(5.46)

see the relevant terms in (4.15) and (4.17). In Fig. 10 we show the saturation
density $\rho_{\mathrm{sat},\epsilon}$ as a function of the perturbative
dimension $\epsilon$ through the two approaches, namely one from $P_{0}=0$ and
the other from $\dot{E}_{0}\equiv\mathrm{d}E_{0}/\mathrm{d}\rho=0$. As
mentioned previously, the saturation densities from these two approaches are
generally different due to the perturbative calculation, but the qualitative
feature is the same: as the dimension $d$ is downward perturbed, the
saturation density is effectively increased. In particular, the critical
dimension above which there is no saturation point for $E_{0}(\rho)$ is found
to be about 4.77 for the toy model.

We now switch to calculating the symmetry energy in $d$D perturbatively, see
(4.16). Since the symmetry potential in the toy model has no momentum
dependence, we immediately obtain the following expression

$\displaystyle E_{\mathrm{sym}}(\rho)=$
$\displaystyle\frac{\overline{k}_{\mathrm{F}}^{2}}{6M}+\left.\frac{\overline{k}_{\mathrm{F}}}{6}\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right|_{|\mathbf{k}|=\overline{k}_{\mathrm{F}}}+\frac{1}{2}U_{\mathrm{sym}}(\rho,\overline{k}_{\mathrm{F}})$
$\displaystyle+\epsilon\left[\frac{\overline{k}_{\mathrm{F}}^{2}}{6M}\left(2\sigma-\frac{1}{3}\right)+\frac{\overline{k}_{\mathrm{F}}}{6}\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\left(\left.{\displaystyle\sigma\overline{k}_{\mathrm{F}}\frac{\partial^{2}U_{0}}{\partial|\mathbf{k}|^{2}}}\right/{\displaystyle\frac{\partial
U_{0}}{\partial|\mathbf{k}|}}+\sigma-\frac{1}{3}\right)\right]_{|\mathbf{k}|=\overline{k}_{\mathrm{F}}}.$
(5.47)

After evaluating the derivatives, one obtains

$\displaystyle E_{\mathrm{sym}}(\rho)=$
$\displaystyle\frac{\overline{k}_{\mathrm{F}}^{2}}{6M}+\frac{1}{6}\beta_{\mathrm{IS}}\left(\frac{\rho}{\rho_{0}}\right)^{4/3}+\frac{1}{3}\lambda_{\mathrm{IS}}\left(\frac{\rho}{\rho_{0}}\right)^{5/3}-\frac{1}{2}\mu\left(\frac{\rho}{\rho_{0}}\right)\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}\left/\left[1+\theta\left(\frac{k_{\mathrm{F}}}{\Lambda}\right)^{3}\right]^{2}\right.$
$\displaystyle+\epsilon\left[\frac{\overline{k}_{\mathrm{F}}^{2}}{6M}\left(2\sigma-\frac{1}{3}\right)-\frac{1}{2}\mu\left(\frac{\rho}{\rho_{0}}\right)\frac{x}{(1+x)^{2}}\left(\frac{2\sigma(1-2x)}{1+x}+\sigma-\frac{1}{3}\right)\right].$
(5.48)

For small $x$, the $\epsilon$-term becomes,

$\displaystyle\epsilon\left[\frac{\overline{k}_{\mathrm{F}}^{2}}{6M}\left(2\sigma-\frac{1}{3}\right)-\frac{1}{2}\mu\left(\frac{\rho}{\rho_{0}}\right)\left[\left(3\sigma-\frac{1}{3}\right)x-\left(12\sigma-\frac{2}{3}\right)x^{2}+(27\sigma-1)x^{3}\right]\right].$
(5.49)

Since $3\sigma-1/3>0$, $2\sigma-1/3$¿0 and $\mu<0$, the factor in the square
bracket of (5.49) is positive definite, indicating that the symmetry energy in
lower dimensions (e.g., 2D) with negative $\epsilon$ is reduced compared with
its 3D counterpart. This conclusion is consistent with the analysis given in
section 4 (see discussion given after Eq. (4.23)). Specifically, we have
$\overline{k}_{\mathrm{F}}^{2}/6M\cdot(2\sigma-1/3)\approx 1.8\,\mathrm{MeV}$
and $-2^{-1}\mu(\rho/\rho_{0})(3\sigma-1/3)x\approx 1.3\,\mathrm{MeV}$ for
$\rho=\rho_{0}$, and thus the linear $\epsilon$ term is about
$3.1\epsilon\,\mathrm{MeV}$. It is totally different from the one considering
the momentum dependence of the symmetry potential giving about
$-2.1\epsilon\,\mathrm{MeV}$, as discussed after the Eq. (4.23).

| linear-$\epsilon$ (potential) | linear-$\epsilon$ (total) | 2D $E_{\mathrm{sym}}(\rho_{0})$ | reference
---|---|---|---|---
optical model fitting | $-3.9\epsilon\,\mathrm{MeV}$ | $-2.1\epsilon\,\mathrm{MeV}$ | enhanced | Ref. [92]
Skyrme | $+0.4\epsilon\,\mathrm{MeV}$ | $+2.2\epsilon\,\mathrm{MeV}$ | reduced | Ref. [102]
MDI | $-2.9\epsilon\,\mathrm{MeV}$ | $-1.1\epsilon\,\mathrm{MeV}$ | enhanced | Ref. [145]
ImMDI | $2.6\epsilon\,\mathrm{MeV}+0.14y\epsilon$ | $4.4\epsilon\,\mathrm{MeV}+0.14y\epsilon$ | undetermined | Ref. [146]
toy model | $+1.3\epsilon\,\mathrm{MeV}$ | $+3.1\epsilon\,\mathrm{MeV}$ | reduced | $/$
Tab. 4: Linear-$\epsilon$ correction to the symmetry energy from different
models, the kinetic contribution is $1.8\epsilon\,\mathrm{MeV}$.

Fig. 11: Symmetry energy under different $L$ for five perturbative dimensions
$\epsilon$, $E_{\mathrm{sym}}(\overline{\rho}_{0})=32\,\mathrm{MeV}$ is fixed.

Besides using our toy model potentials discussed above, we mention another two
examples in the following. In the first example, we adopt the Skyrme-type
single-nucleon potential to order $|\mathbf{k}|^{6}k_{\mathrm{F}}^{3}$ and
$k_{\mathrm{F}}^{9}$ from Ref. [102]. In such a model, the analytical
expressions for the $U_{0}$ and $U_{\mathrm{sym}}$ are given and the relevant
parameters were fitted to fulfill similar empirical constraints as we
discussed earlier. The potential-related part of the linear-$\epsilon$
correction in (4.16) can thus be evaluated straightforwardly to be about
$0.4\,\mathrm{MeV}$ for the SP6s (SP6m or SP6h) set [102]. Consequently, the
linear-$\epsilon$ correction to the total symmetry energy is about
$2.2\epsilon\,\mathrm{MeV}$ which is close to our toy model calculation. In
the second example, we adopt an isospin- and momentum-dependent interaction
(MDI) potential that has been used extensively to explore heavy-ion reaction
dynamics within transport models [145, 147]. With the MDI potential, the
potential-related part of the linear-$\epsilon$ correction in (4.16) is found
to be about $-2.9\epsilon\,\mathrm{MeV}$, leading to a total correction of
about $-1.1\epsilon\,\mathrm{MeV}$, i.e., the 2D symmetry energy would be
enhanced. Furthermore, for a comparison the prediction from using an improve
MDI (ImMDI) potential [146] can also be obtained similarly. In the ImMDI
potential model, the momentum-dependence of $U_{0}$ is improved to better fit
the experimentally constrained high energy behavior of the nucleon optical
potential. The $y$ parameter (in unit of MeV) characterizes the momentum
dependence of the $U_{\mathrm{sym}}$, i.e., $\partial
U_{\mathrm{sym}}/\partial|\mathbf{k}|>0$ if $y>0$. Tab. 4 gives a summary of
the linear-$\epsilon$ correction to the symmetry energy from different models
investigated in this work. We notice that the sign of the total correction
using the ImMDI potential is not deterministic. For comparisons, in section 7
we shall extend the perturbative calculations to order $\epsilon^{2}$ for the
symmetry energy and investigate the corresponding effects in 2D, see Tab. 6.

| linear-$\epsilon$ (potential) | linear-$\epsilon$ (total) | 2D $L\equiv L(\rho_{0})$ | reference
---|---|---|---|---
Skyrme (SP6s) | $-8.1\epsilon\,\mathrm{MeV}$ | $-12.7\epsilon\,\mathrm{MeV}$ | enhanced | Ref. [102]
Skyrme (SP6m) | $-9.0\epsilon\,\mathrm{MeV}$ | $-13.6\epsilon\,\mathrm{MeV}$ | enhanced | Ref. [102]
Skyrme (SP6h) | $-9.9\epsilon\,\mathrm{MeV}$ | $-14.5\epsilon\,\mathrm{MeV}$ | enhanced | Ref. [102]
MDI | $-20.1\epsilon\,\mathrm{MeV}$ | $-24.7\epsilon\,\mathrm{MeV}$ | enhanced | Ref. [145]
ImMDI | $18.4\epsilon\,\mathrm{MeV}+0.80y\epsilon$ | $13.8\epsilon\,\mathrm{MeV}+0.80y\epsilon$ | undetermined | Ref. [146]
toy model | $+0.5\epsilon\,\mathrm{MeV}$ | $-4.1\epsilon\,\mathrm{MeV}$ | enhanced | $/$
Tab. 5: Linear-$\epsilon$ correction to the slope of the symmetry energy from
different models, the kinetic contribution is $-4.6\epsilon\,\mathrm{MeV}$.

The relevant contribution to the slope parameter parameter $L$ of the symmetry
energy in the toy model could also be evaluated (see formula (4.20)) by
inspecting the following expression

$\displaystyle\left(2\sigma-\frac{2}{3}\right)\left(\frac{\overline{k}_{\mathrm{F}}^{2}}{3M}\right)\epsilon+\left[\frac{\sigma}{6}\frac{\partial^{3}U_{0}}{\partial|\mathbf{k}|^{3}}\cdot\overline{k}_{\mathrm{F}}^{3}+\frac{1}{6}\left(3\sigma-\frac{2}{3}\right)\frac{\partial^{2}U_{0}}{\partial|\mathbf{k}|^{2}}\cdot\overline{k}_{\mathrm{F}}^{2}+\frac{1}{6}\left(\sigma-\frac{2}{3}\right)\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\cdot\overline{k}_{\mathrm{F}}\right]\epsilon$
$\displaystyle=$
$\displaystyle\left(2\sigma-\frac{2}{3}\right)\left(\frac{\overline{k}_{\mathrm{F}}^{2}}{3M}\right)\epsilon-\mu\left(\frac{\rho}{\rho_{0}}\right)\left[\frac{1}{2}\left(\sigma-\frac{2}{3}\right)\frac{x}{(x+1)^{2}}+\left(3\sigma-\frac{2}{3}\right)\frac{x(1-2x)}{(x+1)^{3}}+\frac{\sigma
x(10x^{2}-16x+1)}{(x+1)^{4}}\right]\epsilon$ $\displaystyle\approx$
$\displaystyle\left[\left(2\sigma-\frac{2}{3}\right)\left(\frac{\overline{k}_{\mathrm{F}}^{2}}{3M}\right)-\frac{1}{2}\mu\left(\frac{\rho}{\rho_{0}}\right)\left[\left(9\sigma-2\right)x-8(9\sigma-1)x^{2}\right]\right]\epsilon,~{}~{}\mathrm{for~{}small~{}}x.$
(5.50)

As the first term (characterized by $2\sigma-2/3$) in the square bracket is
negative, while the second term (characterized by the $\mu$ constant) is
positive since $\mu$ is negative, the overall sign of the linear
$\epsilon$-term depends on the model parameter $\mu$. Numerically we have
$\overline{k}_{\mathrm{F}}^{2}/3M\cdot(2\sigma-2/3)\approx-4.6\,\mathrm{MeV}$,
and $-2^{-1}\mu(\rho/\rho_{0})(9\sigma-2)x\approx 0.5\,\mathrm{MeV}$ at
$\rho=\rho_{0}$ in the toy model calculation, leading to a linear-$\epsilon$
correction of about $-4.1\epsilon\,\mathrm{MeV}$. Since the $\mu$ parameter in
the current model characterizes the momentum-dependence of the single-nucleon
potential, we find that the latter could effectively affect the behavior of
$L$ when a perturbed expansion $\epsilon$ is applied. This should be compared
with the perturbed $L$ parameter without momentum dependence, see formula
(4.34), or the first term in the above expression in which a negative
$\epsilon$ (e.g., 2D) will always positively increase the slope parameter,
i.e., hardening the symmetry energy as the density increases. We see once
again the non-trivial role played by the momentum dependence of the symmetry
potential.

In Fig. 11, we show two cases (corresponding to two different values of the
$L$ parameter) in which the symmetry energy is calculated perturbatively
around $d=3$ from $d=1$ to $d=5$. It is seen that as the dimension $d$
decreases, the symmetry energy decreases correspondingly. Of course, the
momentum dependence of the symmetry potential $U_{\mathrm{sym}}$ missing in
our toy model potential, as mentioned and briefly discussed in section 4 (see
the paragraph after formula (4.21) and the analysis after (4.23)), may
introduce important effects on the density dependence of
$E_{\mathrm{sym}}(\rho)$ when considering a finite $\epsilon$. These effects
are left for further studies. As listed in Tab. 5, here we compare the toy
model prediction on the $L$ parameter with the results from using the Skyrme
and MDI potentials. For the Skyrme-type pseudo potential developed in Ref.
[102], we find that the linear-$\epsilon$ correction of (4.20) is about
$-12.7\epsilon\,\mathrm{MeV}$, $-13.6\epsilon\,\mathrm{MeV}$, or
$-14.5\epsilon\,\mathrm{MeV}$ for the SP6s, SP6m, or the SP6h parameter set,
respectively. For the MDI model [145, 147], the linear-$\epsilon$ correction
is about $-24.7\epsilon\,\mathrm{MeV}$. Unlike the circumstance for the
magnitude of the symmetry energy listed in Tab. 4, except for the ImMDI
potential (which strongly depends on the $y$ parameter) the sign of the
linear-$\epsilon$ correction for the $L$ parameter is definite, i.e., the
symmetry energy in 2D tends to be hardened (i.e., the correction is positive)
at higher densities (since $\epsilon<0$).

Fig. 12: The $d$-dependence of the function $\sigma(d)$ and $\sigma(d)-1/2d$.

Finally, we want to point out that all the above perturbative calculations on
the EOS of ANM are based on the conventional 3D. However, there is no
fundamental reasons why this should be the choice. In fact, there exist
examples in which the basic dimension to be perturbed is not 3 [50, 51]. In
situations where one needs to consider perturbation based on other dimensions,
the correction $\sigma$ should be generalized correspondingly,

$\sigma(d_{\mathrm{f}})=\frac{1}{2d_{\mathrm{f}}^{2}}\left(d_{\mathrm{f}}\Psi\left(\frac{d_{\mathrm{f}}}{2}\right)+2(d_{\mathrm{f}}+2)\ln
2+d_{\mathrm{f}}\ln\pi+2\right)-\frac{1}{d_{\mathrm{f}}^{2}}\ln\left(2^{d_{\mathrm{f}}}\pi^{d_{\mathrm{f}}/2}\Gamma\left(\frac{d_{\mathrm{f}}}{2}+1\right)\right),$
(5.51)

where $\Psi(x)$ is defined as the derivative of the $\ln\Gamma(x)$, i.e.,
$\Psi(x)\equiv\mathrm{d}\ln\Gamma(x)/\mathrm{d}x$, and
$\sigma(3)\equiv\sigma\approx 0.2396$, see (4.2). As an example, we work out
the perturbed symmetry energy to linear $\epsilon$-order by neglecting the 2nd
derivative of $U_{0}$ with respect to $|\mathbf{k}|$,

$\displaystyle E_{\mathrm{sym}}(\rho)\approx$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{2d_{\mathrm{f}}M}+\left.\frac{k_{\mathrm{F}}}{2d_{\mathrm{f}}}\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right|_{|\mathbf{k}|=k_{\mathrm{F}}}+\frac{1}{2}U_{\mathrm{sym}}(\rho,k_{\mathrm{F}})$
$\displaystyle+\epsilon\left[\left(\sigma_{\mathrm{f}}-\frac{1}{2d_{\mathrm{f}}}\right)\frac{k_{\mathrm{F}}^{2}}{d_{\mathrm{f}}M_{\mathrm{s}}^{\ast}}+\frac{k_{\mathrm{F}}}{2}\sigma_{\mathrm{f}}\left(\frac{\partial
U_{\mathrm{sym}}}{\partial|\mathbf{k}|}-\frac{1}{d_{\mathrm{f}}}\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right)_{|\mathbf{k}|=k_{\mathrm{F}}}\right],$
(5.52)

where $\sigma_{\mathrm{f}}\equiv\sigma(d_{\mathrm{f}})$ and the second line is
a direct generalization of (4.23). If we assume that the sign of the momentum-
dependence of the single potential is the same as that in 3D, i.e., $\partial
U_{\mathrm{sym}}/\partial|\mathbf{k}|<0$ and $\partial
U_{0}/\partial|\mathbf{k}|>0$, then the first term in the $\epsilon$-order
contribution, i.e.,
$(\sigma_{\mathrm{f}}-1/2d_{\mathrm{f}})k_{\mathrm{F}}^{2}/d_{\mathrm{f}}M_{\mathrm{s}}^{\ast}$
eventually becomes smaller (but still positive) as $d_{\mathrm{f}}$ increases,
see Fig. 12 for the $d$-dependence of the function $\sigma(d)$ and that of
$\sigma(d)-1/2d$. While the second term in the $\epsilon$-order contribution
eventually saturates to $2^{-1}k_{\mathrm{F}}\sigma_{\mathrm{f}}\partial
U_{\mathrm{sym}}/\partial|\mathbf{k}|$ as $d_{\mathrm{f}}$ increases,
indicting that in the large $d$-limit the density dependence of the (first-
order) perturbed symmetry energy is dominated by the momentum-dependence of
the symmetry potential, i.e., the symmetry energy tends to be reduced compared
with the one in $d_{\mathrm{f}}$D if it was upwardly perturbed (with
$\epsilon>0$) since $\partial U_{\mathrm{sym}}/\partial|\mathbf{k}|<0$ as
assumed, see the relevant discussion in section 4. In particular, for
$d_{\mathrm{f}}=1$, we can rewrite the above expression as,

$\displaystyle E_{\mathrm{sym}}(\rho)\approx$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{2M}+\left.\frac{k_{\mathrm{F}}}{2}\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right|_{|\mathbf{k}|=k_{\mathrm{F}}}+\frac{1}{2}U_{\mathrm{sym}}(\rho,k_{\mathrm{F}})$
$\displaystyle+\epsilon\left[\frac{k_{\mathrm{F}}^{2}}{2M_{\mathrm{s}}^{\ast}}\left(\sigma_{1}-1\right)+\sigma_{1}\frac{k_{\mathrm{F}}}{2}\left(\frac{k_{\mathrm{F}}}{M}+\frac{\partial
U_{\mathrm{sym}}}{\partial|\mathbf{k}|}\right)_{|\mathbf{k}|=k_{\mathrm{F}}}\right].$
(5.53)

If we assume the combination $[k_{\mathrm{F}}/M+\partial
U_{\mathrm{sym}}/\partial|\mathbf{k}|]_{|\mathbf{k}|=k_{\mathrm{F}}}$ behaves
similarly as in 3D, then this term is positive [92]. Combining it with the
first term namely $(\sigma_{1}-1)k_{\mathrm{F}}^{2}/2M_{\mathrm{s}}^{\ast}$
which is positive since $\sigma_{1}>1$ (as shown in Fig. 12), one then finds
that the overall $\epsilon$-order correction to the symmetry energy is
positive if $\epsilon>0$, indicating that the $E_{\mathrm{sym}}(\rho)$ should
be enhanced if its is upwardly perturbed. From these discussions, we can find
once again that the dimensionality could change essentially the conclusion to
certain problems though more accurate (quantitative) conclusions need further
investigations. Finally, the relatively smaller $\sigma_{\mathrm{f}}$ for
$d_{\mathrm{f}}\approx 2$ or $3$ than $\sigma_{1}\approx 1.5253$ indicates the
$\epsilon$-expansion around the reference dimension $d_{\mathrm{f}}=1$ may
introduce difficulties in certain problems.

## 6\. Kinetic EOS of neutron-rich matter with high-momentum nucleons induced
by short-range correlations

### 6.1. Single-nucleon momentum distribution function in the presence of
short-range correlations

In this section, we incorporate the short-range correlations (SRCs) induced
high momentum tail (HMT) into the calculations of the kinetic EOS of ANM in a
space of general dimension $d$. Here we first describe briefly the single-
nucleon momentum distribution function $n_{\mathbf{k}}^{J}$ in ANM in 3D.
Based on predictions of microscopic nuclear many-body theories and relevant
experimental findings [95, 96, 97, 98], the single-nucleon momentum
distribution in ANM can be parametrized as [148, 149, 150, 151],

$n^{J}_{\mathbf{k}}(\rho,\delta)=\left\\{\begin{array}[]{ll}\Delta_{J},{}{}&0<|\mathbf{k}|<k_{\mathrm{F}}^{J},\\\
\displaystyle{C}_{J}\left({k_{\mathrm{F}}^{J}}/{|\mathbf{k}|}\right)^{4},{}{}&k_{\mathrm{F}}^{J}<|\mathbf{k}|<\phi_{J}k_{\mathrm{F}}^{J}.\end{array}\right.$
(6.1)

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

$\Delta_{J}+x_{J}^{\mathrm{HMT}}=1,~{}~{}x_{J}^{\mathrm{HMT}}=\left.\int_{k_{\mathrm{F}}^{J}}^{\phi_{J}k_{\mathrm{F}}^{J}}n_{\mathbf{k}}^{J}\mathrm{d}\mathbf{k}\right/{\displaystyle\int_{0}^{\phi_{J}k_{\mathrm{F}}^{J}}n_{\mathbf{k}}^{J}\mathrm{d}\mathbf{k}}=3C_{{J}}\left(1-\frac{1}{\phi_{{J}}}\right),$
(6.2)

and the normalization condition
$[{2}/{(2\pi)^{3}}]\int_{0}^{\infty}n^{J}_{\mathbf{k}}(\rho,\delta)\mathrm{d}\mathbf{k}=\rho_{J}=(k_{\mathrm{F}}^{J})^{3}/3\pi^{2}$.
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,
$Y_{J}=Y_{0}(1+Y_{1}\tau_{3}^{J}\delta)$ with $Y=C$ or $\phi$ [148], the
associated parameters were correspondingly 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 [95, 152], see, e.g., Ref. [101] for a recent review on related
issues [153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166,
167, 168, 169, 170].

When generalizing the $n_{\mathbf{k}}$ to arbitrary dimensions, we need to
make certain assumptions on the parameters involved. The contact coefficient
$C_{J}$ thus far explored in nuclear system using different theoretical
approaches, or in ultra-cold gases adopting different types of trapped
potentials, at zero or finite temperatures all show good universality [104,
171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184], e.g.,
see Chap. 18 of Ref. [41] for a review. In this work, we thus assume that the
$C_{J}$ (including its $C_{0}$ and $C_{1}$ terms) is also (near) universal
with respect to dimension $d$. On the other hand, to our best knowledge, there
is no fundamental constraints on the high momentum cutoffs in lower or higher
dimensions (different from 3). We thus assume that they are also universal and
investigate the kinetic EOS of ANM based on these assumptions. Although the
SRC-induced HMT appears only in the kinetic EOS (through the single-nucleon
momentum distribution function) in the study here, it should be thought as an
effective nucleon-nucleon interaction. Here we focus on its effects on the
kinetic EOS of ANM in $d$D via the $\epsilon$-expansion.

For nucleons in ANM in $d$D, the relation (6.2) should be generalized to,

$\Delta_{J}+\frac{dC_{J}}{d-4}\left(\phi_{J}^{d-4}-1\right)=1,$ (6.3)

where the second term on the left hand side is the corresponding
$x_{J}^{\mathrm{HMT}}$ in $d$D. Consequently, we obtain the zero-momentum
depletion $\Delta_{J}$ in the first four dimensions as,

$\displaystyle\Delta_{J}^{(\mathrm{1D})}=$ $\displaystyle
1-\frac{C_{J}}{3}\left(1-\frac{1}{\phi_{J}^{3}}\right),$ (6.4)
$\displaystyle\Delta_{J}^{(\mathrm{2D})}=$ $\displaystyle
1-C_{J}\left(1-\frac{1}{\phi_{J}^{2}}\right),$ (6.5)
$\displaystyle\Delta_{J}^{(\mathrm{3D})}=$ $\displaystyle
1-3C_{J}\left(1-\frac{1}{\phi_{J}}\right),$ (6.6)
$\displaystyle\Delta_{J}^{(\mathrm{4D})}=$ $\displaystyle
1-4C_{J}\ln\phi_{J}.$ (6.7)

A main feature of these expressions is that the $\phi_{J}$ in $d=1,2,3$ could
be mathematically taken to be infinity, while that in 4D has a natural limit,
i.e., $\phi_{J}\leq\exp(1/4C_{J})$. Similarly, the kinetic EOS of ANM is given
by

$E^{\mathrm{kin}}(\rho,\delta)=\sum_{J=\mathrm{n,p}}\frac{d}{d+2}\frac{k_{\mathrm{F}}^{2}}{2M}\left(1+\tau_{3}^{J}\delta\right)^{1+2/d}\left[1+C_{J}\left(\frac{8}{(d-2)(d-4)}+\frac{d+2}{d-2}\phi^{d-2}_{J}-\frac{d}{d-4}\phi^{d-4}_{J}\right)\right].$
(6.8)

The dependence of $E^{\mathrm{kin}}(\rho,\delta)$ on the dimension $d$ is
rather non-trivial due to the complicated combination of $C_{J},\phi_{J}$ and
$d$.

### 6.2. Kinetic EOS of SNM and the $\epsilon$-expansion

According to the general formula (6.8), we can immediately obtain the kinetic
part of SNM EOS in $d$D as,

$\displaystyle E_{0}^{\mathrm{kin}}(\rho)=$
$\displaystyle\frac{d}{d+2}\frac{k_{\mathrm{F}}^{2}}{2M}\left[1+C_{0}\left(\frac{8}{(d-2)(d-4)}+\frac{d+2}{d-2}\phi^{d-2}_{0}-\frac{d}{d-4}\phi^{d-4}_{0}\right)\right]\equiv\frac{k_{\mathrm{F}}^{2}}{2M}\Lambda(d,C_{0},\phi_{0}),$
(6.9)

where the last relation defines the function $\Lambda(d,C_{0},\phi_{0})$,
which encapsulates the dependence of the EOS on the dimension $d$. We notice
that the $\Lambda(d,C_{0},\phi_{0})$ apparently diverge with
$d=d_{\mathrm{f}}=2$ and/or $d=d_{\mathrm{f}}=4$. Nevertheless, away from the
poles with $d=d_{\mathrm{f}}+\epsilon$, the diverging terms all cancel
completely and naturally in the $\epsilon$ expansion. For example, for the 2D
situation, one can obtain the $\epsilon$-expansion

$\frac{8}{(d-2)(d-4)}+\frac{d+2}{d-2}\phi^{d-2}_{0}-\frac{d}{d-4}\phi^{d-4}_{0}\approx-1+4\ln\phi_{0}+\frac{1}{\phi_{0}^{2}}+\epsilon\left(1+\ln\phi_{0}+2\ln^{2}\phi_{0}+\frac{\ln\phi_{0}+1}{\phi_{0}^{2}}\right),$
(6.10)

to order $\epsilon=d-2$, using the formula $a^{x}\approx 1+x\ln
a+2^{-1}x^{2}\ln^{2}a$ for small $x$.

Expanding the EOS of SNM around the dimension $d_{\mathrm{f}}$ (where
$d_{\mathrm{f}}=1,2,3,4$) to order $\epsilon$ gives

$\displaystyle E_{0,d_{\mathrm{f}}=1}^{\mathrm{kin}}(\rho)\approx$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{6M}\left[1+C_{0}\left(\frac{8}{3}-\frac{3}{\phi_{0}}+\frac{1}{3\phi_{0}^{3}}\right)\right]+\epsilon\cdot\frac{k_{\mathrm{F}}^{2}}{2M}\left[\frac{2}{9}+C_{0}\left(\frac{16}{9}-\frac{\ln\phi_{0}}{\phi_{0}}-\frac{2}{\phi_{0}}+\frac{\ln\phi_{0}}{9\phi_{0}^{3}}+\frac{2}{9\phi_{0}^{3}}\right)\right]$
$\displaystyle+\epsilon^{2}\cdot\frac{k_{\mathrm{F}}^{2}}{2M}\left[-\frac{2}{27}+C_{0}\left(\frac{152}{81}-\frac{\ln^{2}\phi_{0}}{2\phi_{0}}-\frac{2\ln\phi_{0}}{\phi_{0}}-\frac{2}{\phi_{0}}+\frac{\ln^{2}\phi_{0}}{18\phi_{0}^{3}}+\frac{2\ln\phi_{0}}{9\phi_{0}^{3}}+\frac{10}{81\phi_{0}^{3}}\right)\right],$
(6.11) $\displaystyle E_{0,d_{\mathrm{f}}=2}^{\mathrm{kin}}(\rho)\approx$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{4M}\left[1+C_{0}\left(4\ln\phi_{0}+\frac{1}{\phi_{0}^{2}}-1\right)\right]+\epsilon\cdot\frac{k_{\mathrm{F}}^{2}}{2M}\left[\frac{1}{8}+C_{0}\left(\ln^{2}\phi_{0}+\ln\phi_{0}-\frac{5}{8}+\frac{\ln\phi_{0}}{2\phi_{0}^{2}}+\frac{5}{8\phi_{0}^{2}}\right)\right]$
$\displaystyle+\epsilon^{2}\cdot\frac{k_{\mathrm{F}}^{2}}{2M}\left[-\frac{1}{32}+C_{0}\left(-\frac{11}{32}+\frac{\ln^{3}\phi_{0}}{3}+\frac{\ln^{2}\phi_{0}}{2}+\frac{\ln^{2}\phi_{0}}{4\phi_{0}^{2}}+\frac{5\ln\phi_{0}}{8\phi_{0}^{2}}+\frac{11}{32\phi_{0}^{2}}\right)\right],$
(6.12)

and

$\displaystyle E_{0,d_{\mathrm{f}}=3}^{\mathrm{kin}}(\rho)\approx$
$\displaystyle\frac{3}{5}\frac{k_{\mathrm{F}}^{2}}{2M}\left[1+C_{0}\left(5\phi_{0}+\frac{3}{\phi_{0}}-8\right)\right]+\epsilon\cdot\frac{k_{\mathrm{F}}^{2}}{2M}\left[\frac{2}{25}+C_{0}\left(3\phi_{0}\ln\phi_{0}-2\phi_{0}+\frac{9\ln\phi_{0}}{5\phi_{0}}+\frac{66}{25\phi_{0}}-\frac{16}{25}\right)\right]$
$\displaystyle+\epsilon^{2}\cdot\frac{k_{\mathrm{F}}^{2}}{2M}\left[-\frac{2}{125}+C_{0}\left(-\frac{584}{125}+\frac{3\phi_{0}\ln^{2}\phi_{0}}{2}-2\phi_{0}\ln\phi_{0}+2\phi_{0}+\frac{9\ln^{2}\phi_{0}}{10\phi_{0}}+\frac{66\ln\phi_{0}}{25\phi_{0}}+\frac{334}{125\phi_{0}}\right)\right],$
(6.13) $\displaystyle E_{0,d_{\mathrm{f}}=4}^{\mathrm{kin}}(\rho)\approx$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{3M}\left[1+C_{0}\left(3\phi_{0}^{2}-4\ln\phi_{0}-3\right)\right]+\epsilon\cdot\frac{k_{\mathrm{F}}^{2}}{2M}\left[\frac{1}{18}+C_{0}\left(\frac{1}{2}+2\phi_{0}^{2}\ln\phi_{0}-\frac{\phi_{0}^{2}}{2}-\frac{4\ln^{2}\phi_{0}}{3}-\frac{8\ln\phi_{0}}{9}\right)\right]$
$\displaystyle+\epsilon^{2}\cdot\frac{k_{\mathrm{F}}^{2}}{2M}\left[-\frac{1}{108}+C_{0}\left(-\frac{1}{4}+\phi_{0}^{2}\ln^{2}\phi_{0}-\frac{\phi_{0}^{2}\ln\phi_{0}}{2}+\frac{\phi_{0}^{2}}{4}-\frac{4\ln^{3}\phi_{0}}{9}-\frac{4\ln^{2}\phi_{0}}{9}-\frac{\ln\phi_{0}}{54}\right)\right].$
(6.14)

It is obvious that the factors in the square brackets of (6.15) to (6.18)
reduce to 1 naturally if $\phi_{0}=1$ is taken (i.e., no high momentum nucleon
above $k_{\mathrm{F}}$). In addition, the $\epsilon^{0}$-order terms in the
above expansions are the EOS of SNM in dimensions $d$, i.e.,

$\displaystyle E_{0,{(\mathrm{1D})}}^{\mathrm{kin}}(\rho)=$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{6M}\left[1+C_{0}\left(\frac{8}{3}-\frac{3}{\phi_{0}}+\frac{1}{3\phi_{0}^{3}}\right)\right],$
(6.15) $\displaystyle E_{0,{(\mathrm{2D})}}^{\mathrm{kin}}(\rho)=$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{4M}\left[1+C_{0}\left(4\ln\phi_{0}+\frac{1}{\phi_{0}^{2}}-1\right)\right],$
(6.16) $\displaystyle E_{0,{(\mathrm{3D})}}^{\mathrm{kin}}(\rho)=$
$\displaystyle\frac{3}{5}\frac{k_{\mathrm{F}}^{2}}{2M}\left[1+C_{0}\left(5\phi_{0}+\frac{3}{\phi_{0}}-8\right)\right],$
(6.17) $\displaystyle E_{0,{(\mathrm{4D})}}^{\mathrm{kin}}(\rho)=$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{3M}\left[1+C_{0}\left(3\phi_{0}^{2}-4\ln\phi_{0}-3\right)\right],$
(6.18)

where the nucleon Fermi momentum $k_{\mathrm{F}}$ in dimensions $d$ is related
to the nucleon density by (2.18).

Fig. 13: Perturbative calculations on
$\Lambda(d_{\mathrm{f}}+\epsilon,C_{0},\phi_{0})$ (left) and the kinetic EOS
$E_{0}^{\mathrm{kin}}(\rho_{0})$ (right) for different $d_{\mathrm{f}}$. Only
the second-order results are shown in the right panel.

In the left panel of Fig. 13 we show the values of $\Lambda(d,C_{0},\phi_{0})$
from the perturbative calculations. The dotted, dash-dotted and solid lines
are the results from the zeroth-order, first-order and second-order
approximation, respectively. The dashed cyan lines are the exact results for
each $d_{\mathrm{f}}$. It is seen clearly that the difference between the
first-order and the second-order results are smaller compared to that between
the zeroth-order and the first-order approximations (e.g., within the range of
$-1\lesssim\epsilon\lesssim 1$), and this difference increases as $d$
increases. This feature shows the convergence of the $\epsilon$-expansion even
though the perturbative dimension $\epsilon$ could take sizable values like 1
or $-1$. In addition, the results from equating $d_{\mathrm{f}}$ to 1, 2 or 3
are all reasonable. For example, taking $d_{\mathrm{f}}=1$ and $\epsilon=1$ or
$\epsilon=2$ to order $\epsilon$ or order $\epsilon^{2}$ reasonably gives the
corresponding $\Lambda$ function (with the HMT included) in 2D or 3D, see the
dash-dotted black line shown in the left panel of Fig. 13. In particular, we
have $\Lambda(1+1,C_{0},\phi_{0})\approx 0.728$ to linear order in $\epsilon$
and $\Lambda(1+1,C_{0},\phi_{0})\approx 0.683$ to quadratic order in
$\epsilon$, while the exact value for the $\Lambda$ function in 2D is 0.713.
Here $C_{0}\approx 0.161$ and $\phi_{0}\approx 2.38$ are adopted, see the
solid and dot-dashed black lines and the red diamond. Similarly, starting from
$d_{\mathrm{f}}=2$ and taking $\epsilon=-1$ or $\epsilon=1$ (to order
$\epsilon^{2}$) gives the $\Lambda$ function reasonably well in 1D and 3D, see
the dash-dotted magenta line shown in the left panel of Fig. 13. The reference
dimension $d_{\mathrm{f}}=3$ could also be used to make reliable perturbative
calculations, as shown by the blue lines in the left panel of Fig. 13. In
particular, taking $d_{\mathrm{f}}=3$ and $\epsilon=-1$ gives
$\Lambda(3-1,C_{0},\phi_{0})\approx 0.607$ and $0.754$, to linear order of
$\epsilon$ and quadratic order of $\epsilon^{2}$, respectively, and both are
close to the exact value 0.713. These features of the $\epsilon$-expansion
indicate that although $\epsilon=1$ or $\epsilon=2$ is not small in
conventional sense, the structure of the mathematical expressions makes the
expansions effective and reasonable. On the other hand, one can not
effectively obtain the $\Lambda$ function in 3D from $d_{\mathrm{f}}=4$ with
$\epsilon=-1$. The perturbations behave well for the $\Lambda$ function for
$1\lesssim d_{\mathrm{f}}\lesssim 3$ with the perturbation $\epsilon$ in the
range of about $-1\lesssim\epsilon\lesssim 1$. For even better
approximation/extrapolation, e.g., from $d_{\mathrm{f}}=1$ to $d=3$ (i.e.,
with $\epsilon=2$), the Pade approximation [185] may be more useful.

In the above, we only considered the perturbative expansion of the function
$\Lambda(d,C_{0},\phi_{0})$. In order to estimate the kinetic EOS of SNM, one
also needs to consider the Fermi energy $E_{\mathrm{F}}=k_{\mathrm{F}}^{2}/2M$
in $d$D. Let’s do here the relevant calculations for the 2D situation, where
$k_{\mathrm{F}}^{(\mathrm{2D})}=\sqrt{\pi\rho_{2}}$ with
$\rho_{2}=\rho^{2/3}$. Taking the conventional density in 3D as $\rho\approx
0.16\,\mathrm{fm}^{-3}$, then we obtain $k_{\mathrm{F}}^{(\mathrm{2D})}\approx
189.88\,\mathrm{MeV}$ and $E_{\mathrm{F}}^{(\mathrm{2D})}\approx
19.20\,\mathrm{MeV}$, and consequently the exact kinetic EOS
$E_{0,(\mathrm{2D})}^{\mathrm{kin}}(\rho_{0})$ of SNM is found to be about
$E_{\mathrm{F}}^{(2\mathrm{D})}\Lambda(2,C_{0},\phi_{0})\approx
13.69\,\mathrm{MeV}$ (using the exact $\Lambda(2,C_{0},\phi_{0})\approx
0.713$). We can estimate this value by starting from the kinetic EOS in 1D or
3D to linear order of $\epsilon$. Specifically, we have for the Fermi momentum
$k_{\mathrm{F}}^{(\mathrm{1D})}=\pi\rho_{1}/4\approx 84.13\,\mathrm{MeV}$
where $\rho_{1}=\rho^{1/3}$, and Fermi momentum in 2D could be obtained as
$k_{\mathrm{F}}^{(\mathrm{2D})}\approx
k_{\mathrm{F}}^{(\mathrm{1D})}(1+\sigma_{1}\epsilon)\approx
212.46\,\mathrm{MeV}$ where $\epsilon=1$ is applied and
$\sigma_{1}\equiv\sigma(1)=-\gamma_{\mathrm{E}}/2+2^{-1}\ln\pi-\ln(\pi/4)\approx
1.5253$ (see definition (5.51)), and thus the approximation for
$E_{0,(\mathrm{2D})}^{\mathrm{kin}}(\rho_{0})$ is about $17.50\,\mathrm{MeV}$
(adopting $\Lambda(1+\epsilon,C_{0},\phi_{0})\approx 0.728$ for $\epsilon=1$).
Similarly, if we start from the setting in 3D, then the Fermi momentum is
approximated as $k_{\mathrm{F}}^{(\mathrm{2D})}\approx
k_{\mathrm{F}}^{(\mathrm{3D})}(1-\sigma)\approx 200.02\,\mathrm{MeV}$ and
consequently $E_{0,(\mathrm{2D})}^{\mathrm{kin}}(\rho_{0})\approx
12.93\,\mathrm{MeV}$, where $\sigma\approx 0.2396$ is given in (4.2) and
$\Lambda(3-1,C_{0},\phi_{0})\approx 0.607$ is used. We have seen both the
perturbative expansions based on the 1D and 3D correspondences are effective.
More results are shown in the right panel of Fig. 13, from which we find again
that $1\lesssim d_{\mathrm{f}}\lesssim 3$ with $-1\lesssim\epsilon\lesssim 1$
is reasonable. In fact, one obtains that
$E_{0,(\mathrm{2D})}^{\mathrm{kin}}(\rho_{0})\approx 14.36\,\mathrm{MeV}$ and
$E_{0,(\mathrm{2D})}^{\mathrm{kin}}(\rho_{0})\approx 15.21\,\mathrm{MeV}$ from
the corresponding EOS in 3D and 1D, respectively. Obviously, the approximation
for $E_{0,(\mathrm{2D})}^{\mathrm{kin}}(\rho_{0})$ is better from the 3D
correspondence than that from the 1D correspondence. It is partially due to
the smaller value of $\sigma(d_{\mathrm{f}})$ with $d_{\mathrm{f}}=3$, i.e.,
$\sigma(3)\approx 0.2396$ while $\sigma(1)\approx 1.5253$. Thus in order to
investigate the 2D EOS, the starting point from $d_{\mathrm{f}}=3$ is
reasonable (as we already have done in the previous sections).

Fig. 14: Dependence of the function $\Lambda$ on $\phi_{0}$.

As an application of the $\epsilon$-expansion, next we estimate the kinetic
EOS of SNM in 2D as $\phi_{0}\to\infty$ using the corresponding kinetic EOS in
1D. First, let us examine the dependence of the kinetic EOS on the high
momentum cutoff $\phi_{0}$. In particular, in Fig. 14, the dependence of the
function $\Lambda(d,C_{0},\phi_{0})$ on the high momentum cutoff $\phi_{0}$ is
shown. Moreover, we have analytically

$\displaystyle\frac{\partial\Lambda(d,C_{0},\phi_{0})}{\partial\phi_{0}}=$
$\displaystyle-\frac{C_{0}d\phi_{0}^{d-3}}{d+2}\frac{1}{d+2}\left(\frac{d}{d+2}\frac{1}{\phi_{0}^{2}}-1\right).$
(6.19)

Since the high momentum cutoff $\phi_{0}\geq 1$, the derivative (6.19) should
always be larger than zero, indicating that $\Lambda(d,C_{0},\phi_{0})$ is an
increasing function of $\phi_{0}$, regardless of the dimension $d$ of the
space where the nucleons live. The kinetic EOS of SNM diverges if the high
momentum cutoff $\phi_{0}$ (unphysically) approaches infinity, except in 1D.
As the dimension $d$ increases the order of divergence also increase, i.e., it
changes from $\ln\phi_{0}$ (logarithmic divergence) in 2D, to $\phi_{0}$
(linear divergence) in 3D and to $\phi_{0}^{2}$ (quadratic divergence) in 4D.
More generally, for $d\geq 3$, the order of divergence of the kinetic EOS of
SNM is $d-2$, as clearly shown in the expression (6.9). When
$\phi_{0}\to\infty$ in 1D, the high momentum part of the nucleon distribution
function contributes a non-trivial term, i.e., $8C_{0}/3$ (compared with 1),
to the kinetic EOS, as shown in Eq. (6.15) and demonstrated as the dashed line
in Fig. 14. By taking $\phi_{0}\to\infty$ in (6.12), we obtain to linear order
of $\epsilon$ that,

$\displaystyle E_{0,d_{\mathrm{f}}=1,\epsilon}^{\mathrm{kin}}(\rho)\approx$
$\displaystyle\frac{k_{\mathrm{F}}^{(\mathrm{1D}),2}(1+2\sigma_{1}\epsilon)}{6M}\left[1+\frac{8}{3}C_{0}+\left(\frac{2}{3}+\frac{16}{3}C_{0}\right)\epsilon\right]+\mathcal{O}(\epsilon^{2})$
$\displaystyle+\sum\mathrm{term\;approaches\;zero\;as\;}\phi_{0}\to\infty.$
(6.20)

Consequently,

$E_{0,\mathrm{(2D)}}^{\mathrm{kin},\phi_{0}\to\infty}(\rho)\approx
E_{0,d_{\mathrm{f}}=1,\epsilon=1}^{\mathrm{kin}}(\rho)\approx\frac{k_{\mathrm{F}}^{(\mathrm{1D}),2}(1+2\sigma_{1})}{6M}\left(\frac{5}{3}+8C_{0}\right).$
(6.21)

Taking $\rho\approx 0.16\,\mathrm{fm}^{-3}$, this gives approximately
$E_{0,\mathrm{(2D)}}^{\mathrm{kin},\phi_{0}\to\infty}(\rho)\approx
15.04\,\mathrm{MeV}$. It is slightly large than the exact
$E_{\mathrm{F}}^{(2\mathrm{D})}\Lambda(2,C_{0},\phi_{0})\approx
13.69\,\mathrm{MeV}$ if a finite high momentum cutoff $\phi_{0}\approx 2.38$
is used. The FFG model prediction on the 2D kinetic EOS of SNM is simply
$k_{\mathrm{F}}^{(\mathrm{2D}),2}/4M=\pi\rho_{2}/4M\approx 9.60\,\mathrm{MeV}$
where $\rho_{2}=\rho^{2/3}$. It is also very interesting to notice that if
$\epsilon=2$ is applied in (6.2), then one obtains
$E_{0,\mathrm{(3D)}}^{\mathrm{kin},\phi_{0}\to\infty}(\rho)\approx
39.97\,\mathrm{MeV}$ which is very close to the value of about 40.45 MeV [148]
if $\phi_{0}\approx 2.38$ and $C_{0}\approx 0.161$ are adopted in 3D.
Physically one can not extend the cutoff $\phi_{0}$ to infinity in
conventional 3D situation. This is because above certain momentum the $k^{-4}$
form of the HMT in the single-nucleon momentum distribution is no longer
reasonable again due to the higher-order correlations such as the three-
nucleon interactions [152, 186, 187, 188, 189, 190]. In this case, other HMT
forms in $n_{\mathbf{k}}^{0}$ should be necessarily considered.

The above results indicate that the $\epsilon$-expansion of the kinetic EOS
from certain dimension $d_{\mathrm{f}}$ (e.g., $d_{\mathrm{f}}=1$) is
reasonable and effective. However, as the expansion order $n$ of $\epsilon$
increases, the extrapolation becomes less believable, since $(1+1)^{n}=2^{n}$
could be very large and even become divergent.

### 6.3. Momentum fluctuations of nucleons in symmetric nuclear matter in $d$D
spaces

Once the nucleon momentum distribution is specified, one can investigate its
moments. In particular, the variance of nucleon momentum $k$ namely $\langle
k^{2}\rangle-\langle k\rangle^{2}$ characterizing the strength of nucleon
momentum fluctuations is very informative. We define the (relative) nucleon
momentum fluctuation $\Upsilon_{k}$ in SNM in $d$D spaces as,

$\Upsilon_{k}(d)\equiv\sqrt{\frac{\langle k^{2}\rangle-\langle
k\rangle^{2}}{\langle k\rangle^{2}}}=\sqrt{\frac{\langle k^{2}\rangle}{\langle
k\rangle^{2}}-1},$ (6.22)

with the moment $\langle k^{\gamma}\rangle=\langle\mathbf{k}^{\gamma}\rangle$
being defined with respect to the momentum distribution $n_{\mathbf{k}}^{0}$,
i.e.,

$\langle\mathbf{k}^{\gamma}\rangle=\left.\int
n_{\mathbf{k}}^{0}\mathbf{k}^{\gamma}\mathrm{d}^{d}\mathbf{k}\right/\int
n_{\mathbf{k}}^{0}\mathrm{d}^{d}\mathbf{k}.$ (6.23)

For our purposes here, $\gamma=1$ and $\gamma=2$ are relevant,

$\displaystyle\left.\langle k\rangle\right/k_{\mathrm{F}}=$
$\displaystyle\frac{d}{d+1}\left[1+C_{0}\left(\frac{4}{(d-3)(d-4)}+\frac{d+1}{d-3}\phi_{0}^{d-3}-\frac{d}{d-4}\phi_{0}^{d-4}\right)\right],$
(6.24) $\displaystyle\left.\langle k^{2}\rangle\right/k_{\mathrm{F}}^{2}=$
$\displaystyle\frac{d}{d+2}\left[1+C_{0}\left(\frac{8}{(d-2)(d-4)}+\frac{d+2}{d-2}\phi_{0}^{d-2}-\frac{d}{d-4}\phi_{0}^{d-4}\right)\right],$
(6.25)

from which the fluctuation factor $\Upsilon_{k}(d)$ in $d$D could be obtained.

For the nucleons in $d=1,2,3,4$ with HMT, one has,

$\displaystyle\mathrm{1D}:~{}~{}\left.\langle k\rangle\right/k_{\mathrm{F}}=$
$\displaystyle\frac{1}{2}\left[1+C_{0}\left(\frac{2}{3}+\frac{1}{3\phi_{0}^{3}}-\frac{1}{\phi_{0}^{2}}\right)\right],~{}~{}\left.\langle
k^{2}\rangle\right/k_{\mathrm{F}}^{2}=\frac{1}{3}\left[1+C_{0}\left(\frac{8}{3}+\frac{1}{3\phi_{0}^{3}}-\frac{3}{\phi_{0}}\right)\right],$
(6.26) $\displaystyle\mathrm{2D}:~{}~{}\left.\langle
k\rangle\right/k_{\mathrm{F}}=$
$\displaystyle\frac{2}{4}\left[1+C_{0}\left(2+\frac{1}{\phi_{0}^{2}}-\frac{3}{\phi_{0}}\right)\right],~{}~{}\left.\langle
k^{2}\rangle\right/k_{\mathrm{F}}^{2}=\frac{1}{2}\left[1+C_{0}\left(4\ln\phi_{0}+\frac{1}{\phi_{0}^{2}}-1\right)\right],$
(6.27) $\displaystyle\mathrm{3D}:~{}~{}\left.\langle
k\rangle\right/k_{\mathrm{F}}=$
$\displaystyle\frac{3}{4}\left[1+C_{0}\left(4\ln\phi_{0}+\frac{3}{\phi_{0}}-3\right)\right],~{}~{}\left.\langle
k^{2}\rangle\right/k_{\mathrm{F}}^{2}=\frac{3}{5}\left[1+C_{0}\left(5\phi_{0}+\frac{3}{\phi_{0}}-8\right)\right],$
(6.28) $\displaystyle\mathrm{4D}:~{}~{}\left.\langle
k\rangle\right/k_{\mathrm{F}}=$
$\displaystyle\frac{4}{5}\left[1+C_{0}\left(5\phi_{0}-4\ln\phi_{0}-5\right)\right],~{}~{}\left.\langle
k^{2}\rangle\right/k_{\mathrm{F}}^{2}=\frac{2}{3}\left[1+C_{0}\left(3\phi_{0}^{2}-4\ln\phi_{0}-3\right)\right].$
(6.29)

For example, we have for 3D,

$\Upsilon_{k}^{\mathrm{HMT}}(3)=\sqrt{\frac{16\phi_{0}}{15}\frac{5C_{0}\phi_{0}^{2}-8C_{0}\phi_{0}+3C_{0}+\phi_{0}}{4C_{0}\phi_{0}\ln\phi_{0}-3C_{0}\phi_{0}+3C_{0}+\phi_{0}}-1}\to\Upsilon_{k}^{\mathrm{FFG}}(3)=\sqrt{\frac{1}{15}},~{}~{}\mathrm{as}~{}~{}\phi_{0}\to
1.$ (6.30)

For the FFG model, one has $\Upsilon_{k}^{\mathrm{FFG}}(d)=1/\sqrt{d^{2}+2d}$,
and thus $\Upsilon_{k}^{\mathrm{FFG}}(3)\approx 0.258$ and
$\Upsilon_{k}^{\mathrm{FFG}}(2)\approx 0.354$.

Fig. 15: Fluctuation of the momentum $\Upsilon_{k}$ at saturation density
$\rho_{0}$ as a function of $d$.

In Fig. 15 we show the first and the second fluctuations in momentum
$\mathbf{k}$ as functions of the spacial dimension $d$ (left) and the relative
momentum fluctuation $\Upsilon_{k}$, both in the FFG and the HMT models. One
can see from the figure: (a) as the dimension $d$ decreases both the first and
the second fluctuations decrease, while the relative fluctuation
$\Upsilon_{k}$ becomes larger, i.e., low-dimensional nucleonic matter has
larger (relative) momentum fluctuation compared with the high dimensional
situations, for either the FFG or the HMT model; (b) when the SRC-induced HMT
is considered, the fluctuation at a fixed $d$ increases, i.e., the
interactions strengthen the momentum fluctuations as now nucleons are allowed
to move into the high-momentum states above the Fermi surface. Obviously the
moment $\langle k^{\gamma}\rangle^{1/\gamma}$ is limited by the Fermi momentum
$k_{\mathrm{F}}$ in the FFG model since no nucleons are allowed to be above
the Fermi surface. The finding that fluctuations due to interactions are
stronger in low dimensions is well known in solid state physics, as the
Mermin-Wagner theorem [191] states that continuous symmetries cannot be
spontaneously broken at finite temperatures in systems with short-range
interactions in dimensions $d\leq 2$. More intuitively, this means that long-
range fluctuations can be created with small energy costs. In addition, we
also find that the relative fluctuation $\Upsilon_{k}$ in low dimensions
approaches its FFG counterpart (although both are large quantitatively), i.e.,
$\Upsilon_{k}$ approaches the free model prediction. More quantitatively, we
have $\Upsilon_{k}^{\mathrm{HMT}}(4)\approx 0.406$,
$\Gamma_{k}^{\mathrm{HMT}}(3)\approx 0.442$,
$\Gamma_{k}^{\mathrm{HMT}}(2)\approx 0.467$ and
$\Gamma_{k}^{\mathrm{HMT}}(1)\approx 0.632$. Compared with
$\Upsilon_{k}^{\mathrm{FFG}}(4)\approx 0.204$,
$\Upsilon_{k}^{\mathrm{FFG}}(3)\approx 0.258$,
$\Upsilon_{k}^{\mathrm{FFG}}(2)\approx 0.354$ and
$\Gamma_{k}^{\mathrm{FFG}}(1)\approx 0.577$, the HMT enhancements are about
99%, 71%, 32% and 9.5%, respectively as $d$ decreases from 4 to 1.

### 6.4. Kinetic symmetry energy in $d$D spaces with SRC-induced high-momentum
nucleons

By expanding the kinetic EOS of ANM and extract the coefficient of the
$\delta^{2}$ term, we obtain the kinetic symmetry energy for $d=$1-4 as,

$\displaystyle E_{\mathrm{sym}}^{\mathrm{kin},(\mathrm{1D})}(\rho)=$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{2M}\Bigg{\\{}1+\frac{8C_{0}}{3\phi_{0}^{3}}\Bigg{[}\left(C_{1}+1\right)\phi_{0}^{3}+\Bigg{(}\left(\frac{3C_{1}}{8}+\frac{9}{8}\right)\phi_{1}-\frac{3\phi_{1}^{2}}{8}-\frac{9C_{1}}{8}-\frac{9}{8}\Bigg{)}\phi_{0}^{2}$
$\displaystyle\hskip
170.71652pt-\frac{(\phi_{1}-1)(-2\phi_{1}+C_{1}+1)}{8}\Bigg{]}\Bigg{\\}},$
(6.31) $\displaystyle E_{\mathrm{sym}}^{\mathrm{kin},(\mathrm{2D})}(\rho)=$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{4M}\Bigg{\\{}1+\frac{4C_{0}}{\phi_{0}^{2}}\Bigg{[}2\left(C_{1}+\frac{1}{2}\right)\phi_{0}^{2}\ln\phi_{0}+\Bigg{(}(C_{1}+2)\phi_{1}-\frac{\phi_{1}^{2}}{2}-\frac{C_{1}}{2}-\frac{1}{4}\Bigg{)}\phi_{0}^{2}$
$\displaystyle\hskip
170.71652pt-\frac{1}{2}(\phi_{1}-1)\left(C_{1}-\frac{3\phi_{1}}{2}+\frac{1}{2}\right)\Bigg{]}\Bigg{\\}},$
(6.32) $\displaystyle E_{\mathrm{sym}}^{\mathrm{kin},(\mathrm{3D})}(\rho)=$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{6M}\Bigg{\\{}1+\frac{9C_{0}}{\phi_{0}}\Bigg{[}\frac{1}{3}+C_{1}-\left(1+\frac{3C_{1}}{5}\right)\phi_{1}+\frac{3\phi_{1}^{2}}{5}-\left(\frac{8}{9}+\frac{8C_{1}}{3}\right)\phi_{0}$
$\displaystyle\hskip
170.71652pt+\left(\frac{5}{9}+\frac{5C_{1}}{3}+\left(\frac{5}{3}+C_{1}\right)\phi_{1}\right)\phi_{0}^{2}\Bigg{]}\Bigg{\\}},$
(6.33) $\displaystyle E_{\mathrm{sym}}^{\mathrm{kin},(\mathrm{4D})}(\rho)=$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{8M}\Bigg{\\{}1-16\left(C_{1}+\frac{1}{4}\right)C_{0}\ln\phi_{0}$
$\displaystyle\hskip
28.45274pt+16C_{0}\Bigg{[}\Bigg{(}\frac{\phi_{0}^{2}}{2}+\frac{1}{3}\Bigg{)}\phi_{1}^{2}+\left(\phi_{0}^{2}-\frac{2}{3}\right)\left(C_{1}+\frac{3}{2}\right)\phi_{1}+\frac{3}{4}\left(\phi_{0}^{2}-1\right)\left(C_{1}+\frac{1}{4}\right)\Bigg{]}\Bigg{\\}}.$
(6.34)

Similarly, we also list the analytical expressions for the fourth-order
symmetry energy for $d=1-4$ as,

$\displaystyle E_{\mathrm{sym,4}}^{\mathrm{kin},(\mathrm{1D})}(\rho)=$
$\displaystyle\frac{4k_{\mathrm{F}}^{2}}{9M}\frac{C_{0}}{\phi_{0}^{3}}\left[\left(-\frac{9\phi_{0}^{2}}{8}+\frac{15}{8}\right)\phi_{1}^{4}+\frac{9(C_{1}+3)\phi_{1}^{3}}{8}\left(\phi_{0}^{2}-\frac{10}{9}\right)-\frac{27(C_{1}+1)\phi_{1}^{2}}{8}\left(\phi_{0}^{2}-\frac{2}{3}\right)\right.$
$\displaystyle\hskip
85.35826pt\left.+\frac{27\phi_{1}}{8}\left(C_{1}+\frac{1}{3}\right)\left(\phi_{0}^{2}-\frac{1}{3}\right)+\phi_{0}^{3}C_{1}-\frac{9\phi_{0}^{2}C_{1}}{8}+\frac{C_{1}}{8}\right],$
(6.35) $\displaystyle E_{\mathrm{sym,4}}^{\mathrm{kin},(\mathrm{2D})}(\rho)=$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{3M}\frac{C_{0}\phi_{1}}{\phi_{0}^{2}}\Bigg{[}\Bigg{(}-\frac{3\phi_{0}^{2}}{4}+\frac{15}{4}\Bigg{)}\phi_{1}^{3}+\left(\phi_{0}^{2}-3\right)(C_{1}+2)\phi_{1}^{2}$
$\displaystyle\hskip
85.35826pt\left.-3\left(C_{1}+\frac{1}{2}\right)\left(\phi_{0}^{2}-\frac{3}{2}\right)\phi_{1}+3\phi_{0}^{2}C_{1}-\frac{3C_{1}}{2}\right],$
(6.36) $\displaystyle E_{\mathrm{sym,4}}^{\mathrm{kin},(\mathrm{3D})}(\rho)=$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{162M}\Bigg{\\{}1+\frac{135C_{0}}{\phi_{0}}\Bigg{[}\frac{27\phi_{1}^{4}}{25}-\left(\frac{27C_{1}}{25}+\frac{9}{5}\right)\phi_{1}^{3}+\left(\frac{9C_{1}}{5}+\frac{3}{5}\right)\phi_{1}^{2}$
$\displaystyle\hskip
85.35826pt+\left(\phi_{0}^{2}-\frac{3}{5}\right)\left(C_{1}-\frac{1}{9}\right)\phi_{1}-\frac{\phi_{0}-1}{9}\left(C_{1}-\frac{1}{3}\right)\left(\phi_{0}-\frac{3}{5}\right)\Bigg{]}\Bigg{\\}},$
(6.37) $\displaystyle E_{\mathrm{sym,4}}^{\mathrm{kin},(\mathrm{4D})}(\rho)=$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{128M}\left\\{1+\frac{32C_{0}\ln\phi_{0}}{3}\left(C_{1}-\frac{3}{8}\right)+192C_{0}\left[\frac{2\phi_{1}^{4}}{9}-\left(\frac{8C_{1}}{27}+\frac{4}{9}\right)\phi_{1}^{3}+\left(\phi_{0}^{2}+\frac{2}{3}\right)\left(C_{1}+\frac{1}{4}\right)\phi_{1}^{2}\right.\right.$
$\displaystyle\hskip
85.35826pt\left.\left.+\frac{\phi_{1}}{2}\left(C_{1}-\frac{1}{6}\right)\left(\phi_{0}^{2}-\frac{2}{3}\right)-\frac{\phi_{0}^{2}-1}{24}\left(C_{1}-\frac{3}{8}\right)\right]\right\\}.$
(6.38)

As a reference, here we also give analytical expressions for the kinetic
symmetry energy as well as the fourth-order kinetic symmetry energy in a space
of general dimensions $d$ (not necessarily integers) as

$\displaystyle
E_{\mathrm{sym}}^{\mathrm{kin},d}(\rho)=\frac{1}{2d(d-4)(d-2)(d+2)}\frac{k_{\mathrm{F}}^{2}}{2M}\Bigg{\\{}2(d+2)\left[8C_{0}(dC_{1}+1)+(d-4)(d-2)\right]$
$\displaystyle\hskip
56.9055pt+C_{0}(d-4)(d-2)\Big{[}\phi_{1}^{2}d^{4}+\left(2C_{1}-5\phi_{1}+2\right)\phi_{1}d^{3}$
$\displaystyle\hskip
56.9055pt+2\left(C_{1}-2C_{1}\phi_{1}+3\phi_{1}^{2}\right)d^{2}+2(2C_{1}-4\phi_{1}+1)d+4\Big{]}\phi_{0}^{d-2}$
$\displaystyle\hskip 56.9055pt-
C_{0}d(d-2)\Big{[}\phi_{1}^{2}d^{4}+(2C_{1}-9\phi_{1}+2)\phi_{1}d^{3}+2\left[10\phi_{1}^{2}-2(2C_{1}+1)\phi_{1}+C_{1}\right]d^{2}$
$\displaystyle\hskip
56.9055pt+2(2C_{1}-8\phi_{1}+1)d+4\Big{]}\phi_{0}^{d-4}\Bigg{\\}},$ (6.39)
$\displaystyle
E_{\mathrm{sym},4}^{\mathrm{kin},d}(\rho)=\frac{1}{24d^{3}(d-4)(d+2)}\frac{k_{\mathrm{F}}^{2}}{2M}\Bigg{\\{}dC_{0}\phi_{0}^{d-4}\Bigg{[}-(d-4)(d-5)(d-6)(d-7)d^{4}\phi_{1}^{4}$
$\displaystyle\hskip 56.9055pt+4d^{3}(d-4)(d-5)(d-6)(dC_{1}+d+2)\phi_{1}^{3}$
$\displaystyle\hskip
56.9055pt-12d^{2}(d-4)(d-5)(d+2)(dC_{1}+1)\phi_{1}^{2}-24d(d-4)(d+2)\left(dC_{1}-\frac{d}{3}+\frac{2}{3}\right)\phi_{1}$
$\displaystyle\hskip
56.9055pt+8(d-2)(d+2)\left(dC_{1}-\frac{d}{2}+\frac{1}{2}\right)\Bigg{]}$
$\displaystyle\hskip
28.45274pt+(d-2)\Bigg{[}C_{0}d^{4}(d-4)^{2}(d-3)(d-5)\phi_{1}^{4}\phi_{0}^{d-2}+4C_{0}d^{3}(d-4)^{2}(d-3)(dC_{1}+d+2)\phi_{1}^{3}\phi_{0}^{d-2}$
$\displaystyle\hskip
56.9055pt+12C_{0}d^{2}(d-4)(d-3)(d+2)(dC_{1}+)\phi_{1}^{2}\phi_{0}^{d-2}$
$\displaystyle\hskip
56.9055pt+24C_{0}d(d-4)(d+2)\left(dC_{1}-\frac{d}{3}+\frac{2}{3}\right)\phi_{1}\phi_{0}^{d-2}$
$\displaystyle\hskip
56.9055pt-8dC_{0}C_{1}\left(d^{2}\phi_{0}^{d-2}-2d\phi_{0}^{d-2}-8\phi_{0}^{d-2}+8\right)$
$\displaystyle\hskip
56.9055pt+(d-1)\left[4d^{2}\left(C_{0}\phi_{0}^{d-2}+1\right)-8d\left(C_{0}\phi_{0}^{d-2}+3\right)-32C_{0}\left(\phi_{0}^{d-2}-1\right)+3\right]\Bigg{]}\Bigg{\\}}.$
(6.40)

If one takes the high momentum cutoff to $\phi_{0}=1$ and $\phi_{1}=0$, all
the kinetic symmetry energies reduce to the FFG predictions e.g.,
$E_{\mathrm{sym},d}^{\mathrm{kin},\mathrm{FFG}}(\rho)={k_{\mathrm{F}}^{2}}/{2dM}$,
see (2.36). Moreover, the $E_{\mathrm{sym},4}^{\mathrm{kin},d}(\rho)$ in
dimensions $d=1$ and $d=2$ are zero consistent with the FFG prediction given
by the expression (2.38).

Fig. 16: Kinetic EOS of ANM in different dimensions.

To have more quantitative evaluations of the kinetic symmetry energies in $d$D
spaces, we now adopt $C_{0}\approx 0.161\pm 0.015$, $C_{1}\approx-0.25\pm
0.07$, $\phi_{0}\approx 2.38\pm 0.56$, $\phi_{1}\approx-0.56\pm 0.10$ [148] as
constrained in the conventional 3D situation and $\rho_{0}\approx 0.16\pm
0.02\,\mathrm{fm}^{-3}$. Then using the $k_{\mathrm{F}}$ given by (2.18) where
$\rho_{d}=\rho^{d/3}$ and $\rho$ is the conventional 3D density, we find that
the fourth-order symmetry energies in $d$D are
$E_{\mathrm{sym},4}^{\mathrm{kin},\mathrm{(1D)}}(\rho_{0})\approx-0.38\pm
0.14\,\mathrm{MeV}$,
$E_{\mathrm{sym},4}^{\mathrm{kin},\mathrm{(2D)}}(\rho_{0})\approx 0.12\pm
0.44\,\mathrm{MeV}$,
$E_{\mathrm{sym},4}^{\mathrm{kin},\mathrm{(3D)}}(\rho_{0})\approx 7.12\pm
2.36\,\mathrm{MeV}$ [148], and
$E_{\mathrm{sym},4}^{\mathrm{kin},\mathrm{(4D)}}(\rho_{0})\approx 20.81\pm
10.91\,\mathrm{MeV}$, respectively. Similarly, one can also obtain the values
of the kinetic EOS of SNM and the quadratic kinetic symmetry energy in $d$D
spaces. Quantitatively, for the former we have
$E_{0,\mathrm{(1D)}}^{\mathrm{kin}}(\rho_{0})\approx 1.55\pm
0.14\,\mathrm{MeV}$, $E_{0,\mathrm{(2D)}}^{\mathrm{kin}}(\rho_{0})\approx
13.61\pm 1.79\,\mathrm{MeV}$,
$E_{0,\mathrm{(3D)}}^{\mathrm{kin}}(\rho_{0})\approx 40.47\pm
9.48\,\mathrm{MeV}$ [148], and
$E_{0,\mathrm{(4D)}}^{\mathrm{kin}}(\rho_{0})\approx 97.53\pm
42.30\,\mathrm{MeV}$, respectively. While for the quadratic kinetic symmetry
energy, we have
$E_{\mathrm{sym}}^{\mathrm{kin},\mathrm{(1D)}}(\rho_{0})\approx 3.98\pm
0.41\,\mathrm{MeV}$,
$E_{\mathrm{sym}}^{\mathrm{kin},\mathrm{(2D)}}(\rho_{0})\approx 5.41\pm
1.45\,\mathrm{MeV}$,
$E_{\mathrm{sym}}^{\mathrm{kin},\mathrm{(3D)}}(\rho_{0})\approx-13.80\pm
9.63\,\mathrm{MeV}$ [148], and
$E_{\mathrm{sym}}^{\mathrm{kin},\mathrm{(4D)}}(\rho_{0})\approx-74.04\pm
52.50\,\mathrm{MeV}$, respectively. In addition, the sixth-order kinetic
symmetry energy could also be given. Due to its generally complicated nature
of the long analytical expression, here we do not give its explicit form.
Numerically, they are found to be
$E_{\mathrm{sym},6}^{\mathrm{kin},\mathrm{(1D)}}(\rho_{0})\approx-0.06\pm
0.05\,\mathrm{MeV}$,
$E_{\mathrm{sym},6}^{\mathrm{kin},\mathrm{(2D)}}(\rho_{0})\approx 0.10\pm
0.14\,\mathrm{MeV}$,
$E_{\mathrm{sym},6}^{\mathrm{kin},\mathrm{(3D)}}(\rho_{0})\approx 1.01\pm
0.51\,\mathrm{MeV}$, and
$E_{\mathrm{sym},6}^{\mathrm{kin},\mathrm{(4D)}}(\rho_{0})\approx 4.51\pm
2.25\,\mathrm{MeV}$, respectively.

Fig. 17: Kinetic EOS of PNM to order $\delta^{6}$ in the FFG/HMT model.

For comparisons, we notice that the FFG model predicts
$E_{0}^{\mathrm{kin}}(\rho)=dk_{\mathrm{F}}^{2}/(d+2)$,
$E_{\mathrm{sym}}^{\mathrm{kin}}(\rho)=k_{\mathrm{F}}/2dM$,
$E_{\mathrm{sym,4}}^{\mathrm{kin}}(\rho)=(2/d-1)(2/d-2)k_{\mathrm{F}}^{2}/24dM$,
and
$E_{\mathrm{sym},6}^{\mathrm{kin},d}(\rho)=(3d-2)(d-1)(d-2)(2d-1)k_{\mathrm{F}}^{2}/180d^{5}M$,
respectively. Numerically, we then have
$E_{0,d}^{\mathrm{kin}}(\rho_{0})\lesssim 36.20\,\mathrm{MeV}$,
$E_{\mathrm{sym},2}^{\mathrm{kin},d}(\rho_{0})\lesssim 13.57\,\mathrm{MeV}$,
$E_{\mathrm{sym},4}^{\mathrm{kin},d}(\rho_{0})\lesssim 0.85\,\mathrm{MeV}$,
and $E_{\mathrm{sym},6}^{\mathrm{kin},d}(\rho_{0})\lesssim
0.12\,\mathrm{MeV}$, respectively, for $1\leq d\leq 4$ based on the FFG model
predictions. Fig. 16 shows the components of the kinetic EOS considering the
SRC-induced HMT in comparison with the FFG model predictions . It is seen that
for $d=1$ and $d=2$, all components of the kinetic EOS with the SRC/HMT are
very close to their FFG counterparts. In addition, the fourth-order and the
sixth-order symmetry energies from the FFG model in 1D or 2D are exactly zero.
Even when the SRC-induced HMT is considered in 1D and 2D, the predicted
kinetic symmetry energies are still very close to the FFG model predictions.
Interestingly, however, at higher dimensions the SRC/HMT effects on all the
kinetic energies studied become more apparent. Fig. 17 shows the kinetic EOSs
of PNM to order $\delta^{6}$ in the FFG and HMT models. It is seen that the
two models predict very similarly for $d\leq 4$ that the kinetic energy in PNM
increases with the dimension $d$. The negligible difference between the
kinetic energies in PNM calculated with the two models is expected. This is
because the HMT model parameters were based on the experimental finding that
the HMT fraction is about 20% in SNM while it is less than about 2% in PNM
[148].

Fig. 18: Kinetic EOS of ANM as a function of $d$ at saturation density
$\rho_{0}$. The upper boundary of the red area in the right panel corresponds
to the FFG prediction (to order $\delta^{6}$), while the blue dash line is for
the exact kinetic EOS of PNM from the FFG model.

If one treats the dimension $d$ as a continuous variable, then we can plot
different-order terms of the kinetic EOS as a continuous function of $d$, see
the left panel of Fig. 18, where the red segment on each shaded area
corresponds to the kinetic EOS including the SRC-induced HMT. The kinetic
fourth-order symmetry energy and the sixth-order symmetry energy are
comparatively smaller than the kinetic EOS of SNM and the quadratic kinetic
symmetry energy. The maximum value of the
$E_{\mathrm{sym},4}^{\mathrm{kin},d}(\rho_{0})$ occurs at about $d\approx 4$,
see also the third panel of Fig. 16. The kinetic EOS of SNM and the kinetic
symmetry energy as well as the fourth- (sixth-) order symmetry energy are all
analytical functions of $d$ (as $d\to 0$). In addition, at $d=2$ and/or $d=4$,
both $E_{\mathrm{sym}}^{\mathrm{kin},d}(\rho_{0})$ and
$E_{\mathrm{sym},4}^{\mathrm{kin},d}(\rho_{0})$ are continuous (with respect
to $d$), showing that the apparent singular points of the general expressions
of $E_{\mathrm{sym}}^{\mathrm{kin},d}(\rho)$ and
$E_{\mathrm{sym},4}^{\mathrm{kin},d}(\rho)$ at $d=2$ and/or $d=4$ are unreal
as we discussed earlier. Moreover, for $d\lesssim 2$, consistent with the
results shown in Fig. 16 all the kinetic EOSs shown in Fig. 18 deviate only
slightly from their FFG correspondences (i.e., ideal EOS without interaction).
This may indicate that the $d=2$ is perhaps a critical dimension. This is also
consistent with the relative momentum fluctuation $\Upsilon_{k}$ investigated
in the last subsection, i.e., the prediction on $\Upsilon_{k}$ from the HMT
and the FFG models becomes very close as $d$ decreases, see the right panel of
Fig. 15.

In the right panel of Fig. 18, we show the corresponding results for the
kinetic EOS of PNM. Here the blue dash line and the magenta long dash line are
exact kinetic EOS of PNM for the FFG and the HMT models, respectively. The red
region corresponds to the approximation to order $\delta^{6}$ (with the upper
(lower) boundary corresponding to the FFG (HMT) model). It is seen that the
kinetic EOS of PNM is less affected by the SRC-induced HMT at least for
$d\lesssim 3$, see the blue dash line (exact FFG prediction) and the magenta
long dash line (exact HMT prediction). As we mentioned earlier, this is
because there are few HMT nucleons in PNM as indicated by experiments for the
conventional 3D case.

It is instructive and educational to examine the PNM results for several
limiting cases. As a reference, in 3D we notice that the
$E_{\mathrm{n}}^{\mathrm{kin}}(\rho_{0})=2^{1/3}\cdot
3k_{\mathrm{F}}^{2}/10M\approx 35.1\,\mathrm{MeV}$ for the exact FFG model.
One can show that as $d\to 0$,

$\displaystyle E_{\mathrm{n}}^{\mathrm{kin}}(\rho)\to$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{2M}\frac{d}{d+2}\cdot
2^{1+2/d}\left[1+C_{\mathrm{n}}\left(1-\frac{1}{\phi_{\mathrm{n}}^{2}}\right)\right]$
$\displaystyle\approx$
$\displaystyle\frac{k_{\mathrm{F}}^{2}}{2M}\frac{d}{d+2}\left(2+\frac{4\ln
2}{d}\right)\left[1+C_{\mathrm{n}}\left(1-\frac{1}{\phi_{\mathrm{n}}^{2}}\right)\right]$
$\displaystyle\approx$ $\displaystyle\frac{k_{\mathrm{F}}^{2}\ln
2}{M}\left[1+C_{\mathrm{n}}\left(1-\frac{1}{\phi_{\mathrm{n}}^{2}}\right)\right]\to
0,$ (6.41)

since $k_{\mathrm{F}}\approx\rho_{d}2^{1-2/d}\sqrt{\pi}\to 0$ as $d\to 0$
($\rho_{d}=\rho^{d/3}$ is a constant as $d\to 0$), where
$C_{\mathrm{n}}=C_{0}(1+C_{1})$ and $\phi_{\mathrm{n}}=\phi_{0}(1+\phi_{1})$.
On the other hand, the kinetic EOS of PNM to order $\delta^{6}$ from the HMT
model (lower boundary of the red region indicated by cyan solid line) is
somewhat smaller than the exact HMT prediction (magenta long dash line)
especially for $d\gtrsim 3$. For example,
$E_{\mathrm{n}}^{\mathrm{kin}}(\rho_{0})\approx 51.8\,\mathrm{MeV}$ adopting
the exact formula while it is about $48.5\,\mathrm{MeV}$ for the EOS up to
$\delta^{6}$ in 4D. It indicates that even higher order kinetic symmetry
energies beyond order 6 are still sizable (in the presence of the HMT) in
spaces with higher dimensions.

Finally, similar to the kinetic EOS of SNM in dimension 1, there would be an
extra contribution from the HMT to the kinetic symmetry energy if
$\phi_{0}\to\infty$ is taken. This term is
$k_{\mathrm{F}}^{2}/2M\cdot[8C_{0}(1+C_{1})/3]$, see the expression (6.31),
which is greater than zero since $1+C_{1}>0$. However, the corresponding extra
term contributing to the fourth-order kinetic symmetry energy is
$4C_{0}C_{1}k_{\mathrm{F}}^{2}/9M$, see the expression (6.35). It is generally
negative if $C_{1}<0$. Combining the three extra terms associated with the
kinetic EOS of SNM, the quadratic kinetic symmetry energy and the kinetic
fourth-order symmetry energy, we have

$\Delta
E^{\mathrm{kin}}_{\mathrm{n},(\mathrm{1D})}(\rho)=({16k_{\mathrm{F}}^{2}}/{9M})\cdot
C_{\mathrm{n}},~{}~{}\mathrm{as}~{}~{}\phi_{0}\to\infty.$ (6.42)

This then gives us the EOS of PNM in 1D as
$E_{\mathrm{n}}(\rho)=2k_{\mathrm{F}}^{2}/3M+({16k_{\mathrm{F}}^{2}}/{9M})\cdot
C_{\mathrm{n}}$, since the sixth-order and even higher order symmetry energies
in dimension 1 approach zero as $\phi_{0}\to\infty$. Numerically one find in
1D that $E_{\mathrm{n}}(\rho_{0})\approx 5.03\,\mathrm{MeV}$ and $\Delta
E^{\mathrm{kin}}_{\mathrm{n},(\mathrm{1D})}(\rho_{0})\approx
1.62\,\mathrm{MeV}$ adopting the conventional 3D saturation density as
$\rho_{0}\approx 0.16\,\mathrm{fm}^{-3}$, i.e., the density in 1D is about
$0.16^{1/3}\,\mathrm{fm}^{-1}\approx 0.54\,\mathrm{fm}^{-1}$.

## 7\. The $E_{\mathrm{sym}}(\rho)$ in $\mathrm{2D}$ with perturbative
contribution from $\epsilon^{2}$-order

In this section, we perturb the symmetry energy in 3D to second order of
$\epsilon=d-d_{\mathrm{f}}=d-3$ and study its preliminary features. In
particular, we have by generalizing (4.16),

$\displaystyle E_{\mathrm{sym}}(\rho)=$
$\displaystyle\frac{\overline{k}_{\mathrm{F}}^{2}}{6M}+\left.\frac{\overline{k}_{\mathrm{F}}}{6}\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right|_{|\mathbf{k}|=\overline{k}_{\mathrm{F}}}+\frac{1}{2}U_{\mathrm{sym}}(\rho,\overline{k}_{\mathrm{F}})$
$\displaystyle+\epsilon\left[\frac{\overline{k}_{\mathrm{F}}^{2}}{6M}\left(2\sigma-\frac{1}{3}\right)+\frac{\overline{k}_{\mathrm{F}}}{6}\left({\displaystyle\sigma\overline{k}_{\mathrm{F}}\frac{\partial^{2}U_{0}}{\partial|\mathbf{k}|^{2}}}+\left(\sigma-\frac{1}{3}\right)\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right)+\frac{\sigma}{2}\frac{\partial
U_{\mathrm{sym}}}{\partial|\mathbf{k}|}\cdot\overline{k}_{\mathrm{F}}\right]_{|\mathbf{k}|=\overline{k}_{\mathrm{F}}}$
$\displaystyle+\epsilon^{2}\left[\frac{\sigma^{2}\overline{k}_{\mathrm{F}}^{2}}{12M}\left(\overline{k}_{\mathrm{F}}M\frac{\partial^{3}U_{0}}{\partial|\mathbf{k}|^{3}}+2M\frac{\partial^{2}U_{0}}{\partial|\mathbf{k}|^{2}}+3M\frac{\partial^{2}U_{\mathrm{sym}}}{\partial|\mathbf{k}|^{2}}+2\right)-\frac{\sigma\overline{k}_{\mathrm{F}}^{2}}{18M}\left(M\frac{\partial^{2}U_{0}}{\partial|\mathbf{k}|^{2}}+\frac{M}{\overline{k}_{\mathrm{F}}}\frac{\partial
U_{0}}{\partial|\mathbf{k}|}+2\right)\right.$ $\displaystyle\hskip
56.9055pt+\left.\frac{\sigma^{\prime}\overline{k}_{\mathrm{F}}^{2}}{6M}\left(M\frac{\partial^{2}U_{0}}{\partial|\mathbf{k}|^{2}}+\frac{M}{\overline{k}_{\mathrm{F}}}\frac{\partial
U_{0}}{\partial|\mathbf{k}|}+\frac{3M}{\overline{k}_{\mathrm{F}}}\frac{\partial
U_{\mathrm{sym}}}{\partial|\mathbf{k}|}+2\right)+\frac{\overline{k}_{\mathrm{F}}^{2}}{54M}\left(\frac{M}{\overline{k}_{\mathrm{F}}}\frac{\partial
U_{0}}{\partial|\mathbf{k}|}+1\right)\right]_{|\mathbf{k}|=\overline{k}_{\mathrm{F}}},$
(7.1)

where $\sigma^{\prime}\approx-0.0307$ characterizes the second-order
correction of the Fermi momentum, i.e., via
$k_{\mathrm{F}}\approx\overline{k}_{\mathrm{F}}(1+\sigma\epsilon+\sigma^{\prime}\epsilon^{2})$.
Tab. 6 gives the contributions from different models at order $\epsilon^{2}$
to the symmetry energy. In the ImMDI model, if the $y$ parameter is taken to
be about $-115\,\mathrm{MeV}$ [146], then all the contributions to
$E_{\mathrm{sym}}(\rho_{0})$ at order $\epsilon^{2}$ are negative. Another
feature is that for each model, the second-order contribution can not change
the qualitative conclusion from the linear-order. For example, in the Skyrme
model [102] although the second-order contribution is negative the overall
result is still positive determined by the linear term. In this sense, the
second-order term is really a “perturbation” (small correction) although
$\epsilon=-1$ is applied for the 2D symmetry energy. One can similarly define
the effective form of the symmetry energy in 2D as in (4.28).

|  $\epsilon$-contribution |  $\epsilon^{2}$-contribution | sign of $\epsilon^{2}$-term | reference
---|---|---|---|---
Skyrme | $+2.2\epsilon\,\mathrm{MeV}$ | $-0.82\epsilon^{2}\,\mathrm{MeV}$ | negative | Ref. [102]
MDI | $-1.1\epsilon\,\mathrm{MeV}$ | $-0.92\epsilon^{2}\,\mathrm{MeV}$ | negative | Ref. [145]
ImMDI | $4.4\epsilon\,\mathrm{MeV}+0.14y\epsilon$ | $5.3\epsilon^{2}\,\mathrm{MeV}+0.05y\epsilon^{2}$ | undetermined | Ref. [146]
toy model | $+3.1\epsilon\,\mathrm{MeV}$ | $-0.48\epsilon^{2}\,\mathrm{MeV}$ | negative | $/$
Tab. 6: Second order of $\epsilon$ correction to the symmetry energy from
different models.

If one neglects the momentum dependence of the $U_{0}$ to second order and
third order, i.e., $\partial^{2}U_{0}/\partial|\mathbf{k}|^{2}\approx 0$ and
$\partial^{3}U_{0}/\partial|\mathbf{k}|^{3}\approx 0$, as well as the momentum
dependence of the $U_{\mathrm{sym}}$ to second order, i.e.,
$\partial^{2}U_{\mathrm{sym}}/\partial|\mathbf{k}|^{2}\approx 0$, then the
perturbed symmetry energy to $\epsilon^{2}$-order could be approximated as,

$\displaystyle E_{\mathrm{sym}}(\rho)=$
$\displaystyle\frac{\overline{k}_{\mathrm{F}}^{2}}{6M}+\left.\frac{\overline{k}_{\mathrm{F}}}{6}\frac{\partial
U_{0}}{\partial|\mathbf{k}|}\right|_{|\mathbf{k}|=\overline{k}_{\mathrm{F}}}+\frac{1}{2}U_{\mathrm{sym}}(\rho,\overline{k}_{\mathrm{F}})$
$\displaystyle+\epsilon\left[\frac{\overline{k}_{\mathrm{F}}^{2}}{6M}\left(2\sigma-\frac{1}{3}\right)+\frac{\overline{k}_{\mathrm{F}}}{6}\left(\sigma-\frac{1}{3}\right)\frac{\partial
U_{0}}{\partial|\mathbf{k}|}+\frac{\sigma}{2}\frac{\partial
U_{\mathrm{sym}}}{\partial|\mathbf{k}|}\cdot\overline{k}_{\mathrm{F}}\right]_{|\mathbf{k}|=\overline{k}_{\mathrm{F}}}$
$\displaystyle+\epsilon^{2}\left[\frac{\sigma^{2}\overline{k}_{\mathrm{F}}^{2}}{6M}-\frac{\sigma\overline{k}_{\mathrm{F}}^{2}}{18M}\left(\frac{M}{\overline{k}_{\mathrm{F}}}\frac{\partial
U_{0}}{\partial|\mathbf{k}|}+2\right)\right.$ $\displaystyle\hskip
28.45274pt+\left.\frac{\sigma^{\prime}\overline{k}_{\mathrm{F}}^{2}}{6M}\left(\frac{M}{\overline{k}_{\mathrm{F}}}\frac{\partial
U_{0}}{\partial|\mathbf{k}|}+\frac{3M}{\overline{k}_{\mathrm{F}}}\frac{\partial
U_{\mathrm{sym}}}{\partial|\mathbf{k}|}+2\right)+\frac{\overline{k}_{\mathrm{F}}^{2}}{54M}\left(\frac{M}{\overline{k}_{\mathrm{F}}}\frac{\partial
U_{0}}{\partial|\mathbf{k}|}+1\right)\right]_{|\mathbf{k}|=\overline{k}_{\mathrm{F}}}.$
(7.2)

Then the $\epsilon^{2}$-order contribution to the symmetry energy become
$-0.46\epsilon^{2},-0.36\epsilon^{2},8.2\epsilon^{2}\,\mathrm{MeV}-0.003y\epsilon^{2}$,
and $-0.65\epsilon^{2}$, respectively, for the four models listed in Tab.6. In
addition, the correction to the kinetic symmetry energy in 2D to order
$\epsilon^{2}$ is given as
$[{\overline{k}_{\mathrm{F}}^{2}}/{54M}]\cdot[(3\sigma-1)^{2}+18\sigma^{\prime}]\approx-0.65\,\mathrm{MeV}$,
and consequently $E_{\mathrm{sym}}^{\mathrm{kin}}(\rho_{0})\approx
9.84\,\mathrm{MeV}$. Compared with the result of the linear-order calculation
of about 10.49 MeV (see the estimate (4.39)), the second-order correction
makes the final result much closer to the exact value of about 9.60 MeV. Even
higher order corrections (i.e., those beyond $\epsilon^{2}$) are expected to
be small for $|\epsilon|\lesssim 1$.

## 8\. Summary and discussions of some unresolved issues for future studies

Freeing up the spatial dimension $d$ for neutron-rich matter, we derived the
general expression of its EOS in terms of the nucleon isoscalar and isovector
potentials based on the generalized HVH theorem. We find that the SNM EOS is
more bounded at a higher 3D-equivalent saturation density while the symmetry
energy becomes smaller in spaces with lower dimensions compared to the
conventional 3D case. While we speculated that there are objects and/or sub-
systems of nucleons that may be considered as living in spaces with reduced
dimensions, implications of our findings for nuclear physics and astrophysics
remain to be studied. In the following we summarize our main results and point
out a few unresolved issues deserving further studies.

  1. (a)

Analytical expressions of the nucleon specific energy $E_{0}(\rho)$, pressure
$P_{0}(\rho)$, incompressibility coefficient $K_{0}(\rho)$ and skewness
coefficient $J_{0}(\rho)$ of SNM are derived in a general $d$D space. The
corresponding expressions for the quadratic symmetry energy
$E_{\mathrm{sym}}(\rho)$, its slope parameter $L(\rho)$ and curvature
coefficient $K_{\mathrm{sym}}(\rho)$ as well as the fourth-order symmetry
energy $E_{\mathrm{sym,4}}(\rho)$ are also given.

  2. (b)

The general features of the EOS of ANM in 2D are analyzed in some details. In
particular, we found that the EOS of ANM (including the symmetry energy and
the EOS of SNM) could be obtained from an effective expansion based on
$\sqrt{\rho}$. Moreover, there are no kinetic terms appearing in the general
expressions for $K_{0}(\rho),J_{0}(\rho)$ and $E_{\mathrm{sym,4}}(\rho)$.
Based on this finding, we have shown that the quartic term
$E_{\mathrm{sym,4}}(\rho)$ is generally far smaller than the quadratic
symmetry energy $E_{\mathrm{sym}}(\rho)$ in the sense that
$E_{\mathrm{sym,4}}(\rho_{0})/E_{\mathrm{sym}}(\rho_{0})\lesssim 1\%$,
validating the conventional parabolic approximation for the EOS of ANM in 2D.
Furthermore, the fourth-order symmetry energy in other dimensions (from 1D to
higher dimensions) is also shown to be small due to the specific structure of
$E_{\mathrm{sym,4}}(\rho)$ where the dimension $d$ plays an essential role
(see the expression (2.38)). Thus, from the viewpoint in a $d$D space in which
the nucleons live, the effectiveness of the parabolic approximation of its EOS
is consequently natural and the conventional 3D case is not unique.

  3. (c)

The EOSs in $d$D spaces are derived from the $\epsilon$-expansion (using an
assumed small perturbative dimension $\epsilon=d-d_{\mathrm{f}}$ with
$d_{\mathrm{f}}$ being a reference dimension) based on the EOS in 3D. In
particular, we investigated the kinetic EOS both for the FFG model (section 4
with $d_{\mathrm{f}}=3$ to linear order of $\epsilon$) and for the HMT model
(section 6 with general $d_{\mathrm{f}}$). We found that starting from the
conventional 3D kinetic EOS, one can reasonably approximate the corresponding
kinetic EOS of SNM and the kinetic symmetry energy in 2D, see (4.38) and
(4.39). In addition, the $\epsilon$-expansion based on $d_{\mathrm{f}}\approx
1\sim 3$ even with the SRC-induced HMT included is effective, see results of
Fig. 13. Moreover, the tendency of the EOS in a perturbed dimension is
analyzed using the relevant analytical expressions. Specifically, we found
that the nucleon specific energy and pressure in SNM will be reduced
(enhanced) if the perturbative dimension $\epsilon$ is negative (positive)
compared with the 3D case, e.g., the EOS of SNM in 2D will be correspondingly
softened compared with the 3D EOS of SNM, see the formula (4.15). The
correction to the symmetry energy from the $\epsilon^{2}$ contribution does
not qualitatively change the prediction from the linear approximation, as
shown in section 7. In a nutshell, the 3D EOS of ANM in fact encapsulates the
very relevant/useful information on the EOS in $d_{\mathrm{f}}$D with
$d_{\mathrm{f}}$ being near 3, and the $\epsilon$-expansion provides a useful
technique to explore the low-$d$ EOS of ANM from its 3D counterpart.

  4. (d)

A toy model for the single-nucleon potential is constructed by enforcing the
corresponding EOS of ANM to fulfill all the available empirical constraints,
including the one on the pressure in SNM from studying nuclear collective flow
data in relativistic heavy-ion collision [128], the constraint on the single-
nucleon potential in SNM from optical model analyses of nucleon-nucleus
scattering data [94], the scalar nucleon effective mass around the saturation
density [101], the causality constraint at densities $\lesssim 6\rho_{0}$, the
isobaric analog state constraint on the symmetry energy near the saturation
density [130] as well as the constraint on the EOS of PNM from chiral
effective field theories [144]. The constructed model (which owns certain
momentum dependence) is then applied to study the EOS of ANM in $d$D spaces.
Specifically, we found that the EOS of SNM is enhanced (reduced) if $d$ is
upwardly (downward) perturbed with respect to the 3D case, verifying the
general conclusion found in section 4. This means that the many-nucleon system
tends to be more bounded in spaces with reduced dimensions, while it may
become completely unbounded in spaces with $\gtrsim 4$. The latter phenomenon
in fact reflects some deep principles from the viewpoint of the high-
dimensional geometry regarding the $d$-dimensional sphere. In addition, the
symmetry energy in 2D is found to be reduced compared to its correspondence in
3D.

  5. (e)

The kinetic EOS of ANM in the presence of SRC-induced HMT in the single-
nucleon momentum distribution function in $d$D is studied. Besides the
effectiveness of the $\epsilon$-expansion of the kinetic EOS of SNM, we also
found that the first two moments of the nucleon momentum distribution in lower
dimensions become smaller in both the FFG and HMT models, while the relative
nucleon momentum fluctuation significantly increases as $d$ decreases below 2,
see Fig. 15. More interestingly, the prediction on the relative momentum
fluctuation $\Upsilon_{k}$ from the HMT and the FFG model becomes closer as
$d$ decreases, indicating that the many-nucleon system in low dimensions
behaves like a free system in terms of their kinetic EOSs. The same phenomenon
is also found through the calculations on the isospin-expansion of the kinetic
EOS of ANM, i.e., the predictions on them from the FFG and the HMT model
become very similar in low dimensions (e.g., in 1D or 2D), as shown in the
left panel of Fig. 18. The nearly free property of the kinetic EOS of ANM in
low dimensions may provide a useful tool to explore the EOS in 3D from the
corresponding counterpart in 2D or 1D, or vice versa. Furthermore, the
predictions on the kinetic EOS of PNM by combining the
$E_{0}^{\mathrm{kin}}(\rho)$, $E_{\mathrm{sym}}^{\mathrm{kin}}(\rho)$,
$E_{\mathrm{sym,4}}^{\mathrm{kin}}(\rho)$ and
$E_{\mathrm{sym,6}}^{\mathrm{kin}}(\rho)$ from the HMT and the FFG models are
very close for $d\lesssim 3$, verifying the conventional picture in 3D that
there exist very few high-momentum neutrons in PNM. However, as the dimension
$d$ increases beyond 3, the above predictions from the HMT and the FFG models
tend to deviate, see the right panel of Fig. 18, indicating that the EOS of
PNM in high dimensions may behave very differently from the one in $d\lesssim
3$.

There is currently no systematic study on the EOS of ANM in a space of general
dimension $d$, partially due to the experimental limitation, theoretical
inability or lack of interest. While our work presented here is exploratory in
nature, we found the exploration very interesting and feel confident that the
results summarized above are useful. Certainly, much more work is needed to
further investigate many of the issues we have touched on in the present work.
There are also many other interesting issues to be studied. To stimulate
further studies along this line we list below a few possible such issues:

  1. (a)

We studied in this work the kinetic EOS (both with and without the HMT) in
some details in the general dimension $d$, the next important step is to
effectively/reasonably incorporate the nucleon-nucleon interactions/potentials
in a self-consistent way, either at the phenomenological or at the microscopic
levels. For example, one needs to consider how to determine the model
parameters in $d$D, and the basic principles and/or symmetries to be
considered, etc. Before detailed schemes (fitting schemes and/or the
principles) are available, the $\epsilon$-expansion provides a practical
starting point, e.g., one can use (4.28) to investigate how the 3D knowledge
on the symmetry energy could induce relevant information in dimensions
$\epsilon+d_{\mathrm{f}}$.

  2. (b)

Considering the HMT issue, how can one obtain the EOS of ANM based on the HVH
theorem in general $d$D, i.e., the close relation between the partial
derivative of the energy density with respect to the density and the single
energy at $k_{\mathrm{F}}$, since when the momentum distribution has depletion
as well as the HMT, one does not have
$\partial\varepsilon/\partial\rho=e(k_{\mathrm{F}})$, where $e$ is the single-
nucleon energy as the sum of the kinetic part and the potential part, and
$\varepsilon$ is the energy density of the system, i.e., $e(k_{\mathrm{F}})$
is generally not the chemical potential of the nucleons (which however holds
for the FFG model). This is in fact a fundamental problem in (nuclear) many-
body theories [192].

  3. (c)

As discussed in the last section, considering that the many-nucleon system in
low dimensions is close to its FFG counterpart, can one rely on it (e.g., via
the higher-order $\epsilon$-expansions) to extract useful information for the
corresponding EOS in 3D? A related issue is on the correlation between the
symmetry energy and its slope parameter. It is known that the latter (at
$\rho_{0}$) in 3D is constrained to be about $L\approx fS$ with $f$ being
about 2-3 [122] and $S\equiv E_{\mathrm{sym}}(\rho_{0})$, which is in fact
quite close to its FFG prediction $L=3S$. If the low-dimensional system is
close to the free model, can one establish the corresponding correlation (even
with the effective potential included) to be near $L\approx 3S$ in 2D or
$L\approx 6S$ in 1D, or more generally $L\approx 6S/d$?

  4. (d)

In this work, we study the EOS of ANM in a space of general dimension $d$
totally adopting the non-relativistic calculations. Since the relativistic
kinematics, the nucleon-nucleon interactions, and/or the dimensionality
together may effectively affect the EOS of ANM [100], it would be extremely
interesting to combine the relativistic kinematics and the fruitful nucleon-
nucleon interactions (we have already found that the momentum dependence of
the single-nucleon potential could effectively influence certain problems) in
the general dimension $d$ and to explore how they influence each other
coherently. See the following overall scheme (see (2.21) for the definition of
$\theta_{\mathrm{f}}$ and $e_{J}$ is the general relativistic single-nucleon
energy [63]) on its inner relations.

$\boxed{\begin{CD}\begin{array}[]{c}e_{J}(\rho,\delta,|\mathbf{k}|)~{}\mbox{in~{}$d$D}\\\
\left(\begin{array}[]{c}\mathrm{NN~{}interactions}\\\
\mathrm{dimensionality}\end{array}\right)\end{array}@>{\delta\approx
0}>{\mathrm{HVH/others}}>\begin{array}[]{l}\mbox{Relativistic}\\\
\mathrm{characteristics:}\\\
E_{\mathrm{sym}}(\rho),L(\rho),\cdots\end{array}@>{|\mathbf{k}|,k_{\mathrm{F}}\ll
M}>{\epsilon=d-d_{\mathrm{f}}}>\left\\{\begin{array}[]{l}\mathrm{NR~{}formulas}\\\
\mbox{$d_{\mathrm{f}}$D~{}quantities}\end{array}\right.\\\
@V{}V{|\mathbf{k}|,k_{\mathrm{F}}\ll M}V@V{\theta_{\mathrm{f}}\approx
0}V{}V\\\
\begin{array}[]{c}|\mathbf{k}|^{2}/2M+U_{J}(\rho,\delta,|\mathbf{k}|)\\\
(\mathrm{nucleon~{}potential})\end{array}\begin{array}[]{l}\mathrm{Characteristics:}\\\
P(\rho),K_{0}(\rho),J_{0}(\rho),\cdots\end{array}\\\
@V{}V{}V@V{{|\mathbf{k}|,k_{\mathrm{F}}\ll M}}V{\epsilon=d-d_{\mathrm{f}}}V\\\
\mathrm{NR~{}formulas}@>{\delta\approx 0,\theta_{\mathrm{f}}\approx
0}>{\epsilon=d-d_{\mathrm{f}}\approx
0}>\begin{array}[]{l}\mbox{NR~{}formulas~{}in $d_{\mathrm{f}}$D:}\\\
P(\rho),K_{0}(\rho),J_{0}(\rho),\cdots\end{array}\\\ \end{CD}}$

  5. (e)

The role played by the momentum-dependences of the single-nucleon potentials
especially that of the symmetry potential $U_{\mathrm{sym}}$ on the symmetry
energy is revealed. Currently, we can not determine whether an enhancement or
a reduction may be induced to the $E_{\mathrm{sym}}(\rho)$ if $d$ is
perturbed, see the formula (4.16) and Tab. 4, the discussions after Eq. (4.23)
and those after Eq. (5.49). To be deterministic, more detailed information
about the momentum dependence of the symmetry potential $U_{\mathrm{sym}}$ and
more accurate calculations (e.g., to higher orders of $\epsilon$) are needed.

  6. (f)

The last but not least, it is really important to find applications where the
low-dimensional EOS of ANM could be used. As we have speculated in the
Introduction, the crust of a neutron star may be treated as a quasi-2D system
while its non-rotating core may be consider as a 1D system, and certain sub-
systems of particles in heavy-ion reactions may be also treated as a low-
dimensional manifold. Finding novel signatures of these problems adopting the
general $d$D EOS of ANM is a big challenge and also an exciting task. For
example, the radial equation relating the crust EOS of neutron star matter
with observables should also be derived starting from a suitable 2D setting
[193, 194] (e.g., via the 2D Christoffel coefficients). Thus, the conventional
Tolman-Oppenheimer-Volkoff equation [195] and its input EOS may both need to
be modified to account for the changes from 3D to $d$D spaces.

## Acknowledgment

This work is supported in part by the U.S. Department of Energy, Office of
Science, under Award No. DE-SC0013702, the CUSTIPEN (China- U.S. Theory
Institute for Physics with Exotic Nuclei) under US Department of Energy Grant
No. DE-SC0009971. The authors would like to thank Zhen Zhang for helpful
discussions on the possible low-dimensional nuclear systems created in
experiments or in astrophysics.

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