# Ultracold spin-balanced fermionic quantum liquids with renormalized $P$-wave
interactions

J. M. Alarcón<EMAIL_ADDRESS>Universidad de Alcalá, Grupo de Física
Nuclear y de Partículas, Departamento de Física y Matemáticas, 28805 Alcalá de
Henares (Madrid), Spain  J. A. Oller<EMAIL_ADDRESS>Departamento de Física,
Universidad de Murcia, E-30071 Murcia, Spain.

###### Abstract

We consider a spin-balanced degenerate gas of spin-1/2 fermions whose dynamics
is governed by low-energy $P$-wave interactions, characterized by the
scattering volume $a_{1}$ and effective momentum $r_{1}$. The energy per
particle ${\bar{\cal{E}}}$ in the many-body system is calculated by resumming
the ladder diagrams comprising both particle-particle and hole-hole
intermediate states, following the novel advances recently developed by us in
Ann.Phys.437,168741(2022). This allows to obtain a renormalized result for
${\bar{\cal{E}}}$ within generic cutoff regularization schemes, with
${\bar{\cal{E}}}$ directly expressed in terms of the scattering parameters
$a_{1}$ and $r_{1}$, once the cut off is sent to infinity. The whole set of
possible values of $a_{1}$ and $r_{1}$ is explored for the first time in the
literature looking for minima in the energy per particle with
${\bar{\cal{E}}}$ given as described. They are actually found, but a further
inspection reveals that the associated scattering parameters give rise to
resonance poles in the complex momentum-plane with positive imaginary part,
which is at odds with the Hermiticity of the Hamiltonian. We also determine
that these conflictive poles, with a pole-position momentum that is smaller in
absolute value than the Fermi momentum of the system, clearly impact the
calculation of ${\bar{\cal{E}}}$. As a result, we conclude that unpolarized
spin-1/2 fermionic normal matter interacting in $P$-wave is not stable. We
also study three universal parameters around the unitary limit. Finally, the
whole set of values for the parameters $a_{1}$, $r_{1}$ is characterized
according to whether they give rise to unallowed poles and, if so, by
attending to their pole positions relative to the Fermi momentum of the system
explored.

##  I Introduction

The stability of liquids, whether classical or quantum-mechanical, stems from
the competition between kinetic energy (temperature), repulsive and attractive
forces [1, 2]. Near the absolute zero, attractive forces allow the constituent
particles to stay close to each other – forming bound states – while
repulsion, usually at short distances [3], avoids the collapse of the system
in the thermodynamic limit. In the quantum case, it was recently shown
theoretically [4], and subsequently observed experimentally [5, 6, 7], that
certain bosonic systems can stably liquify at low temperatures thanks to one
of the competing interactions arising from quantum fluctuations. Fermions, on
the other hand, obey the Pauli principle, which acts as a natural mechanism
for short-distance repulsion [8]. Therefore, attractive interactions on their
own may stabilize fermionic quantum liquids.

Over the past decade or so, there have been tremendous advances in the
manipulation and characterization of low-energy atom-atom interactions in the
$P$-wave channel [9, 10, 11, 12, 13, 14], their $S$-wave counterparts being
well controlled and understood for some time now [15]. The effective low-
energy $P$-wave interactions can be tuned essentially at will, including near
resonance, where it may become the dominant interaction in the system, and
certainly does for spin-polarized systems. An important, yet difficult
question due to the non-perturbative nature of the interaction, is whether
fermions near a $P$-wave resonance can form a stable liquid. In Ref. [16] a
polarized degenerate gas of identical fermions is studied near an isotropic or
anisotropic Feshbach resonance [9], and a controlled expansion is developed
for the case of narrow resonances characterized by $|k_{F}/r_{1}|\ll 1$. A
much richer structure of phases is predicted [16] as compared with the case of
$S$-wave interactions.

In this work, we consider instead the case of a spin-balanced, or unpolarized,
degenerate Fermi gas interacting in $P$-wave and address the question whether
such system can be stable. Because of the Pauli exclusion principle the
$P$-wave interactions of two fermions take place necessarily in the triplet
spin channel. For the $P$ wave in a spin-balanced Fermi system to be more
relevant at low energies than the $S$ wave a suppression is required for the
spin singlet interactions in which the $S$ wave takes place. It is beyond the
scope of our research to determine which conditions could drive to this
situation both in the laboratory or in Nature. Nonetheless, we consider that
it is an interesting theoretical question to pursuit whether such system can
bound. It is also worth noticing that $P$-wave interactions provide important
contributions to the equation of state of fermionic systems in nature which
naturally occur unpolarized, like neutron matter [17, 18] that is of key
importance in Astrophysics for the study of neutron stars. Indeed, the
interest in these objects has been boosted by the recent observation of
gravitational waves emitted from the coalescence of a compact binary made by a
black hole and a neutron star [19].

In our study we describe elastic fermion-fermion interaction in $P$-wave by
the effective-range expansion (ERE) up to and including the scattering volume,
$a_{1}$, and the effective momentum, $r_{1}$. At least two low-energy
scattering parameters are needed to end with a renormalized $P$-wave fermion-
fermion interaction [20, 21] within cutoff regularization when sending the
cutoff to infinity. The many-body calculation of the energy density
${\cal{E}}$ is based on the application of the formalism recently developed by
[20], where a compact expression is derived by resumming the ladder diagrams
for a spin-1/2 fermionic many-body system and, for the first time in the
literature, providing renormalized results within a general cutoff
regularization scheme after the cutoff is sent to infinity. The theory
developed in Ref. [20] rests in the reformulation of many-body quantum field
theory undertaken in Ref. [22], which determines the generating functional of
Green functions in the fermionic many-body environment. The interested reader
can also consult Ref. [23] for perturbative calculations up to next-to-leading
order, and Ref. [24] for a recent review, considering also non-perturbative
applications. We give here a brief account of the main features of this
approach and refer to Ref. [20] for a detailed exposition.

We would like to stress now that resumming the two-body interactions with
intermediate particle-particle and hole-hole states, $i.e.$ the ladder
diagrams [25], is the leading interacting contribution for the calculation of
the energy density ${\cal{E}}$ as required by the in-medium non-perturbative
power-counting of Ref. [18], and reviewed in Ref. [24]. If the high-energy
scale of the interactions is $\Lambda$ the expansion of in-medium diagrams is
in powers of $k_{F}/\Lambda_{\xi}$ with $\Lambda_{\xi}\lesssim\Lambda$, so
that subleading contributions suppressed by at least one extra factor of
${\cal O}(k_{F}/\Lambda_{\xi})$. E.g. for the case of neutron matter, with
pions included as explicit degrees of freedom in the theory, the expansion
scale is around $\sqrt{3}\pi f_{\pi}\simeq 0.5$ GeV or $2.53$ fm-1,
corresponding to a density of 0.55 fm-3 [24].  This is large enough to address
the equations of state for pure neutron and symmetric neutral matter in the
density regions of interest for many applications, like for studying neutron
stars [26] or saturation of nuclear matter [18], respectively, among many
others. See Ref. [24] for a recent review. The importance to achieve the
resummation of the ladder diagrams for studying ${\bar{\cal{E}}}$, as a
starting point on top of which to build perturbative many-body corrections,
has been stressed in many instances in the literature. Indeed, as originally
explained by Thouless in his seminal paper [25], the Brueckner theory is a
particular case of the ladder resummation when the intermediate states are
only of the particle-particle type. For more recent papers along this
direction, emphasizing an effective-field theory point of view, we can quote
Refs. [27, 28, 29, 30, 18, 31, 32]. It also follows from these discussions and
reasons that the evaluation of ${\bar{\cal{E}}}$ within the ladder resummation
is certainly an interesting result. This calculation and its proper
interpretation are the main aim of our present research.

The ladder resummation was finally performed algebraically for arbitrary
vacuum fermion-fermion interactions in Ref. [20]. The case of contact
interactions was explicitly solved, providing renormalized results for a
general cutoff regularization scheme. As commented above the approach of Ref.
[20] is the basis for our present research, in which we extend the results
given in the previous reference for a many-body system of spin 1/2 fermions
that interact in $P$ wave. This is so because we now study the energy per
particle ${\bar{\cal{E}}}$ in the whole set of scattering parameters and the
Fermi momentum $k_{F}$ of the system by employing two dimensionless parameters
$u=\tanh(-1/a_{1}r_{1}^{3})$ and $v=\tanh(k_{F}/r_{1})$ with $u,\,v\in(-1,1)$.
In the square $(-1,1)\otimes(-1,1)$ inside the $uv$-plane we find four
continuum one-dimensional sets of $u$ and $v$ values in which the zero-
temperature equation of state exhibits minima, which would correspond to the
equilibrium energy per particle. Other properties of these minima, like the
speed of sound and the compressibility coefficient, are calculated in App. A.

However, we further scrutinize the feasibility of such minima by studying the
pole content of the vacuum $P$-wave scattering amplitude sharing the same
$a_{1}$ and $r_{1}$ parameters. Unfortunately, we find that all these minima
correspond to a vacuum pole structure which includes two resonant poles whose
associated momenta have a positive imaginary part. These configurations are
excluded by general principles, since the momenta in the resonance pole
positions should have negative imaginary part as required by the Hermiticity
of the Hamiltonian [33].

Nonetheless, one should keep in mind that such unallowed poles could be
artifacts of the ERE that lie beyond its radius of convergence. Then, we have
still explored the possibility that some of these conflicting poles, whose
distance from the origin of momentum establishes an upper bound for the radius
of convergence of the ERE, could lie at momenta larger than $2k_{F}$. In this
way, the presence of such forbidden poles does not exclude that the in-medium
momenta involved in the calculation of ${\bar{\cal{E}}}$ may be smaller than
the ERE radius of convergence. This has been done by evaluating the absolute
value of the ratio between $k_{F}$ and the resonant momentum, and it is always
found that this ratio is larger than 1, so that these poles are relevant in
the momentum range of the many-body system and clearly impact the calculation
of ${\bar{\cal{E}}}$.

As a result of these considerations, we conclude that it is not possible that
a degenerate unpolarized fermionic system interacting in $P$ wave reaches a
stable configuration in normal matter for sufficiently low density (such that
one can apply the ERE for characterizing the scattering amplitude in vacuum).
This conclusion is complementary to that of Ref. [16] for a single-species
polarized fermion gas, where it is obtained that the normal matter state is
not stable.

The structure of the manuscript is as follows. After this Introduction we
discuss in Sec. II the formalism employed to calculate the energy density
${\cal{E}}$ and plot the energy per particle ${\bar{\cal{E}}}$ for the whole
set of scattering parameters $a_{1}$ and $r_{1}$ looking for its minima as a
function of the Fermi momentum $k_{F}$. Section III is dedicated to discuss
the pole structure of a $P$-wave partial-wave amplitude characterized by its
scattering volume and effective momentum, and its implications for the
feasibility of calculated ${\bar{\cal{E}}}$, paying special attention to its
minima. Conclusions are gathered in Sec. IV. There is also an appendix
dedicated to other properties of the minima in ${\bar{\cal{E}}}$.

##  II Calculation of the energy per particle

Reference [20] performs the resummation of the Hartree and Fock diagrams in
terms of an arbitrary vacuum $T$ matrix, called $t_{V}$, by resumming the
ladder diagrams [25], shown in Fig. 1. This resummation includes both
particle-particle and hole-hole intermediate states interacting by insertions
of $t_{V}$. We follow the notation of Ref. [20] and employ the resulting
expression for the energy density,

$\displaystyle{\cal{E}}$
$\displaystyle=\frac{k_{F}^{5}}{10m\pi^{2}}-\frac{i}{2}{\rm
Tr}\ln\left(I-t_{m}L_{d}\right)\,.$ (1)

Above, the first term on the right-hand side is the kinetic-energy density of
a spin-$1/2$ Fermi gas, with fermion mass $m$ and Fermi momentum $k_{F}$. The
second term is the interaction-energy density expressed in terms of a trace
that is taken over all two-fermion states inside the Fermi sea, that is, with
momenta under $k_{F}$ (this limitation is encoded in the loop function $L_{d}$
to be specified below). In Eq. (1), $t_{m}$ is an on-shell two-fermion
scattering amplitude that we define after some preliminaries are introduced. A
detailed derivation of this formula is given Section 2.2 of Ref. [20].

The expression for $L_{d}$ is

$\displaystyle L_{d}(p,{\mathbf{a}})$
$\displaystyle\\!=\\!i\frac{mp}{16\pi^{2}}\sum_{\sigma_{1,2}}\\!\int\\!d\hat{{\mathbf{k}}}\,\theta({k_{F}}-|{\mathbf{a}}+p\hat{{\mathbf{k}}}|)\theta({k_{F}}-|{\mathbf{a}}-p\hat{{\mathbf{k}}}|)$
$\displaystyle\times|p\hat{{\mathbf{k}}}\sigma_{1}\sigma_{2}\rangle_{A}\,{{}_{A}\langle}p\hat{{\mathbf{k}}}\sigma_{1}\sigma_{2}|\,.$
(2)

We denote by $|{\mathbf{p}}\sigma_{1}\sigma_{2}\rangle_{A}$ an antisymmetric
two-fermion state. The two fermions have spin 1/2, and third components of
spin $\sigma_{i}$, $i=1,2$. Their relative momentum is
${\mathbf{p}}\equiv({\mathbf{p}}_{1}-{\mathbf{p}}_{2})/2$, where
${\mathbf{p}}_{i}$ is the momentum of the $i_{\rm th}$ particle, and the total
momentum is $2{\mathbf{a}}\equiv{\mathbf{p}}_{1}+{\mathbf{p}}_{2}$. The
Heaviside functions in Eq. (II) guarantee that the two fermions have momentum
below $k_{F}$. The total energy of the on-shell pair is
$2a^{0}=({\mathbf{p}}_{1}^{2}+{\mathbf{p}}_{2}^{2})/2m=({\mathbf{a}}^{2}+{\mathbf{p}}^{2})/m$,
which fixes $p\equiv|{\mathbf{p}}|$ for a given total momentum (since
${\mathbf{a}}$ is conserved in the scattering process).

The in-medium scattering amplitude $t_{m}({\mathbf{a}})$ stems from the
iteration of $t_{V}$ with mixed two-fermion intermediate states in which one
of them has momentum below $k_{F}$ while the momentum of the other one is
unconstrained. The loop function $L_{m}(p,{\mathbf{a}})$ associated to these
intermediate states obeys the expression

$\displaystyle
L_{m}(p,{\mathbf{a}})\\!=\\!-\frac{m}{2}\\!\sum_{\sigma_{1,2}}\\!\int\\!\frac{d{\mathbf{k}}}{(2\pi)^{3}}\big{[}\theta({k_{F}}-|{\mathbf{a}}+{\mathbf{k}}|)$
(3)
$\displaystyle+\theta({k_{F}}-|{\mathbf{a}}-{\mathbf{k}}|)\big{]}\frac{|{\mathbf{k}}\sigma_{1}\sigma_{2}\rangle_{A}\,{{}_{A}\langle}{\mathbf{k}}\sigma_{1}\sigma_{2}|}{{\mathbf{k}}^{2}-{\mathbf{p}}^{2}-i\epsilon}\,.$

The amplitude $t_{m}({\mathbf{a}})$ satisfies the following equation,

$\displaystyle
t_{m}({\mathbf{a}})=t_{V}+t_{V}L_{m}(p,{\mathbf{a}})t_{m}({\mathbf{a}})\,.$
(4)

In Fig. 1, we draw the diagrams resummed by the interacting part of
${\cal{E}}$ in Eq. (1). The Hartree diagrams are shown in Fig. 1(a) and the
Fock ones in 1(b). The double lines (one in black and another in red running
parallel to each other) are fermions inside the Fermi sea, while a single
solid line corresponds to an in-medium interaction $t_{m}$. Notice that in the
Hartree diagrams two facing sets of double lines made up a $L_{d}$, while for
the Fock diagrams this is so for radially opposite double lines. The
intermediate states in $L_{d}$ interact by $t_{m}$, and the three dots
indicate further iterations of the product $t_{m}L_{d}$.

---
Figure 1: Resummation of the Hartree (a) and Fock (b) diagrams. The black and
red lines running in parallel are on-shell fermions with momenta less than
$k_{F}$, and the $L_{d}$ loops are given by two sets of such lines facing each
other (Hartree) or radially opposite (Fock). The inwards single solid lines
are $t_{m}$, and the dots indicate extra insertions of the interaction.

The evaluation of the interaction energy is done using the partial-wave
expansion developed in Ref. [20]. The states in the partial-wave basis are
denoted by $|J\mu\ell Sp\rangle$, with definite total angular momentum $J$,
third component $\mu$, orbital angular momentum $\ell$ and total spin $S$. By
taking the trace in Eq. (1) in the plane-wave basis one has

$\displaystyle{{\cal{E}}}$
$\displaystyle=\frac{k_{F}^{5}}{10m\pi^{2}}-\frac{2i}{m\pi^{3}}\\!\\!\sum_{J\mu\ell
S}\\!\\!\chi(S\ell)^{2}\\!\\!\int_{0}^{k_{F}}\\!\\!a^{2}da\int_{0}^{\sqrt{k_{F}^{2}-a^{2}}}\\!\\!\\!\\!pdp$
(5) $\displaystyle\times\langle J\mu\ell
Sp|\ln\left(I-t_{m}(a{\hat{\mathbf{z}}})L_{d}(p,a{\hat{\mathbf{z}}})\right)|J\mu\ell
Sp\rangle\,,$

where rotational invariance allows to take ${\mathbf{a}}$ along the $z$ axis.
The factor $\chi(S\ell)^{2}$ selects even values for the sum $S+\ell$ because
of the Pauli exclusion principle, being 2 in that case.

The integral equation satisfied by the partial-wave amplitudes (PWAs) in the
many-body environment is more involved than in the vacuum case because of the
extra mixing between PWAs due to the dependence of scattering on the total
momenta. This extra mixing is accounted for within the formalism by the matrix

$\displaystyle{\cal
B}_{J_{2}\mu\ell_{2},J_{1}\mu\ell_{1}}=-2\chi(S\ell_{2})\chi(S\ell_{1})\\!\sum_{m_{3}s_{3}}(m_{3}s_{3}\mu|\ell_{2}SJ_{2})$
(6)
$\displaystyle\times(m_{3}s_{3}\mu|\ell_{1}SJ_{1})\\!\int\\!d{\mathbf{\hat{\mathbf{k}}}}\,Y_{\ell_{2}}^{m_{3}}({\mathbf{\hat{\mathbf{k}}}})^{*}Y_{\ell_{1}}^{m_{3}}({\mathbf{\hat{\mathbf{k}}}})\theta({k_{F}}-|{\mathbf{k}}-a{\hat{\mathbf{z}}}|)\,.$

However, $\mu$ and $S$ are conserved in the scattering process [20]. In the
previous equation $Y_{\ell}^{m}({\mathbf{\hat{\mathbf{k}}}})$ is a spherical
harmonic and $(m_{1}m_{2}m_{3}|j_{1}j_{2}j_{3})$ is the Clebsh-Gordan
coefficient for the addition of the angular momenta
$\mathbf{j}_{1}+\mathbf{j}_{2}=\mathbf{j}_{3}$.

For the case of contact interactions it is shown in Ref. [20] that it is
possible to derive an algebraic expression for the vacuum off-shell PWAs
${t_{V}(p^{\prime},p)}_{\alpha\beta}$ for $p$, $p^{\prime}$ bounded (this is
the case in the integral equation of Eq. (4), where $p$ and
$p^{\prime}<2k_{F}$). Using a cutoff regularization in a generic scheme these
PWAs are renormalized by reproducing the effective-range expansion (ERE) up to
some order, with the cutoff finally sent to infinity. Within a matrix
notation, so that the set of coupled PWAs is denoted by the matrix
${\mathsf{t}_{V}}(p^{\prime},p)$, one can show that

$\displaystyle{\mathsf{t}_{V}}(p^{\prime},p)=\frac{4\pi}{m}(p^{\prime})^{\ell}\tau(p)(p)^{\ell}\,,$
(7) $\displaystyle(p)^{\ell}={\rm
diag}({p}^{\ell_{1}},\ldots,{p}^{\ell_{n}})\,,$

and analogously for $(p^{\prime})^{\ell}$. In the previous equation,
$\ell_{i}$ ($i=1,\ldots,n$) is the orbital angular momentum of the $n$ coupled
PWAs, and

$\displaystyle\tau(p)^{-1}\\!=\\!-(a)^{-1}\\!+\\!\frac{1}{2}(r)p^{2}\\!+\\!\sum_{i=2}^{M}(v^{(2i)}_{\ell})p^{2i}\\!-\\!i(p^{\ell})^{2}(p)\,,$
(8)

with $M$ the order up to which the ERE is reproduced. In this equation $(a)$,
$(r)$ and $(v^{(2i)}_{\ell})$ are $n\times n$ matrices corresponding to the
scattering length, effective range and shape parameters,
respectively.111Depending of the PWAs the dimension of the entries in $(a)$,
$(r)$, etc, are different. We note that a minimum value of $M$, depending on
the PWAs studied, is needed to achieve renormalizability. E.g. in the case of
$P$-wave scattering, considered here, one needs at least $M=2$, as also
obtained in Ref. [21].

In terms of this expression, and using an analogous matrix notation for the
set of PWAs in the many-body environment,

$\displaystyle{\mathsf{t}_{m}}(p,p)=\frac{4\pi}{m}(p^{\ell})\left(\tau(p)^{-1}+[{\cal
G}_{m}(p)]\right)^{-1}(p^{\ell})\,,$ (9) $\displaystyle[{\cal G}_{m}(p)]$
$\displaystyle=-\frac{1}{\pi}\int_{0}^{\infty}\frac{k^{2}dk}{k^{2}-p^{2}-i\epsilon}(k^{\ell})\cdot{\cal
B}\cdot(k^{\ell})\,.$

Notice that the integral over $k$ in $[{\cal G}_{m}(p)]$ is bounded because of
the Heaviside function in Eq. (6) with $a\leq k_{F}$, cf. Eq. (5), such that
$k<2k_{F}$.

Now, we proceed to apply the previous formalism to a fermion liquid
interacting via a spin-independent $P$-wave potential in momentum space

$\displaystyle
v(k,p)={\mathbf{k}}\cdot{\mathbf{p}}(d_{0}+d_{2}(k^{2}+p^{2}))~{},$ (10)

with the parameters $d_{0}$ and $d_{2}$ given in terms of the cutoff to
reproduce the ERE up to and including the effective range. For this one-
channel interaction the ERE simplifies since $(a)$, $(r)$, and $(v_{1}^{2i})$
in Eq. (8) are just numbers equal to $a_{1}$, $r_{1}$ and $v_{1}^{2i}$,
respectively, as the scattering in vacuum is uncoupled. Had we used
dimensional regularization, as in Refs. [29, 28], instead of cutoff
regularization [34], a renormalized result is possible reproducing only the
scattering volume. However, as we argue in [20], we interpret it as an
artifact of dimensional regularization in non-perturbative calculations
because, as it is well known, the power-like divergences are set to zero in
this particular regularization method.222See also Refs. [35, 36] for further
discussions and more examples in vacuum scattering.

In the case of a $P$-wave interaction ($\ell=1$) of two identical fermions,
Fermi statistics requires that $S=1$, i.e. the spin-triplet channel is
selected, and for each $\mu$ only the PWAs with $J\geq|\mu|$ contribute. For
$\mu=0$ we have the mixing between the PWAs ${}^{3}P_{0}$ and ${}^{3}P_{2}$,
while ${}^{3}P_{1}$ does not couple. In the case with $\mu=\pm 1$ the mixing
is between ${}^{3}P_{1}$ and ${}^{3}P_{2}$, and for $\mu=\pm 2$ only the
${}^{3}P_{2}$ contributes. One can solve directly for ${\mathsf{t}_{m}}(p,p)$
in this case and in terms of it calculate the energy density ${\cal{E}}$
applying Eq. (1). The evaluation of the trace of the $\log$ is performed by
diagonalizing the matrix in its argument which is possible because it is a
unitary matrix [20].

We study the energy per particle $\bar{{\cal{E}}}={\cal{E}}/\rho$, where
$\rho=k_{F}^{3}/3\pi^{2}$ is the number density, as a function of the
scattering volume $a_{1}$ and the effective momentum $r_{1}$. The
dimensionless variables

$\displaystyle x$ $\displaystyle=-1/a_{1}r_{1}^{3}~{},$ (11) $\displaystyle y$
$\displaystyle=k_{F}/r_{1}$

are introduced, so that

$\displaystyle\bar{{\cal{E}}}=\frac{3k_{F}^{2}}{10m}f(x,y)=r_{1}^{2}\frac{3y^{2}}{10m}f(x,y)~{}.$
(12)

To look for the minima of ${\bar{\cal{E}}}$ when varying $k_{F}$ for given
values of $a_{1}$ and $r_{1}$ it is advantageous to consider the new
coordinates

$\displaystyle u$ $\displaystyle=\tanh(x)~{},$ (13) $\displaystyle v$
$\displaystyle=\tanh(y)~{},$

and then the whole $xy$-plane is compressed to the finite extent
$(-1,1)\otimes(-1,1)$ in the $uv$-plane. The minimum of ${\bar{\cal{E}}}$ has
to be searched along the $v$ axis, because when $k_{F}$ changes $u$ stays put.
We look for them numerically in steps of $10^{-3}$ in the $v$ variable for a
given $u$. Numerical inaccuracy prevents us from taking smaller steps, which
could a priori limit our search for potentially narrow minima.

---
Figure 2: The energy per particle ${\bar{\cal{E}}}$ in units of kelvin is
plotted in the $uv$-plane. The positions of the minima are also indicated by
the dots on top of the surface.

The energy per particle, in kelvin, ${\bar{\cal{E}}}/k_{B}$, with $k_{B}$ the
Boltzmann constant, is plotted in Fig. 2 in the $uv$-plane. The filled dots in
the same figure mark the positions of the minima. The value of
${\bar{\cal{E}}}$ in absolute terms results by taking, for concreteness,
$|r_{1}|=0.28\,a_{B}^{-1}$ [13] and $m$ for ${}^{6}\mathrm{Li}$ scattering,
with $a_{B}$ the Bohr radius 333Within our conventions the effective range is
multiplied by a factor $-2$ compared to [13].. Notice that for given values of
$x$ and $y$ the sign of $r_{1}$ does not affect ${\bar{\cal{E}}}$ since, as
indicated above, ${\bar{\cal{E}}}=r_{1}^{2}\frac{3y^{2}}{10m}f(x,y)$. In Fig.
2, we observe four branches of minima within the whole set of parametric
values for $a_{1}$ and $r_{1}$, two with ${\bar{\cal{E}}}>0$ and the other two
with ${\bar{\cal{E}}}<0$. Among all of them, and within our numerical
precision, only one branch can approach arbitrarily close to the unitary limit
at $u=v=0$ (i.e. $|a_{1}|,\,|r_{1}|\to\infty$, and $k_{F}\to 0$), though never
strictly on it. This is the longest branch in the top right corner in Fig. 2.
Indeed, as we have checked explicitly, the minimum for $u=0$ happens at
$k_{F}=0$. The other branches only occur for values of $|u|$ large enough
that, for a given $r_{1}$, implies a maximum value for $|a_{1}|$, and stay
away from the unitary limit.444We expect that temperature would not affect our
results on ${\bar{\cal{E}}}$ shown in Fig. 2 as long as it is much smaller
than ${\bar{\cal{E}}}$ expressed in kelvin degrees. This is the case in
ultracold fermionic atom experiments where temperature is typically around a
few tens of $\mu K$ (the only possible exception would be to stay almost on
top the unitary limit where ${\bar{\cal{E}}}$ also tends to vanish). The
values of ${\bar{\cal{E}}}$ along the minima can be better read from Figs. 7
and 8. Other properties of the minima of ${\bar{\cal{E}}}$ are discussed in
the Appendix A. They are relegated there because the presence of these minima
is very much affected by untenable pole structures associated to the ERE used
for their calculation, as discussed in Sec. III.

Let us consider closely the interesting unitary limit with $x=0$ and work out
the leading behavior in powers of $y$ of
$f(0,y)=\bar{{\cal{E}}}\frac{10m}{3k_{F}^{2}}$ as introduced above. To
calculate ${\cal{E}}$ we have the dimensionless combination
$1-{\mathsf{t}_{m}}L_{d}$ inside a $\log$, cf. Eq.(1). In
$k_{F}^{3}\tau(k_{F})$ we can extract a global dimensionless factor
$y^{3}=(k_{F}/r_{1})^{3}$ and the rest adopts the form
$\left(x+\frac{1}{2}y^{2}+v_{1}^{(2)}r_{1}y^{4}-iy^{3}+{\cal
O}(y^{4})\right)^{-1}$, with $x=0$ in the unitary limit. We indicate the real
and imaginary parts of ${\mathsf{t}_{m}}$ as ${\mathsf{t}_{m}^{r}}$,
${\mathsf{t}_{m}^{i}}$, respectively, and $L_{d}$ is written as $iL_{d}^{i}$.
Therefore, ${\mathsf{t}^{r}_{m}}L_{d}^{i}$ is proportional to $y$ while
$1-i{\mathsf{t}_{m}^{i}}L_{d}^{i}$ behaves as $1+{\cal O}(y^{2})$, so that
${\rm
arctan}\\!\left({\mathsf{t}_{m}^{r}L_{d}^{i}}/(1+{\mathsf{t}^{i}_{m}}L_{d}^{i})\right)\sim
y$. We can then write $\bar{{\cal{E}}}=\frac{3k_{F}^{2}}{10m}(1+\xi_{1}y)$,
for $x=0$ and $|y|\ll 1$. Because of the non-perturbative contribution of the
effective-range parameter the unitary limit depends on the order in which the
limits $x\to 0$ and $y\to 0$ are taken. Following an analogous procedure as
before it results that for $|y/x|\ll 1$ the function $f(x,y)$ behaves as
$(1+\xi_{2}y^{3}/x+\xi_{3}y^{5}/x^{2})$, where both $\xi_{2}$ and $\xi_{3}$
are universal since the higher-order shape parameters in the ERE, starting
from $v_{1}^{(2)}$, are ${\cal O}(y^{7})$. Our calculation predicts the
following values for these universal parameters

$\displaystyle\xi_{1}$
$\displaystyle=-6.4054\,,~{}\xi_{2}=-0.9549\,,~{}\xi_{3}=0.2049\,.$ (14)

At this point we would like to dedicate some discussions to the corrections to
our results for normal matter based on resumming the ladder diagrams in Fig. 1
for arbitrary vacuum interactions. For that we apply the power-counting
developed in Ref. [18], and more recently reviewed in Ref. [24]. This power
counting requires to resum the two-body interactions with intermediate
particle-particle and hole-hole states as the leading interacting contribution
for the calculation of the energy density ${\cal{E}}$. If the high-energy
scale of the interactions is $\Lambda$ the expansion of in-medium diagrams is
in powers of $k_{F}/\Lambda_{\xi}$ with $\Lambda_{\xi}\lesssim\Lambda$, so
that subleading contributions suppressed by at least one extra factor of
${\cal O}(k_{F}/\Lambda_{\xi})$. E.g. for the case of neutron matter the
expansion scale is around $\sqrt{3}\pi f_{\pi}\simeq 0.5$ GeV or $2.53$ fm-1,
corresponding to a density of 0.55 fm-3 [24]. Therefore, according to the
power counting of Ref. [18], $\xi_{1}$ and $\xi_{2}$ above should not receive
further contributions from more complicated in-medium Feynman diagrams. Thus,
our study around the unitary limit of the energy per particle for small values
of $k_{F}$ is particularly interesting and solid.

##  III Constraints from the pole structure in vacuum

The secular equation for the poles of $\tau(p)$ in the uncoupled $P$-wave
scattering reads

$\displaystyle-\frac{1}{a_{1}}+\frac{1}{2}r_{1}p^{2}-ip^{3}=0~{}.$ (15)

This equation was already studied in the Appendix E of our previous paper [20]
and applied for specific sets of values of the scattering parameters in $P$
wave. Here we explore it for the full set of $a_{1}$ and $r_{1}$ parametric
values, already considered in Fig. 2 for the calculation of ${\bar{\cal{E}}}$.

Another important aspect not discussed in Ref. [20] is the possibility that a
priori forbidden pole dispositions could drive to an acceptable calculation of
${\bar{\cal{E}}}$. The reason is because the unallowed poles that stem from
the use of the ERE should occur at values of $p$ beyond the radius of
convergence of the ERE. So, if $k_{F}$ is small enough so that 2$k_{F}$ is
smaller than the radius of convergence of the ERE, one could still use that
calculation for ${\bar{\cal{E}}}$ even if the ERE generates unacceptable poles
in the complex $p$-plane beyond its radius of convergence. Notice that in the
calculation of $t_{m}$, Eq. (4), in-medium momenta ${\mathbf{k}}$ up to
$2k_{F}$ are involved in the function $L_{m}(p,a)$ of Eq. (3).

The zeroes in Eq. (15) are

$\displaystyle p_{1}$
$\displaystyle=-\frac{ir_{1}}{6}\left(1+z+z^{-1}\right)~{},$ (16)
$\displaystyle p_{2}$
$\displaystyle=\frac{r_{1}}{6}\left(-i+e^{i\pi/6}z^{-1}-e^{-i\pi/6}z\right)~{},$
$\displaystyle p_{3}$
$\displaystyle=\frac{r_{1}}{6}\left(-i-e^{-i\pi/6}z^{-1}+e^{i\pi/6}z\right)~{},$

where,

$\displaystyle\alpha_{1}$ $\displaystyle\equiv a_{1}^{1/3}=|a_{1}|^{1/3}{\rm
sgn}(a_{1})~{},$ (17) $\displaystyle t$
$\displaystyle\equiv\frac{\alpha_{1}r_{1}}{3\,2^{2/3}}=\frac{\alpha_{1}r_{1}}{108^{1/3}}~{},$
$\displaystyle z$
$\displaystyle\equiv\frac{t}{(1+t^{3}+\sqrt{1+2t^{3}})^{1/3}}~{}.$

From these definitions we have that $z\in\mathbb{R}$, for
$t\geq-1/2^{\frac{1}{3}}$.

One can also work out straightforwardly the residues of the $P$-wave $S$
matrix at the different poles, which read

$\displaystyle\lim_{k\to p_{i}}(k-p_{i})S_{1}(k)$ $\displaystyle=\lim_{k\to
p_{i}}(k-p_{i})\frac{2ik^{3}}{-\frac{1}{a_{1}}+\frac{1}{2}r_{1}k^{2}-ik^{3}}$
$\displaystyle=\frac{-2p_{i}^{3}}{(p_{i}-p_{k})(p_{i}-p_{l})}~{},$ (18)

with $k,l\neq i$. It is well known since Heisenberg’s papers on the $S$ matrix
that the residue of $iS_{\ell}(k)$ at the bound-state poles must be positive.
There is a factor $i$ in front of the $S$ matrix because
${\color[rgb]{0,0,0}{2\pi i{\rm{Res}}S_{\ell}(p_{n})=}}\left.\oint
dkS_{\ell}(k)\right|_{p_{i}}=|C_{i}|^{2}$, integrating around the bound-state
pole $p_{i}$ in a counterclockwise sense, where $C_{i}$ is a normalization
constant multiplying the bound-state wave function [37]. Hence, the residue of
$iS_{\ell}(k)$ must be positive at the bound-state pole. In the following we
introduce the shorter notation ${\rm Res}_{i}$ to refer to the residue of
$iS(k)$ at the $i_{\rm th}$ pole position, cf. Eq. (III).555The requirement
${\rm Res}_{i}>0$ is equivalent to demand that the residue of the $P$-wave
scattering amplitude be positive, i.e. the coupling square $g^{2}$ of a bound
state to the continuum must be positive. The relation is ${\rm
Res}_{i}=g^{2}/2{\rm Im}p_{i}$, with ${\rm Im}p_{i}\geq 0$ for a bound-state
pole.

Our interest for the residue as a disclaimer for the acceptance or rejection
of a bound-state pole rests on whether ${\rm Res}_{i}$ is positive or
negative, respectively. In this sense we can drop the factor $ip_{i}^{3}$ in
${\rm Res}_{i}$ because it is positive for a bound-state pole.666To see this
let us denote by $i\lambda_{i}$, $\lambda_{i}\in\mathbb{R}$, the pole position
for a bound or virtual state, then $ip_{i}^{3}=\lambda_{i}^{3}$, being
positive for a bound state ($\lambda_{i}\geq 0$) and negative for a virtual
state ($\lambda_{i}<0$). Therefore, the sign of ${\rm Res}_{i}$ is not
affected by the factor $ip_{i}^{3}$ for a bound state, while its sign changes
for a virtual state. As a result, keeping $ip_{i}^{3}$ from the numerator in
Eq. (III) has not effects whatsoever in our considerations that follow and we
drop it. Then, we introduce the quantity $\tau_{i}$ defined as

$\displaystyle\tau_{i}=\frac{{\rm
Res}_{i}}{ip_{i}^{3}}=\frac{-2}{(p_{i}-p_{k})(p_{i}-p_{l})}~{},~{}k,l\neq
i~{},$ (19)

as it follows from Eq. (III).

A summary for pole structure in the different ranges of values of $t$ and
$r_{1}$ is presented in Table 1. Its derivation is discussed in detail in the
Appendix B. It is found that the excluded regions have values for the
scattering parameters such that $tr_{1}>0$, which is equivalent to having
$a_{1}>0$, see Eq. (17).

Case | $r_{1}<0$ | $r_{1}>0$
---|---|---
$t<-1/2^{1/3}$ |  $p_{1}$ v.s. | b.s. PR
|  $p_{2}$ b.s. PR | v.s.
|  $p_{3}$ b.s. NR | v.s.
$t=-1/2^{1/3}$ |  $p_{1}=p_{2}$ Double b.s. pole | Double v.s. pole
|  $p_{3}$ v.s. | b.s. PR
$-1/2^{1/3}<t<0$ |  $p_{1}$ v.s. | b.s. PR
|  positive ${\rm Im}p_{2,3}$ | negative ${\rm Im}p_{2,3}$
$0<t$ |  $p_{1}$ b.s. PR | v.s.
| negative ${\rm Im}p_{2,3}$ |  positive ${\rm Im}p_{2,3}$
Table 1: Pole structure for the different regions in $t$ and $r_{1}$. The
values that are excluded are those with some boldface letter in them and
happen for $tr_{1}>0$ or, equivalently, $a_{1}>0$. In the Table, PR(NR) is
positive(negative) residue, and b.s.(v.s.) is bound(virtual) state pole.

###  III.1 Checking the distance to the non allowed poles

Inspection of Fig. 2 clearly reveals that the minima for ${\bar{\cal{E}}}$
occur in the region of scattering parameters driving to a pole content in
conflict with general principles, as summarized in Table 1. This is due to
fact that in the minima $uv<0$, or equivalently $tr_{1}>0$ because of the
relations $x=\operatorname{arctanh}u=-1/108t^{3}$ and
$r_{1}=k_{F}/\operatorname{arctanh}v$, with $k_{F}\geq 0$. However, to
directly rule out because of this reason the set of minima explored in Sec.
II, cf. Fig. 2, would not be justified. The point is that the ERE has a radius
of convergence given by the position of the nearest singularity to threshold,
so that out of this convergence region it could give rise to artifacts, like
the presence of unacceptable poles. However, within its radius of convergence
it is perfectly legitimate to keep using the ERE. This reasoning was not taken
into account in our recent paper [20], which directly rejected the results for
${\bar{\cal{E}}}$ when using the ERE with parameters given rise to
unacceptable poles. Therefore, a deeper analysis on the use of the ERE is also
needed to fully appreciate the range of applicability of the results in Ref.
[20].

${{}^{2S+1}L_{J}}$ |  $a_{1}$ [fm3] |  $r_{1}$ [fm-1] | $t$ |  $p_{3}$ [fm-1] |  $\tau_{3}$ [fm2]
---|---|---|---|---|---
${{}^{1}P}_{1}$ | 2.759 | $-6.54$ | $-1.926$ | $i\,0.352$ | $-1.035$
${{}^{3}P}_{1}$ | 1.536 | $-8.40$ | $-2.059$ | $i\,0.412$ | $-0.668$
Table 2:  Uncoupled nucleon-nucleon $P$-wave PWAs with positive $a_{1}$,
giving rise to excluded pole structures when considering the ERE up to and
including $a_{1}$ and $r_{1}$. We give $a_{1}$, $r_{1}$, $t<-1/2^{1/3}$, the
conflictive bound-state pole $p_{3}$ and its residue $\tau_{3}$ for each PWA.

As illustration of these points let us consider the application of the ERE in
nucleon-nucleon scattering, where one can find uncoupled $P$-wave PWAs with
positive scattering volume, like the ${{}^{1}P}_{1}$ and ${{}^{3}P}_{1}$
ones.777We use the standard notation ${{}^{2S+1}L}_{J}$ to denote PWAs. The
central values of $a_{1}$ and $r_{1}$ for these $P$ waves, taken from Ref.
[38], are shown in Table 2. In the same table we also give the resulting
values of $t$ (which are smaller than $-1/2^{1/3}$), the problematic pole
position $p_{3}$, Eq. (B.12), and the unacceptable $\tau_{3}<0$, Eq. (B.13).
In the case of nucleon-nucleon scattering the radius of convergence of the ERE
is $|p|<m_{\pi}/2\approx 0.35$ fm-1, which signals the onset of the left-hand
cut due to one-pion exchange for $p^{2}<-m_{\pi}^{2}/4$, with $m_{\pi}$ the
pion mass. We observe that for the ${{}^{1}P}_{1}$ partial-wave amplitude the
value of $|p_{3}|$ is almost coincident with this upper limit. For the
${{}^{3}P}_{1}$ PWA the value of $|p_{3}|$ is a bit larger, around 0.41 fm-1.
We then learn from these examples extracted from nucleon-nucleon scattering
that physical values of $a_{1}$ and $r_{1}$ can drive to non-allowed pole
structures, with the conflicting poles happening (barely) out of the
convergence region of the ERE. They cannot lie inside the radius of
convergence for finite-order ERE, if this expansion is reasonable and well-
behaved in its application.888This is perfectly the case for the ${}^{1}P_{1}$
and ${}^{3}P_{1}$ nucleon-nucleon PWAs since higher shape parameters in the
ERE are given in Ref. [38], up to $v_{4}$, and we have checked that the
unallowed bound state for each PWA remains stable if higher-order terms in the
ERE are considered.

Contrarily to the case of nucleon-nucleon scattering we do not know the radius
of convergence of the ERE for the results of ${\bar{\cal{E}}}$ shown in Fig.
2. However, for each set of ERE parameters we can determine the position of
the unacceptable poles for $a_{1}>0$ in the complex $p$-plane. Attending to
Table 1, it turns out that in the regions of the $uv$-plane where the minima
lie the modulus of these non-allowed poles positions is $|p_{2}|$. For our
next considerations it is important to realize that $|p_{2}|$ is larger or
equal than the radius of convergence of the ERE (accepting that it is well
behaved with a progressive improvement in the results as the order is
increased). That is, $|p_{2}|$ corresponding to an unallowed pole is an upper
bound for the radius of convergence of the ERE. We then compare it to $k_{F}$,
and if $2k_{F}>|p_{2}|$ then the results obtained for ${\bar{\cal{E}}}$ are
calculated by employing the ERE beyond its convergence radius, cf. Eq. (9).

In this regard, let us notice that a resonant pole $p_{R}$ in the complex
momentum-space with positive imaginary part also implies a positive imaginary
part of the pole position in energy, $E_{R}\equiv M_{R}+i\Gamma/2$, with
$M_{R}$ and $\Gamma$ real positive numbers. In the $P$-wave scattering
amplitude this pole gives rise to a term of the form
$-g^{2}/(E-M_{R}-i\Gamma/2)$ that produces phase shifts according to the
argument of $M_{R}-E-i\Gamma/2$. Therefore, the phase shifts are negative and
cross $-90$ degrees at the resonance mass. However, a proper resonance
behavior must give rise to positive phase shifts because of the delay in the
phase of the scattering state trapped by the pseudo-stationary resonance.

####  III.1.1 $u<0$ and $v>0$

Let us first consider the region $u<0$ ($t>0$) and $v>0$ in Fig. 2. According
to Table 1 the problem arises from the fact that ${\rm Im}p_{2,3}$ is
positive, so that we evaluate the ratio $k_{F}/|p_{2}|$ as a function $t$
along the two curves where the minima lie. The procedure is the following:
Given $t$ we can calculate along each curve $u(t)$ and $v(t)$ so that the
sought ratio, cf. Eq. (16), is

$\displaystyle\frac{k_{F}}{|p_{2}(t)|}$
$\displaystyle=\frac{6y(t)}{\left|-i+e^{i\pi/6}u(t)^{-1}-e^{-i\pi/6}u(t)\right|}~{}.$
(20)

If this ratio were much smaller than 1 then we could argue that the poles
$p_{2}$ and $p_{3}$ are not directly relevant for the densities involved in
the many-body system, not providing resonant contributions. Conversely, if
this ratio were larger than 1 then the unphysical resonance phenomenon would
be relevant and the associated minimum should be excluded.

Indeed, let us notice that from Eq. (B.16) it follows that

$\displaystyle\frac{2{\rm Re}p_{2}-{\rm
Im}p_{2}}{r_{1}}=\frac{1-z}{12z}\left(2\sqrt{3}-1+(2\sqrt{3}+1)z\right)>0$
(21)

since $0<z<1$. Therefore, the Laurent series in momentum of the $P$-wave
amplitude provided by the ERE around the resonance pole $p_{2}$ or $p_{3}$,
selecting the one with positive real part, converges along the physical real
axis for

$\displaystyle 0<p<|{\rm Re}p_{2}|\left(1+2\sqrt{1-\left(\frac{{\rm
Im}p_{2}}{2{\rm Re}p_{2}}\right)^{2}}\right)~{}.$ (22)

Thus, one has resonant contributions from this pole affecting the PWA for
physical values of $p$.

To be more definitive let us show the typical behavior of the $P$-wave
partial-wave amplitude for the minima with $uv<0$ in Fig. 2. The phase shift,
$\delta$, and the modulus squared of the resulting $P$-wave scattering
amplitudes, $(\sin^{2}\delta)/p^{2}$, are shown in the left and right panels
of Fig. 3 (we take units such that $r_{1}=1$), respectively. There, the top
row is for the minimum with $u=-0.2$, $v=0.67$ (belonging to the branch having
${\bar{\cal{E}}}<0$), and the bottom row corresponds to the minimum with
$u=-0.50$, $v=0.71$ (with ${\bar{\cal{E}}}>0$). The values for $a_{1}$ and
$k_{F}$ in the minima selected are: $a_{1}=4.93~{}r_{1}^{-3}$ and
$k_{F}=0.79~{}r_{1}$ for $u=-0.2,\,v=0.67$; $a_{1}=1.82~{}r_{1}^{3}$ and
$k_{F}=0.90~{}r_{1}$ for $u=-0.50,\,v=0.71$. We can observe clearly marked
resonant behaviors, with strong peaks in the square of the amplitudes and
rapid varying phase shifts happening in both cases for $p<k_{F}$. This
unacceptable strong resonance signal should clearly affect the calculation of
${\bar{\cal{E}}}$, which then becomes non trustable and it should be rejected.
Notice that these phase shifts have an inverse resonance shape because of the
reason explained above associated with the wrong sign of the imaginary part of
the momentum in the pole position. Both minima drive to qualitatively similar
behaviors of the $P$-wave scattering amplitude.

Region around the unitary limit: One has then that $u\to 0^{-}$,
$t\to+\infty$, $z(t)\to 1$, and

$\displaystyle
p_{2}=r_{1}\left(\frac{1}{3\sqrt{6}t^{3/2}}+i\frac{1}{54t^{3}}\right)+{\cal
O}(t^{-9/2})~{}.$ (23)

It is not possible to get rid of this pole in the unitary limit since

$\displaystyle\frac{k_{F}}{|p_{2}|}\to 3\sqrt{6}yt^{3/2}~{},$ (24)

and it diverges for $t\to+\infty$.

####  III.1.2 $u>0$ and $v<0$

We can also proceed similarly for the two curves of minima in the region with
$u>0$ ($t<0)$ and $v<0$. We distinguish two further regions:

The region $-1/2^{1/3}<t<0$, $r_{1}<0$, is excluded according to Table 1. The
criterion to be studied is the same as already expressed in Eq. (20). In this
region $-1<z<0$, and within it along $-1<z<-(2\sqrt{3}-1)/(2\sqrt{3}+1)$ the
difference $2|{\rm Re}p_{2}|-|{\rm Im}p_{2}|<0$. However, this requires that
$0.0185<u<0.055$ where there are no minima according to Fig. 2. Therefore, the
Laurent series around the resonance pole still converges within the interval
in Eq. (22) for the minima in the lower left quadrant of the $uv$-plane in
Fig. 2.

Analogously as for the case above with $u<0$, $v>0$, let us show the typical
behavior of the $P$-wave scattering amplitude that is driven by the presence
of these unacceptable resonance poles in the complex $p$-plane with positive
imaginary part. We take two typical minima with $u=0.70$, $v=-0.91$ in the
lower branch (with ${\bar{\cal{E}}}<0$), and $u=0.80$, $v=-0.84$ in the upper
branch (with ${\bar{\cal{E}}}>0$). The resulting scattering volume $a_{1}$ and
Fermi momentum $k_{F}$ are: $a_{1}=1.15~{}|r_{1}|^{-3}$ and
$k_{F}=1.54~{}|r_{1}|$ for the minimum with ${\bar{\cal{E}}}<0$;
$a_{1}=0.91~{}|r_{1}|^{-3}$ and $k_{F}=1.23~{}|r_{1}|$ for the minimum with
${\bar{\cal{E}}}>0$. The phase shifts and amplitude squared that follow are
shown in the top and bottom rows in Fig. 4 for the lower- and upper-branch
minima, respectively. In the figure we take units such that $r_{1}=-1$. We
observe again a clear resonance shape in the phase shifts and modulus squared
of the $P$-wave partial-wave amplitude, though now the resonance is wider as
compared with the case in Fig. 4. This is due to the larger imaginary part of
$p_{2}$ for the poles with $u>0$ than for those with $u<0$. For instance, for
the lower-branch minimum with $u<0$ in Fig. 3 we have
$p_{2}=0.48+i\,0.15~{}r_{1}$, while for the analogous one with $u>0$ in Fig. 4
the pole position is $p_{2}=0.80+i\,0.66~{}|r_{1}|$. The peak positions in
Fig. 4 are around $k_{F}$ in both cases and hence the calculations of
${\bar{\cal{E}}}$ are affected by the presence of this wrong strong resonance
signals and should be discarded.

|
---|---
|
Figure 3: Phase shifts and modulus squared of the $P$-wave scattering amplitude as a function of momentum in units of $r_{1}$ for two minima with $u<0$ and $v>0$. The upper two panels correspond to the minimum $u=-0.20,\,v=0.67$ with ${\bar{\cal{E}}}<0$, and the lower panels are for the minimum $u=-0.50,\,v=0.71$ with ${\bar{\cal{E}}}>0$. The interval of values shown in the $p$ axis runs from 0 up to $2k_{F}$. |
---|---
|
Figure 4: Phase shifts and modulus squared of the $P$-wave scattering
amplitude as a function of momentum in units of $|r_{1}|$ for two minima with
$u>0$ and $v<0$. The upper two panels correspond to the minimum
$u=0.70,\,v=-0.91$ with ${\bar{\cal{E}}}<0$, and the lower ones are for the
minimum $u=0.80,\,v=-0.84$ with ${\bar{\cal{E}}}>0$. The interval of values
shown in the $p$ axis runs from 0 up to $2k_{F}$.

For the other region with $t<-1/2^{1/3}$ the problem according to Table 1 is
that the bound-state pole $p_{3}$ has a negative residue. The quotient is now

$\displaystyle\frac{k_{F}}{|p_{3}(t)|}$
$\displaystyle=\frac{6|y(t)|}{\left|-i-e^{-i\pi/6}u(t)^{-1}+e^{i\pi/6}u(t)\right|}$
(25) $\displaystyle=\frac{6|y(t)|}{1-\cos\phi-\sqrt{3}\sin\phi}~{}.$

However, this region is not relevant for the minima in Fig. 2 because
$t\leq-1/2^{1/3}$ implies that $u\leq 0.01852$, which is out of the region
with minima, as one can check by simple inspection of Fig. 2.

###  III.2 Discussion

---
Figure 5: The ratio $k_{F}/|p_{2}|$, Eq. (20), is plotted for the minima of
${\bar{\cal{E}}}$ shown in Fig. 2. The panel (a) corresponds to the branches
of minima with ${\bar{\cal{E}}}<0$, and the panel (b) to those with
${\bar{\cal{E}}}>0$. We observe that the ratio is always greater than 1 for
all the minima pinned down.

For the four branches of minima of ${\bar{\cal{E}}}$ in Fig. 2 we check the
value of the ratio $k_{F}/|p_{2}|$ in Eq. (20), which is the relevant one for
determining the positions of the problematic poles with respect to $k_{F}$.
The resulting values for this ratio are plotted in Fig. 5, with the panel (a)
for the minima with ${\bar{\cal{E}}}<0$ and the panel (b) for
${\bar{\cal{E}}}>0$. We observe from the resulting curves that
$k_{F}/|p_{2}|>1$ and hence the non legitimate poles $p_{2}$ and $p_{3}$ are
relevant for the densities involved in the degenerate gas of identical
spin-1/2 fermions interacting in $P$-wave. In particular, the most important
one for $r_{1}>0(<0)$ is $p_{2}(p_{3})$ which has positive real part.

Therefore, we conclude that a degenerate fermion gas at $T=0$ with spin-
independent $P$-wave interactions, characterized by scattering volume $a_{1}$
and effective momentum $r_{1}$, is not stable. No acceptable minima of
${\bar{\cal{E}}}$ are found in the whole range of values for the scattering
parameters, Fig. 5. This conclusion is complementary to that reached in Ref.
[16] for polarized Fermi gas with identical fermions interacting in $P$-wave,
so that no minima were found for normal matter. We have now extended this
result also for the spin-balanced case.

---
Figure 6: The shaded regions in the quadrants with $uv<0$ comprise those
values of $u$ and $v$ for which $k_{F}/|p_{2}|>0.5$, such that the calculation
of ${\bar{\cal{E}}}$ requires to use the ERE beyond its applicability when
expressed in terms of $a_{1}$ and $r_{1}$. The white regions either do not
have unacceptable poles ($uv>0$) or if they do ($uv<0$) then the ratio
$k_{F}/|p_{i}|<0.5$, with $|p_{i}|$ the absolute value of the pole position(s)
of the problematic pole(s).

We also move on with these considerations and apply them to the quadrants with
$uv<0$ and evaluate for every value of $u$ and $v$ the ratio between $k_{F}$
and the modulus of the unacceptable pole(s), a ratio that we call
$w(u,v)$.999It is clear from Table 1 that if there are more than one unallowed
pole the moduli of their pole positions are common to all of them. We divide
the $uv$-plane in two regions as shown in Fig. 6: i) The one with $uv>0$,
which are free of unacceptable poles, and those with $uv<0$ such that
$w(u,v)<0.5$. The reason for this upper bound is because in the calculation of
$t_{m}$, Eq. (4), in-medium momenta up to $2k_{F}$ are involved in the
function $L_{m}(p,a)$ of Eq. (3). ii) The rest of the quadrants with $uv<0$
are shaded and for them the calculations are based on the use of ERE out of
its radius of convergence. Without further information a conservative attitude
requires its exclusion. Of course, the set of minima discussed above in more
detail lie within the shaded areas of the $uv$-plane in Fig. 6, because
$k_{F}/|p_{2}|>1$ as already shown in Fig. 5.

Finally, we also consider the inspection of ${\bar{\cal{E}}}$ as a function of
$-1/a_{1}k_{F}^{3}$ for $r_{1}=0$. The ERE $P$-wave scattering amplitude
simplifies then to $k^{2}(-1/a_{1}-ik^{3})^{-1}$. The issue here is again the
raise of unacceptable poles for $a_{1}\geq 0$. Taking the limit $r_{1}\to 0$
in the expression for $p_{i}$, $i=1,2,3$, we have

$\displaystyle p_{1}$ $\displaystyle=-i\frac{1}{\alpha_{1}}~{},$ (26)
$\displaystyle p_{2}$ $\displaystyle=\frac{\sqrt{3}+i}{2\alpha_{1}}~{},$
$\displaystyle p_{3}$ $\displaystyle=\frac{-\sqrt{3}+i}{2\alpha_{1}}~{}.$

For $\alpha_{1}>0$ it follows that $p_{1}$ is a virtual state but $p_{2}$ and
$p_{3}$ constitute a couple of resonance poles with positive imaginary part,
which is not allowed. This settles that the radius of convergence of a well-
behaved ERE should be smaller than $1/\alpha_{1}$ for $\alpha_{1}>0$.
Therefore, in the perturbative regime in which $|a_{1}|\to 0$ this upper bound
to the radius of convergence tends to infinity and poses no problem. However,
as $a_{1}\to+\infty$ the radius of convergence tends to zero at least as fast
as $1/\alpha_{1}$.

Imposing then that $2k_{F}<1/\alpha_{1}$ for $a_{1}>0$, the condition
$8\lesssim 1/a_{1}k_{F}^{3}$ is required for an ERE study with only the
scattering volume included to make sense. This translates at the level of the
density of the system into the upper bound $\rho<1/24\pi^{2}a_{1}$.

##  IV Conclusions

In summary, we have performed a non-perturbative renormalized calculation for
unpolarized fermionic quantum liquids interacting with pure $P$-wave spin-
independent interactions, which are characterized by the scattering volume
$a_{1}$ and the effective range $r_{1}$. We have then applied the novel
formula of Ref. [20] that allows one to calculate the energy density of a
spin-1/2 fermion many-body system directly in terms of the partial-wave
amplitudes in vacuum by resumming the Hartree-Fock series in the ladder
approximation. The minima of the energy per particle ${\bar{\cal{E}}}$ are
searched in the whole space of $P$-wave scattering parameters and a rich
structure of minima is observed. Out of the four branches of minima that
result, one of them reaches values close to (but not strictly at) the unitary
limit.

Further constraints are imposed on the existence of these minima by
establishing whether the pole structure of the vacuum scattering amplitudes is
permissible. It is then observed that all the observed minima in
${\bar{\cal{E}}}$ correspond to scattering parameters that give rise to
resonance poles with positive imaginary part in the complex momentum-plane,
which is not acceptable by general arguments based on the Hermitian character
of the Hamiltonian. Nonetheless, this is not sufficient reason to reject those
minima because the ERE can give rise to artifacts beyond its radius of
convergence, like generating unallowed poles, as we have exemplified with the
${{}^{1}P}_{1}$ and ${{}^{3}P}_{1}$ nucleon-nucleon $P$-wave scattering
amplitudes. We have further inquired whether these problematic poles are
directly affecting the interactions for the many-body system and found that
this is actually the case, because the referred poles are located with a
momentum smaller than $k_{F}$ in absolute value. Then, we conclude that no
stable minima result for spin-1/2 balanced Fermi gas in normal matter. Other
regions of possible values of $a_{1}$, $r_{1}$ and $k_{F}$ have been also
identified as lying beyond the radius of expansion of the ERE.

In addition, we have studied the energy per particle of this system around the
unitary limit and given values for the leading universal parameters in an
expansion in powers of $k_{F}$. It would be also interesting to apply these
methods to other systems such as unbalanced-spin quantum liquids and/or
interacting with higher partial waves.

## Acknowledgements

We are grateful to Manuel Valiente for bringing this problem to our attention,
for many useful discussions, his participation in the writing of the
introduction and a critical reading of parts of the manuscript. This work has
been supported in part by the MICINN AEI (Spain) Grants
PID2019-106080GB-C21/AEI/10.13039/501100011033 and
PID2019-106080GB-C22/AEI/10.13039/501100011033.

##  Appendix A Properties of the minima

From the calculation of ${\bar{\cal{E}}}$ we can obtain other thermodynamic
and mechanical properties of the minima by employing well-known thermodynamic
identities. At the minimum the pressure $P$ vanishes, while the chemical
potential $\mu$ is simply ${\bar{\cal{E}}}$. We also consider the sound
velocity $c_{s}$ and the coefficient of compressibility $K$. At a minimum of
${\bar{\cal{E}}}$ these magnitudes fulfill the relations

$\displaystyle c_{s}^{2}$ $\displaystyle=\frac{\partial
P}{\partial\rho(m+{{\bar{\cal{E}}}})}=\frac{1}{m+{\bar{{\cal{E}}}}}\frac{\partial}{\partial\rho}\left(\rho^{2}\frac{\partial{{\bar{\cal{E}}}}}{\partial\rho}\right)\,,$
(A.1) $\displaystyle K^{-1}$
$\displaystyle=c_{s}^{2}\rho(m+{\bar{\cal{E}}})\,.$

Another interesting relation is that
$K^{-1}=\rho^{2}\frac{\partial\mu}{\partial\rho}$, so that the derivative of
the chemical potential with respect to the density is positive at the minima.
The calculated properties of the minima are plotted in Figs. 7
(${\bar{\cal{E}}}<0$) and 8 (${\bar{\cal{E}}}>0$), calculated with the sign of
$r_{1}$ chosen such that $r_{1}v>0$, because then
$k_{F}=r_{1}\operatorname{arctanh}v>0$ as it must (notice that for the minima
$uv<0$). From top to bottom and left to right, we plot $a_{1}$, $\rho$ (both
in units of $a_{B}^{-3}$), ${\bar{\cal{E}}}$ in kelvin, $c_{s}$ in $m/s$, the
ratio $c_{s}/v_{F}$, where $v_{F}=k_{F}/m$ is the Fermi velocity, and
$K/K_{\rm free}$, with $K_{\rm free}=\frac{k_{F}^{5}}{9\pi^{2}}$ the
compressibility coefficient for a free Fermi gas.

Figure 7: Properties for the minima with ${\bar{\cal{E}}}<0$. From top to
bottom and left to right $a_{1}\,[a_{B}^{-3}]$, $\rho\,[a_{B}^{-3}]$,
${\bar{\cal{E}}}\/k_{B}[K]$, $c_{s}\,[m/s]$, $c_{s}/v_{F}$ and
$K/K_{\text{free}}$ are plotted.

In Fig. 7 the minima with ${\bar{\cal{E}}}<0$ are considered. The left branch
when approaching the unitary limit has a clear gas behavior with diverging $K$
and vanishing $\rho$, ${\bar{\cal{E}}}$ and $c_{s}$ in that limit. The other
branch has a maximum value for $a_{1}$ of $67.2\,a_{B}^{-1}$. For the branches
with ${\bar{\cal{E}}}>0$ drawn in Fig. 8 their scattering volumes are also
bounded from above with the maxima values of $a_{1}=101.8\,a_{B}^{-1}$ for
$u<0$, and $49.0\,a_{B}^{-1}$ for $u>0$. In both figures we observe that the
density of particles has typical values of a few fermions in a volume of
$10^{3}\,a_{B}^{3}$. The sound velocity shows typical values of hundreds of
$m/s$ and, except in the case of diverging $a_{1}$, it is larger than $v_{F}$.

Figure 8: Properties for the minima with ${\bar{\cal{E}}}>0$. The light shaded
areas reflect the remaining numerical uncertainty in our calculation. For the
rest see the caption in Fig. 7.

##  Appendix B Pole content as a function of $t$ and $r_{1}$

We now analyze in detail the pole content from Eq. (15) as a function of $t$
and $r_{1}$, indicating the regions that are allowed and excluded according
whether the properties of these poles are allowed or not.

###  B.1 Pole content for $t=-1/2^{\frac{1}{3}}$

At $t=-1/2^{\frac{1}{3}}$ it follows from Eq. (17) that

$\displaystyle r_{1}\alpha_{1}=-54^{1/3}~{}.$ (B.1)

The resulting pole positions are

$\displaystyle p_{1}$ $\displaystyle=-\frac{ir_{1}}{3}~{},$ (B.2)
$\displaystyle p_{2}$ $\displaystyle=p_{1}=-\frac{ir_{1}}{3}~{},$
$\displaystyle p_{3}$ $\displaystyle=\frac{ir_{1}}{6}~{},$

Region excluded $r_{1}<0$: we have a double bound-state pole which is excluded
based on general considerations [37, 39].

Region allowed $r_{1}>0$: we have a double virtual-state pole and a bound
state pole ($p_{3}$). The residue of the bound state is

$\displaystyle\tau_{3}$ $\displaystyle=\frac{8}{r_{1}^{2}}>0~{},$ (B.4)

The borderline value $r_{1}=0$ is excluded because all poles are degenerate
$p_{i}=0$ and they could be considered as a higher-order bound-state pole.

###  B.2 Pole content for $t<-1/2^{\frac{1}{3}}$

In this case $|z(t)|=1$ because

$\displaystyle z(t)$
$\displaystyle=\frac{t}{(1+t^{3}+i\sqrt{-1-2t^{3}})^{1/3}}~{},$ (B.5)
$\displaystyle|z(t)^{3}|$
$\displaystyle=\sqrt{\frac{t^{6}}{(1+t^{3})^{2}-1-2t^{3}}}=1~{},$

and we write $z(t)$ in terms of its phase $\phi(t)$,

$\displaystyle z(t)$ $\displaystyle\equiv e^{i\phi(t)}~{},~{}t<-1/2^{1/3}~{}.$
(B.6)

For $t\to-\infty$ then $u\to e^{i2\pi/3}$ and for $t\to-1/2^{1/3}$ one has
$u\to-1$. Then, its phase moves from $\phi=\pi$ ($t\to-1/2^{1/3}$) to
$\phi=2\pi/3$ ($t\to-\infty$).

Let us now analyze the different pole positions. For the pole position $p_{1}$
we have

$\displaystyle p_{1}=-\frac{ir_{1}}{6}(1+2\cos\phi)~{}.$ (B.7)

For $\phi=2\pi/3$ we have $1+2\cos\phi=0$. For larger values of $\phi<\pi$
then

$\displaystyle 1+2\cos\phi<1+2\cos 2\pi/3=0~{}.$ (B.8)

Thus, if $r_{1}>0$ then $p_{1}$ is a bound state, otherwise ($r_{1}<0$) it is
a virtual state. Regarding $\tau_{1}$ we have

$\displaystyle\tau_{1}$ $\displaystyle=\frac{24}{r_{1}^{2}(1+2\cos
2\phi)}>0~{}.$ (B.9)

This is so because $4\pi/3<2\phi<2\pi$ and then $0<1+2\cos 2\phi$.

Pole position $p_{2}$:

$\displaystyle p_{2}=\frac{ir_{1}}{6}(-1+\cos\phi-\sqrt{3}\sin\phi)~{}.$
(B.10)

For $\pi<\phi<2\pi/3$ it is trivial to conclude that
$-1+\cos\phi-\sqrt{3}\sin\phi<0$ and then $p_{2}$ is a bound state for
$r_{1}<0$ and a virtual state for $r_{1}>0$. Regarding its residue,

$\displaystyle\tau_{2}$ $\displaystyle=\frac{12{\rm
cosec}^{2}\phi}{r_{1}^{2}(1-\sqrt{3}\cot\phi)}>0~{},$ (B.11)

because $\cot\phi<0$ for $\pi<\phi<2\pi/3$.

Pole position $p_{3}$:

$\displaystyle p_{3}$
$\displaystyle=\frac{ir_{1}}{6}(-1+\cos\phi+\sqrt{3}\sin\phi)~{}.$ (B.12)

For $\phi=\pi$ then $-1+\cos\phi+\sqrt{3}\sin\phi=-2$. As $\phi$ decreases
towards the limit value of $2\pi/3$ then $\cos\phi+\sqrt{3}\sin\phi$ becomes
larger since both $\cos\phi$ and $\sin\phi$ increases and for $\phi=2\pi/3$
one has that $-1+\cos\phi+\sqrt{3}\sin\phi=0$. Therefore,
$-1+\cos\phi+\sqrt{3}\sin\phi<0$ for $\pi<\phi<2\pi/3$. Thus, $p_{3}$ is bound
state for $r_{1}<0$ and a virtual state for $r_{1}>0$. The $p_{3}$ pole is
shallower than $p_{2}$. Regarding $\tau_{3}$,

$\displaystyle\tau_{3}$ $\displaystyle=\frac{12{\rm
cosec}^{2}\phi}{r_{1}^{2}(1+\sqrt{3}\cot\phi)}<0~{},$ (B.13)

since $1+\sqrt{3}\cot\phi<0$ for $2\pi/3<\phi<\pi$, being zero for
$\phi=2\pi/3$.

Region Excluded: $r_{1}<0$ because $p_{3}$ is a bound state with negative
residue.

Region Allowed: $r_{1}>0$ since $p_{3}$ becomes a virtual state pole.

The borderline value $r_{1}=0$ should be excluded because all poles are
degenerate at $p_{i}=0$, and they could be considered bound states.

###  B.3 Pole content for $t>-1/2^{\frac{1}{3}}$

For these values of $t$ then $z(t)\in\mathbb{R}$. We have that $z<0$ for
$-1/2^{1/3}<t<0$ and $z>0$ for $t>0$.

Pole position $p_{1}$:

$\displaystyle p_{1}$ $\displaystyle=-\frac{ir_{1}}{6}(1+z+z^{-1})~{}.$ (B.14)

For $-1/2^{1/3}<t<0$, i.e. $-1<z<0$, then $h(z)\equiv 1+z+z^{-1}$ has values
within the interval $(-\infty,-1)$, while for $0<t$, i.e. $0<z$, then $h(z)$
decreases from $+\infty$ to 0 as $z$ increases. As a result, $p_{1}$ is either
a bound state or a virtual state depending on the range of values of $t$ for a
given $r_{1}$ according to:

a) $-1/2^{1/3}<t<0$: $p_{1}$ is a bound state for $r_{1}>0$, and a virtual
state if $r_{1}<0$.

b) $0<t$: $p_{1}$ is a virtual state for $r_{1}>0$, while $p_{1}$ is a bound
state when $r_{1}<0$.

Regarding its residue $\tau_{1}$ ,

$\displaystyle\tau_{1}$
$\displaystyle=\frac{24z^{2}}{r_{1}^{2}(1+z^{2}+z^{4})}>0$ (B.15)

for non-vanishing $z$, and no extra constraint stems from here.

The poles $p_{2}$ and $p_{3}$ are resonance poles that share the same
imaginary part and have real parts with different sign. Their imaginary and
real parts are, respectively,

$\displaystyle{\rm Im}p_{2}$ $\displaystyle={\rm
Im}p_{3}=\frac{r_{1}}{12z}(1-z)^{2}~{},$ (B.16) $\displaystyle{\rm Re}p_{2}$
$\displaystyle=-{\rm Re}p_{3}=\frac{r_{1}\sqrt{3}(1-z^{2})}{12z}~{}.$

Then, in the range $-1/2^{1/3}<t<0$ ($z>0$) we have ${\rm Im}p_{2}<0$ for
$r_{1}>0$, while ${\rm Im}p_{2}>0$ if $r_{1}<0$, which is excluded. In turn,
for $0<t$ ($0<z$) it follows that ${\rm Im}p_{2}>0$ for $r_{1}>0$, which is
excluded, while ${\rm Im}p_{2}<0$ for $r_{1}<0$.

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