aainstitutetext: Research and Education Center for Natural Sciences, Keio
University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japanbbinstitutetext:
Department of Physics, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa
223-8521, Japanccinstitutetext:  International Institute for Sustainability
with Knotted Chiral Meta Matter(SKCM2), Hiroshima University, 1-3-2
Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8511, Japan

# Baryonic Vortex Phase and Magnetic Field Generation in QCD with Isospin and
Baryon Chemical Potentials

Zebin Qiu<EMAIL_ADDRESS>a,b,c and Muneto Nitta<EMAIL_ADDRESS>
###### Abstract

We propose a novel baryonic vortex phase in low energy dense QCD with finite
baryon and isospin chemical potentials. It is known that the homogeneous
charged pion condensate emerges as a ground state at finite isospin chemical
potential, and therein arises the Abrikosov vortex lattice with an applied
magnetic field. We first demonstrate that a vortex with the same quantized
magnetic flux as the conventional Abrikosov vortex, carries a baryon number
captured by the third homotopy group of Skyrmions, once we take into account a
modulation of the neutral pion inside the vortex core. Such a vortex-Skyrmion
state is therefore dubbed the baryonic vortex. We further reveal that when the
baryon chemical potential is above a critical value, the baryonic vortex has
negative tension measured from the charged pion condensation. It implies that
the phase, in which such vortices emerge spontaneously without an external
magnetic field, would take over the ground state at high baryon density. Such
a new phase contributes to the comprehension of QCD phase diagram and relates
to the generation of magnetic fields inside neutron stars.

##  1 Introduction

Quantum Chromodynamics (QCD) phase at finite density is a long-standing hot
issue in nuclear physics Eichmann  _et al._ (2016); Schmidt and Sharma (2017);
Fischer (2019); Rothkopf (2020); Guenther (2021); Braguta (2023); Adam  _et
al._ (2023). In the interested case of 2-flavor ($N_{f}=2$) QCD with $u$ and
$d$ quarks, the finite density effects can be captured via the isospin
chemical potential $\mu_{I}=(\mu_{u}-\mu_{d})/2$ and the baryon chemical
potential $\mu_{B}=(\mu_{u}+\mu_{d})/2$. Finite $\mu_{I,B}$ coexist in the
early universe with lepton flavor asymmetry, astrophysical objects such as
neutron stars, and needless to say, heavy-ion collisions. Despite immense
research interest over decades, the phase structure of dense QCD is still
challenging to fully understand given the formidable theoretical study on
strongly correlated systems from first principles as well as the technical
difficulty typified by the sign problem of Lattice QCD (LQCD) at finite baryon
density. However, in the low energy regime, one can resort to effective field
theories (EFTs) Holt  _et al._ (2016); Hammer  _et al._ (2020); Drischler  _et
al._ (2021). Even within long wavelength physics, novel hadronic phases with
distinct topology have been discovered, which are the main interest of the
present paper. The infrared dynamics of QCD is based on the confinement and
the chiral symmetry breaking, which are common grounds shared by both 2-flavor
and 3-flavor QCD. A specific formalism is the chiral perturbation theory
(ChPT) Gasser and Leutwyler (1984); Scherer (2003) in terms of pion fields
which arise as Nambu-Goldstone (NG) bosons as a result of chiral symmetry
breaking. The topology in ChPT framework is encoded in the conserved
Goldstone-Wilczek (GW) current $j_{B}$ Goldstone and Wilczek (1981), which
could be expressed as a 3-form constructed with the $SU(N_{f})$ current. The
coupling of such current with gauge fields via the triangle anomaly is
incarnated in the Wess-Zumino-Witten (WZW) term Witten (1983), where the gauge
field could be either the electromagnetic $A_{\mu}$ or the baryonic
$A_{\mu}^{B}$, and $\mu_{I}$ and $\mu_{B}$ are nothing but zeroth components
of $A_{\mu}$ and $A_{\mu}^{B}$, respectively.

QCD phase structure at finite $\mu_{B}$ becomes especially intriguing when a
magnetic field is present, as reviewed in Refs. Kharzeev (2015); Miransky and
Shovkovy (2015); Andersen  _et al._ (2016); Yamamoto (2021); Cao (2021);
Iwasaki  _et al._ (2021). Especially, the magnetic catalysis Klimenko (1991)
is considered a general effect at zero temperature, i.e., magnetic field
either induces the quark condensate or enhances an already existent
condensate. More recently, an opposite tendency at finite temperature is
uncovered as the inverse magnetic catalysis Bruckmann  _et al._ (2013), which
is more pronounced when the temperature is approaching the critical
temperature of the crossover Aoki  _et al._ (2006) between hadronic and quark
phases. In reality, strong magnetic fields are prevalent in aforementioned
systems like magnetars Duncan and Thompson (1992); Turolla  _et al._ (2015);
Kaspi and Beloborodov (2017) and quark gluon plasma Shuryak (1978); Skokov
_et al._ (2009); McLerran and Skokov (2014). In LQCD, a magnetic field makes
the sign problem moderated Fukuda  _et al._ (2014) or even bypassed Brauner
_et al._ (2019). All these facts keep steadfast attention to studying QCD with
the magnetic field as a probe. For the record, we nominate magnetically
induced states in quark matter, chiral magnetic spiral Basar  _et al._ (2010),
and dual density wave Nakano and Tatsumi (2005), for interested readers.

More pertinent to the hadronic sector that we focus on are the proposals of a
neutral pion $\pi^{0}$ domain wall Son and Stephanov (2008) and its stacking
offshoot, chiral soliton lattice (CSL) Eto  _et al._ (2013); Brauner and
Yamamoto (2017). In short, CSL is a result of the dimensional reduction that
occurs when an axial magnetic field breaks the symmetry $SU(2)$ further into
$U(1)$. Meanwhile the still conserved topological charge decreases the free
energy through the WZW term and breaches the vacuum when the magnetic field is
above critical value $B\geq B_{\text{CSL}}=16\pi m_{\pi}f^{2}_{\pi}/e\mu_{B}$
(with $m_{\pi}$ the effective pion mass and $f_{\pi}$ the pion decay
constant). In the wake of such studies, there has been a trend of thought that
the ground state of dense matter under strong $B$ could be pure pionic Brauner
(2018), in contrast to familiar baryonic states. A similar CSL composed of the
$\eta^{\prime}$ meson (or $\eta$ meson for the two-flavor case) Huang  _et
al._ (2018); Nishimura and Yamamoto (2020); Chen  _et al._ (2021) or its
decomposition to a non-Abelian CSL Eto  _et al._ (2022, 2024a) was found to be
the ground state under rapid rotation.

In view of the underlying topology, from the commonly defined GW current
$j_{B}$, one can manifest that the two kinds of states feature distinguished
homotopy, $\pi_{1}[U(1)]\simeq{\mathbb{Z}}$ for the $\pi^{0}$ domain wall and
$\pi_{3}[SU(2)]\simeq{\mathbb{Z}}$ for the baryon arising as a Skrymion Adkins
_et al._ (1983). In the latter case, the physical meaning of $j_{B}$ is
nothing but the baryon charge current. The conserved baryon number is
sustained in a finite size system guaranteed by the quartic Skyrme term that
overcomes Derrick’s scaling law Derrick (1964). Along this line, CSL could be
further put into the same framework with crystalline baryons under $B$ via the
simplest extension of ChPT, i.e., the Skyrme model Skyrme (1961); Zahed and
Brown (1986). Interestingly, a phase diagram of CSL and baryonic crystal is
established with parameters of baryon area density and magnetic field Chen
_et al._ (2022), stipulating the CSL lives in the region with lower density
and higher magnetic field. Furthermore, when both magnetic field and density
are high, a more sophisticated ground state has been found as the domain-wall
Skyrmion Eto  _et al._ (2023a, b) (see also Ref. Eto  _et al._ (2024b) for a
rotation counterpart), which is essentially a topological lump supported by
$\pi_{2}(S^{2})\simeq{\mathbb{Z}}$ lodged on top of a domain wall Nitta
(2013a, b); Gudnason and Nitta (2014a, b), understood in view of the
decomposition $SU(2)/U(1)\simeq S^{2}$. Moreover, when the density becomes
sufficiently high for $\eta$ mesons to come into play, we have proposed in
preceding work Qiu and Nitta (2023) the idea of mixed soliton lattice, which
comprise $\pi^{0}$ and $\eta$ mesons while carrying lower energy than their
separate lattices. Our results lead to the conjecture, for the first time, a
quasicrystal in QCD context. These studies substantiate the rich phase
structure of hadronic matter with $\mu_{B}$ and $B$.

Yet possibilities of hadronic phases could be even more opulent when the
isospin chemical potential $\mu_{I}$ is taken into account. It has been known
for a long time that a Bose-Einstein condensation of charged pions $\pi^{\pm}$
Ruck  _et al._ (1976); Migdal (1978) occurs when $\mu_{I}\geq m_{\pi}$. Later
on, the deconfined superconducting phase has been proposed at high $\mu_{I}$
and sketched hypothetically on the phase diagram with regard to QCD crossover
in terms of $\mu_{I}$ and temperature Son and Stephanov (2001). Remarkably,
unlike $\mu_{B}$, finite $\mu_{I}$ does not incur the sign problem so there
have been numerous LQCD efforts Kogut and Sinclair (2002a, b, 2004); Brandt
_et al._ (2018a, 2023); Abbott  _et al._ (2023) reinforcing theoretical
arguments. The relevance of isospin asymmetry branches out to astrophysics,
where cutting-edge subjects based on the $\pi^{\pm}$ condensate include the
newly proposed compact stars called pion stars Brandt  _et al._ (2018b);
Stashko  _et al._ (2023); Kirichenkov  _et al._ (2024), as well as the pion
condensate in early universe with lepton flavor asymmetry signified by
gravitational waves Vovchenko  _et al._ (2021).

Particularly to our interest is the meanwhile baryon-rich context, e.g.,
neutron stars. Therein the $\mu_{I}$ is responsible for the nuclear symmetry
energy, whose constraining has been actively discussed Drischler  _et al._
(2020); Adhikari  _et al._ (2021); Reed  _et al._ (2021); Neill  _et al._
(2023). The origin of magnetic fields in neutron stars remains an open
question. The answer shall lie in the innermost structure of the neutron star.
A popular proposal is the proton superconductivity that occurs during the
cooling via neutrino emission Baym  _et al._ (1969); Graber  _et al._ (2017),
which could last over $10^{6}$ yr Ho  _et al._ (2017). It has been indicated
that for magnetic field $B\lesssim 10^{15}$ G the majority of neutron star
interior is in the Meissner state expelling magnetic fluxes whilst for
stronger $B$ a type-II superconducting phase is possible. The latter case
could retain magnetic fields inside the neutron star core. Exploring the
generation of magnetic fields in such a context is one of our motivations.

To this end, we target a low energy hadronic regime under $B$ with both finite
$\mu_{B}$ and $\mu_{I}$, adopting the method of ChPT and the Skyrme model. We
are interested in phases containing vortices that carry the magnetic flux. The
vortex formed by pions with finite $\mu_{I}$ and $B$ but at $\mu_{B}=0$ has
been studied in Refs. Adhikari  _et al._ (2015); Adhikari (2019); Adhikari
_et al._ (2023); Grønli and Brauner (2022). In particular, Ref. Grønli and
Brauner (2022) yields a phase diagram of CSL and an Abrikosov vortex lattice
(AVL). In terms of incorporating baryons, Ref. Canfora  _et al._ (2021)
considers a baryon arising in a pionic crystal, but without effects from
$\mu_{B}$. On the other hand, the authors in refs. Evans and Schmitt (2022,
2024) have considered an AVL with $\mu_{B}$ and $B$ whereas neglected
$\mu_{I}$. Such a scenario appears as a fate of the instability of CSL under
$\pi^{\pm}$ condensate at large $B$ and $\mu_{B}$ Brauner and Yamamoto (2017).
Distinctively, in the present study we turn on both $\mu_{B}$ and $\mu_{I}$,
and innovatively introduce a spatially dependent phase angle $\varphi(x)$ of
$\pi^{0}$, on top of the azimuthal $\phi$ of $\pi^{\pm}$. Intuitively, the
added $\pi^{0}$ winding would decrease the vortex energy via the WZW term just
as in CSL so our updated vortex configuration could feature lower energy and
conquer the ground state.

In our scenario, the essential difference from aforementioned proposals
consists in that our vortex carries a conserved baryon number, i.e., the
spatial integration of $j_{B}^{0}$. Thus our vortex is of baryonic nature,
entitled the “baryonic vortex”. Yet such an idea has its clue from the
findings of Refs. Gudnason and Nitta (2014b, 2015, 2016); Nitta (2015) that
twisting $U(1)$ modulus of a vortex yields a baryon number in the ChPT or
Skyrme model, where Skyrmions are realized as sine-Gordon solitons inside a
vortex, and such composites are called vortex-Skyrmions. Importantly, we
couple a baryonic vortex within the Skyrme model to the Maxwell
electrodynamics. In this way, the magnetic field is solved self-consistently.
Eventually, by evaluating the energy of the baryonic vortex we determine the
critical $\mu_{B}$, above which the baryonic vortex bears lower energy than
the uniform $\pi^{\pm}$ condensate, under a certain $\mu_{I}$. In effect, we
stipulate the phase boundary between the $\pi^{\pm}$ condensate phase where a
vortex could be created but as an excited state under an external magnetic
field, and the baryonic vortex phase where magnetic fluxes are created
spontaneously without external magnetic fields.

The present paper is organized as follows. We begin with a recap of our
theoretical framework in Sec. 2; the $SU(2)$ ChPT coupled with Maxwell
electrodynamics and the extension to the Skyrme model. Next, in Sec. 3 we lay
out our scenario, from the innovative vortex Skyrmion Ansatz tailored for a
self-generated axial magnetic field to the expression of the action in terms
of such Ansatz. Then naturally we present the soliton solution and its
properties in Sec. 4. Quintessential results are encompassed in Sec. 5, i.e.,
the phase diagram of the $\pi^{\pm}$ condensate and the baryonic vortex phase
parameterized by $\mu_{I}$ and $\mu_{B}$. In Sec. 6, we explore our scenario
in ChPT without the Skyrme term, commenting on the validity and necessity of
our proposal. Finally, Sec. 7 is devoted to the summary of the present study
and the outlook for explaining magnetic fields in neutron stars. In the
appendix, we give details of the equation of motion (EOM) for our Ansatz.

##  2 Chiral Lagrangian and the Skyrme Model

In the present work, the targeted physics regime is the low energy hadronic
phase in the chiral limit. We input a homogeneous isospin chemical potential
$\mu_{I}$ and then presume the ground state is the charged pion condensation.
The chiral Lagrangian in terms of the pionic $SU\left(2\right)$ field $\Sigma$
reads

$\mathcal{L}_{\text{chiral}}=\frac{f_{\pi}^{2}}{2}\mathrm{Tr}\left(D_{\mu}\Sigma^{\dagger}D^{\mu}\Sigma\right),$
(1)

where $f_{\pi}$ is the pion decay constant and $D_{\mu}$ is a covariant
derivative promoted by the $U\left(1\right)$ electromagnetic (EM) gauge field
$A_{\mu}$, together with the gauge field for the third component of isospin
$A_{\mu}^{I}$, i.e.,

$D_{\mu}\Sigma=\partial_{\mu}\Sigma-i\left(A_{\mu}+A_{\mu}^{I}\right)\left[Q,\Sigma\right].$
(2)

In our notation the elementary charge $e$ is absorbed into the definition of
the gauge fields. The electric charge matrix $Q$ amounts to
$\mathbb{I}/6+\tau^{3}/2$ with the Pauli matrices $\tau^{i}$. The $\mu_{I}$
comes into play via $A_{\mu}^{I}=\delta_{\mu 0}\mu_{I}$, acting as an electric
potential in the covariant derivative, but contributing differently to the
gauged Wess-Zumino-Witten (WZW) term:

$\mathcal{L}_{\text{WZW}}=\left(qA_{\mu}+A_{\mu}^{B}\right)j_{B}^{\mu},$ (3)

which originates from the triangle anomaly. In the present case $q=1/2$ and
the topological current $j_{B}^{\mu}$ is known as

$\displaystyle j_{B}^{\mu}$
$\displaystyle=\frac{1}{24\pi^{2}}\epsilon^{\mu\nu\alpha\beta}\mathrm{Tr}\left[L_{\nu}L_{\alpha}L_{\beta}-\frac{3iQ}{2}\left(F_{\nu\alpha}+F_{\nu\alpha}^{I}\right)\left(L_{\beta}-R_{\beta}\right)\right],$
$\displaystyle\equiv\frac{1}{24\pi^{2}}\epsilon^{\mu\nu\alpha\beta}\mathrm{Tr}\bigg{\\{}l_{\nu}l_{\alpha}l_{\beta}-3iQ\partial_{\nu}\left[\left(A_{\alpha}+A_{\alpha}^{I}\right)\left(l_{\beta}-r_{\beta}\right)\right]\bigg{\\}},$
(4)

where the $SU\left(2\right)$ currents are
$l_{\mu}=\Sigma^{\dagger}\partial_{\mu}\Sigma$ and
$r_{\mu}=\Sigma\partial_{\mu}\Sigma^{\dagger}$, whose covariant versions read
$L_{\mu}=\Sigma^{\dagger}D_{\mu}\Sigma$ and $R_{\mu}=\Sigma
D_{\mu}\Sigma^{\dagger}$. Apparently,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength
of EM gauge field. So forth defined is the isospin counterpart
$F_{\mu\nu}^{I}$. In short, though affecting the constitution of
$j_{B}^{\mu}$, $A_{\mu}^{I}$ is absent in the bracket among Eq. (3) because
$\mu_{I}$ is associated with the $SU\left(2\right)$, rather than $U(1)$,
gauging of the WZW term.

We are to deal with a dynamical magnetic field so the contribution from the
self-coupling of EM fields to the Lagrangian should be included

$\mathcal{L}_{\text{EM}}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu},$ (5)

in which $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field
strength. Keeping in mind the chiral perturbation theory (ChPT) is an
effective theory based on momentum $p$ expansion, we remark that there exist
two kinds of power counting schemes, concerning the gauge field; one is to
count $e\sim\mathcal{O}\left(p\right)$, and the other is to count
$A_{\mu}\sim\mathcal{O}\left(p\right)$. The bifurcation would influence the
order of $\mathcal{L}_{\mathrm{EM}}$ which actually contains the abbreviated
$1/e^{2}$ as a factor. In the former scheme
$\mathcal{L}_{\mathrm{EM}}\sim\mathcal{O}\left(p^{2}\right)$ so higher order
terms in the Lagrangian typified by the Skyrme term could be regarded as the
next leading order while in the latter scheme,
$\mathcal{L}_{\mathrm{EM}}\sim\mathcal{O}\left(p^{4}\right)$ makes it
necessary to go beyond the $\mathcal{O}(p^{2})$ chiral Lagrangian to evaluate
EM fields consistently. Moreover, one can check that a similar nuance of power
counting applies to the $\mathcal{L}_{\text{WZW}}$. We would further address
the consequence brought by such a nuance in Sec. 6. Our choice of the
$\mathcal{O}\left(p^{4}\right)$ term would be the Skyrme term

$\mathcal{L}_{\text{Skyrme}}=\frac{1}{32s^{2}}\mathrm{Tr}\left(\left[L_{\mu},L_{\nu}\right]\left[L^{\mu},L^{\nu}\right]\right).$
(6)

The dimensionless parameter $s$ was originally fixed by matching the
experiments and would be taken as a constant in our study.

##  3 Vortex Skyrmion Ansatz

For simplicity, we specify our scenario to be a static vortex line in a
longitudinal magnetic field, which means that we shut down time dependence and
retain a cylindrical symmetry in the space. The vortex Skyrmion Ansatz is
written as

$\Sigma=\sigma+i\tau^{i}\pi^{i}=\left(\begin{array}[]{cc}e^{i\varphi}\cos\alpha&e^{-i\phi}\sin\alpha\\\
e^{i\phi}\sin\alpha&e^{-i\varphi}\cos\alpha\end{array}\right).$ (7)

The $\phi$ is the azimuthal angle among the cylindrical coordinate system
$\left(\rho,\phi,z\right)$. $\alpha$ and $\varphi$ are functions of $\rho$ and
$z$, which are to be solved. To produce the magnetic field
$\boldsymbol{B}=B\left(\rho\right)\hat{z}$, the needed spatial component of
the gauge field is $\boldsymbol{A}=A_{\phi}\left(\rho\right)\hat{\phi}$.
Meanwhile, we remind the temporal component $A_{0}=\mu_{I}$ being the isospin
chemical potential. We would further simply our setup by presuming
$\varphi=\varphi\left(z\right)$ and $\alpha\left(\rho\right)$. We remind that
our Ansatz does NOT guarantee the configuration with the lowest energy, but
represents generally an excited state on top of the $\pi^{\pm}$ condensate
background. Hence at spatial infinity, the Ansatz should recover the state of
the superconducting $\pi^{\pm}$ ring with vanishing $\pi^{0}$, prescribing the
boundary condition:

$\alpha\left(0\right)=0,\quad\alpha\left(\infty\right)=\frac{\pi}{2}.$ (8)

The condition at origin is instead a pure $\pi^{0}$ sector winding with
$\varphi(z)$, which is reminiscent to CSL.

Intuitively, when there is a finite baryon chemical potential $\mu_{B}$, such
a nontrivial $\pi^{0}$ winding could contribute to a topologically preserved
baryon number which is absent in $\pi^{\pm}$ condensate. The energy could
therefore be reduced by the coupling of the baryon number to $\mu_{B}$ through
the WZW term. If such a reduction is large enough, we conjecture that our
vortex Skyrmion could be of lower energy than the $\pi^{\pm}$ condensate. This
is our original motivation to consider the vortex Skyrmion Ansatz (7). To this
end, we scrutinize the constitution of the baryon density in terms of our
Ansatz:

$\displaystyle j_{B}^{0}$
$\displaystyle=\frac{1}{4\pi^{2}}\left[\left(\frac{1}{\rho}+A_{\phi}\right)\sin
2\alpha\partial_{\rho}\alpha\partial_{z}\varphi-\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho
A_{\phi}\right)\cos^{2}\alpha\partial_{z}\varphi\right]$
$\displaystyle=\frac{1}{4\pi^{2}}\left[\frac{1}{\rho}\sin
2\alpha\partial_{\rho}\alpha\partial_{z}\varphi-\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho
A_{\phi}\cos^{2}\alpha\right)\partial_{z}\varphi\right].$ (9)

Notably, the second portion in the bracket, a surface term after the spatial
integral, vanish in our scenario given $\alpha\left(\infty\right)=\pi/2$. This
is in sheer contrast to the often discussed $\pi^{0}$ domain walls, or the
CSL, in which the surface term yields the familiar
$-\boldsymbol{B}\cdot\nabla\varphi/4\pi^{2}$ whilst the first term in the
bracket of Eq. (9) vanishes due to the absence of charged $\pi^{\pm}$.
Evaluating the topological baryon charge stipulates the boundary conditions
for $\varphi\left(z\right)$, i.e.,

$\displaystyle\int d^{3}xj_{B}^{0}$ $\displaystyle=\frac{1}{4\pi^{2}}\int\rho
d\rho d\phi dz\frac{1}{\rho}\sin
2\alpha\partial_{\rho}\alpha\partial_{z}\varphi=\frac{N_{B}}{2\pi}\varphi\left(z\right)\big{|}_{z=0}^{z=L},$
(10)

where we have used the boundary condition of $\alpha\left(\rho\right)$ in Eq.
(8). Naturally, from the longitudinal periodicity we set one baryon in each
period $L$ and denote the total baryon number as $N_{B}$. Then the boundary
condition can be written as:

$\varphi\left(0\right)=0,\quad\varphi\left(L\right)=2\pi,$ (11)

so that $\int d^{3}xj_{B}^{0}\equiv N_{B}$. From here one can clearly see such
a vortex Skyrmion carries a conserved baryon number. This is the connotation
that we call our vortex Skyrmion a “baryonic vortex”. Under such a setup, we
note apart from $j_{B}^{0}$, the baryon current features nontrivial azimuthal
component relying on the $\mu_{I}$:

$j_{B}^{\phi}=\frac{1}{4\pi^{2}}\frac{\partial}{\partial\rho}\left(\mu_{I}\cos^{2}\alpha\right)\partial_{z}\varphi.$
(12)

Together with Eq. (9) it leads to the following WZW term:

$\displaystyle\mathcal{L}_{\text{WZW}}$
$\displaystyle=\mu_{B}j_{B}^{0}+qA_{\phi}j_{B}^{\phi}.$ (13)

We clarify that throughout this work, as a first step, we consider homogenous
$\mu_{I}$ and $\mu_{B}$ for simplicity.

To concretely solve these configurations, we need the last boundary condition
for $A_{\phi}.$ It can be defined in view of the chiral Lagrangian, whose
specific expression reads:

$\mathcal{L}_{\text{chiral}}=-\frac{f_{\pi}^{2}}{2}\left\\{\left(\partial_{\rho}\alpha\right)^{2}+\left[\left(\frac{1}{\rho}+A_{\phi}\right)^{2}-\mu_{I}^{2}\right]\sin^{2}\alpha+\cos^{2}\alpha\left(\partial_{z}\varphi\right)^{2}\right\\}.$
(14)

At $\rho\rightarrow\infty$, the kinetic energy $-\mathcal{L}_{\text{chiral}}$
density needs to vanish (except for a constant level brought by $\mu_{I}^{2}$
term), which requires $A_{\phi}=-1/\rho$, a pure gauge field. For numerical
convenience we define $a\left(\rho\right)\equiv\rho A_{\phi}\left(\rho\right)$
and dictate its boundary condition as follows

$a\left(0\right)=0,\quad a\left(\infty\right)=-1.$ (15)

For completeness, we also display the expression of the Skyrme term under our
Ansatz

$\displaystyle\mathcal{L}_{\text{Skyrme}}=$
$\displaystyle-\frac{1}{2s^{2}}\left\\{\left(\partial_{\rho}\alpha\right)^{2}\cos^{2}\alpha\left(\partial_{z}\varphi\right)^{2}\right.$
$\displaystyle\left.+\left[\left(\frac{1}{\rho}+A_{\phi}\right)^{2}-\mu_{I}^{2}\right]\sin^{2}\alpha\left[\left(\partial_{\rho}\alpha\right)^{2}+\cos^{2}\alpha\left(\partial_{z}\varphi\right)^{2}\right]\right\\}.$
(16)

As we will show soon, the Skyrme term plays the role to guarantee a finite
size of the soliton, overcoming the Derrick’s scaling law, as it did in the
Skyrme model description of a baryon.

In summary, what we need to solve in the current study are the functions
$\alpha\left(\rho\right)$, $a\left(\rho\right)$ and $\varphi\left(z\right)$
under boundary conditions Eqs. (8), (11) and (15). As long as we do not
consider the time evolution, solving the EOM is tantamount to finding the
soliton configuration that minimizes the energy functional corresponding to
$\mathcal{L}=\mathcal{L}_{\text{chiral}}+\mathcal{L}_{\text{Skyrme}}+\mathcal{L}_{\text{WZW}}+\mathcal{L}_{\text{EM}}$,
specifically

$H=\int d^{3}x\mathcal{H}=\int 2\pi\rho d\rho
dz\left[\frac{1}{2}\left(\frac{1}{\rho}\partial_{\rho}a\right)^{2}-\mathcal{L}_{\text{chiral}}-\mathcal{L}_{\text{Skyrme}}-\mathcal{L}_{\text{WZW}}\right].$
(17)

In practice, we would employ the variational principle $\delta H=0$ to solve
the system. We remark that the variational principle with respect to $A_{\mu}$
is equivalent to the Maxwell equation $\partial_{\mu}F^{\mu\nu}=j_{Q}^{\nu}$
with the electric current $j_{Q}^{\mu}$ defined by

$j_{Q}^{\mu}=qj_{B}^{\mu}+j_{I}^{\mu},$ (18)

among which the non-anomalous part is dubbed the isospin current

$\displaystyle j_{I}^{\mu}\equiv$ $\displaystyle\frac{\delta}{\delta
A_{\mu}}\left(\mathcal{L}_{\text{chiral}}+\mathcal{L}_{\text{Skyrme}}\right)$
$\displaystyle=$
$\displaystyle\delta^{\mu\phi}\left(\frac{1}{\rho}+A_{\phi}\right)\sin^{2}\alpha\left\\{f_{\pi}^{2}+\frac{1}{s^{2}}\left[\left(\partial_{\rho}\alpha\right)^{2}+\left(\partial_{z}\varphi\right)^{2}\cos^{2}\alpha\right]\right\\}.$
(19)

The specific form of the EOMs, two coupled second order ordinary differential
equations, is clarified in the Appendix.

It is worth mentioning that the pion mass term, if added, would impact not
only the Lagrangian by
$\mathcal{L}_{\text{mass}}=(1/2)f_{\pi}^{2}m_{\pi}^{2}\mathrm{ReTr}\Sigma=f_{\pi}^{2}m_{\pi}^{2}\cos\varphi\cos\alpha,$
but also the validity of the above boundary conditions Eqs. (8) and (11). That
is, when $\mu_{I}>m_{\pi}$ the $\pi^{\pm}$ condensate coexists with a finite
$\pi^{0}$ density on the spatial boundary;
$\alpha\left(\rho\rightarrow\infty\right)\rightarrow\cos^{-1}\left(m_{\pi}^{2}/\mu_{I}^{2}\right)$.
Moreover, in this case $\varphi$ would necessarily feature nontrivial
dependence on $\rho$, invalidating our simplified $\varphi=\varphi(z)$. The
result should contain richer topological structures, which would be presented
shortly in a followup work. Nevertheless, as we know the $\pi^{\pm}$
condensate occurs only when $\mu_{I}>m_{\pi}$, we would tune the input
$\mu_{I}$ from $m_{\pi}$, rather than zero, to larger. So that the
compatibility of the simplified boundary conditions is validated.

##  4 Baryonic Vortex Configurations

We solve the vortex Skyrmion (our ansatz for the baryonic vortex) numerically.
For such purpose, physical quantities are rescaled to be dimensionless, e.g.,

$\tilde{\rho}\equiv\rho f_{\pi}s,\quad\tilde{z}\equiv
zf_{\pi}s,\quad\tilde{A_{\mu}}\equiv\frac{A_{\mu}}{f_{\pi}s},\quad\tilde{H}\equiv\frac{H}{f_{\pi}s^{-1}}.$
(20)

Eventually, we will present our results with units made of $f_{\pi}=54\text{
MeV}$ and $s=5.45$, whose specific values are taken from earlier literature
Adkins  _et al._ (1983)111If we include pion mass $m_{\pi}=138\text{ MeV}$, we
could use $s=4.84$. Moreover, as we clarified though we would consider the
chiral limit, our input $\mu_{I}$ would start from the effective
$m_{\pi}=139\text{ MeV}$, which reads $\tilde{\mu_{I}}>0.47$ via our
dimensionless unit. With these conventions specified, we proceed to preset the
vortex Skyrmion solutions.

One can immediately observe that the $z$-dependence is separated due to our
Ansatz. The EOM of $\varphi$ is nothing but $\partial_{z}^{2}\varphi=0$,
leading to

$\varphi=kz;\quad k\equiv\frac{2\pi}{L}.$ (21)

Here the $k$ marks a typical momentum scale of the system. Obviously,
$\mu_{B}$ would not enter into the transverse EOMs. Then the profile
($\rho$-dependence) of the baryonic vortex is up to $\mu_{I}$ and $k$. We pick
one fixed $\mu_{I}=0.5f_{\pi}s$ to delineate the profile and illustrate how
its properties depend on the longitudinal wave vector $k$. In Fig. 1, we show
the exemplary profile where
$B\left(\rho\right)=\partial_{\rho}a\left(\rho\right)/\rho$.

Figure 1: Vortex Skyrmion (left) and magnetic field (right) at
$\mu_{I}=0.5f_{\pi}s$ and $k=2.0f_{\pi}s$.

It has been a consensus that at an asymptotic distance, the solution is
approaching the $\pi^{\pm}$ condensate. Therein the magnetic field should
behave as in a superconductor. Then one important physical scale is the
penetration depth of the magnetic field. Rigorously speaking, for a general
range of $\rho$, the $B\left(\rho\right)$ in our scenario is not subject to
the London equation. Thus the penetration depth applies only in asymptotic
regions. But in such a spirit, based on the wave packet shape of
$B\left(\rho\right)$ seen in Fig. 1 (right), one can always define the “$1/e$
width” which amounts to $2\lambda$ with $B\left(\lambda\right)=B_{0}/e$, to
characterize the transverse scale of the vortex. We exhibit in Fig. 2 (right)
the $\lambda$ at different $k$, demonstrating the positive correlation between
longitudinal and transverse sizes of the vortex. Such a correlation could be
understood intuitively way by quoting the concept of London penetration depth
$\lambda_{L}$ after the asymptotic expansion

$\displaystyle\lambda_{L}^{-1}$ $\displaystyle\sim
m_{A}\equiv\sin\alpha\sqrt{f_{\pi}^{2}+\frac{1}{s^{2}}\left[\left(\partial_{\rho}\alpha\right)^{2}+k^{2}\cos^{2}\alpha\right]},$
(22)

where $m_{A}$ is the effective mass of the gauge field which can be read off
from the quadratic term of $A_{\phi}$ among the energy functional. At spatial
infinity, $\lambda_{L}$ reaches its known result for the $\pi^{\pm}$
condensate within ChPT framework
$\lambda_{L}\left(\rho\rightarrow\infty\right)\rightarrow f_{\pi}^{-1}$
Adhikari  _et al._ (2015). Here the heuristic generalization of Eq. (22) is
intended to show $\lambda_{L}$ could receive corrections which are positively
correlated with $L=2\pi/k$, and so should $\lambda$. This analysis reasonably
explains qualitative features of Fig. 2 (right). The correlation between
$\lambda$ and $k$ is essential for the Skyrme term to maintain at
$\mathcal{O}\left(p^{4}\right)$ in Lagrangian density so it competes with
lower order terms to guarantee a finite size soliton.

Figure 2: Half $1/e$ width of the vortex at $\mu_{I}=0.5f_{\pi}s$.

So far to illustrate the profile we keep $k$ tunable. It needs to be
determined by minimizing the total energy where the integration boils down to

$H=\int d^{3}x\mathcal{H}=2\pi
N_{B}\int^{L}_{0}dz\int^{R\rightarrow\infty}_{0}\rho d\rho\mathcal{H}.$ (23)

We shall point out a direct spatial integral of the kinetic energy
$-\mathcal{L}_{\text{chiral}}$ would diverge. Nevertheless, the divergent part
can be easily subtracted as the level of the potential energy set by the
$\pi^{\pm}$ condensate background i.e.,

$H_{0}\equiv\int
d^{3}x\left(-\frac{1}{2}\mu_{I}^{2}\right)=-\frac{1}{2}\mu_{I}^{2}\pi
R^{2}N_{B}L\bigg{|}_{R\rightarrow\infty},$ (24)

which is derived by taking $\alpha=\pi/2$ in $-\mathcal{L}_{\text{chiral}}$.
$R\rightarrow\infty$ of the transverse boundary in our current study describes
a single vortex line while $R$ could be finite in our future study of a vortex
lattice. For now, we treat $R$ properly as a large cutoff on the numerical
stage. Consequently, we do find the finite $E=H-H_{0}$, whose physical meaning
is the energy cost to create a vortex on top of the homogeneous $\pi^{\pm}$
condensate. Whether the baryonic vortex could emerge as a stable state relies
on the dependence of $E$ on $k$. In a thermodynamic perspective, we are
employing the grand canonical ensemble to study a system with fixed total
volume $V=\pi R^{2}LN_{B}$. Given that $R$ is an irrelevant constant,
minimizing $E$ is equivalent to minimizing the string tension of the system:

$T=\frac{E}{V}\pi R^{2},$ (25)

which is a function of only $k$ (for fixed $\mu_{I,B}$). In this way we have
substituted the variable $N_{B}$. The stable (meta-stable) vortex is reached
at a global (local) minimum of $T(k)$, if exists, i.e.,

$k_{v}=\mathrm{argmin}\;T\left(k\right).$ (26)

Below, we will further discover that depending on the value of $\mu_{B}$, such
a stable vortex does exist as either an excited state or even a ground state.

##  5 Phase Diagram with Isospin and Baryon Chemical Potentials

Succeeding to the last section, we adopt the same exemplary
$\mu_{I}=0.5f_{\pi}s$ and exhibit the $T(k)$ at different $\mu_{B}$.
Remarkably, as shown in Fig. 3 (left), even when $\mu_{B}=0$, there is a
global minimum of $T\left(k\right)$ with $k\neq 0$, which amounts to
$86.75f_{\pi}^{2}$ at $k_{v}=0.0065f_{\pi}s$. In other words, such a vortex
carries energy $2\pi T\left(k_{v}\right)/k_{v}=831\text{ GeV}$ per baryon
number, which is over a hundred times of the heaviest baryon mass, far from a
ground state. However, when the $\mu_{B}$ is enhanced, referring to the
typical case exhibited in Fig. 3 (middle), we witness a decreasing
$T\left(k_{v}\right)$ with an increasing vortex momentum $k_{v}$.

Figure 3: String tension for $\mu_{I}=0.5f_{\pi}s$, $\mu_{B}$ amounts to $0$
(left), $1.3f_{\pi}s$ (middle) and $2.6f_{\pi}s$ (right). The grey vertical
lines pinpoint the minima $T(k_{v})$.

Importantly, when $\mu_{B}$ is large enough, specifically $\mu_{B}\gtrsim
2.518f_{\pi}s\equiv\mu_{B}^{c}$, the global minimum becomes negative, as shown
in 3 (right), which signifies a new ground state replacing the homogeneous
$\pi^{\pm}$ condensate. This case is crucially different from the case of
$\mu<\mu_{B}^{c}$, in which a positive $T(k_{v})$ means that baryonic vortices
remain excited states and require external magnetic fields to arise, as in
conventional type-II superconductors. By contrast, when
$\mu_{B}\geq\mu_{B}^{c}$, baryonic vortices are spontaneously created without
applying magnetic fields, which implies the spontaneous generation of magnetic
fields.

There is such a critical baryon chemical potential $\mu_{B}^{c}$ for each
value of $\mu_{I}$ above $m_{\pi}$. The $\mu_{B}^{c}$ as the function of
$\mu_{I}$ spans a phase diagram of the two ground states: baryonic vortex and
the $\pi^{\pm}$ condensate. Such a phase diagram is laid out in Fig. 4 (left).
There is a vortex longitudinal density $k_{c}\equiv
k_{v}(\mu_{I},\mu_{B}^{c})$ uniquely determined by the input $\mu_{I}$ and the
corresponding $\mu_{B}^{c}$. In Fig. 4 (right), we show $k_{c}$ tends to blow
up for large $\mu_{I}$. A related comment is that though
$\mu_{B}^{c}(\mu_{I})$ seems to go all the way down with an increasing
$\mu_{I}$, the deduction that $\mu_{B}^{c}$ drops to zero for large enough
$\mu_{I}$ cannot be audaciously made. The reason is that $k_{c}$ could
severely exceed its valid range dictated by the ChPT scheme. Specifically, the
nature of momentum expansion stipulates $k\ll 4\pi f_{\pi}$ which is converted
to $2.31$ with our momentum unit $f_{\pi}s$. Thus even within the current
parameter region of $\mu_{I}$ and $\mu_{B}$, the $k_{c}$ applies only
marginally. We expect the feasibility of our study to be improved after taking
into account fully the pion mass in the dynamics, which would be soon updated
in our sequential work.

Figure 4: Phase diagram parameterized by the isospin chemical potential
$\mu_{I}$ and baryon chemical potential $\mu_{B}$ (left), where the blue
dotted line is the critical $\mu_{B}^{c}\left(\mu_{I}\right)$ that separates
the baryonic vortex phase in blue shade ($\mu_{B}>\mu_{B}^{c}$) and the
$\pi^{\pm}$ condensate in the blank ($\mu_{B}<\mu_{B}^{c}$). Together plotted
is the corresponding longitudinal density $k_{c}$ at each $\mu_{I}$ (right).

Despite the caveat, here our simplified analysis already allows us to
demonstrate for the first time that in finite isospin QCD, a baryonic vortex
with extra $\pi^{0}$ winding on top of a $\pi^{\pm}$ condensate could be of
lower energy than the homogeneous condensate. Remarkably, such a phenomenon
occurs at a reasonable critical baryon chemical potential bounded from above
by $\mu_{B}^{c}(\mu_{I}=m_{\pi})\sim 743\text{ MeV}$, which resides within the
regime of nuclear matter. In other words, in the density window between
$\mu_{B}^{c}$ and nucleon masses around $940\text{ MeV}$, a finite $\mu_{I}$
could yield a ground state made of the baryonic vortex we proposed. In this
way, the self-generated magnetic field could be sustained in nucleon-rich
contexts with isospin asymmetry, especially neutron stars. To further estimate
the strength of such magnetic fields, we need to generalize the current
scenario to a baryonic vortex lattice so that we could attain the transverse
density of the magnetic flux carried by the vortex.

To this end, Refs. Evans and Schmitt (2022, 2024) are inspiring yet
distinguished from our study in that their field profile is solved based on a
perturbative expansion near the critical point of the superconducting phase
within ChPT, while ours is from the Skyrme model with no approximation. Our
baryonic vortex preexists the determination of the critical magnetic field,
allowing for a spontaneous generation of the magnetic field. According to our
power counting scheme, the Skyrme term is essential for the global minimum of
string tension to exist with a finite $k_{v}$. We will further substantiate
this point in the next section.

##  6 Remark on $\mathcal{O}(p^{2})$ Results

To unveil the role played by the Skyrme term and the valid range of our
proposal, we compare our scenario with vortices in ${\cal O}(p^{2})$ ChPT
without the Skyrme term. In this case, the rescaling in Eq. (20) shall be
modified, with momentum in unit $f_{\pi}$. A counterpart of Fig. 3 is
exhibited in Fig. 5, where we maintain the common input $\mu_{I}=0.5\times
5.45f_{\pi}$. Comparing Fig. 5 (left) and Fig. 3 (left), we observe that for
$\mu_{B}=0$, the behavior of $T\left(k\right)$ is similar between ${\cal
O}(p^{2})$ ChPT and the Skyrme model. It features a global minimum where
$k_{v}$ with and without the Skyrme term are $0.0065f_{\pi}s$ and
$0.042f_{\pi}$ respectively, close to each other. The $T\left(k_{v}\right)$
are almost the same as well; $86.75f_{\pi}^{2}$ with the Skyrme term, and
$86.25f_{\pi}^{2}$ without it. So we affirm that for the previous studies
Adhikari  _et al._ (2015); Adhikari (2019); Adhikari  _et al._ (2023); Grønli
and Brauner (2022) without $\mu_{B}$, ${\cal O}(p^{2})$ ChPT used by them is
sufficient and the Skyrme term in our proposal would only produce minor
corrections on their results. Moreover, their scenarios correspond to the
$k=0$ case in our study. As one can see from either Fig. 3 or Fig. 5, the well
of $T(k)$ is very shallow. Thus although our newly introduced finite $k_{v}$
configuration lowers the minimum vortex energy, such an effect is not
prominent for the case with only $\mu_{I}$. However, the situation is
drastically different for $\mu_{B}\neq 0$.

Figure 5: String tension evaluated from ChPT without the Skyrme term for
$\mu_{I}=0.5f_{\pi}s$ while $\mu_{B}$ amounts to $0$ (left), $2.2f_{\pi}$
(middle) and $3.3f_{\pi}$ (right).

We highlight that once a finite $\mu_{B}$ is turned on, Fig. 5 (middle)
exemplifies the eccentric behavior of $T(k)$: the minimum becomes local and
$T\left(k\right)$ decreases with increasing $k$ when $k$ is large. Such an
ultraviolet behavior seems to indicate the instability of the system; a
runaway ground state with infinitely large baryon density. This indication is
even more pronounced when $\mu_{B}$ is larger, as shown in Fig. 5 (right). The
local minimum of $T\left(k\right)$ disappears when $\mu_{B}\gtrsim 3.3f_{\pi}$
and the $T\left(k\right)$ becomes a monotonically decreasing function. We
regard such an infinitely dense vortex as an artifact brought by the the
breakdown of ChPT expansion hierarchy. The negative contribution to $T(k)$ is
from the $\mathcal{O}\left(p^{3}\right)$ WZW term (13) which leads to a factor
$-\mu_{B}k$ after the spatial integration. For ultraviolet $k$, such negative
contribution would always surpass the positive contribution from the
$\mathcal{O}\left(p^{2}\right)$ chiral Lagrangian. This is exactly why we must
add the Skyrme term, an $\mathcal{O}\left(p^{4}\right)$ positive contribution,
for a stable baryonic vortex of finite longitudinal density. Then once the
vortex with finite $k_{v}$ stands as a solution, it carries the baryon number,
leading to the WZW coupling. This is the physical mechanism of a negative
string tension when $\mu_{B}$ is large, which is exclusive to our proposal
that goes beyond $\mathcal{O}(p^{2})$ order.

##  7 Conclusion

In this study, we have proposed a baryonic vortex phase as a novel phase for
dense hadronic matter with finite $\mu_{B}$ and $\mu_{I}$. We determine the
boundary $\mu_{B}=\mu_{B}^{c}(\mu_{I})$ between such a phase and the charged
pion $\pi^{\pm}$ condensate in the phase diagram in Fig. 4. In the presumed
regime $\mu_{I}\gtrsim m_{\pi}$, the critical $\mu_{B}^{c}$ starts from around
743 MeV and decreases with an increasing $\mu_{I}$. When $\mu_{B}>\mu_{B}^{c}$
the baryonic vortex is energetically preferred over the homogeneous
$\pi^{\pm}$ condensate.

Vortices emerging from $\pi^{\pm}$ condensate is not a new story. They form
AVL under an external magnetic field, similar to metallic type-II
superconductors, as studied in the previous works Adhikari  _et al._ (2015);
Adhikari (2019); Adhikari  _et al._ (2023); Grønli and Brauner (2022). In such
cases, the optimal lattice spacing is determined by the strength of the
applied magnetic field, given magnetic flux is quantized for each vortex. The
detail of the interaction between vortices is unimportant as far as the
vortices repel each other. Our results for $0\leq\mu_{B}<\mu_{B}^{c}$ update
these works in the sense of lowering the vortex energy via a nontrivial WZW
coupling of the baryon current to the $\mu_{B}$, which means a less demanding
applied magnetic field to excite the vortices from $\pi^{\pm}$ condensate.

In contrast, our scenario of the baryonic vortex phase at
$\mu_{B}>\mu_{B}^{c}$ is completely original, in which a “baryonic vortex
lattice (BVL)” is expected to emerge even with NO applied magnetic field. In
such a circumstance, the self-generated magnetic field, according to our
evaluation, could reach $10^{-1}f_{\pi}^{2}s^{2}\sim 10^{17}\text{ Gauss}$
near the vortex core. Nevertheless, the magnitude of the macroscopic magnetic
field is still to be quantified, which hinges on the lattice spacing, or the
baryon density of the BVL. Unlike in the AVL, the lowest energy configuration
of a BVL must be achieved from the balance between the negative string tension
of each vortex and the repulsion between vortices. Hence the detail of the
interaction does matter. This part would be figured out in our subsequent
work, which shall illuminate the modification on the critical magnetic field
needed when $\mu_{B}<\mu_{B}^{c}$, and the quantification of baryon density
together with the spontaneously generated magnetic field when
$\mu_{B}>\mu_{B}^{c}$.

The significance of such researches is twofold. The first is the relevance in
astrophysics. We conceive that the self-generated and conserved magnetic flux
of our baryonic vortex possibly explains the origin of the long lived
($>10^{6}$ Yrs) magnetic field inside magnetars. Such a prospect is consistent
with Ref. Ho  _et al._ (2017) yet based on a distinguished nature of
condensate. In addition to the crystalline configuration, a fluid of baryonic
vortex is also considerable, depending on the applicable crust of neutron
stars. Decisive conclusions would be drawn after coupling the baryonic vortex
with gravity and solving the equation of state.

On the other hand, our study caters to the interest in QCD phase diagram.
Along the axis of $\mu_{B}$, at higher density regime where strange quarks
become relevant, we shall attempt to compare our spontaneous baryonic vortex
with quark matter phases such as the crystalline color superconductivity
Alford  _et al._ (2001) and color-flavor-locking, in order to delve into the
quark-hadron continuity. In addition, within the hadronic regime, for the
outlook of a more comprehensive digram of topological phases at finite
$\mu_{B}$ and magnetic field, we shall further incorporate CSL, Skyrme
crystal, domain-wall Skyrmions, mesonic quasicrystal etc, after figuring out
the many-body system comprising of multiple baryonic vortices.

###### Acknowledgements.

This work is supported in part by JSPS KAKENHI [Grants No. JP22H01221] and the
WPI program “Sustainability with Knotted Chiral Meta Matter (SKCM2)” at
Hiroshima University (MN).

## Appendix: Details of the Equation of Motion

After some algebra, the explicit form of energy functional (17) before spatial
integration with our Ansatz (7) applied reads

$\displaystyle\mathcal{H}=$
$\displaystyle\frac{f_{\pi}^{2}}{2}\left\\{\left(\partial_{\rho}\alpha\right)^{2}+\left[\left(\frac{1}{\rho}+A_{\phi}\right)^{2}-\mu_{I}^{2}\right]\sin^{2}\alpha+\cos^{2}\alpha\left(\partial_{z}\varphi\right)^{2}\right\\}$
$\displaystyle+\frac{1}{2s^{2}}\left\\{\left(\partial_{\rho}\alpha\right)^{2}\cos^{2}\alpha\left(\partial_{z}\varphi\right)^{2}+\left[\left(\frac{1}{\rho}+A_{\phi}\right)^{2}-\mu_{I}^{2}\right]\sin^{2}\alpha\left[\left(\partial_{\rho}\alpha\right)^{2}+\cos^{2}\alpha\left(\partial_{z}\varphi\right)^{2}\right]\right\\}$
$\displaystyle-\frac{q}{4\pi^{2}}A_{\phi}\frac{\partial}{\partial\rho}\left(\mu_{I}\cos^{2}\alpha\right)\partial_{z}\varphi-\mu_{B}j_{B}^{0}+\frac{1}{2}\left[\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho
A_{\phi}\right)\right]^{2}.$ (27)

Then with the direct corollary $\varphi=kz$, by the argument around Eq. (21),
we spell out the variational principle of $H=\int d^{3}x\mathcal{H}$ with
respect to the transverse radius $\rho$, first for $\alpha$:

$\displaystyle\frac{\partial}{\partial\rho}\left\\{\rho\left\\{f_{\pi}^{2}+\frac{1}{s^{2}}\left(k^{2}\cos^{2}\alpha+\left[\left(\frac{1}{\rho}+A_{\phi}\right)^{2}-\mu_{I}^{2}\right]\sin^{2}\alpha\right)\right\\}\partial_{\rho}\alpha\right\\}$
$\displaystyle=$
$\displaystyle\frac{\rho}{2}\Bigg{\\{}\left[\left(\frac{1}{\rho}+A_{\phi}\right)^{2}-\mu_{I}^{2}-k^{2}\right]\sin
2\alpha\left[f_{\pi}^{2}+\frac{1}{s^{2}}\left(\partial_{\rho}\alpha\right)^{2}\right]$
$\displaystyle+\frac{1}{s^{2}}\sin 2\alpha\cos
2\alpha\left[\left(\frac{1}{\rho}+A_{\phi}\right)^{2}-\mu_{I}^{2}\right]k^{2}\Bigg{\\}}$
(28)

Beware that the variation shall not be performed on the conserved current
$j_{B}^{\mu}$. So the WZW term gets involved in EOMs only when we consider
$\delta A_{\phi}$, but not $\delta\alpha$. Next, The variational principle in
terms of $A_{\phi}$ would reproduce the Maxwell equation,
$\partial_{\mu}F^{\mu\nu}=j_{Q}^{\nu}$:

$\displaystyle\frac{\partial}{\partial\rho}\left[\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho
A_{\phi}\right)\right]=$
$\displaystyle-\frac{qk}{4\pi^{2}}\frac{\partial}{\partial\rho}\left(\mu_{I}\cos^{2}\alpha\right)$
$\displaystyle+\left(\frac{1}{\rho}+A_{\phi}\right)\sin^{2}\alpha\left\\{f_{\pi}^{2}+\frac{1}{s^{2}}\left[\left(\partial_{\rho}\alpha\right)^{2}+k^{2}\cos^{2}\alpha\right]\right\\},$
(29)

among which the first term on the r.h.s does originate from the WZW term and
here is the only portal where the WZW term enters the transverse EOMs. In
other words, $\mu_{B}$ would not influence the profile under arbitrary $k$ but
it takes part in determining the specific $k$ that leads to a minimized total
free energy.

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