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

# Quasicrystals in QCD

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

We study the ground state of the low energy dense QCD with the assumption of
chiral condensates of quarks. Under an external magnetic field, mesons could
form soliton lattices via the chiral anomaly. For such scenarios, we present a
unified description of pions and $\eta$ meson with a $U(2)$ field in the
framework of the chiral perturbation theory. Our result shows the ground state
is a mixture of the magnetized domain walls formed by neutral pion $\pi^{0}$
and $\eta$ meson when they coexist. The winding number of the ground state
would alter according to the strength of the magnetic field. When the magnetic
field is strong or the chemical potential is large, the proportion of the
mixture is determined by the decay constants and the contributions to the
anomalous action of $\pi^{0}$ and $\eta$ meson. The resulting configuration is
either a mixed soliton lattice or a quasicrystal which could be dubbed a
“chiral soliton quasicrystal”.

##  1 Introduction

The phase structure of QCD is of tremendous research interest for decades.
Especially, under external fields such as magnetic field or rotation, richer
phase structures with nontrivial topology arise due to the coupling of these
external fields with the baryon chemical potential via the chiral anomaly.
Recently, there has been a challenging new idea that the ground state of dense
QCD matter in the presence of a strong magnetic field could be the chiral
soliton lattice (CSL) made of bosonic neutral pion $\pi^{0}$ instead of
fermionic baryons Son and Stephanov (2008); Eto  _et al._ (2013); Brauner and
Yamamoto (2017); Brauner  _et al._ (2019a, 2021). Following this lead, related
phases involving charged pions have then been studied, including the charged
pion condensate Brauner and Yamamoto (2017) and its competition with an
Abrikosov vortex lattice Evans and Schmitt (2022), for which one can draw the
phase diagram parameterized by the magnetic field and the isospin chemical
potential Grønli and Brauner (2022). Further studies of CSLs are devoted to
thermal fluctuations Brauner and Kadam (2017a, b); Brauner  _et al._ (2021);
Brauner and Kolešová (2023), rapid rotation Huang  _et al._ (2018); Nishimura
and Yamamoto (2020); Eto  _et al._ (2022); Chen  _et al._ (2021), and quantum
nucleation Eto and Nitta (2022); Higaki  _et al._ (2022), see also Refs.
Yamada and Yamamoto (2021); Brauner  _et al._ (2019b) for related issues.

These novel phenomena could all be well described by the chiral perturbation
theory (ChPT) which is a low-energy effective theory of QCD. It is based on
the chiral symmetry breaking $U(1)_{V}\times SU(N_{f})_{L}\times
SU(N_{f})_{R}\rightarrow U(1)_{V}\times SU(N_{f})_{V}$ that leaves bound
states of quarks as the Nambu-Goldstone (NG) bosons, specifically pions. The
effective action of ChPT is therefore in terms of the $SU(N_{f})$ field,
constructed by a perturbative expansion with a power counting scheme on
momentum. Effects from chiral anomaly are encoded in the Wess-Zumino-Witten
(WZW) term Son and Zhitnitsky (2004); Son and Stephanov (2008). Describing
physics on the same scales of density and magnetic field, the simplest
extension of the ChPT could be the addition of a gauged Skyrme term which
stabilises a soliton solution describing the baryon. In such a framework, the
aforementioned pionic phases could be compared with a multi-baryon crystal
called Skryme crystal under a magnetic field. Latest efforts have found the
CSL could be taken over by the magnetized Skyrme crystal in a realm of higher
density and/or lower magnetic field among the phase diagram Chen  _et al._
(2022, 2023) (see also Ref. Kawaguchi  _et al._ (2019)). More recently, the
higher density region of the CSL phase has been found to transit to a domain-
wall Skyrmion phase, in which a Skyrmion is decomposed as a baby skyrmion
accommodated in a pion domain wall Eto  _et al._ (2023), attracting its own
research interests in the sense of the topology Nitta (2013, 2022); Ross and
Nitta (2023).

So far the phases mentioned above are centred on pions as building blocks. But
there could be a similar anomalous coupling of “heavier” NG bosons. In the
baryon density window next to (higher than) that of the CSL, there comes the
$\eta$ meson, resulting from the $U(1)_{A}$ axial symmetry breaking. Strictly,
the $U(1)_{A}$ symmetry is not a symmetry of QCD but is violated by anomalous
fluctuations of gluons. However at the large density where the color Deybe
screening applies and instanton effects are suppressed, the $U(1)_{A}$
symmetry could be treated as a symmetry approximately. Its spontaneous
breaking gives rise to the $\eta$ meson whose mass is heavier than pions yet
still light compared to the chemical potential. In such a density regime, the
$\eta$ meson could couple with a magnetic field and form a similar axial
domain wall as $\pi^{0}$ Son and Zhitnitsky (2004); Son and Stephanov (2008).
Interestingly, while the $\pi^{0}$ pion does not couple with a rotation, the
$\eta$ meson does. The result is a rotational counterpart of the $\pi^{0}$
CSL, the $\eta$ CSL Huang  _et al._ (2018); Nishimura and Yamamoto (2020); Eto
_et al._ (2022). Natural playgrounds to study anomalous effects of $\eta$
mesons are the 2-flavor superconducting (2SC) Alford  _et al._ (1998); Rapp
_et al._ (1998) and the color-flavor locked (CFL) Alford  _et al._ (1999)
phases.

For the physics we would like to discuss in the present paper, $N_{f}=2$
(2-flavor) case suffices. We are interested in the effect of a magnetic field
that couples with both the pions and the $\eta$ meson. Though usually these
particles are considered at separate density scales, in principle they can
coexist in nuclear matter. Lately, there have been attempts to put them in the
same theoretical framework of the ChPT Aoki and Creutz (2014); Eto  _et al._
(2022), where the field is augmented to $U(2)=[U(1)\times
SU(2)]/\mathbb{Z}_{2}$, encompassing the $U(1)$ of $\eta$ and the $SU(2)$ of
pions. In such a scenario, more intriguing phase structures appear. It was
pointed out in Ref. Aoki and Creutz (2014) that one can realize the
$CP$-violating Dashen phase by tuning the masses of $u,d$ quarks. In that
paper, neither external fields nor interactions of pions and $\eta$ are
considered. It is important for us to understand that in the $U(2)$ ChPT even
mass terms alone could yield nontrivial solutions other than vacuum.
Furthermore, in Ref. Eto  _et al._ (2022) the coupling of the $\eta$ meson to
a rotation is included. Then, depending on the magnitude of the rotation and
coupling constants, there exist three possible ground state phases, $\eta$ CSL
and two kinds of non-Abelian CSLs, called “dimer” and “deconfined” phases.
Here, the term “non-Abelian” implies that a soliton has non-Abelian moduli
coming from spontaneous symmetry breaking of the vacuum symmetry in the
vicinity of the soliton. When the sine-Gordon soliton is embedded in $SU(N)$,
it is a non-Abelian soliton with ${\mathbb{C}}P^{N-1}$ moduli Nitta (2015);
Eto and Nitta (2015). Such a phenomenon is one direct inspiration for the
present work.

Crystalline configurations with periodicity have been widely studied in the
context of QCD; apart from chiral soliton lattices which we focus on in this
paper, extensive investigations have been conducted on modulated phases in
Nambu-Jona Lasino or Gross-Neveu models Nakano and Tatsumi (2005); Nickel
(2009); Basar  _et al._ (2009); Buballa and Carignano (2015); Hidaka  _et al._
(2015), crystalline color supercondutors Casalbuoni and Nardulli (2004);
Anglani  _et al._ (2014), and so on. On the other hand, in condensed matter
materials, there are many examples of quasicrystals. Quasicrystals are
aperiodic incommensurate lattice structures Steinhardt (2019); Janssen (1988);
DiVincenzo and Steinhardt (1999); Janot (1997); Janssen  _et al._ (2007);
Stadnik (2012); Fan (2016); Scott and Clark (2012); Jaric  _et al._ (1988).
See Refs. Baggioli and Landry (2020); Surówka (2021) for recent studies of
quasicrystals based on field theory.

In this paper, we make the first step of the study on quasicrystal in QCD, by
considering the effect of an external magnetic field on the $U(2)$ NG fields
in the chiral symmetry breaking phase, particularly the $\pi^{0}$ pion and
$\eta$ meson. Our scenario with an external magnetic field brings an essential
difference in the anomalous action compared to that of a rotation. The
anomalous action receives contributions from both $\pi^{0}$ pion and $\eta$
meson, and the resulting magnetic energy determines the ground state. We
witness that pure $\pi^{0}$ solitons often have the lowest energy (being
ground states) and a pure $\eta$ soliton always has much higher energy.
Nevertheless, somewhat surprisingly, the ground state can also be a “mixed”
soliton lattice with the nontrivial winding of both mesons, rather than a
separate $\pi^{0}$ or $\eta$ CSL. More generally, the ground state could even
be a one-dimensional quasicrystal along the direction of the magnetic field.
In the strong magnetic field and/or large chemical potential, we find that the
ratio of the decay constants of pions and $\eta$ meson plays a crucial role.
Specifically, when the ratio happens to be rational, we would have a lattice
or crystal. Otherwise, we would have a quasicrystal. To the best of our
knowledge, this is the first quasicrystal in a realistic setup of QCD.

This paper is organized as follows. In Sec. 2, our setup of the $U(2)$ ChPT
featuring pions and $\eta$ meson is given. In sec. 3, the simplest mixed
soliton lattice is investigated in comparison to pure $\pi_{0}$ and $\eta$
CSLs. In Sec. 4, the ground states varying with the magnetic field are
determined assuming a periodicity up to a certain soliton number per lattice
period. In Sec. 5, The limit of a strong magnetic field and/or large chemical
potential is discussed, where the quasicrystal emerges. Sec. 6 is devoted to a
conclusive summary and outlook discussions. Through this work we adopt
$g^{\mu\nu}=\left(+,-,-,-\right)$.

##  2 $U(2)$ Chiral Perturbation Theory in a Magnetic Field

We consider non-Abelian chiral soliton lattices constructed by the two-flavor
$U(2)$ field:

$U=\exp\left(i\eta/f_{\eta}\right)\Sigma;\quad\Sigma=\exp\left(i\tau^{k}\pi_{k}/f_{\pi}\right);\quad
k=1,2,3.$ (1)

The $f_{\eta}$ and $f_{\pi}$ are the decay constants of the eta meson $\eta$
and pions $\pi^{k}$, respectively. It suffices to adopt the low-energy
effective theory based on the chiral symmetry breaking for our purpose.

The chiral Lagrangian featuring the kinetic and the mass terms reads:

$\displaystyle\mathcal{L}_{\text{chiral}}=$
$\displaystyle\frac{1}{2}\partial_{\nu}\eta\partial^{\nu}\eta+\frac{f_{\pi}^{2}}{4}\mathrm{Tr}\left(\partial_{\mu}\Sigma\partial^{\mu}\Sigma^{\dagger}\right)$
$\displaystyle+\left\\{\frac{a}{2}\left(\det
U-1\right)+b\,\text{Tr}\left[M\left(U-1\right)\right]+\text{h.c.}\right\\}.$
(2)

Here, $b$ is a constant associated with the usual mass matrix $M$. $M$ is
taken to be simply $M=m\mathbb{I}$ because we consider the regime with
chemical potential much larger than quark masses so the masses of $u,d$ quarks
can be regarded as approximately the same $m$. On top of that, $a$ represents
the extra mass given to the $\eta$ meson. We have quoted the form of
$\mathcal{L}_{\text{chiral}}$ from Ref. Eto  _et al._ (2022) which studied
thoroughly, within the same framework, the anomalous coupling of the $U(2)$
field to a rotation. In the present work, differently, we consider an external
magnetic field instead of the rotation.

It is well known that charged pions $\pi_{1},\pi_{2}$ become very “massive”
under a strong magnetic field while the neutral pion $\pi_{3}$ remains the
only NG boson among the pions. Focusing on the ground state properties, we
ignore charged pions and set $\pi_{1,2}=0$ throughout this work.

Whilst rotation couples to the $\eta$ meson only, a magnetic field couples
with both the $\eta$ meson and neutral pion $\pi_{3}$ via the axial anomaly,
contributing an extra term in Lagrangian Son and Zhitnitsky (2004); Son and
Stephanov (2008):

$\mathcal{L}_{B}=\frac{1}{8\pi^{2}}\epsilon^{\mu\nu\alpha\beta}\sum_{i=0,3}C_{i}\partial_{\mu}\phi_{i}A_{\nu}^{\text{B}}F_{\alpha\beta},$
(3)

where the $\phi_{i}$ with $i=0,3$ denote relevant NG bosons respectively,
i.e., $\eta$, $\pi_{3}$ in our case. In other words, $\phi_{i}$ are the
dimensionless

$\phi_{0}\equiv\frac{\eta}{f_{\eta}},\quad\phi_{3}\equiv\frac{\pi_{3}}{f_{\pi}},$
(4)

appearing in the exponentials in Eq. (1). The $F_{\alpha\beta}$ is the
electromagnetic field strength. Without loss of generality, we set up a
homogeneous background magnetic field along longitudinal direction $\hat{z}$,
specifying $F_{xy}=-F_{yx}=-B$. The
$A_{\nu}^{\text{B}}=\left(\mu,\boldsymbol{0}\right)$ is a common choice of
baryon gauge field to feature the chemical potential $\mu$. Then importantly,
Eq. (3) $C_{i}$ are the anomalous coefficients signifying the interplay
between $U(1)_{\text{EM}}$ and $U(1)_{\text{B}}$ gauge fields via the chiral
anomaly:

$C_{i}=\sum_{a=u,d}Q_{5,i}^{a}Q_{i}^{a},$ (5)

in which $Q_{i}^{a}$ and $Q_{5,i}^{a}$ are charges and axial charges of quarks
comprising the relevant NG bosons labeled by index $i$. In our 2-flavor model,
it certainly reads

$\displaystyle\eta\text{: }$ $\displaystyle Q_{5}^{u}=Q_{5}^{d}=1,$ (6)
$\displaystyle\pi_{3}\text{: }$ $\displaystyle Q_{5}^{u}=-Q_{5}^{d}=1.$ (7)

Needless to say $Q^{u}=2/3$ and $Q^{d}=-1/3$. With such combinations, we
finally arrive at the anomalous Lagrangian in our setup:

$\mathcal{L}_{B}=\frac{\mu}{4\pi^{2}}\boldsymbol{B}\cdot\left(\boldsymbol{\nabla}\phi_{3}+\frac{1}{3}\boldsymbol{\nabla}\phi_{0}\right),$
(8)

which should be added to Eq. (2) to complete our effective theory

$\mathcal{L}=\mathcal{L}_{\text{chiral}}\big{|}_{\pi_{1,2}=0}+\mathcal{L}_{B}.$
(9)

We remark that usually the $\eta$ meson becomes relevant in the density regime
above the low-density nuclear matter, e.g., the 2SC or CFL phase. For
sufficiently large $\mu\gg\Lambda_{\text{QCD}}$, both $f_{\eta}$ and $f_{\pi}$
can be evaluated in terms of $\mu$. However, in this work, we regard them as
independent and tunable parameters.

As indicated by the established studies on the CSL lattices, derivative terms
involving transverse directions $\partial_{x,y}$ should vanish in the ground
state because they contribute to the soliton energy functional in a positive-
definite way. The same argument applies to our model when considering soliton
solution with null $\partial_{0}$. Therefore the issue we tackle is
essentially 1D, depending on the longitudinal coordinate $z$ selected by
$\boldsymbol{B}=B\hat{z}$. The equation of motion (EOM) is equivalent to the
variational principle minimizing the Hamiltonian or energy functional $H$
corresponding to $\mathcal{L}$. To be specific, we write the energy
functional:

$\displaystyle\tilde{H}=$
$\displaystyle\frac{1}{2}\left(\alpha\phi_{3}^{\prime 2}+\phi_{0}^{\prime
2}\right)-\frac{\gamma}{2\pi}\left(\phi_{3}^{\prime}+\frac{1}{3}\phi_{0}^{\prime}\right)$
$\displaystyle+\sin\beta\left(1-\cos
2\phi_{0}\right)+\cos\beta\left(1-\cos\phi_{0}\cos\phi_{3}\right),$ (10)

where we have employed dimensionless quantities similar to those in Ref. Eto
_et al._ (2022):

$\tilde{z}\equiv\frac{\left[a^{2}+\left(4mb\right)^{2}\right]^{1/4}}{f_{\eta}}z,\quad\tilde{H}=\frac{H}{\sqrt{a^{2}+\left(4mb\right)^{2}}},$
(11)

with $\prime$ implying $\partial/\partial\tilde{z}$. With the help of Eq.
(11), the parameters $a,b$ are rephrased into

$\alpha\equiv\frac{f_{\pi}^{2}}{f_{\eta}^{2}},\quad\beta\equiv\arctan\frac{a}{4mb},$
(12)

and the magnetic field $B$ is encoded in

$\gamma\equiv\frac{\mu
B}{2\pi}\frac{\left[a^{2}+\left(4mb\right)^{2}\right]^{-1/4}}{f_{\eta}}.$ (13)

In what follows, we abbreviate the tildes on top of $H$ and $z$ and deal with
above-mentioned dimensionless quantities by default. The EOM is derived from
the variational principle exerted on $H$:

$\displaystyle\alpha\phi_{3}^{\prime\prime}$
$\displaystyle=\cos\beta\cos\phi_{0}\sin\phi_{3}$
$\displaystyle\phi_{0}^{\prime\prime}$
$\displaystyle=\cos\beta\cos\phi_{3}\sin\phi_{0}+2\sin\beta\sin 2\phi_{0}.$
(14)

Obviously if one of $\phi_{0}$ and $\phi_{3}$ is absent, the EOM reduces
simply to the sine-Gordon equation whose solution is analytically known. In
our work, the highlight is the mixture of $\phi_{0}$ and $\phi_{3}$ obeying a
coupled sine-Gordon equation (14).

##  3 Mixed Soliton Lattice

In this section, we start with the simplest mixed soliton lattice solution in
our model with assuming a periodicity. In general, we intend to solve the
$U\in[U(1)\times U(1)]/\mathbb{Z}_{2}$ field over the distance $d$ along the
$z$-direction. The winding is depicted by the boundary condition:

$\left(\phi_{3},\phi_{0}\right)\bigg{|}_{z=0}=\left(0,0\right),\quad\left(\phi_{3},\phi_{0}\right)\bigg{|}_{z=d}=\left(p\pi,q\pi\right),$
(15)

where the single-value condition $\Sigma\left(d\right)=\Sigma\left(0\right)$
restricts the choice of $p$ and $q$:

$\frac{p\pm q}{2}\in\mathbb{Z}.$ (16)

The familiar winding $\pi_{1}[U(1)]\simeq\mathbb{Z}$ applies to each separate
$U(1)$ in $\Sigma$. The corresponding topological charges are $\pi^{0}$ and
$\eta^{\prime}$ numbers respectively. In our framework, we prove numerically
that they are reproduced by the following boundary conditions:

$\pi_{0}$: $\displaystyle\left(p,q\right)=\left(2,0\right),$ (17)
$\displaystyle\eta\text{: }$ $\displaystyle\left(p,q\right)=\left(0,2\right).$
(18)

In comparison, we innovatively introduce the concept of a “mixed” soliton with
both $p,q\neq 0$. The simplest case abiding by Eq. (16) is

$\text{(Simplest) Mixed Soliton: }\left(p,q\right)=\left(1,1\right).$ (19)

Before proceeding, to give a better perception of the mixed soliton, let us
mention a special case of $\beta=\pi/2$, which is analytically solvable. The
EOMs (14) reduce to

$\alpha\phi_{3}^{\prime\prime}=0,\quad\phi_{0}^{\prime\prime}=2\sin
2\phi_{0},$ (20)

whose solution reads

$\phi_{3}=\frac{p\pi z}{d},\quad\phi_{0}=\text{am}\left(\nu
z,-\frac{4}{\nu^{2}}\right),$ (21)

where am is the Jacobi amplitude and $\nu$ satisfies
$\phi_{0}\left(d\right)=q\pi$, dictated by the boundary conditions (15). Such
a solution, with nonzero $p$ and $q$, exemplifies the mixed soliton solution.

Now we come to the case with general $\alpha$ and $\beta$, the EOMs are solved
numerically over distance $d$. We apply the solutions into $H$, evaluating the
soliton energy. Notably, in the context of a soliton lattice, $d$ can be
regarded as the lattice period. For a fixed total lattice size, we need to
find the $d$ that minimizes the total lattice energy density

$\mathcal{E}=\frac{1}{d}\int dzH.$ (22)

If $\mathcal{E}$ has a global minimum,
$d_{L}=\text{argmin}\mathcal{E}\left(d\right)$ would be the lattice period of
the ground state. Whereas $\mathcal{E}$ as a function of $d$ does not always
feature a minimum. It depends on the value of the magnetic field, or in other
words, $\gamma$. There should be the critical value of $\gamma$ only above
which can $d_{L}$ exist. In this section, we discuss the critical $\gamma$
physics by limiting our attention to the three building block configurations
Eqs. (17), (18), and (19). Quantities associated with each configuration would
be dressed by lower index “$\pi$”, “$\eta$” and “m”, respectively. We observe
for all three configurations, there are critical $\gamma_{\pi,\eta,\text{m}}$
for $\mathcal{E}_{\pi,\eta,\text{m}}$ to exhibit a global minimum.

$\text{For }\gamma>\gamma_{\pi,\eta,\text{m}},\quad\exists
d_{L}\in\left(0,d\right):\quad\mathcal{E}_{\pi,\eta,\text{m}}^{\prime}\left(d_{L}\right)=0.$
(23)

$\gamma_{\pi,\eta,\text{m}}$ are located numerically and would be detailed in
what follows.

First, we find $\gamma_{\eta}$ is independent of $\alpha$. This is easy to
understand because the only field involving $\alpha$ in $H$ is the $\phi_{3}$,
and it proves numerically that $\phi_{3}\equiv 0$ for
$\left(p,q\right)=\left(0,2\right)$. We therefore display $\gamma_{\eta}$
under several different values of $\beta\in\left(0,\pi/2\right)$ in Fig. 1 ,
which is qualified as a summary of our results about the separate $\eta$
soliton lattice under magnetic field. We shall remind the $\beta=\pi/2$ case
is excluded since it has already been solved analytically by Eq. (21).

Figure 1: Critical $\gamma$ for $\eta^{\prime}$ soliton lattice

Next, we shall report the observation that
$\gamma_{\eta}\sim\mathcal{O}\left(3\gamma_{\pi}\right)$ which is reminiscent
of the discrepancy of anomalous coefficients in Eq. (5) by the prefactor
$1/3$. More explicitly, this can be seen from the topological part, Eq. (8),
among $\mathcal{E}$, depending on the $\gamma$, i.e.,

$\mathcal{E}_{\text{topo}}=-\frac{\gamma}{d}\cdot\begin{cases}1&\text{for
}\pi^{0}\\\ 1/3&\text{for }\eta\\\ 2/3&\text{for ``mixed"}\end{cases}.$

In other words, comparing the two separate soliton lattices under a common
magnetic field, we can naturally expect $\eta$ CSL to bear higher energy than
$\pi^{0}$-lattices. This explains why the magnetic effect of the $\eta$
soliton lattices is often overlooked in lower density/energy regimes; the
$\pi^{0}$-lattice demands a lower critical magnetic field to emerge as the
ground state. Along this line, one might intuitively guess that the mixed
soliton lattice, as a mixture of $\pi^{0}$ and $\eta$ soliton lattices, would
feature intermediate energy. But astonishingly we find that it is not the
case. A mixed soliton lattice can have the lowest energy in a certain
parameter region!

Ultimately we come to address such a duel between $\gamma_{\text{m}}$ and
$\gamma_{\pi}$. Which one is lower depends on parameters $\alpha$ and $\beta$.
We illustrate the dependence of $\gamma_{\pi,\text{m}}$ on $\alpha$ in the
case with $\beta=0$, as shown in Fig. 2. Importantly, for
$\alpha>\alpha_{c}=0.4$, one can witness $\gamma_{\text{m}}<\gamma_{\pi}$ and
during this window $\gamma\in\left(\gamma_{\text{m}},\gamma_{\pi}\right)$ the
mixed soliton lattice arises while neither does the $\pi^{0}$ CSL nor the
$\eta$ CSL.

Figure 2: Critical $\gamma_{\text{m},\pi}$ at $\beta=0$ and varied $\alpha$.

In general, $\alpha_{c}=\alpha_{c}\left(\beta\right)$ depends on $\beta$,
which is displayed in Fig. 3. Therein, for the parameter region above the
$\alpha_{c}\left(\beta\right)$ line, we conclude with
$\gamma_{\text{m}}<\gamma_{\pi}$ that the mixed soliton lattice is the ground
state that demands a lower critical magnetic field and features a lower
energy, in comparison to the stand-alone $\pi^{0}$ or $\eta$ CSL.

Figure 3: Critical $(\alpha_{c},\beta)$ for the ground state to be the mixed
soliton lattice. The dots split the $\alpha$-$\beta$ plane into two regions.
In the upper region above the $\alpha_{c}\left(\beta\right)$, we have
$\gamma_{\text{m}}<\gamma_{\pi}$.

Nevertheless, this is not the end of the story, first, there can be other
(actually numerous) modes of mixing solitons, as dictated by the $p,q$ in
Eq.(15). The one demonstrated above is only the simplest one with $p=q=1$.
Moreover, even in the parameter region where $\gamma_{\text{m}}<\gamma_{\pi}$,
if we tune the magnetic field to $\gamma>\gamma_{\pi}>\gamma_{\text{m}}$, we
find that $\pi^{0}$ CSL could win the competition against the mixed soliton
lattice and take up the ground state, depending on the specific value of
$\gamma$. We will further explain these points in the next section. Before
that, it is worth thinking about the mechanism that makes mixed soliton
lattice energetically favorable, for relatively small $\beta$ and large
$\alpha$ according to Fig. 3. The favor of large $\alpha$ for a mixed soliton
lattice is explained by the kinetic term of Hamiltonian
$\tilde{H}_{\text{kin}}=\frac{\alpha}{2}\phi_{3}^{\prime
2}+\frac{1}{2}\phi_{0}^{\prime 2}$ because nonzero variation of $\phi_{0}$
will take the load off $\phi_{3}^{\prime}$ if the weight $\alpha$ is heavy.
The leverage of small $\beta$ can be seen from the mixed mass term in the
Lagrangian
$\tilde{H}_{\text{mix}}=\cos\beta\left(1-\cos\phi_{0}\cos\phi_{3}\right)$
which could be lower for $\text{$\phi_{0,3}$}\in\left(0,\pi\right)$ than that
for $\phi_{0}=0$ or $\phi_{3}\in\left(0,2\pi\right)$. In this way, we have
figured out the three basic lattice configurations with the $\pi^{0}$, $\eta$,
and the mixed soliton of $\left(p,q\right)=\left(1,1\right)$. Later on,
configurations of higher $p,q$ can be regarded as a further mixture using
these three ingredients with different proportions.

##  4 Ground State Alternation

After figuring out critical $\gamma_{\text{m},\pi,\eta}$, we readily discuss
the phases of ground state under a varying $\gamma$. Definitely we are more
interested in the parameter region above the
$\alpha_{c}\left(\beta\right)$-line where a mixed soliton lattice can be the
ground state with the lowest $\gamma$. An exemplary case is with
$\beta=\pi/16$ and $\alpha=0.7$ given that common phenomenology results
concur111Of course, the specific value depends on additional information of
flavor symmetry breaking and mixture of $\eta$ and $\eta^{\prime}$, which are
beyond our consideration and cannot be obtained in the current framework. Thus
we just choose an exemplary value. In contrast, there are other choices from
theoretical studies, e.g., in high density limit (though strictly it is the
3-flavor CFL case which does not apply here), one will have
$f_{\eta}=0.87f_{\pi}$ instead. on $f_{\eta}\simeq 1.2f_{\pi}$. In this case
we locate $\gamma_{\text{m}}=6.5<\gamma_{\pi}=6.7$, letting alone the much
higher $\gamma_{\eta}=26.7$. It is confirmed numerically that for the full
range of the magnetic field, $\eta$-solitons always have the highest energy,
compared to the others. Therefore the $\eta$ soliton lattices can be taken out
of our study for now. However, the duel between the $\pi^{0}$ and the mixed
soliton lattices for the ground state brings us another magnetic field scale
$\gamma_{\pi\text{m}}=7.2$, which flips the sign of
$\mathcal{E_{\pi}-\mathcal{E}_{\text{m}}}$. That is to say, for
$\gamma\in\left(\gamma_{\pi},\gamma_{\pi\text{m}}\right)$ both the $\pi^{0}$
and mixed soliton lattices arise, and the latter has lower energy
$\mathcal{E}_{\text{m}}<\mathcal{E}_{\pi}$, being the ground state. While for
$\gamma>\gamma_{\pi\text{m}}$, $\mathcal{E}_{\pi}<\mathcal{E}_{\text{m}}$
indicates the realm of the ground state is lost to the $\pi^{0}$-soliton
lattice. We call such a phenomenon the ground state alternation, among
different soliton lattice configurations.

Such ground state alternations can be generalized to more configurations
dictated by varied $\left(p,q\right)$. Actually, the three configurations
presented so far are those winding $U$ by $2\pi$, or $p+q=2$ in the language
of Eq. (16). The next winding class is naturally $p+q=4$. It yields two new
independent configurations $\left(p,q\right)=\left(1,3\right)$ and
$\left(3,1\right)$, for which we employed indices “4” and “5” to mark their
physical quantities respectively. Also, we clarify the general rule of naming
$\gamma_{\\#*}$ is based on the connotation that
$\mathcal{E}_{\\#}-\mathcal{E}_{*}$ turns from positive to negative when
$\gamma$ surpasses $\gamma_{\\#*}$ from below. Now we are ready to present the
booming magnetic field scales when just two new configurations 4 and 5 are
added into our consideration. We then have the following sequences of
inequalities of energies of various configurations:

$\displaystyle\gamma$
$\displaystyle\in\begin{cases}\left[\gamma_{\text{m}},\,\gamma_{\pi}\right):&\left(p,q\right)=\left(1,1\right)\text{
only}\\\
\left[\gamma_{\pi},\,\gamma_{\pi\text{m}}\right):&\mathcal{E}_{\text{m}}<\mathcal{E}_{\pi}\\\
\left[\gamma_{\pi\text{m}},\,\gamma_{4}=7.3\right):&\mathcal{E}_{\pi}<\mathcal{E}_{\text{m}}\\\
\left[\gamma_{4},\,\gamma_{4\text{m}}=8.8\right):&\mathcal{E}_{\pi}<\mathcal{E}_{\text{m}}<\mathcal{E}_{4}\\\
\left[\gamma_{4\text{m}},\,\gamma_{5}=12.8\right):&\mathcal{E}_{\pi}<\mathcal{E}_{4}<\mathcal{E}_{\text{m}}\\\
\left[\gamma_{5},\,\gamma_{4\pi}=13.0\right):&\mathcal{E}_{\pi}<\mathcal{E}_{4}<\mathcal{E}_{\text{m}}<\mathcal{E}_{5}\\\
\left[\gamma_{4\pi},\,\gamma_{\eta}\right):&\mathcal{E}_{4}<\mathcal{E}_{\pi}<\mathcal{E}_{\text{m}}<\mathcal{E}_{5}\\\
\left[\gamma_{\eta},\,\infty\right):&\mathcal{E}_{4}<\mathcal{E}_{\pi}<\mathcal{E}_{\text{m}}<\mathcal{E}_{5}<\mathcal{E}_{\eta}\;.\end{cases}$
(24)

The leftmost energies after the colon are of the ground states, followed by
those of several metastable states. Each transition is a first order
transition.

If one only focuses on the ground state, the analysis can be a bit simplified
and the conclusion is: when $\gamma$ is increased from zero, the ground state
is taken up by lattices of $\left(p,q\right)=\left(1,1\right)$,
$\left(2,0\right)$ and $\left(3,1\right)$ sequentially:

$\displaystyle\text{Ground State }\left(p,q\right)$
$\displaystyle=\begin{cases}\left(1,1\right)&\gamma\in\left[\gamma_{\text{m}},\,\gamma_{\pi\text{m}}\right)\\\
\left(2,0\right)&\gamma\in\left[\gamma_{\pi\text{m}},\,\gamma_{4\pi}\right)\\\
\left(3,1\right)&\gamma\in\left(\gamma_{4\pi},\,\infty\right)\;.\end{cases}$
(25)

Actually the configuration $\left(3,1\right)$ (or $\left(1,3\right)$) could be
regarded as a further mixture between the simplest mixed soliton and a pure
$\pi^{0}$ or $\eta$. In such a way, one can imagine infinite possibilities of
merging different mixed solitons to form different new types of lattices. The
more possibilities one takes into account, the more byzantine ground state
alternations will emerge. Certainly, we cannot exhaust all possible
configurations of mixed solitons but we do find a certain pattern of the
ground state in strong magnetic field and/or high density limit
$\gamma\rightarrow\infty$, which would be the topic of the subsequent section.

Before that, as a finale of the present section, we would quantify the ground
state alternation by the magnetic moment density (per unit transverse area),
which can be derived from its relation to the free energy from

$\mathrm{M}\equiv-\frac{\delta}{\delta B}\int dzH=-\frac{\delta}{\delta
B}\left(\mathcal{E}_{\text{topo}}d\right)=\tilde{\mu}\left(\frac{p}{2}+\frac{q}{6}\right),$
(26)

where $\tilde{\mu}=\gamma/B$ is the rescaled chemical potential which can be
regarded as a constant for fixed $\alpha$ and $\beta$. Such magnetization of
axial domain walls has been well understood for e.g., separate $\pi^{0}$ and
$\eta$ solitons in a magnetic field, from the anomalous action coupling
$A_{\mu}^{\text{B}}$ and $A_{\mu}$. Here we generalize this idea to our mixed
soliton lattices allowed by $U$. The nuance is, different $\left(p,q\right)$
feature different winding numbers and occupy different spatial lengths. Hence
it is more appropriate to consider the average of Eq. (26) over the lattice
period $d_{L}$, defined by $\bar{\mathrm{m}}\equiv\mathrm{M}/d_{L}$. If we
regard $\bar{\mathrm{m}}$ as a function of $\gamma$, it would be a piecewise
function with each jump representing a certain alternation of ground states
with changed $p$ and $q$. Again we take the five configurations appearing in
Eq. (24) as examples, demonstrating the behavior of
$\bar{\mathrm{m}}/\tilde{\mu}$ in Fig. 4. Such a figure manifests the first
order nature of the phase transition between varied ground states. The
$\bar{\mathrm{m}}(\gamma)$ diagram could exhibit finer structures when more
possibilities of $(p,q)$ are considered. We would further delve into this
point in future works as briefed in Sec. 6.

Figure 4: Magnetic moment density averaged over lattice period of alternating
ground states.

##  5 Quasicrystals in Strong Magnetic Field and/or High Density Limit

Hinted by the ground state alternation presented in the previous section, a
nontrivial phenomenon is, despite the weight factor $1/3$ of $q$ in comparison
to $p$ among the contribution to $\mathcal{E}_{\text{topo}}\propto-\gamma$,
the ground state for $\gamma\rightarrow\infty$ is not
$\left(p,q\right)=\left(2,0\right)$. In conclusion, we discover the ground
state in the strong magnetic field and/or high density (both mean large
$\gamma$) limit is the mixed soliton configuration satisfying

$\text{Ground State at }\gamma\rightarrow\infty\text{:}\quad
r\equiv\frac{p}{q}\simeq\frac{3}{\alpha}$ (27)

while being independent of $\beta$. We manifest this fact by comparing soliton
lattice configurations of different ratios $r\equiv p/q$ and finding the $r$
with the lowest $\mathcal{E}$ under a very large input of $\gamma$. To be
specific, we make our enumeration of $p$ and $q$ limited (and therefore
practical) by setting the precision goal of $r$ to
$\mathcal{O}\left(1\right)$222The higher precision is achieved by the larger
number of $p$ and $q$. For example, $r=4.2$ and $4.3$ are realized by
$\left(p,q\right)=\left(42,10\right)$ and $\left(86,20\right)$ while $r=4.28$
is realized by $\left(p,q\right)=\left(428,100\right)$.. We would explain our
result Eq. (27) via a semi-analytical approximation scheme that disentangles
the mixed soliton EOMs.

The first important observation relying on our numerical results is: $d_{L}$
tends to be small when $\gamma$ is large. We illustrate such a tendency with
the case of the configuration $\left(3,1\right)$ under again $\alpha=0.7$ and
$\beta=\pi/16$ in Fig. 5. The shape of $d_{L}\left(\gamma\right)$ remains
qualitatively similar for other values of $\alpha,\beta,p,q$. This fact could
already explain the irrelevance of $\beta$ in Eq. (27). Looking into the
energy functional $H$ in Eq. (10), one would find the $\beta$ related term is
or order $\mathcal{O}\left(1\right)$ while the kinetic term and the
$\gamma$-related term scale as $\mathcal{O}\left(d^{-2}\right)$ and
$\mathcal{O}\left(d^{-1}\right)$ respectively. Therefore when $d$ is squeezed
to a very small quantity by the large $\gamma$, the $\beta$-related
contribution to the energy becomes negligible. Physically, given the
$\beta$-related term is induced by the $\pi^{0}$ and $\eta$ masses, its effect
is naturally negligible under a large chemical potential $\mu\gg m$.

Figure 5: Dependence of the lattice period of configuration $(p,q)=(3,1)$ on
$\gamma$.

In all, we make the approximation of dropping the
$\mathcal{O}\left(d^{0}\right)$ term in the Hamiltonian in Eq. (10):

$H\left(\gamma\rightarrow\infty\right)\simeq\frac{1}{2}\left(\alpha\phi_{3}^{\prime
2}+\phi_{0}^{\prime
2}\right)-\frac{\gamma}{2\pi}\left(\phi_{3}^{\prime}+\frac{1}{3}\phi_{0}^{\prime}\right)\equiv
H_{\infty}.$ (28)

Such an approximate $H_{\infty}$ can be minimized in the following way on a
functional level directly

$\displaystyle H_{\infty}$
$\displaystyle=\frac{1}{2}\left[\left(\sqrt{\alpha}\phi_{3}^{\prime}-\frac{\gamma}{2\sqrt{\alpha}\pi}\right)^{2}+\left(\phi_{0}^{\prime}-\frac{\gamma}{6\pi}\right)^{2}\right]-\frac{\gamma^{2}}{8\pi^{2}}\left(\frac{1}{\alpha}+\frac{1}{9}\right)$
$\displaystyle\geq-\frac{\gamma^{2}}{8\pi^{2}}\left(\frac{1}{\alpha}+\frac{1}{9}\right)\equiv
E_{\min}.$ (29)

The minimization is realized when the $\phi_{0,3}$ are solved by

$\phi_{3}=\frac{\gamma}{2\alpha\pi}z,\quad\phi_{0}=\frac{\gamma}{6\pi}z.$ (30)

One can further derive the lattice period $d_{L}$ by matching the boundary
conditions of $\phi_{0,3}$:

$d_{L}=\frac{2\pi^{2}}{\gamma}\cdot p\alpha=\frac{2\pi^{2}}{\gamma}\cdot
3q\quad\Rightarrow\quad\frac{p}{q}=\frac{3}{\alpha},$ (31)

which in the meantime manifests the ground state ratio stated in Eq. (27).
Such a semi-analytic proof yields an interesting interpretation that in the
strong $\gamma$ limit, the mixed soliton lattice, is actually decoupled to two
separate CSLs of $\phi_{0,3}$ whose EOMs are solved independently. This is
because the nontrivial coupling between them comes from the $\beta$-related
terms that prove irrelevant for large $\gamma$. What binds them together is
the common lattice period $d$, i.e., the mixed soliton configuration
stipulated by the underlying torus topology $T^{2}=S^{1}\times S^{1}$. Then
naturally, the mixture proportion of two CSLs signified by $r$ depends on the
proportion of their energy contribution in $H_{\infty}$, which is ultimately
determined by the proportion of coefficients $\alpha=f_{\pi}^{2}/f_{\eta}^{2}$
in quadratic terms and the $1/3$ in $\mathcal{L}_{B}$.

Figure 6: Ground state quasicrystal demonstrated on the $p$-$q$ plane. Gray
dots represent legitimate mixed soliton lattice satisfying Eq. (16). The blue
line has a slope $3/\alpha$ with $\alpha=0.3e$. The light orange band bounded
by orange and green dash lines indicates nearby $(p,q)$ dots. These dots are
highlighted with burnt orange color. The solid orange lines mark the distances
between the $(p,q)$ dots and the $p=3q/\alpha$ slope. The closer such distance
is, the more approximate the $(p,q)$ configuration is to the ground state.

The subtlety of Eq. (31) lies in that $p/q$ is obviously a rational number
while the $3/\alpha$ on the right hand side of the equation is generally
irrational. That means, subject to the condition Eq. (16), the ground state
could only select the optimal $p$ and $q$ that produces the ratio nearest to
(irrational) $3/\alpha$. The result is thereby a 1D quasicrystal which can be
demonstrated vividly via the projection onto a 2D grid spanned by the
$p,q$-axes. We call such a quasicrystal the “chiral soliton quasicrystal” and
show an exemplary case in Fig. 6 with the choice $\alpha=0.3e$. Each gray dot
represents a certain type of mixed soliton lattice. The nearest dot (apart
from the origin) to the blue line $p=10q/e$ is actually at the infinity of the
plane. As one summons larger $p$ and $q$, the rational number $p/q$
approximates the irrational number $3/\alpha$ more accurately. Or in other
words, the periodicity with finite $p$ and $q$ could be realized only in an
approximate sense, while the true ground state does not have a periodicity.
That is why such a configuration is referred to as a quasicrystal. To the best
of our knowledge, this discovery is the first incarnation of a quasicrystal in
QCD.

In this way, with our innovative mixed solitons taken into account, Eq. (31)
informs us, at least in strong magnetic field limit, the ground state is a
mixed soliton lattice (for rational $\alpha$) or quasicrystal (for irrational
$\alpha$), whose critical $\gamma$ could be lower than that of the
$\pi^{0}$-CSL, letting alone the $\eta$-CSL. The set of irrational numbers is
dense and while the set of rational numbers is not dense. Thus, the ground
state is generally a chiral soliton quasicrystal. Such a conclusion also
applies to the case with a magnetic field of intermediate strength, depending
on $\alpha$ and $\beta$, as substantiated in Sec. 4. These lead to a
revolutionary idea that the density for $\eta$ to become physically relevant
could be lower than thought, as long as its mixture with $\pi^{0}$ in our
scenario is accounted.

Finally, we shall remark on the applicability of our scenario with respect to
the magnitude of $\gamma$. Our theoretical framework is based on the
perturbative expansion in terms of a small momentum (spatial derivative)
compared to the scale of the decay constants. Specifically, such a scale is
the cutoff momentum $\Lambda\simeq 4\pi f_{\pi,\eta}$ brought in by
considering loop effects beginning with the quartic term of momentum. On the
other hand, one can easily read off the characteristic momentum scale of our
mixed soliton lattice/quasicrystal in strong $\gamma$ limit from Eq. (30). We
should guarantee the characteristic momenta to be much smaller than the cutoff
momentum so that the derivative terms of higher order in the Lagrangian are
irrelevant compared to $\mathcal{L}_{B}$, maintaining the perturbation
hierarchy. In our dimensionless definitions, especially with the
$f_{\pi,\eta}$ absorbed into $\pi_{3}$ and $\eta$ fields (c.f. Eq. (4)), the
constraint on $\gamma$ reads

$\max\left[\frac{\gamma}{2\alpha\pi},\frac{\gamma}{6\pi}\right]\ll 4\pi.$ (32)

One can easily check that, for instance, the special values of $\gamma$ in Eq.
(24) satisfy $\gamma\ll 5.6\pi^{2}$ indeed. Such a constraint could be
translated with physical units, $\mu B\ll
16\pi^{3}\min[f_{\pi}^{3},f_{\eta}/3]$, which indicates a window of density
and/or magnetic field for the validity of our study.

##  6 Conclusion and Outlook

In this work, we have introduced the idea of the mixed soliton lattices and
quasicrystals in the context of magnetized dense hadronic matter. It has been
known that the $\pi^{0}$ and $\eta$ mesons could emerge as domain walls in
their corresponding density regimes. Nonetheless, unlike the previous
discussions on them at separate energy scales, we discover that the ground
state could be a mixture of the two kinds of solitons with a certain
proportion described by $r=p/q$. Such an $r$ would alter depending on the
value of $\mu B$, dubbed “ground state alternation”. The magnetization of such
a mixed soliton lattice is also quantified, yielding a piecewise function of
$\mu B$. In the high density and/or strong magnetic field limit, we have
highlighted that the ground state tends to be with
$r_{g}=3/\alpha=3f_{\eta}^{2}/f_{\pi}^{2}$. If such $r_{g}$ happens to be a
rational number, then a specific type of a mixed soliton lattice with the
corresponding finite $p,q\neq 0$ would take up the ground state. If $r_{g}$ is
irrational, the ground state is a mixed soliton quasicrystal. Given the
densities of (ir)rational numbers, the ground state is in general a chiral
soliton quasicrystal. In either case, the critical magnetic field could be
lower than that of a separate $\pi^{0}$ or $\eta$ CSL.

We have several outlooks for further exploration. One is to consider charged
pions which are ignored portions among $U(2)$ in our current proposal. Of
course, they form excitations with higher energy due to electromagnetic
interactions. They would bring in richer topological structures from a
different homotopy class. See Ref. Eto  _et al._ (2023) for domain-wall
Skyrmions. Another point we would further delve into is a more quantitative
explanation of the underlying mechanism that reduces the energy of mixed
soliton compared to its individual ingredients. That relies on the analysis of
the interaction between $\pi^{0}$ and $\eta$. Relevant discussions in Ref. Eto
_et al._ (2022) can be followed.

Moreover, we have gripping results under discussion for forthcoming works. The
winding number in our scenario reads $\left(p+q\right)/2$. Our scenario of the
ground state alternation implies the ground state winding number as a function
of the magnetic field could be a Devil’s Staircase (Cantor function), which is
novel and peculiar in the context of the QCD ground state. We would delve into
this point with concrete evaluations of mass parameters $a$ and $bm$, chasing
a realistic context for experimental observations.

Also significant is the extension study of our quasicrystal, especially the
gapless modes. In the case of usual lattices, the translational symmetry is
spontaneously broken to a discrete symmetry, and consequently there is a
Nambu-Goldstone mode corresponding to the spontaneously broken translational
symmetry, a phonon. On the other hand, the translational symmetry is
completely broken in quasicrystals. Quasicrystals are known to allow diffusive
Nambu-Goldstone modes, called phasons Baggioli and Landry (2020). In order to
study further aspects of quasicrystals in field theory, one should refer Refs.
Baggioli and Landry (2020); Surówka (2021). In our case, in addition to them,
there should be also ${\mathbb{C}}P^{1}$ modes corresponding to the
spontaneously broken the $SU(2)_{\rm V}$ symmetry down to $U(1)$, as the case
of the non-Abelian CSL in a rotation Eto  _et al._ (2022). Whether such non-
Abelian modes are also diffusive is an interesting problem to explore.

###### Acknowledgements.

We thank Thomas Brauner for useful comments. 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. Z.Q was supported
by JSPS KAKENHI, Grant-in-Aid for Scientific Research No. JP20J20974 (PD).

## References

  * Son and Stephanov (2008) D. T. Son and M. A. Stephanov, Phys. Rev. D 77, 014021 (2008), arXiv:0710.1084 [hep-ph] .
  * Eto  _et al._ (2013) M. Eto, K. Hashimoto, and T. Hatsuda, Phys. Rev. D 88, 081701 (2013), arXiv:1209.4814 [hep-ph] .
  * Brauner and Yamamoto (2017) T. Brauner and N. Yamamoto, JHEP 04, 132 (2017), arXiv:1609.05213 [hep-ph] .
  * Brauner  _et al._ (2019a) T. Brauner, G. Filios, and H. Kolešová, JHEP 12, 029 (2019a), arXiv:1905.11409 [hep-ph] .
  * Brauner  _et al._ (2021) T. Brauner, H. Kolešová, and N. Yamamoto, Phys. Lett. B 823, 136767 (2021), arXiv:2108.10044 [hep-ph] .
  * Evans and Schmitt (2022) G. W. Evans and A. Schmitt, JHEP 09, 192 (2022), arXiv:2206.01227 [hep-th] .
  * Grønli and Brauner (2022) M. S. Grønli and T. Brauner, Eur. Phys. J. C 82, 354 (2022), arXiv:2201.07065 [hep-ph] .
  * Brauner and Kadam (2017a) T. Brauner and S. V. Kadam, JHEP 11, 103 (2017a), arXiv:1706.04514 [hep-ph] .
  * Brauner and Kadam (2017b) T. Brauner and S. Kadam, JHEP 03, 015 (2017b), arXiv:1701.06793 [hep-ph] .
  * Brauner and Kolešová (2023) T. Brauner and H. Kolešová, (2023), arXiv:2302.06902 [hep-ph] .
  * Huang  _et al._ (2018) X.-G. Huang, K. Nishimura, and N. Yamamoto, JHEP 02, 069 (2018), arXiv:1711.02190 [hep-ph] .
  * Nishimura and Yamamoto (2020) K. Nishimura and N. Yamamoto, JHEP 07, 196 (2020), arXiv:2003.13945 [hep-ph] .
  * Eto  _et al._ (2022) M. Eto, K. Nishimura, and M. Nitta, JHEP 08, 305 (2022), arXiv:2112.01381 [hep-ph] .
  * Chen  _et al._ (2021) H.-L. Chen, X.-G. Huang, and J. Liao, Lect. Notes Phys. 987, 349 (2021), arXiv:2108.00586 [hep-ph] .
  * Eto and Nitta (2022) M. Eto and M. Nitta, JHEP 09, 077 (2022), arXiv:2207.00211 [hep-th] .
  * Higaki  _et al._ (2022) T. Higaki, K. Kamada, and K. Nishimura, Phys. Rev. D 106, 096022 (2022), arXiv:2207.00212 [hep-th] .
  * Yamada and Yamamoto (2021) A. Yamada and N. Yamamoto, Phys. Rev. D 104, 054041 (2021), arXiv:2107.07074 [hep-ph] .
  * Brauner  _et al._ (2019b) T. Brauner, G. Filios, and H. Kolešová, Phys. Rev. Lett. 123, 012001 (2019b), arXiv:1902.07522 [hep-ph] .
  * Son and Zhitnitsky (2004) D. T. Son and A. R. Zhitnitsky, Phys. Rev. D 70, 074018 (2004), arXiv:hep-ph/0405216 .
  * Chen  _et al._ (2022) S. Chen, K. Fukushima, and Z. Qiu, Phys. Rev. D 105, L011502 (2022), arXiv:2104.11482 [hep-ph] .
  * Chen  _et al._ (2023) S. Chen, K. Fukushima, and Z. Qiu, (2023), arXiv:2303.04692 [hep-th] .
  * Kawaguchi  _et al._ (2019) M. Kawaguchi, Y.-L. Ma, and S. Matsuzaki, Phys. Rev. C 100, 025207 (2019), arXiv:1810.12880 [nucl-th] .
  * Eto  _et al._ (2023) M. Eto, K. Nishimura, and M. Nitta, (2023), arXiv:2304.02940 [hep-ph] .
  * Nitta (2013) M. Nitta, Phys. Rev. D 87, 025013 (2013), arXiv:1210.2233 [hep-th] .
  * Nitta (2022) M. Nitta, Phys. Rev. D 105, 105006 (2022), arXiv:2202.03929 [hep-th] .
  * Ross and Nitta (2023) C. Ross and M. Nitta, Phys. Rev. B 107, 024422 (2023), arXiv:2205.11417 [cond-mat.mes-hall] .
  * Alford  _et al._ (1998) M. G. Alford, K. Rajagopal, and F. Wilczek, Phys. Lett. B 422, 247 (1998), arXiv:hep-ph/9711395 .
  * Rapp  _et al._ (1998) R. Rapp, T. Schäfer, E. V. Shuryak, and M. Velkovsky, Phys. Rev. Lett. 81, 53 (1998), arXiv:hep-ph/9711396 .
  * Alford  _et al._ (1999) M. G. Alford, K. Rajagopal, and F. Wilczek, Nucl. Phys. B 537, 443 (1999), arXiv:hep-ph/9804403 .
  * Aoki and Creutz (2014) S. Aoki and M. Creutz, Phys. Rev. Lett. 112, 141603 (2014), arXiv:1402.1837 [hep-lat] .
  * Nitta (2015) M. Nitta, Nucl. Phys. B 895, 288 (2015), arXiv:1412.8276 [hep-th] .
  * Eto and Nitta (2015) M. Eto and M. Nitta, Phys. Rev. D 91, 085044 (2015), arXiv:1501.07038 [hep-th] .
  * Nakano and Tatsumi (2005) E. Nakano and T. Tatsumi, Phys. Rev. D 71, 114006 (2005), arXiv:hep-ph/0411350 .
  * Nickel (2009) D. Nickel, Phys. Rev. Lett. 103, 072301 (2009), arXiv:0902.1778 [hep-ph] .
  * Basar  _et al._ (2009) G. Basar, G. V. Dunne, and M. Thies, Phys. Rev. D 79, 105012 (2009), arXiv:0903.1868 [hep-th] .
  * Buballa and Carignano (2015) M. Buballa and S. Carignano, Prog. Part. Nucl. Phys. 81, 39 (2015), arXiv:1406.1367 [hep-ph] .
  * Hidaka  _et al._ (2015) Y. Hidaka, K. Kamikado, T. Kanazawa, and T. Noumi, Phys. Rev. D 92, 034003 (2015), arXiv:1505.00848 [hep-ph] .
  * Casalbuoni and Nardulli (2004) R. Casalbuoni and G. Nardulli, Rev. Mod. Phys. 76, 263 (2004), arXiv:hep-ph/0305069 .
  * Anglani  _et al._ (2014) R. Anglani, R. Casalbuoni, M. Ciminale, N. Ippolito, R. Gatto, M. Mannarelli, and M. Ruggieri, Rev. Mod. Phys. 86, 509 (2014), arXiv:1302.4264 [hep-ph] .
  * Steinhardt (2019) P. Steinhardt,  _The Second Kind of Impossible: The Extraordinary Quest for a New Form of Matter_ (Simon $\&$ Schuster, 2019).
  * Janssen (1988) T. Janssen, Physics Reports 168, 55 (1988).
  * DiVincenzo and Steinhardt (1999) D. DiVincenzo and P. Steinhardt,  _Quasicrystals: The State of the Art_ , Series on directions in condensed matter physics (World Scientific, 1999).
  * Janot (1997) C. Janot,  _Quasicrystals: A Primer_ , Monographs on the physics and chemistry of materials (Clarendon Press, 1997).
  * Janssen  _et al._ (2007) T. Janssen, G. Chapuis, and M. D. B. et al.,  _Aperiodic crystals: from modulated phases to quasicrystals_ , Vol. 20 (Oxford University Press, 2007).
  * Stadnik (2012) Z. Stadnik,  _Physical Properties of Quasicrystals_ , Springer Series in Solid-State Sciences (Springer Berlin Heidelberg, 2012).
  * Fan (2016) T. Fan,  _Mathematical Theory of Elasticity of Quasicrystals and Its Applications_ , Springer Series in Materials Science (Springer Singapore, 2016).
  * Scott and Clark (2012) J. Scott and N. Clark,  _Incommensurate Crystals, Liquid Crystals, and Quasi-Crystals_ , Nato Science Series B (Springer US, 2012).
  * Jaric  _et al._ (1988) M. Jaric, M. Jaric, P. Bak, and D. Gratias,  _Introduction to Quasicrystals_ , Advances in Veterinary Medicine (Academic Press, 1988).
  * Baggioli and Landry (2020) M. Baggioli and M. Landry, SciPost Phys. 9, 062 (2020), arXiv:2008.05339 [hep-th] .
  * Surówka (2021) P. Surówka, Phys. Rev. B 103, L201119 (2021), arXiv:2101.12234 [cond-mat.str-el] .