*Renxin Xu, School of Physics, Peking University, Beijing 100871, China.

# Neutron Star versus Neutral Star:
On the 90th anniversary of Landau’s publication in astrophysics

Renxin Xu  School of Physics and State Key Laboratory of Nuclear Physics and
Technology, Peking University, Beijing 100871, China  Kavli Institute for
Astronomy and Astrophysics, Peking University, Beijing 100871, China
<EMAIL_ADDRESS>
(13 October 2020; 18 October 2020; 22 October 2020)

###### Abstract

In the late age of developing quantum mechanics, Lev Landau, one of the
distinguished players, made great efforts to understand the nature of matter,
even stellar matter, by applying the quantum theory. Ninety years ago, he
published his idea of “neutron” star, which burst upon him during his visit
over Europe in the previous year. The key point that motivated Landau to write
the paper is to make a state with lower energy for “gigantic nucleus”,
avoiding extremely high kinematic energy of electron gas due to the new Fermi-
Dirac statistics focused hotly on at that time. Landau had no alternative but
to neutronize/neutralize by “combining a proton and an electron”, as electron
and proton were supposed to be elementary before the discovery of neutron.
However, our understanding of the Nature has fundamentally improved today, and
another way (i.e., strangeonization) could also embody neutralization and thus
a low-energy state that Landau had in mind, which could further make
unprecedented opportunities in this multi-messenger era of astronomy.
Strangeon matter in “old” physics may impact dramatically on today’s physics,
from compact stars initiated by Landau, to cosmic rays and dark matter. In
this essay, we are making briefly the origin and development of neutron star
concept to reform radically, to remember Landau’s substantial contribution in
astrophysics and to recall those peculiar memories.

###### keywords:

pulsar, neutron star, dense matter, elementary particles

††articletype: Article Type

##  1 Introduction

The Mandela effect111 Interestingly, a similar effect has already been
included in a Chinese idiom, “to circulate erroneous reports by someone
wrongly informed” (yǐ é chuán é) in as early as the southern Song dynasty, to
be very popular in the social contacts nowadays.  is miraculous in psychology
and media studies, which could be the result of wrong combination of memory
fragments under memory reconstruction. Unfortunately, it affects also
tremendously the historical memory of the neutron star concept, as will be
explained briefly here. Researchers are encouraged to think seriously and
independently to read the literatures, drawing a lesson from the mix-up
memory, especially during this information age.

Yes, it should be recommended for one to focus on real things rather than only
fashions in the mainstream, since the latter could be strongly coupled with
the media transportation. Nonetheless, there is some fundamental differences
between natural sciences and social actions: the laws of nature are objective,
never minding the social emotions! In this sense, the publication of Landau’s
idea about neutron stars could also be an excellent example for educational
meaning.

Let’s make an effort in this aspect. In order to clarify the fact and its
implications, the sciences relevant to the publication by LD. Landau (1932) on
stars, including the past and future, would be reformed from the bottom in
this contribution, on the 90th anniversary of Landau’s presenting his original
idea.

##  2 The magnificent era: physics from macroscopic to microscopic, and
eventually to the quantum

The developments of social civilization are driven by the curiosity of human
beings. What’s the nature of our material world (e.g., the rock shown in Fig.
1)? Why does it exist?

Figure 1: A rock containing copper (Malachite). Normal atom matter at pressure
free is condensed by the electromagnetic (or simply electric) force, while the
strangeon matter, to be explained in §6, is by the fundamental strong
interaction. Multiscale forms can exist for both kinds of condensed matter,
the electric and the strong ones.

The basic unit of normal matter was speculated even in the pre-Socratic period
of the Ancient era (the basic stuff was hypothesized to be indestructible222
Oppositely, substances in our daily life are thought to be destructible in
ancient China, as was recorded in Tianxia Pian by Zhuang Zi ($\sim$
369BC$-$286BC): “ Taking a stick of wood with finite extent, one cuts half
each day; it is expected to last for an infinitely long time”.  “atoms” by
Democritus, $\sim$ 460BC$-$370BC), but it was a belief that symmetry, which is
well-defined in mathematics, should play a key role in understanding the
material structure, such as the Platonic solids (i.e., the five regular convex
polyhedrons).

The philosophical thinking above became gradually related to practical
sciences of understanding the thermal phenomenon more than two thousand years
later ($\sim 1700$AD$-1900$AD), resulting in the first industrial revolution.
Thermodynamics focuses on the thermal properties of matter from a macro
perspective, while statistical physics studies the global behavior of a huge
number of particles in molecular models. Both approaches reach the same end
from different directions. Many forerunners offered wisdom for understanding
atom matter (e.g., Fig. 1), especially the realist among the main figures,
Ludwig Edward Boltzmann ($1844-1906$), who “stands as a link between two other
great theoretical physicists: James Clerk Maxwell in the 19th century and
Albert Einstein in the 20th” (Cercignani Penrose, 2006). This is a pretty big
step to explore the fundamental units of Nature, leading us into a scientific
world from atoms to subatoms.

Ahh…What’s the nature of atom? In fact, atoms are destructible since J.
J.Thomson addressed that the so-called “cathode rays” leaving from atoms are
actually negatively charged but extremely light units (i.e., electrons) in
1897. The other part of atom is believed to be positive, massive and small
nucleus after Ernest Rutherford explained the large angle scattering of the
Geiger–Marsden experiment in 1911. However, such a “nucleus+electron(s)” atom
cannot be stable at all in classical electrodynamics, though Niels Bohr
assumed that an atom should not collapse by introducing a quantum angular
momentum (ie. Bohr’s distinct energy states) in 1913, that calls for a new era
of completely different mechanics.

It is now well known that the revolution of quantum mechanics unveiled the
facts of atom. Beginning with the study of black-body radiation, especially
the Planck’s formulae of spectrum in 1900 (it is worth noting that Boltzmann’s
ideas were central to Planck’s analysis), we know that light is particle-like,
each one with energy (Einstein in 1905) and momentum (Compton in 1923), though
it is wave-like in classical electromagnetism. This peculiar wave-particle
duality of light was extended for all the other “particles”, particularly the
electron, by de Broglie in 1924. Eventually, quantum mechanics (the matrix
mechanics by Heisenberg in 1925 and the wave mechanics by Schrödinger in 1926)
was established, which reproduces quantitatively the eigen energies of the
ground state and the excited ones, providing essentially a solid foundation
for Bohr’s distinct energy states.

However, why don’t all the electrons in a stable atom go to the ground quantum
state? Physicists cannot understand until a “new” statistics (i.e., the third
one) was discovered.

##  3 The “New” Statistics and its implications

By the end of the 19th century and at the beginning of the 20th, Maxwell-
Boltzmann (MB) statistics, successful in explaining the thermodynamics of gas
and sometimes of liquid and solid, was believed without doubt until Einstein
received a letter from Bose in 1924, about revisiting the puzzling black-body
radiation. Today, we know that Bose-Einstein (BE) statistics is for a quantum
system of identical particles (i.e., Bosons) whose states are exchangeable.
Certainly, the thermal system of photons is a typical example.

The third statistics was established soon after Pauli proposed his exclusion
principle for electrons in 1925, explaining the periodic law of chemical
elements. This statistical property of electron, differing from that of
photon, was noted333 It is also said that Pascual Jordan (1902-1980) was the
first one who proposed the new statistics in 1925 (Schucking, 1999). This sad
story happened due to Max Born’s negligence. Jordan referred to his own
brainchild as “the Pauli statistics”.  independently by Fermi and Dirac in
1926, marking the birth of the new statistics, i.e., the Fermi-Dirac (FD) one.

FD statistics is powerful in understanding the Nature, and the first three
examples are listed as following.

1, To solve a problem in the “Heaven” by Fowler (1926): What’s nature of white
dwarfs with extremely high compactness compared to normal main-sequence stars?
In FD statistics, cold electron gas in a white dwarf, even at zero
temperature, could still contribute significant pressure, while the pressure
in main-sequence stars is only of MB statistics. In fact, at low temperature
limit, the pressure of FD statistics is surprisingly larger than that of
traditional MB statistics, because the kinematic energy of dense electrons is
remarkably high, as illustrated by the Heisenberg relation $\Delta
x\cdot\Delta p\sim\hbar$, with $\Delta x$ the separation between electrons,
$\Delta p$ the order of momentum and $\hbar$ the Planck constant.

2, To solve a problem on the “Earth” by Sommerfeld (1927): Why is the specific
heat of electrons in metal negligible? Although the Drude-Lorentz model
(1900-1909) based on MB statistics may explain the electrical and thermal
conductivities of metal, only a small part of electrons near the Fermi
surface, $\sim kT/E_{\rm f}$, would contribute to the heat capacity in FD
statistics, with $k$ the Boltzmann constant, $T$ the temperature and $E_{\rm
f}$ the Fermi energy of electron which is much larger than $kT$ at room
temperature. The reason for this is also that the kinematic energy of an
electron in comparably cold and dense matter is remarkably high. This was
testing the capability of quantum theory on problems beyond atoms, but the
point of extremely high energy in cold and dense electron gas is meaningful
again!

3, To make an effort of understanding the magnetism, to be so familiar a
phenomenon but also a test of FD statistics. The early history of quantum
theory of solids, especially of metals was well summarized by Hoddeson .
(1987). Pauli (1927), for the first time, attempted to explain paramagnetism
with FD statistics, but the others experimentally observed (e.g.,
diamagnetism, magnetoresistance, the Hall effect) were still puzzling. As one
of the visiting fellows, Landau joined in Pauli’s group in 1929, and away from
the group, he published his solution of the full quantum-mechanical problem of
electrons orbiting in a magnetic field, explaining diamagnetism (L. Landau,
1930). This research was related to Landau’s trip to Europe, as will be
discussed in the next section.

##  4 Lev Landau and his scientific trip over Europe

The west side of Europe is a vital source of modern sciences, which was
certainly attracting young students with a strong spirit of inquiry about the
Nature. Although the international exchanges become more and more convenient,
there are still two major ways of reaching the center of science even today.
Two archetypes of the successful Asian-born youngsters of genius, during the
later period of developing quantum theory in 1930s, are presented as
following.

One is for Subrahmanyan Chandrasekhar (1910$-$1995), born in Indian (Saikia
Trimble, 2011) as a part within the British Commonwealth (about six thousand
kilometers away from the center of science). As one of the undergraduates at
Presidency college, Chandra listened to the speech lectured by Sommerfeld in
the autumn of 1928 and discussed afterward about FD statistics, leading him to
publish the first and undergraduate research (Chandrasekhar, 1929) on Compton
scattering of moving electrons which obey FD statistics (communicated by R. H.
Fowler), to supplement Dirac’s work on Compton scattering of moving electrons
with Maxwellian distribution in hot stellar atmosphere (Dirac, 1925).

On his two-and-a-half week journey from India to Cambridge in July 1930,
Chandra recognized that his supervisor’s calculation (Fowler, 1926) was based
on non-relativistic energy momentum relation, but the degenerate electrons
could be relativistic in massive white dwarfs. Chandra submitted this research
titled “The maximum mass of ideal white dwarfs” ($M_{\rm max}=0.91M_{\odot}$,
applying the theory of polytropic gas spheres) to ApJ on 12nd November 1930,
published in the following July (Chandrasekhar, 1931), known as the
“Chandrasekhar limit” in text books. Chandra was very productive in
astrophysics, from hydrodynamics, magnetohydrodynamics, to general relativity
and black hole.

The other is for Lev Landau (1908$-$1968), born in Baku of Azerbaijan
(Lifshitz, 1969), one of the former Soviet Union republics (about four
thousand kilometers away from the center of science). Landau started to travel
abroad in October 1929, initially supported by the People’s Commissariat of
Education, worked in Denmark, Great Britain and Switzerland, and returned to
Leningrad Physicotechnical Institute in March 1931. Besides the theory of the
diamagnetism of an electron gas, Landau was thinking the relativistic
electrons in white dwarfs during his third times of visiting Bohr’s group in
Copenhagen when his Soviet stipend had run out but Bohr helped obtain a
Rockefeller Fellowship which allowed Landau to prolong his stay. During this
time from February to March in 1931, Landau, Bohr and Rosenfeld discussed a
paper written by Landau but not published, nonetheless Landau submitted the
manuscript in January 1932 and published in February (LD. Landau, 1932), the
same month when the discovery of the neutron was announced (Yakovlev ., 2013).

The key point of LD. Landau (1932) is also relevant to the highly kinematic
energy of dense electron gas, as sampled in §3 on FD statistics. Can any
physical mechanism cancel the kinematic energy of electrons (typically $\hbar
c/\Delta x\sim 300$ MeV if electrons keeps there) to form a stable state with
lower energy? Landau provided a way to do this, by combining “protons and
electrons in atomic nuclei very close together”, and he “expect that this must
occur when the density of matter becomes so great that atomic nuclei come in
close contact, forming one gigantic nucleus”. However, he didn’t define
gigantic nucleus: where is the boundary of baryon number, $A_{\rm c}$, between
normal microscopic and gigantic nuclei? Certainly a nucleus with stellar mass
(baryon number $A\sim 10^{57}$) is gigantic since the corresponding number
$A<300$ for atomic nuclei of chemical elements, but the critical number
$A_{\rm c}$ could be somewhere between numbers differing $\sim 55$ orders of
magnitude!

The “neutron star” concept superficially anticipated by Landau is not
surprising, though before the discovery of neutron. Before the establishment
of quantum mechanics, it is a general idea that neutral doublet (combination
of an electron and a proton, thus to differentiate cementing electrons from
planetary ones in an atom) may exist in an atomic nucleus (e.g., Rutherford,
1920; Harkins., 1920), postulating“novel properties”: to move freely through
matter and to be difficult to detect. This specific feature of weak
interaction resembles that of dark matter with indirect evidence by gravity.
Harkins (1921) renamed doublet “neutron”, which was discovered by Chadwick
(1932). Today, neutron imaging are very useful in material science and
engineering, as well as biology (Anderson ., 2009).

Even before the work by Chandrasekhar (1931), Edmund C. Stoner (1899$-$1968),
known for his work on the origin and nature of magnetism, had also noted a
limiting mass of white dwarfs ($M_{\rm max}\sim M_{\odot}$) in the uniform
density approximation (Stoner, 1930), considering that the total energy of
kinetic electrons and stellar gravity should be minimum. LD. Landau (1932)
recalculated the mass limit of white dwarfs ($M_{\rm max}=1.5M_{\odot}$) too,
which could be expressed in terms of fundamental constants. Let’s consider a
self-gravitational star with mass $M$ and radius $R$, consisting of basic unit
with mass $m_{0}$ and total number $N$. With the overall energy, $E\simeq
N\cdot(\hbar cN^{1/3}-Gm_{0}^{2}N)/R$, one has then the mass limit $M_{\rm
max}\sim N_{\rm max}\cdot m_{0}\simeq M_{\rm p}^{3}/m_{0}^{2}$, with the
Planck mass $M_{\rm p}=\sqrt{\hbar c/G}=1.2\times 10^{19}$ GeV, otherwise
$R=0$ if $N>N_{\rm max}$. Estimating the scale of each unit to be the Compton
wavelength $\lambda_{0}\sim\hbar/(m_{0}c)$, we have thus the typical radius
$R\simeq\lambda_{0}\cdot N_{\rm max}^{1/3}$. For $m_{0}\sim 1$ GeV, we have
characteristic magnitudes for compact stars: $N_{\rm max}\sim 10^{57}$ and
$M_{\rm max}\sim M_{\odot}$. It is evident that both quantum physics and
special/general relativity should participant in this study, as was indicated
by the physical constants of $\hbar$, $c$ and $G$.

Landau’s idea develops then, especially after the discovery of pulsars, and
becomes very elaborate models of normal neutron stars in the mainstream. But
there is still a fundamental misunderstanding about the origin of the idea.
Rosenfeld told an anecdote in 1974 that Landau improvised the concept of
neutron stars in a discussion with Bohr and Rosenfeld just after the news of
the discovery of the neutron reached Copenhagen in February 1932, which is
noted in the famous text book by Shapiro Teukolsky (1983). This false memory
of Rosenfeld, presented in a talk 43 years later, matters seriously in the
academic society, that is another example caused by the Mandela effect.

##  5 Landau’s Faith: Neutron Star or Neutral Star?

As discussed above, Landau’s mind was actually to make a neutral star when
density becomes so large that atomic nuclei come in close contact. It is worth
noting, however, that the mass of Landau’s gigantic nucleus (LD. Landau, 1932)
should not always be $>M_{\odot}$, but probably $\ll M_{\odot}$, i.e., maybe
$A_{\rm c}\ll 10^{57}$. Landau noticed that a white dwarf with mass $>M_{\rm
max}$ cannot be stable against collapsing to a gigantic nucleus, but low-mass
such a nucleus, surrounded by normal ions and electrons, could still be
possible if the laws of quantum mechanics (and thus FD statistics) are
violated. Many authors did then follow along the line of Landau (1932,1938),
to model neutron stars tested by astronomical observations, setting chemical
equilibrium between phases in a star, since afterwards Landau published a
special paper focused again on this topic, trying to solve the problem of
stellar energy (L. Landau, 1938).

We have to say that only neutronization can achieve Landau’s goal of
neutralization during the era when proton and neutron are supposed to be
fundamental particles. But a dangerous law, unknown at Landau’s time, lurks in
this option: the nuclear symmetry energy! Although neutronization can remove
energetic electrons, an extremely high asymmetry of the resultant isospin will
contribute significant symmetry energy. Meanwhile, due to the large asymmetry
of isospin, dense electron gas would be necessary to suppress the
$\beta$-decay of neutron to proton, that requires normal matter of stellar
crust to meet the standard of such an electron density.

Should the neutronization be the only way to neutralize in a gigantic nucleus?
Certainly, alternative way to neutralize is possible in the standard model of
particle physics. Due to the strong coupling of $\alpha_{\rm s}\sim 1$, quarks
are bound inside a system (e.g., a nucleon) with scale of $\leq 1$ fm, and the
typical energy is then $E_{\rm scale}\sim\hbar c/\Delta x\simeq 0.5$ GeV, with
$\Delta x$ the separation between quarks. This means that, in the first step,
we may ignore heavy flavours of quarks (c,t,b), and could take advantage of a
triangle diagram (Fig. 2) for the light flavors of quarks (u,d,s). Normal
atomic nuclei are around point “A”. Landau’s way ends at point “n”, while the
other way could be from “A” to “s”. Both ways above are of neutralization, in
response to the original intention of LD. Landau (1932). At the quark level,
the net results of neutronization and strangeonization are $u+e\rightarrow
d+\nu_{e}$, $u+e\rightarrow s+\nu_{e}$, respectively.

Light quark-flavour symmetry is restored at point “s”, but the building units
could be either quarks (e.g., Witten, 1984) or strangeons (RX. Xu, 2003; Lai
Xu, 2017). The former is based on the pertubative strong interaction between
quarks, while the latter on the non-pertubative. It is well-known that the
nature of an atomic nucleus, as microscopic strong matter, is determined by
non-pertubative QCD (quantum chromodynamics), and one may simply argue that a
gigantic nucleus at pressure free could also be the similar. In this sense, a
strangeon is an analogy of a nucleon, but only changing the number of valence-
quark flavours from 2 to 3, i.e., 2-flavoured nucleon while 3-flavoured
strangeon.

In a word, atomic nuclei are 2-flavoured, but a gigantic nucleus could be
3-flavoured. Where is the boundary? One should be embarrassed for answering
the question about $A_{\rm c}$, which is essentially determined by both the
weak and the strong interactions, especially the latter in non-perturbative
regime. Frankly speaking, one should ask Landau for $A_{\rm c}$ since he
proposed “gigantic nucleus” 90 years ago, if he could still be healthy today
and if one does not care about Landau’s possible feeling of awkwardness.
Nonetheless, from the weak side, an electron becomes relativistic if localized
in the Compton wavelength $\hbar/(m_{\rm e}c)\sim 10^{3}$ fm, one may have
$A_{\rm c}\sim 10^{9}$. From the strong side, one could have $A_{\rm c}\geq
10^{2}$ in a liquid drop model. We may then estimate a value of $A_{\rm c}$
between $10^{3}$ and $10^{10}$.

##  6 Neutron versus Strangeon

Although both neutron and strangeon are neutral, two differences between them
are worth noting.

1, Flavour Symmetry. Adopting a phenomenological approach, nuclear physicists
introduce a symmetry energy representing the symmetry between the proton and
neutron of a stable nucleus, which is essentially the balance of two flavors
of quarks (u,d), but the underlying physics is yet to be well understood. For
2-flavoured nuclei, it’s impossible to satisfy both the neutralization and the
flavour-symmetry, but simply a straightforward task for 3-flavoured nuclei.
444The isospin is 1/2 for nucleon, but could be defined as 0 for strangeon. It
seems natural if Nature loves a principle of flavour maximization (Xu, 2018;
Miao ., 2022). This difference carries profound implications: 2-flavoured
strong matter in bulk is too less-massive to be stable against decaying into
atomic nuclei (the minimum mass of conventional neutron star could be $\sim
0.1M_{\odot}$), while the baryon number of stable strangeon matter could be as
low as $10^{10}$ (so-called strangeon nugget).

2, Mass and Wavepacket Size. It might be fine to represent an atomic nucleus
as a liquid drop since the quantum wavepackets of identical nucleons could
overlap. However, the quantum wavelength of massive strangeon is comparably
short, and one may attribute the properties of strangeon matter to classical
physics. Unless during its formation at high temperature, strangeon matter
could be in a solid state as the thermal kenematic energy would be much lower
than the interaction potential (a few $10$ MeV) between strangeons. The free
energy, elastic and gravitational, of solid strangeon stars could power
extraordinary astrophysical bursts, from so-called magnetars (RX. Xu ., 2006),
to fast radio bursts (Wang ., 2022), and even $\gamma$-ray burst (R. Xu Liang,
2009). Further, the non-relativity of strangeons results in a stiffer equation
of state than that of nucleon matter (Lai Xu, 2009), a simple key on solving
the hyperon puzzle.

Is strangeon matter just hyperon matter? No hyperon matter has been
conjectured yet, but strangeon differs from hyperon in two facts at least.
$a$, The baryon number of hyperon is $B=1$, whereas strangeon is baryon-like
but not a real baryon, whose number could be $B>1$, at least $B=2$ (dibaryon)
and maybe $B=6$ (quark alpha, $Q_{\alpha}$). $b$, The interaction between two
$\Lambda$-particles could be attractive, with binding energy of a few MeVs
(Green ., 2021) or higher according to lattice QCD simulations. Certainly, an
attractive interaction inbetween could not support against gravity.

Let’s go back to Fig. 1. The mass spectrum of normal electric matter is
continuous, from molecules (a few atom-units), to dusts (much smaller than the
Avogadro constant), planets ($\sim 10^{52}$), and stars ($\sim 10^{57}$).
However, there could be a gap (around $\sim A_{\rm c}$) in the mass
distribution of strong matter: nucleon matter exists for baryon number
$A<A_{\rm c}$, while strangeon matter for $A_{\rm c}<A<A_{\rm max}$.

Quakes are natural on the solid Earth, and thus on solid strangeon stars too.
Similar to electric matter inside the solid Earth, the geometrical symmetry of
strangeons in solid matter could also not be simple and perfect, and a
quasicrystal structure is possible too. An amorphous (glass-like) structure of
solid strangeon matter could even form if the cooling is so quick that phase
equilibrium can hardly be achieved during liquid-solid transition. Anyway, a
quake as a result of slip should occur along the direction on a fault plane
(Lu ., 2022), as commonly observed on the Earth, with simulated results
illustrated in Fig. 3. Besides strangeon stars, strangeon nuggets formed
through a crossover QCD phase transition in the early Universe before the big
bang nucleosynthesis has also been demons, indicating a strangeon dark matter
candidate without introducing particles beyond the standard model (Wu .,
2022).

Figure 2: An updated triangle of light-quark flavours (Xu, 2020). The points
inside the triangle define the states with certain quark number densities of
three flavours ({$n_{\rm u},n_{\rm d},n_{\rm s}$} for up, down and strange
quarks), measured by the heights of one point to one of the triangle edges.
Axis $S$ denotes strangeness, with perfect isospin symmetry. Normal nuclei are
around point “A”, conventional neutron stars are in point “n”, extremely
unstable proton stars are in point “p”, and strange stars (both strange quark
star and strangeon star) in point “s” in the center of the triangle with
$n_{\rm u}=n_{\rm d}=n_{\rm s}$. The charge-mass-ratio of quarks at point “A”
is $\mathcal{R}\simeq 1/2$, but $\mathcal{R}=0$ on the line of “ns”; the
former is “electronic” and “luminous”, while the latter could hardly be
detected because of neutrality and would take a “dark” role as matter. The
zoomed $\triangle$snA shown in the right indicates that, supernovae explosions
(from point “A” to a point in line “ns”) realize neutronization
($\mathcal{N}$) or strangeonization ($\mathcal{S}$), and kilonovae realize
inverse-neutronization ($\bar{\mathcal{N}}$, from point “n” back to point “A”)
or inverse-strangeonization ($\bar{\mathcal{S}}$, “s$\rightarrow$A”).  Figure
3: A comparison between starquakes in conventional neutron stars (panel “a”,
NS) and in strangeon stars (panel “b”, SS), showing dependence of glitch
magnitude (log ($\Delta\nu/\nu$)) on the starquake strain drop ($\varepsilon$)
and moment ratio ($\chi$). The maximum and minimum glitch magnitude
($10^{-10}\sim 10^{-5}$) are bounded by black lines. The line of $10^{-5}$ is
not given in NS model for it is over the theoretical maximum. The shadow in
panel “a” mark the area that $\chi$ is not allowed in neutron stars. A scaled
earthquake of the largest observed one (i.e. C60: 1960 Chile earthquake) is
marked as white stars in both figures as references, respectively. Grey lines
are isolines of starquake magnitude with corresponding $\Delta\varepsilon$ and
$\chi$. Panel “c” shows the comparison between NS and SS models. The range of
observation glitch amplitude is shown as horizontal dot lines. C60 in the two
models is respectively marked as the blue and orange star. These calculations
presented by Lu . (2022) show that the observed glitch amplitude can be
explained by the starquakes in the strangeon star model, though the required
scaled starquake magnitude is much larger than that occurred on the Earth.

##  7 Summary and Outlooks

This note is my second one in memory of Landau’s original idea on neutron
star, with the first one published around ten years ago (Xu, 2011). Following
Landau’s approach more than 90 years ago, we are explaining a novel kind of
matter at pressure free, the strong matter by strangeons rather than by
nucleon, as an analogy of the normal atom matter illustrated in Fig. 1 for an
example. Today, the nature of pulsar-like stars is focused in both physics and
astrophysics, hopefully to be the first big problem solved in this multi-
messenger era of astronomy. A detailed study in microphysics may eventually
reveal the critical number, $A_{\rm c}$. Although the number can’t yet be
computed from the first principles at present, it does not mean that Nature
will not adopt this idea and refuse to use it.

In fact, a strangeon is a kind of multi-quark states, which is hotly discussed
in high-energy physics, both theoretical and in experimental. It is surely
welcome to investigate the formation of strangeon with either lattice QCD or
effective field theory, that would be conducive to quantitatively describing
the state equation of supranuclear matter. Although one cannot rule out other
possibilities, e.g., that multi-quark states and free quarks could coexist
(Burikham ., 2009), the energy scale of strong matter at pressure free should
not be fine-tuned. Therefore, a consistent state of chemical composition would
be reasonable, but the baryon numbers of strangeons from the surface to the
center may increase slightly nevertheless.

Looking back at the journey to understand the nature of our material world, we
could be trying to know strong matter in an opposite direction to that of
electric matter. The study of electric matter in Fig.1told us the micro-units
and their quantum nature, but we are reproducing the world of strong matter
with the standard model (QCD and the electroweak theory).

## ACKNOWLEDGMENTS

The author would like to thank those involved in the continuous discussions in
the pulsar group at Peking University. This work is supported by the National
SKA Program of China (2020SKA0120100). The author would like to acknowledge
his debt to Mr. Shichuan Chen, who reads the manuscript carefully with
valuable comments.

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# Deep Learning Approach Combining Lightweight CNN Architecture with Transfer
Learning: An Automatic Approach for the Detection and Recognition of
Bangladeshi Banknotes

Ali Hasan Md. Linkon  Computer Science and Engineering
Shahjalal University of Science
and Technology
Sylhet, Bangladesh
<EMAIL_ADDRESS>Md. Mahir Labib  Computer Science and Engineering
Shahjalal University of Science
and Technology
Sylhet, Bangladesh
<EMAIL_ADDRESS>Faisal Haque Bappy  Soumik Sarker  Computer Science
and Engineering
Shahjalal University of Science
and Technology
Sylhet, Bangladesh
<EMAIL_ADDRESS>Computer Science and Engineering
Shahjalal University of Science
and Technology
Sylhet, Bangladesh
<EMAIL_ADDRESS>Marium-E-Jannat  Computer Science and Engineering
Shahjalal University of Science
and Technology
Sylhet, Bangladesh
<EMAIL_ADDRESS>Md Saiful Islam  Computer Science and Engineering
Shahjalal University of Science
and Technology
Sylhet, Bangladesh
<EMAIL_ADDRESS>
###### Abstract

Automatic detection and recognition of banknotes can be a very useful
technology for people with visual difficulties and also for the banks itself
by providing efficient management for handling different paper currencies.
Lightweight models can easily be integrated into any handy IoT based
gadgets/devices. This article presents our experiments on several state-of-
the-art deep learning methods based on Lightweight Convolutional Neural
Network architectures combining with transfer learning. ResNet152v2,
MobileNet, and NASNetMobile were used as the base models with two different
datasets containing Bangladeshi banknote images. The Bangla Currency dataset
has 8000 Bangladeshi banknote images where the Bangla Money dataset consists
of 1970 images. The performances of the models were measured using both the
datasets and the combination of the two datasets. In order to achieve maximum
efficiency, we used various augmentations, hyperparameter tuning, and
optimizations techniques. We have achieved maximum test accuracy of 98.88% on
8000 images dataset using MobileNet, 100% on the 1970 images dataset using
NASNetMobile, and 97.77% on the combined dataset (9970 images) using
MobileNet.

###### Index Terms:

Bangladeshi banknote, Bangladeshi paper currency detection and recognition,
Currency detection and recognition, Deep Learning, Convolutional Neural
Network, Transfer Learning

##  I Introduction

Bangladeshi banknote detection is vital from many points of view. Banknote
detection technology can benefit many people with visual disabilities.
According to WHO, at least 2.2 billion people worldwide are affected by vision
impairment or blindness[1]. In Bangladesh, approximately 0.8 million people
are blind[2]. Visually impaired people may use this form of technology to make
better decisions about differences in notes. ATMs or other currency detection
devices can also use the banknote detection model for better improvement.
However, the lack of a dataset and efficient model reduces Bangladeshi
banknote detection’s chance of improvement. Due to the lack of resources,
limited research work was done on Bangladeshi banknote detection. In order to
avoid this problem, data augmentation and transfer learning can be a possible
solution. In recent times a dataset of 1970 Bangla banknote images has been
available on Kaggle named ‘Bangla Money (Taka recognition dataset)’[3].
Moreover, Murad et al.[6] has also created a vast dataset of 8000 banknote
images.

Figure 1: Simplified architecture of our model

In this paper, we aim to investigate transfer learning combining with deep
learning methods for banknote detection and recognition. Transfer learning has
helped us to overcome the limitations of the dataset. We have trained two
datasets using MobileNet, NASNetMobile, and ResNet152v2 pre-trained on the
ImageNet dataset. We have also used various augmentation techniques, such as
rotating, zooming, shifting, and shearing, to reduce the dataset’s
limitations. In order to achieve maximum efficiency, we have tuned
hyperparameters and used different optimization techniques like reducing
learning rates on plateaus, early stopping, and model checkpoints. We have
achieved the best result of 98.88% in 8000 images dataset using MobileNet,
100% in the 1970 images dataset using NASNetMobile, and 97.77% in the combined
dataset using MobileNet. Figure 1 represents the simplified architecture of
our model. In the ‘CNN feature extractor’ section, we have used different
models such as MobileNet, NASNetMobile, and ResNet152v2. We have shown that
lightweight models like MobileNet and NasNetMobile have achieved better
accuracy than the heavyweight model ResNet152v2.

Section II addresses some notable related works on automatic currency
recognition and detection system. In Section III, we have included a concise
explanation of our experimented datasets. We have discussed our proposed
method in the Section 3. In Section V, we have compared and analyzed the
received results and accuracy of different models. In Section VI, we have
discussed the limitations and scope of potential research in Bangladeshi
banknote recognition.

##  II Related Work

Nadim Jahangir et al.[4] has contributed significantly to the early stage of
Bangladeshi banknote recognition. According to them, they first converted
images to gray-level images and passed through Histogram Equalization. The
equalized image then was converted to a binary image by thresholding. Various
MASKs were imposed on the binary image to get slab values and make it readable
to ANN. They claimed their model’s average accuracy as 98.57% with incorrect
recognition for some old, defective, taint, and worn-out notes.

In M.M. Rahman et al.[5], the dataset was collected from various sources over
the Internet. They applied Oriented FAST and Rotated BRIEF (ORB) on that image
data. They claimed their system could identify Bangladeshi paper currency
notes with 89.4% accuracy on the white background and 78.4% accuracy on the
complex background.

Hasan Murad et al.[6] contributed a vast novel dataset of 8000 images of
Bangladeshi banknotes. For the classification of banknotes, they used
MobileNet[10] deep learning architecture. They claimed to achieve an overall
accuracy of 99.80% in the testing phase. However, their model recognized any
background other than a white one as a valid note as their dataset did not
cover all background variations.

In Shubham Mittal et al.[7], they used the transfer learning technique on 380
images dataset of Indian Currency Rupee. Pre-trained lightweight MobileNet[10]
achieved 96.6% accuracy on the test dataset.

Abhishek Pathak et al.[8] introduced a mobile-based Indian currency detection
model that would allow visually impaired people to check a currency’s value.
K-means clustering was used for the classification. They claimed to use
features like banknote color, ROI, and background.

N. A. J. Sufri et al.[9] used the banknote region and orientation of using
Malaysian Ringgit banknotes. According to their claim, both k-NN and Decision
Tree Classifier achieved 99.7% accuracy while SVM and Bayesian Classifier
achieved 100% accuracy.

Figure 2: Comparison between the Datasets, Bangla Money (Taka recognition
dataset)[3] and Bangla Currency Dataset[6]

##  III Dataset

For this study, we have collected datasets from two different sources. One is
Kaggle’s open-access dataset named as ‘Bangla Money (Taka recognition
dataset)’[3], and another is from Hasan Murad et al.’s[8] with the author’s
approval. ‘Bangla Money (Taka recognition dataset)’[3] by Noushad Sojib et al.
consists of 1970 Bangladeshi banknote images of 9 different categories. This
dataset has 1, 2, 5, 10, 20, 50, 100, 500, 1000 Taka Bangla banknotes. All
images are captured with mobile phone camera, and each image is of
120$\times$250 pixels.

Hasan Murad et al.’s[6] ‘Bangla Currency Dataset’ consists of 8000 images of
Bangladeshi banknotes. Currently, this is the largest Bangla banknote dataset
to our best knowledge. However, this dataset has not contained any 1 Taka
Bangladeshi banknote. This dataset has data of 8 classes. For each class, this
dataset has 1000 images. For making the dataset adjustable to real-life
scenery, all of the images are taken in arbitrary lighting, resolution,
folded, and background. Each of the image sizes is 224$\times$224 pixels.
Figure 2 represents the number of images and categories in the two datasets.
Figure 3 denotes sample images from both datasets.

As both of the dataset’s size is not large enough, data augmentation is
essential for the model improvement. Due to proclaimed efficiency, we have
used online data augmentation to reduce overfitting. Each image is augmented
into 10 more images. As a result, the number of images to be trained increased
10 times. Augmentations techniques are given below:

  * –

Rotating (range = [0,180])

  * –

Width & Height shift (range = 0.1)

  * –

Shearing (range = 0.1)

  * –

Zooming (range = [0.8, 1.5])

  * –

Horizontal flip

  * –

Fill mode (nearest)

##  IV Methodology

Figure 3: Sample images from Bangla Money Dataset and Bangla Currency Dataset
by Hasan Murad et al.

In the Transfer Learning process, a computer can intelligently apply
previously learned knowledge to solve new problems faster and efficiently.
Suppose a deep learning model, trained to detect a cat can be used to detect a
dog. If this model is used as a starting point and retrained to detect a dog,
we can say transfer learning is applied. Fine-tuning is an essential aspect of
transfer learning. The hyperparameters used for the base model with base
dataset may not work for a different dataset. In fine-tuning, hyperparameters
are selected in such a way so that the pre-trained model works with the new
dataset. In our case, we have used 3 state-of-the-art models: MobileNet,
ResNet152v2, and NASNetMobile. The models are pre-trained using ‘ImageNet’
weights.

####  IV-1 MobileNet[10]

In MobileNet, Point convolution and Depthwise convolutions are used. Point
convolution maps an input pixel with an output pixel in all the channels used.

####  IV-2  NASNetMobile[11]

NASNetMobile has two types of convolutional cells used after one another
multiple times. They are normal and reduction cell. Both these return a
feature map.

####  IV-3 Resnet152v2[12]

ResNet uses a skip-connection, which helps the gradient to backpropagate and
train deeper networks. ResNet has two main types of blocks; identity block and
Convolutional block.

All the models are pre-trained with various features from ‘ImageNet’ dataset
and require an input image size of $224\times 224\times 3$. We have set 50
epochs for training. But we get a convergence of our models at random epochs,
as shown in figure 4. We have used a batch size of 32 for our training. Using
Adam optimizer, we have chosen a learning rate at 0.0001. We have decreased
the learning rate on plateaus and saved the best model based on lowest
validation loss. As a result of which, the expected results are obtained
within very few epochs. To avoid underfitting or overfitting, we have set a
monitor so that after every 2 epochs, if the validation loss does not
decrease, we decreased the learning rate by a factor of 0.8.

We have used 4 fully connected layers. Since ImageNet has 1000 categories for
classification, we have used 1024 neurons for first dense layer. Sequentially,
the next dense layers have 512, 512, 256, and 128 neurons, respectively. A
softmax classification layer with 8 classes was used for the ‘Bangla Currency’
dataset by Hasan Murad et al.[6], and with 9 classes was used for the ‘Bangla
Money’[3] dataset. Softmax classification gives us the logits or probabilities
for each category. For multi-class classification, it is wiser to choose
Categorical Cross-Entropy as probabilistic loss function.

##  V Result and Discussion

In this section, we will present the details of our experimental result. All
these programs were implemented using Python 3.6 and Keras framework. The
experiments were carried out on Google Colaboratory using GPU as runtime.
Google Colab provides Tesla K80, P-100, and Tesla T4 randomly.

###  V-A Comparison of Model’s Performance

In Table I, we have provided an overview of the experiments carried out on
‘Bangla Currency’, ‘Bangla Money’, and the combination of the two datasets. We
have used Precision, Recall, Test Accuracy, and F1 score as metrics for
evaluation purposes.

In the ‘Bangla Money’[3] dataset with 1970 images, NASNetMobile gives the best
accuracy 100% in the test set. In the ‘Bangla Currency’[6] dataset with 8000
images, MobileNet gives the best accuracy 98.88% in the test dataset. Both of
the datasets give decent accuracy and less bias & variance on MobileNet
architecture. For this reason, we have selected MobileNet architecture for the
combination of the two datasets. ‘Bangla Currency’ by Hasan Murad et al.[6]
datasets contains arbitrary lighting and background. But ‘Bangla Money’
dataset has no arbitrary background. So, the combined dataset has worse
accuracy than the individual dataset. However, data augmentation and fine-
tuning improve accuracy. In the combined dataset with 9970 images, MobileNet
gives a decent accuracy of 97.77% in the test set. In all of the cases,
lightweight models like MobileNet and NASNetMobile are performing similar to
heavyweight model like ResNet152v2. So, these lightweight models can be easily
implemented in any smartphone or handy gadgets/devices based on IoT
technology.

In figure 4, training and validation accuracy is shown. In some cases, we have
tried to increase the learning rate or decrease it so that our model does not
underfit or overfit.

TABLE I: Comparison of performances of different datasets on different CNN architectures Dataset |  | Model
---
(CNN)
Precision |  | Test
---
Accuracy
F1 Score
| MobileNet | 0.99 | 0.99 | 0.99
Bangla Currency | ResNet152 | 0.95 | 0.95 | 0.95
| NASNetMobile | 0.93 | 0.93 | 0.92
| MobileNet | 0.99 | 0.99 | 0.99
Bangla Money | ResNet152 | 0.98 | 0.98 | 0.98
| NASNetMobile | 1.0 | 1.0 | 1.0
Combined Dataset | MobileNet | 0.98 | 0.98 | 0.98
Figure 4: Comparison of training and validation accuracy of different models
under two datasets

###  V-B Comparison with Others’ Contribution

In table II, we have presented a comparison of our accuracy, dataset, and
approach with others who worked with Bangladeshi banknotes. There is very few
application of CNN in detecting Bangladeshi banknote. We have gained 100%
accuracy on the new dataset ‘Bangla Money’[3]. We have achieved 99.88%
accuracy on the ‘Bangla Currency’ dataset introduced in the paper of Hasan
Murad et al. [6]. We have also combined both of the datasets and achieved
97.77% accuracy. With fewer epochs, we have reached almost similar scenarios
as Hasan Murad et al. [6] without overfitting the model.

TABLE II: Comparison of performance and dataset with others works on Bangladeshi banknotes detection Contributors |  Total images in datasets |  Methodology |  Accuracy
---|---|---|---
Nadim Jahangir et al.[4] |  1700 |  ANN with Axis Symmetrical Masks |  98.57%
M.M. Rahman et al.[5] |  Not mentioned |  Oriented FAST and Rotated BRIEF (ORB) |  78.4%
Hasan Murad et al.[6] |  8000 |  Transfer learning using MobileNet |  99.80%
Ours |  9970 (8000+1970) |  Transfer learning using MobileNet, ResNet152v2 and NASNetMobile |  98.88% in 8000 images, 100% in 1970 images and 97.77% in 9970 images dataset

##  VI Limitations and Future Directions

The shortage of datasets is a major concern in Bangladeshi banknotes detection
and recognition system. Arbitrary lighting and background are still vital
issues for banknote detection. Our models struggle to perform well in several
cases where images have complex background. Our main aim was to enhance
model’s efficiency in the detection of Bangladeshi banknotes, which is shown
in the paper. As we have selected models as lightweight, it can easily be
implemented in any device such as smartphones. We have planned to integrate
fake banknote detection algorithms with our lightweight model to ensure
banknote security on which we are currently working. We are also working on
analyzing the automatic banknote detection system’s impact on visually
impaired people. As all the handy gadgets will be based on IoT technology in
the near future, so a lightweight model for banknote detection and recognition
is needed. We found that our proposed model (based on the experimented models)
can be a promising candidate in this regard.

##  VII Conclusion

In this paper, we have tried to differentiate state-of-the-art architectures
with two datasets containing Bangladeshi banknote images. We have also used
our best model on the combined dataset and achieved a decent result. It has
shown that Transfer Learning approach plays a significant role in Bangladeshi
banknotes detection. We have found out pre-trained lightweight model
MobileNet, and NasNetMobile give better accuracy in detecting banknotes. These
models can easily be deployed on any IoT devices for practical use.

## Acknowledgment

We would like to thank Shahjalal University of Science and Technology (SUST)
and SUST NLP research group for their support.

## References

  * [1] Who.int. 2020. Vision Impairment And Blindness. [online] Available at: https://www.who.int/news-room/fact-sheets/detail/blindness-and-visual-impairment [Accessed 24 July 2020].
  * [2] the Guardian. 2020. Bangladesh Fights To End Blindness. [online] Available at: https://www.theguardian.com/world/2010/sep/28/bangladesh-volunteers-childhood-blindness-treatment [Accessed 24 July 2020].
  * [3] Sojib, N., 2020. Bangla Money. [online] Kaggle.com. Available at: https://www.kaggle.com/nsojib/bangla-money [Accessed 24 July 2020].
  * [4] N. Jahangir and A. R. Chowdhury, “Bangladeshi banknote recognition by neural network with axis symmetrical masks,” 2007 10th international conference on computer and information technology, Dhaka, 2007, pp. 1-5, doi: 10.1109/ICCITECHN.2007.4579423.
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hour). The black solid line represents a fit with a model of a constant
rotation of the polarization angle. The dotted horizontal line indicates null
Stokes parameters.

Figure 11: Variation in the Stokes parameters as a function of time through
all three observations of Mrk 421 taken with IXPE. The black cross denotes the
time-integrated average values of $Q$ and $U$ along with its $1-\sigma$
uncertainty region. The coloured circles denote the measured mean values of
$Q$ and $U$ in each time bin and their $1-\sigma$ uncertainty radii. The inner
and outer dashed circles represent constant polarization degrees of $\Pi=10\%$
and $\Pi=20\%$, respectively, to help guide the eye. The colours of the data
points correspond to the elapsed time relative to the beginning of the
observation (in ks).

#### Searching for the optimal rotation rate via a likelihood analysis

For a detailed treatment of the optimal estimation of the (normalized) Stokes
parameters $q$ and $u$ in the case of unbinned polarimetric data via a
likelihood analysis, we refer the reader to (Marshall, 2021). An unbinned
likelihood analysis is appropriate for fitting X-ray event data because of the
Poisson nature of the counting statistics. This method has the advantage of
being independent of any subjective choice of the binning of the data in
energy or time.

Here, we expand the likelihood analysis method to the case of a rotating
polarization angle. The log-likelihood formula for the polarization parameters
of a set of (Poisson distributed) events depends on the Stokes parameters as

$S(q,u)=-2\sum_{i}\ln(1+q\mu_{i}\cos 2\Psi_{i}+u\mu_{i}\sin 2\Psi_{i}),$ (1)

where the sum is over each event $i$, $\mu_{i}$ is the modulation factor for
event $i$ (at energy $E_{i}$), $\Psi_{i}$ is the event’s instrumental phase
angle, and it is assumed that $q$ and $u$ do not depend on the source flux. In
addition, we are not fitting for the parameters of the spectral model.

Since the time-averaged polarization is small and the apparent rotation is
smooth, we can approximate the time behavior with a simple model that has a
varying polarization vector of constant degree, rotating at constant rate
$\dot{\Psi}$. We also assume that there is no steady non-rotating component of
the polarization (i.e., $q_{0}=u_{0}=0$). For this model, the likelihood can
be written as

$S(\dot{\Psi},q,u)=-2\sum_{i}\ln\\{1+q\mu_{i}\cos[2\dot{\Psi}(t_{i}-t_{0})]+u\mu_{i}\sin[2\dot{\Psi}(t_{i}-t{{}_{0}})]\\},$
(2)

where $t_{0}$ is a reference time, such as the midpoint of the observation
period, and $t_{i}$ is the time of event $i$. For any given rotation rate
$\dot{\Psi}$, one can determine the optimal Stokes parameters $\hat{q}_{r}$
and $\hat{u}_{r}$ by marginalizing over $q$ and $u$ and substituting
$\hat{q}_{r}$ and $\hat{u}_{r}$ back into Eq. 2 to yield
$S^{\prime}(\dot{\Psi})$. The best-fit value of $\dot{\Psi}$,
$\hat{\dot{\Psi}}$, is given by the minimum of $S^{\prime}(\dot{\Psi})$ and,
for Gaussian-distributed errors, uncertainties can be determined using $\Delta
S^{\prime}(\dot{\Psi})=S^{\prime}(\dot{\Psi})-S^{\prime}(\hat{\dot{\Psi}})$,
since $\Delta S^{\prime}(\dot{\Psi})$ is distributed as $\chi^{2}$ with one
degree of freedom, being a function of only one interesting parameter.
Furthermore, we can compute the significance that $\hat{\dot{\Psi}}$ is non-
zero using

$\Delta
S^{\prime}(0)=S^{\prime}(0)-S^{\prime}(\hat{\dot{\Psi}})=S(0,\hat{q}_{0},\hat{u}_{0})-S(\hat{\dot{\Psi}}\mathchar
24635\relax\;\hat{q}_{r},\hat{u}_{r}).$ (3)

Again, by marginalizing over $q$ and $u$ in order to test against the
hypothesis that $\dot{\Psi}=0$, $\Delta S^{\prime}(0)$ is distributed as
$\chi^{2}$ with one degree of freedom.

We searched for the minima of $\Delta S(\dot{\Psi})$ over a grid of values of
$\dot{\Psi}$, which we set to be $-1000\hbox{${}^{\circ}$}/\rm day$ to
$+1000\hbox{${}^{\circ}$}/\rm day$ using the IXPE event files of the three
observations of Mrk 421. Upon determining the best-fit value of $\dot{\Psi}$,
we estimated the uncertainty of the best-fit parameters at a confidence level
of 90 % using $\Delta S^{\prime}(\dot{\Psi})$. The results of this analysis
are displayed in Fig. 8. As expected, for Obs. 1, the estimated rotation rate
is consistent with null rotation ($\dot{\Psi}=-6\pm 9\hbox{${}^{\circ}$}/\rm
day$), while the estimated polarization degree ($\Pi_{\rm x}=15\pm 2\%$) is
consistent with our previous work Di Gesu et al. (2022). Conversely, for Obs.
2 and 3 we find absolute minima of $\Delta S^{\prime}(\dot{\Psi})$ at
$\dot{\Psi}=80\pm 9\hbox{${}^{\circ}$}/\rm day$ and $\dot{\Psi}=91\pm
8\hbox{${}^{\circ}$}/\rm day$, respectively, and that $\dot{\Psi}=0$ is ruled
out at better than $7\sigma$ for both observations. The estimated polarization
degrees for these two data sets were both $\Pi_{\rm x}=10\pm 1\%$, which
suggests that while the polarization angle was rotating, the polarization
degree may have remained similar to that measured when not rotating. The
estimated rotation rates for Obs. 2 and 3 are consistent within the
uncertainties, which suggests that we may have sampled two segments of one
continuous rotation event spanning three days. Combining Obs. 2 and 3, we
found a rotation rate of $R=77.0\pm 2.4\hbox{${}^{\circ}$}/\rm day$ and
$\Pi_{\rm x}=10.1\pm 0.8\%$, indicating that the third observation was
approximately in phase with the second after a day passed.

#### Modeling the time-binned Stokes parameters

As a second test of the hypothesis of polarization angle rotation canceling
the polarization degree in the time-averaged data, we modeled the time series
of the normalized Stokes parameters. As in Di Gesu et al. (2022), in order to
investigate the polarization variability on different time scales, we used the
temporal variations of the normalized Stokes parameters $q(t)$ and $u(t)$ with
various time bin sizes in the range from $\sim$8 ks (corresponding to $N_{\rm
bins}$=20 bins) to $\sim$60 ks ($N_{\rm bins}$=3). We fit all the time series
with (1) a constant model and (2) a model where the polarization angle rotates
at a constant rate $\dot{\Psi}$ while the polarization degree $\Pi$ remains
constant, such that the Stokes parameters vary as trigonometric functions:
$q(t)=\Pi\cos(2(\dot{\Psi}t+\phi))$ and $u(t)=\Pi\sin(2(\dot{\Psi}t+\phi))$
(where $\phi$ is a phase angle). As a test of the goodness of the fits, we
computed, for all the cases tested, the probability for the null hypothesis
($P_{\rm null}$) to obtain by chance a $\chi^{2}$ value at least as large as
that measured if the data are drawn according to the model.

Figure 9 displays, for both IXPE observations and for all the tested time
binning schemes, the $P_{\rm null}$ values obtained for the constant model
(black symbols) and the constant rotation model (red symbols). A constant
model is statistically unacceptable ($P_{\rm null}<1\%$) for any time binning.
A model including a linear time variation of the polarization angle always
provides a better fit (i.e., an increase of $P_{\rm null}$, up to $\sim$99% in
several cases) than a constant model. Figure 10 shows all the fit
realizations, for the case $N_{\rm bins}$ in the range between 3 and 20. In
the fitting exercise of Figure 9, we keep the parameters $\dot{\Psi}$ and
$\Pi$ fixed to those determined from the likelihood analysis. Hence, $\phi$ is
the only free parameter. Moreover, we note that the independent fits of $q(t)$
and $u(t)$ return best-fit values of $\phi$ consistent within each other
(within a typical uncertainty of $\sim 8\hbox{${}^{\circ}$}$). This is an a
posteriori indication that our fits are physical. To further assess the
phenomenology of the polarization variability, we allow, in turn, the rotation
rate $\dot{\Psi}$ and the polarization degree $\Pi$ to be free in our fits. We
find that $\dot{\Psi}$ is unconstrained in these fits. Conversely, the best
fit corresponds to values of $\Pi$ that are marginally inconsistent between
the fit of $q(t)$ (e.g., $\Pi=11\%\pm 2\%$ for the case $N_{\rm bins}$=16) and
that of $u(t)$ (e.g., $\Pi=5\%\pm 2\%$). This suggests that $\Pi$ and/or
$\dot{\Psi}$ are somewhat variable with time about the average values, rather
than constant. The detailed modeling of these variability features is beyond
the scope of our hypothesis testing, and may be assessed in future works in
the context of physically motivated models.

### Fitting a rotation model in the $Q$-$U$ plane

As a third and final test, we modeled the time behaviour of the data in the
$Q$-$U$ plane, with $N_{\rm bins}=24$. We display the time evolution of the
data in Figure 11. In this framework, as a first, straightforward test for
variability, we calculated the $\chi^{2}$ statistic for the mean $Q/U$ Stokes
parameters in each time bin under the null hypothesis that they are all
consistent with the time-integrated mean Stokes parameters ($\bar{Q}$ and
$\bar{U}$). Since the Stokes parameters from each time bin (denoted as $Q_{i}$
and $U_{i}$, respectively, for time bin $i$) are normally distributed and
independent, the quantity

$\chi^{2}=\sum_{i}\frac{\left(Q_{i}-\bar{Q}\right)^{2}+\left(U_{i}-\bar{U}\right)^{2}}{\sigma_{i}^{2}}$
(4)

under this null hypothesis follows a $\chi^{2}$ distribution with $2N_{\rm
bins}-2=46$ degrees of freedom. We can therefore compare the observed value of
$\chi^{2}$ with the expected values. For Obs. 1, we find $\chi^{2}=55$, the
probability of which arising from the null hypothesis of constant Stokes
parameters is $P_{\rm null}\sim 0.17$, suggesting that these data are
consistent with the constant polarization model. Conversely, for Obs. 2 and 3,
we determine $\chi^{2}$ values of 78 and 121, respectively. The probability of
acquiring $\chi^{2}$ exceeding this value by chance under the null hypothesis
of constant polarization is $P_{\rm null}<2\times 10^{-3}$. This implies that,
in these cases, the null hypothesis can be rejected with high confidence.
Using this same statistical framework, we are able to determine the best-fit
rotation rate $\dot{\Psi}_{\rm x}$ for the model by assuming that both
$\dot{\Psi}_{\rm x}$ and the polarization degree are constant (see above for
the mathematical form of this model). When applying these calculations to the
data of Obs. 1, we find no evidence of time-dependent rotation of $\Psi_{\rm
x}$: the best-fit constant rotation rate is $-4\pm 7$ $\hbox{${}^{\circ}$}/\rm
day$). For Obs. 2 and 3, the best-fit rotation rates are $\dot{\Psi}_{\rm
x}=78\pm 8$ $\hbox{${}^{\circ}$}/\rm day$ and $83\pm 6$
$\hbox{${}^{\circ}$}/\rm day$ respectively. These values are consistent with
the findings of the previously described methods using the same model,
indicating that all three methods result in a consistent rate of rotation. The
corresponding $\chi^{2}$ value for each of these fits is 53 for 45 $d.o.f$,
with a corresponding value of $P_{\rm null}\sim 0.18$. We can therefore
conclude that a constant rotation model provides a good fit to the data.
Attempts to fit these data to a more complex elliptical rotation model (i.e.,
where the polarization degree varies with time) do not result in a
statistically significant improvement in the overall value of $\chi^{2}$ given
the reduced degrees of freedom.

### Comparison with a stochastic model

In order to verify further our explanation of the cause of the X-ray
polarization angle rotation in Mrk 421, we tested the hypothesis that the
rotation occurred by chance as a result of a stochastic variation of
$\Psi_{\rm x}$. We considered two possibilities for the involvement of random
processes. First, the variations in $\Psi_{\rm x}$ during the two IXPE
observations of June corresponded to two independent rotations. Second, the
two IXPE observations were part of one continuous rotation, of which we
sampled only two segments. To determine the probability of the rotation(s)
occurring by chance due to the random variations of $\Psi_{\rm x}$, we
simulated the time dependence of the polarization degree and angle that occur
via random walks, and compared them to the IXPE results. As described in
detail in Kiehlmann et al. (2016, 2017), the simulations are based on two
parameters: the number of maximally polarized (i.e., each with uniform
magnetic field) cells, $N_{\rm cell}$, and the number of cells that change per
day, $N_{\rm Var}$. The cells each have the same intensity, hence the total
intensity $I_{\rm total}=N_{\rm cell}I$. The initial Stokes parameters $Q$ and
$U$ for each cell were drawn from Gaussian distributions. At every simulation
time step, one cell was selected randomly, and its $Q$, $U$ values were drawn
anew, then the averages of $Q$, $U$, $\Pi$, and $\Psi$ over all cells were
calculated anew. The number of simulated time steps depends on the observation
duration and ($N_{\rm Var}$); e.g., for $N_{\rm Var}=100$ we simulated 100
data points per day. For each of the aforementioned possibilities, we
performed a grid search in the $N_{\rm cell},~{}N_{\rm Var}$ parameter space
to identify episodes of rotation of $\Psi$.

For the first case, we set the total length of the simulated polarization
light curves to the length of each of the IXPE observations. We created 10,000
simulated light curves, binned to match one of the data binning schemes, and
recorded the average $\Pi$ of the light curve and the longest continuous
variation of $\Psi$ that could be interpreted as a rotation. We considered a
trial successful if the average $\Pi$ and standard deviation of the simulated
time dependence was within 10% of the observed values, and if the variation in
$\Psi$ was equal to or larger than that observed. Figures 12 and 13 show the
results for the parameter search of each event. The binning of the data can
have a significant effect on the number of successful trials. For that reason,
we repeated the analysis for different numbers of bins from 3 to 18. For IXPE
Obs. 2 (first rotation) we found the $p$ significance values to range from
0.0049 to 0.14 (13 bins), with a median of 0.057. For the following rotation
(IXPE Obs. 3), we found $p=0.0013$ to 0.21 (17 bins), with a median of 0.032.
The probability of two independent events occurring by pure chance in a row is
then $p=0.029$.

For the second possibility, we set the length of the simulated light curves
equal to the total duration of the observations from the beginning of IXPE
Obs. 2 to the end of Obs. 3. To test the case of an individual rotation, the
polarization-angle curve of Obs. 3 was shifted by $\pm 180^{\circ}$ as needed
to align the two segments. In this case, we explored from 3 to 26 bins and
additionally required that the rotation rate of the second segment lie within
10% of that of the first segment. We repeated the above procedure and
estimated the number of successful trials (Fig. 14). We find $p=0.0006$ to
0.022 (23 bins) with a median of 0.0053. All of the above results indicate
that it is unlikely that the observed rotation(s) were caused by a purely
stochastic process. Rather, the event has a high probability of being related
to non-stochastic physical processes in the jet.

Figure 12: Random-walk parameter search for the second IXPE observation. The
numbers in the squares indicate the fraction of successful trials. Figure 13:
Random-walk parameter search for the third IXPE observation. The numbers in
the squares indicate the fraction of successful trials. Figure 14: Random-walk
parameter search assuming the second and third IXPE observations are part of
the same rotation. The numbers in the squares indicate the fraction of
successful trials. Figure 15: Archival optical brightness (black triangles:
R-band, Steward observatory, red and cyan triangles: c-band and o-band, ATLAS
observatory), and X-ray flux (lower panel, black triangles) light curves for
Mrk 421. The grey shaded areas represent the duration of the IXPE pointings.

### Activity state of Mrk 421

In Fig. 15 we display the long term light curves of Mrk 421 in optical
brightness (top panel) from the Steward observatory monitoring program (Smith
et al., 2009) and ATLAS (Tonry et al., 2018; Heinze et al., 2018; Shingles et
al., 2021), as well as the X-ray flux (bottom panel) from the Swift XRT. The
optical observations cover a time range from 2006 to 2022 and we highlight as
vertical grey lines the time of the IXPE pointings.
The Swift-XRT observations are in the 0.3-10 keV band. The minimum and maximum
value of the X-ray flux were $0.5\times 10^{-10}$ and $55\times 10^{-10}$ erg
s-1 cm-2, respectively, with a median value of $8.5\times 10^{-10}$. As a
comparison, the flux measured by Swift at the time of the IXPE pointings (see
Table 2) ranged between $7.5\times 10^{-10}$ and $14.1\times 10^{-10}$ erg s-1
cm-2. Thus, we conclude that Mrk 421 was at an average level of X-ray activity
at the time of the IXPE observations.
We display the optical brightness in the top panel of Fig. 15. The Steward
observatory observations are in the R-band, while for ATLAS we show both
c-band and o-band observations. While the Steward and ATLAS observations are
not directly comparable without applying conversion factors between optical
bands, it is evident from the overlap of the two surveys (2016-2018) that any
difference is rather small compared to the overall brightness variations of
Mrk 421. The ATLAS observations in 2022 are $\sim$0.5mag lower than the
average of the Steward observations (2011-2016) and $\sim$1-1.5mag lower than
the flares seen in the Steward data. Therefore, we can also conclude that the
optical brightness of Mrk 421 at the time of the IXPE observations was at an
average state.