2015 Vol. X No. XX, 000–000

11institutetext: Yunnan Observatories, Chinese Academy of Sciences, Kunming
650216, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
22institutetext: Key Laboratory for the Structure and Evolution of Celestial
Objects, Chinese Academy of Sciences, Kunming 650216, China 33institutetext:
University of Chinese Academy of Sciences, Beijing 100049, China
44institutetext: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100012, China
55institutetext: Yunnan Minzu University, Kunming 650031, China

# Is the X-ray pulsating companion of HD 49798 a possible type Ia supernova
progenitor?

D.-D. Liu 112233 W.-H. Zhou 334455 C.-Y. Wu 112233 B. Wang 1122

(Received ; accepted)

###### Abstract

HD 49798 (a hydrogen depleted subdwarf O6 star) with its massive white dwarf
(WD) companion has been suggested to be a progenitor candidate of type Ia
supernovae (SNe Ia). However, it is still uncertain whether the companion of
HD 49798 is a carbon-oxygen (CO) WD or an oxygen-neon (ONe) WD. A CO WD will
explode as an SN Ia when its mass grows approach to Chandrasekhar mass, while
the outcome of an accreting ONe WD is likely to be a neutron star. We followed
a series of Monte Carlo binary population synthesis approach to simulate the
formation of ONe WD + He star systems. We found that there is almost no
orbital period as large as HD 49798 with its WD companion in these ONe WD + He
star systems based on our simulations, which means that the companion of HD
49798 might not be an ONe WD. We suggest that the companion of HD 49798 is
most likely a CO WD, which can be expected to increase its mass to the
Chandrasekhar mass limit by accreting He-rich material from HD 49798. Thus, HD
49798 with its companion may produce an SN Ia in its future evolution.

###### keywords:

binaries: close — stars: individual— stars: evolution — supernovae: general —
white dwarfs

##  1 Introduction

Type Ia supernova (SN Ia) explosions are among the most luminous phenomena in
the Universe, and play an important role in astrophysics. Due to the
remarkable uniformity of their high luminosities, SNe Ia are successfully used
as standard cosmological distance indicators. It has been suggested that the
Universe is expanding at an increasing rate through the observation of SNe Ia,
which reveals the existence of dark energy (e.g., Riess et al. 1998;
Perlmutter et al. 1999). Furthermore, SN Ia explosions are also relatively
important for galactic chemical evolution for the reason that a great amount
of iron can be produced during this process (e.g., Greggio & Renzini 1983;
Matteucci & Greggio 1986), and they are also accelerators of cosmic rays
(e.g., Fang & Zhang 2012). However, SN Ia progenitors and their explosion
mechanisms are still uncertain, which may have an influence on the accuracy of
being the role of distance indicators (e.g., Podsiadlowski et al. 2008; Howell
et al. 2011; Liu et al. 2012; Wang & Han 2012; Wang et al. 2013; Hillebrandt
et al. 2013; Maoz et al. 2014).

A theoretical consensus has been reached that SNe Ia originate from
thermonuclear runaway explosions of carbon-oxygen white dwarfs (CO WDs) in
binaries (see Hoyle & Fowler 1960; Nomoto et al. 1997). When a CO WD increases
its mass approach to the Chandrasekar mass limit in a close binary, an SN Ia
would be produced. However, the key issue is the uncertainty of the companion
star. There are two competing progenitor models which are widely accepted,
i.e., the single-degenerate model and the double-degenerate model. In the
single-degenerate model, a CO WD accretes material from a non-degenerate
companion to get its mass increased. The companion in this model could be a
main sequence (MS) star, a subgiant star, a red giant (RG) star or a heluim
(He) star (e.g., Whelan &Iben 1973; Nomoto et al. 1984; Li & van den Heuvel
1997; Langer et al. 2000; Han & Podsiadlowski 2004; Han & Podsiadlowski 2006;
Meng et al. 2009; Xu & Li 2009; Wang et al. 2009a, 2010; Ablimit et al. 2014;
Chen H.-L. et al. 2014; Geier et al. 2015). In the double-degenerate model,
the merger of two CO WDs produces an SN Ia; the merger is due to the loss of
orbital angular momentum driven by gravitational wave radiation (e.g., Tutukov
& Yungelson 1981; Iben & Tutukov 1984; Webbink 1984; Han 1998; Chen et al.
2012).

The CO WD + He star channel is an emerging variant of the single-degenerate
model which can naturally account for the missing of hydrogen lines in the
most spectra of observed SNe Ia. Wang et al. (2009a) systematically studied
the CO WD + He star channel and derived the space parameters for producing SNe
Ia. They found that SNe Ia from this channel may have a contribution to SNe Ia
with short delay times. In the observations, a number of massive WD + He star
binaries have been found, e.g., V445 Pup, HD 49798 with its companion,
CD$-$30∘ 11223 and KPD 193$+$2752, etc. All of these binaries are possible
candidates of the SN Ia progenitors (e.g., Geier et al. 2007, 2013; Kato et
al. 2008; Woudt et al. 2009; Wang & Han 2010; Mereghetti et al. 2011). Among
these observed WD + He star systems, HD 49798 with its massive compact
companion is one of the relatively well studied binaries composed of a bright
subdwarf star whose hydrogen has been depleted (Jaschek & Jaschek 1963) and a
fast X-ray pulsar source which is a massive WD with a short spin period
(Israel et al. 1997).

HD 49798/RX J0648.0-4418 is a single spectroscopic binary which has been
investigated extensively. Jaschek & Jaschek (1963) first classified HD 49798
as a subdwarf O6 star and obtained the variations of its radial velocity. The
period of this binary was suggested to be 1.548 day (see Thackeray 1970;
Stickland & Lloyd 1994). A soft X-ray emission source was first found by
Einstein Observatory in the position of HD 49798 (see Simon et al. 1979).
According to the detailed data from Roentgen Satellite, Israel et al. (1997)
found that the X-ray source in the HD 49798/RX J0648.0-4418 is very soft and
has a high-energy excess, and that the X-ray source has a pulsating period of
13.2s. Bisscheroux et al. (1997) argued that the companion is a WD but not a
neutron star based on its low X-ray luminosity (see also Hamann et al. 1981;
Mereghetti et al. 2011). Bisscheroux et al. (1997) suggested that the X-ray
emission of this binary is from wind accretion process of the compact object,
in which the accretion rate from the stellar wind of HD 49798 is $\sim
10^{-10}-10^{-11}\rm\,M_{\odot}/yr$. They also speculated that HD 49798 is at
He-shell burning stage, which might be the explanation for its high
luminosity. Recently, Mereghetti et al. (2009) observed the object with the
Newton X-ray Multimirror Mission satellite during the X-ray eclipse. They gave
the parameters of this binary: $R_{\rm He}=1.45\pm 0.25\,R_{\odot}$, $M_{\rm
He}=1.50\pm 0.05\,M_{\odot}$ for HD 49798 and $M_{\rm WD}=1.28\pm
0.05\,M_{\odot}$ for the WD companion.

Wang & Han (2010) recently performed a detailed binary evolutionary
calculation for HD 49798/RX J0648.0-4418. The temperature and luminosity of HD
49798 derived from their calculations are consistent with that of observations
(see Fig. 1 of Wang & Han 2010). Their work indicates that HD 49798 will fill
its Roche lobe after $\sim 4\times\rm 10^{4}\rm\,yr$, and the WD companion of
HD 49798 will increase its mass to Chandrasekar mass by accreting He-rich
material. However, it is still unknown which kind of WD the companion of HD
49798 is. If the compact companion is a CO WD, it will grow to be an SN Ia
when the carbon is ignited in the center (or off-center) of the WD (e.g.,
Nomoto 1982). If the companion of HD 49798 is an ONe WD, Ne and Mg in the WD
start to capture electrons when the WD grows to Chandrasekhar mass. During the
deflagration of O and Ne, too little energy could be released to give rise to
an explosion of the whole WD (Miyaji et al. 1980). Driven by the subsequent
electron capture, the outcome of the WD should be a neutron star with fast
spin but not an SN Ia (e.g., Canal et al. 1980; Nomoto & Kondo 1991).

The purpose of this article is to examine whether the companion of HD 49798 is
an ONe WD or not via a binary population synthesis (BPS) method. The BPS
numerical code and the input physics are described in Section 2. In Section 3,
we present the results of our calculations. The discussion and conclusions are
provided in Section 4.

##  2 Methods

By using Hurley’s rapid binary evolution code (Hurley et al. 2000, 2002), we
performed a series of Monte Carlo simulations to study the formation of ONe WD
+ He star systems. In this work, an ONe WD is formed by the envelope loss of a
thermal pulsing asymptotic giant branch (TPAGB) star with $M_{\rm up}\leq
M\leq M_{\rm ec}$, while a CO WD is formed by the envelope loss of a TPAGB
star with $M<M_{\rm up}$ (see Hurley et al. 2000), where $M$, $M_{\rm up}$ and
$M_{\rm ec}$ are the initial mass of the AGB star, the minimum mass undergoing
non-degenerate C ignition and the minimum mass of an AGB star avoiding
electron capture on Ne and Mg in its core, respectively (see table 1 of Pols
et al. 1998). In each simulation, $1\times 10^{7}$ primordial binary systems
are included. The current parameters of HD 49798 and its WD companion has been
provided by Mereghetti et al. (2009, 2011), that is, the masses of the He star
and the WD are $1.50\pm 0.05\,M_{\odot}$ and $1.28\pm 0.05\,M_{\odot}$,
respectively. The orbital period of this binary is 1.548 day (Thackeray 1970).
Moreover, HD 49798 is a slightly evolved He star which has not filled its
Roche lobe yet; the He star will fill its Roche lobe after about $4\times\rm
10^{4}\rm\,yr$ (see Wang & Han 2010), which means that the masses and the
orbit period of the systems are similar to the parameters at the beginning of
the formation of the WD + He star system. Thus, if the companion of HD 49798
is an ONe WD, the parameters of this binary are more likely to be located
within the contours of ONe WD + He star systems determined by Monte Carlo BPS
simulations.

Figure 1: Four binary evolutionary scenarios to produce ONe WD + He star
systems.

According to the evolutionary phase when the primordial primary first fills
its Roche lobe, there are four scenarios which can produce ONe WD + He star
systems (see Fig. 1), as follows:

Scenario A: When the primordial primary is a subgiant or first giant branch
(FGB) star, it fills its Roche lobe. At this stage, the Roche lobe overflow
(RLOF) is stable. At the end of the RLOF, the primordial primary becomes a He
star. Subsequently, the He star continue to evolve and will fill its Roche
lobe again at the He RG stage. As a result, an ONe WD + MS system is produced
after RLOF. Then the MS star continues to evolve and will fill its Roche lobe
at the subgiant or He-core burning phase. A common envelope (CE) may be formed
due to the dynamically unstable mass transfer. If the CE ejection happens, an
ONe WD + He star system would be produced.

Scenario B: When the primordial primary first fills its Roche lobe at the
early asymptotic giant branch (EAGB) stage, the mass transfer is dynamically
unstable, leading to the formation of a CE. After the CE ejection, the primary
becomes a He RG star and the orbital separation decays sharply. Then the
primary fills its Roche lobe again. In this case, the RLOF is stable. After
the He-shell is exhausted, an ONe WD + MS system is produced. The MS star
continues to evolve and may fill its Roche lobe at the Subgiant or FGB stage.
A CE will be formed for the reason of the dynamically unstable mass transfer.
If the CE can be ejected, an ONe WD + He star will be produced.

Scenario C: The primordial primary first fills its Roche lobe at the TPAGB
stage and a CE may be formed due to the dynamically unstable mass transfer.
After the CE ejection, an ONe WD + Subgiant/FGB star system is formed.
Subsequently, the binary evolution is similar to that of the scenario B above
and an ONe WD + He star system will be formed.

Scenario D: The primordial primary fills its Roche lobe when it is a TPAGB
star and the primordial secondary is at the He-core burning phase. In this
case, the mass transfer is dynamically unstable, leading to the formation of a
CE. If the CE ejection happens, the primary turns to be an ONe WD and the
secondary becomes a He star, i.e., an ONe WD + He star system is produced.

Some basic assumptions in our BPS simulations are listed as follows: (1) The
initial mass function described by Miller & Scalo (1979) is adopted. The
primordial primary mass ranges from $0.1\,M_{\odot}$ to $100\,M_{\odot}$. (2)
The primordial secondary mass is determined by the initial primordial primary
mass and initial mass-ratio. For simplicity, a uniform mass-ratio distribution
is adopted (e.g., Mazeh et al. 1992; Goldberg & Mazeh 1994). (3) The
distribution of the separations of the initial orbit is constant in log($a$)
($a$ is orbital separation), and all stars are set as members of binary
systems with a circular orbit. While a boundary for close and wide binaries is
set to be $10\,R_{\odot}$ (Han et al. 1995). (4) The abundance of solar
metallicity $Z=0.02$ is adopted. (5) The process of RLOF is calculated by the
method described in Tout et al. (1997). (6) Stable mass transfer process
presented by Webbink (1985) is adopted. (7) The output of the CE stage is
calculated by the standard energy equation (e.g., Webbink 1984), in which
there are two uncertain parameters, i.e., the CE ejection efficiency
($\alpha_{\rm CE}$) and a stellar structure parameter ($\lambda$). Following
the work of Wang et al. (2009b), we simply combined these two parameters as a
single free parameter (i.e., $\lambda\alpha_{\rm CE}$). In this work, we set
$\lambda\alpha_{\rm CE}$ to be 0.5, 1.0, 1.5 and 2.0 to examine its influence
on the production of ONe WD + He star systems.111Recent studies showed that
the stellar structure parameter, $\lambda$, may be bigger during some
particular stages (e.g., Xu & Li 2010; Zuo & Li 2014). Thus, we considered an
extreme case with $\lambda\alpha_{\rm CE}=2.0$.

##  3 Results

Table 1: Initial parameters of binaries which can evolve to ONe WD + He star systems. The simulation contains $1\times 10^{7}$ primordial sample binaries, in which we set $\lambda\alpha_{\rm CE}=1.0$. Notes: $M_{\rm 1,0}=$ initial mass of the primordial primary; $M_{\rm 2,0}=$ initial mass of the primordial secondary; $P_{\rm 0}=$ initial orbital period of the primordial binary; Number = the number of ONe WD + He star systems produced from each scenario. Scenario | $M_{\rm 1,0}$ | $M_{\rm 2,0}$ | $P_{\rm 0}$ | Number
---|---|---|---|---
| $(M_{\odot})$ | $(M_{\odot})$ | (day) |
(A) | $8.0-11$ | $2.5-10$ | $2.0-960$ | $2469$
(B) | $6.0-9.0$ | $2.5-8.5$ | $420-1600$ | $1704$
(C) | $6.0-8.5$ | $2.5-7.5$ | $1400-6750$ | $2027$
(D) | $6.0-8.5$ | $6.0-8.5$ | $1500-6000$ | $227$

We conducted a series of Monte Carlo BPS approach to simulate the formation of
ONe WD + He star systems. In Table 1, we presented the initial parameters of
binaries for those four scenarios in Fig. 1 and gave the number of ONe WD + He
star systems produced from each scenario. From this table, we can see that
these four scenarios are mainly distinguished by the difference of initial
orbital periods, while the distinguishing factor between scenario C and
scenario D is the initial mass ratio.

Figure 2: The initial orbital period and WD mass ($\rm log\,P^{i}$ and $M^{\rm
i}_{\rm WD}$) distribution of ONe WD + He star systems based on BPS
simulations, in which He star mass is constrained between 1.45 and 1.55
$\,M_{\odot}$. The dots, triangles, inverted triangles and open circles
represent the cases with $\lambda\alpha_{\rm CE}=0.5$, 1.0, 1.5 and 2.0,
respectively. The black square with error bar represents HD 49798/RX
J0648.0-4418.

Based on the four cases of simulations with different CE ejection parameters
(i.e., $\lambda\alpha_{\rm CE}=0.5$, 1.0, 1.5 and 2.0), we totally obtained
about 28000 ONe WD + He star binaries. From these binary systems, we picked
out binaries with the He star mass between 1.45 and $1.55\,M_{\odot}$ which is
consistent with the possible mass range of HD 49798. Fig. 2 shows the
distribution of 937 ONe WD + He star systems in the initial orbital period and
WD mass ($\rm log\,P^{i}$-$M^{\rm i}_{\rm WD}$) plane with the constraint of
the He star mass. From this figure, we can see that as the value of
$\lambda\alpha_{\rm CE}$ increases, the orbital separation tends to be wider
and more ONe WD + He systems would be produced. This is because, for a larger
value of $\lambda\alpha_{\rm CE}$ the ejection of CE will release less orbital
energy and be easier to happen. Note that, the upper boundaries is determined
by the existence of Chandrasekar mass limit for ONe WDs, while the lower
boundaries are constrained by the minimum mass of WDs for the ignition of
carbon to form ONe WDs. The ONe WD + He star systems beyond the left
boundaries have their He star masses lower than the mass range of HD 49798,
whereas larger than that mass range beyond the right boundary. Considering the
cases with $\lambda\alpha_{\rm CE}=0.5$, 1.0 and 1.5, we can see that no ONe
WD + He star system has a separation (orbital period) as large as HD 49798/RX
J0648.0-4418 for the same masses of He stars. While for the extreme case with
$\lambda\alpha_{\rm CE}=2.0$, HD 49798/RX J0648.0-4418 is located on the
boundary of the contour of the obtained ONe WD + He star systems. However, the
probability is very low. We note that the extreme case with
$\lambda\alpha_{\rm CE}=2.0$ may not be physical in our simulations. Thus, we
speculated that HD 49798 with its companion might not be an ONe WD + He star
system.

In Fig. 3, we present a more comprehensive distribution of orbital periods of
ONe WD + He star systems without the constraints of He star mass. From this
figure, we can see that almost all the ONe WD + He star systems distribute
between 0.032 and 1.0 day. As to the orbital period of HD 49798/RX
J0648.0-4418 (1.548 day), for the cases with $\lambda\alpha_{\rm CE}=0.5$, 1.0
and 1.5, it is still difficult to produce such a wide ONe WD + He star system
even without the constraints of the He star mass; the orbital period of HD
49798/RX J0648.0-4418 can be reproduced by the extreme case with the value of
$\lambda\alpha_{\rm CE}=2.0$, but the probability is very low. Note that there
are two peaks in the curves with $\lambda\alpha_{\rm CE}=1.5$ and 2.0, which
are produced from different forming scenarios (see Fig. 1); binaries near the
left peak are formed from the scenario A, whereas the right peak are from all
those four scenarios.

Figure 3: Distribution of the orbital periods of ONe WD + He star systems with
different values of $\lambda\alpha_{\rm CE}$, in which the mass of He stars
are not restricted. The solid, dashed, dash-dotted and dotted lines represent
the cases with $\lambda\alpha_{\rm CE}=0.5$, 1.0, 1.5 and 2.0, respectively.
Here, every case is normalized to be 1.

##  4 Discussion and conclusions

In this article, we found that there is almost no orbital period as large as
HD 49798/RX J0648.0-4418 in these ONe WD + He star systems, which means that
the companion of HD 49798 may not be an ONe WD under our assumptions.222The
orbital period of HD 49798/RX J0648.0-4418 can be reproduced by the CO WD + He
star model, which has the range of $0.01-5.62\,\rm day$. Thus, the ultimate
fate of the binary can be constrained to a certain extent. By assuming the
compact companion is a CO WD, Wang & Han (2010) recently made an evolutionary
investigation of this binary. In their calculations, the optically thick wind
model is adopted and the mass accumulation efficiency of He-shell flash is
from Kato & Hachisu (2004). Wang & Han (2010) suggested that about $6\times\rm
10^{4}\,\rm yr$ later, the WD companion of HD 49798 will increase its mass to
$1.4\,M_{\odot}$ and an SNe Ia may be produced, leaving a surviving He star as
massive as $1.1817\,M_{\odot}$. Furthermore, Mereghetti et al. (2009)
suggested that the companion of HD 49798 may be a rapidly rotating WD, which
would lead to a CO WD more massive than Chandrasekhar mass (e.g., Yoon &
Langer 2005; Chen & Li 2009). From Fig. 2 of Wang & Han (2010) we can see
that, the WD companion of HD 49798 could eventually increase its mass to
$1.62\,M_{\odot}$ when the differential rotation cannot be maintained. Thus,
we speculate that the companion of HD 49798 may evolve to an overluminous SNe
Ia.

However, if the companion of HD 49798 is a CO WD, a question would be raised
that how such a massive CO WD like the companion of HD 49798 could be formed
with almost no mass transfer from the He star before it fills its Roche lobe.
HD 49798/RX J0648.0-4418 is considered to be the production of a CE ejection
and spiral-in process (e.g., Israel et al. 1997; Bisscheroux et al. 1997).
Iben & Tutukov (1993) claimed that HD 49798 was produced from an
$8-9\,M_{\odot}$ progenitor which had filled its Roche lobe before the He-core
burning stage. They thought that HD 49798 is currently a He-core burning star
(see also Kudritzki & Simon 1978). In contrast, Bisscheroux et al. (1997)
argued that the progenitor of HD 49798 is an EAGB star before the CE is formed
and currently HD 49798 is at He-shell burning stage with a CO core in the
center. Furthermore, Wang & Han (2010) simulated the evolution of the binary
HD 49798/RX J0648.0-4418 and found that the mass of the CO core is about
$0.79\,M_{\odot}$ at the current position of HD 49798.

The compact companion of HD 49798 might also be a hybrid CONe WD. When the
convective boundary mixing is taken into account, a super AGB star may evolve
to form a hybrid CONe core with a relatively large unburned CO core (about
$0.2\,M_{\odot}$), which surrounded by an ONe zone (greater than
$0.85\,M_{\odot}$) and a thin CO layer on the surface (Denissenkov et al.
2013). By considering a series of convective Urca shell flashes and some
different mixing assumptions, Denissenkov et al. (2015) found that an
explosive carbon ignition would be reached when hybrid CONe WDs approach to
the Chandrasekhar mass, leading to a low peak luminosity SNe Ia because of the
low carbon to oxygen radio. Wang et al. (2014) gave the birthrate of SNe Ia of
CONe WD + He star scenario using a BPS approach. Furthermore, Chen et al.
(2014) suggested that hybrid CONe WDs could be as massive as $1.3\,M_{\odot}$
in an extreme case. Thus, the companion of HD 49798, which is as massive as
$1.28\,M_{\odot}$, may also be a hybrid CONe WD. However, it is still unclear
that what is the relationship between the mass of CO core and ONe regions and
what is the smallest CO core mass for thermonuclear explosion happens.
Moreover, the carbon-burning rate, which is still an uncertain parameter,
plays an important role in determining the biggest mass of hybrid CONe WDs
(see Chen M.-C. et al. 2014). The uncertainties of hybrid CONe WDs make it
doubtful that whether such a massive WD like the companion of HD 49798 could
be a hybrid CONe WD.

We also note that, it is still being argued about the predicted ability of the
BPS method. Toonen et al. (2014) recently compared four different BPS codes
and found that the differences of their results are not because of numerical
effects, but due to different Monte Carlo simulation assumptions (e.g.,
initial mass function, initial mass ratio distribution, orbital eccentricity
distribution, etc). Wang et al. (2009b) compared some initial parameters for
producing SNe Ia through the CO WD + He star model. Their work indicates that
the initial assumptions adopted in this article may result in a maximum SN Ia
birthrate. In this work, we performed four sets of simulations. The results
presented here show that a larger value of $\lambda\alpha_{\rm CE}$ could
produce wider and more ONe WD + He star systems. We also presented an extreme
case with $\lambda\alpha_{\rm CE}=2.0$, and found that HD 49798 with its WD
companion may be reproduced in the ONe WD + He star model with an extreme
value of $\lambda\alpha_{\rm CE}$, but the probability is very low.

Aside from HD 49798 with its X-ray pulsating companion, another well studied
WD + He star system is V445 Pup which is the first detected He nova (Ashok &
Banerjee 2003; Kato & Hachisu 2003). The mass of the He star in V445 Pup is
about $1.2-1.3\,M_{\odot}$ derived from the pre-outburst luminosity of the
binary (Woudt et al. 2009). Kato et al. (2008) fitted the light curve of the
binary and estimated that the mass of the WD companion is larger than
$1.35\,M_{\odot}$. They also found that almost half of the material acceted
from the He star remains on the surface of the WD. Recently, Goranskij et
al.(2010) declared that the orbital period of V445 Pup might be 0.65 day.
Morever, the WD companion is more likely a CO WD but not a ONe WD since no
Neon line was detected in the ejected nebula (e.g., Woudt & Steeghs 2005).
Thus, V445 Pup is also a possible candidate of SN Ia progenitors.

In this article, we performed a series of Monte Carlo BPS simulations with
different values of CE ejection parameters. A number of ONe WD + He star
systems were obtained and the companion of HD 49798 is really difficult to be
an ONe WD under our simulations. If we exclude the case of an ONe WD, the
companion of HD 49798 may be a CO WD or a CONe hybrid WD, both of which can be
expected to evolve to an SN Ia. Because of the rapidly rotation of the
companion, it is possible to produce an over-luminos SN Ia. If the companion
of HD 49798 is a CO WD, it would be a challenge for the current binary
evolution theory to explain how such a massive CO WD was formed. We hope that
this work can stimulate more observations on the WD + He star systems so that
more detailed studies of WD + He star channel can be proceeded.

###### Acknowledgements.

We acknowledge useful comments and suggestions from the referee. This study is
supported by the National Basic Research Program of China (973 program,
2014CB845700), the National Natural Science Foundation of China (Nos 11322327
and 11390374), and the Natural Science Foundation of Yunnan Province (Nos
2013FB083 and 2013HB097).

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# Notes on Quantum oscillation for Hatsugai-Kohmoto model

Yin Zhong<EMAIL_ADDRESS>School of Physical Science and Technology $\&$
Key Laboratory for Magnetism and Magnetic Materials of the MoE, Lanzhou
University, Lanzhou 730000, China  Lanzhou Center for Theoretical Physics, Key
Laboratory of Theoretical Physics of Gansu Province

###### Abstract

Motivated by the non-Fermi liquid (NFL) in solvable Hatsugai-Kohmoto (HK)
model and ubiquitous quantum oscillation phenomena observed in strongly
correlated electron systems, we have studied the quantum oscillation in HK
model in terms of a combination of analytic and numerical calculation. The
analytic results indicate the existence of quantum oscillation in NFL state
and its properties can be described by Lifshitz-Kosevich-like formula.
Although numerical simulation from exact diagonolization exhibits certain
oscillation behavior, it is hard to extract its oscillation period and
amplitude. Therefore, more work (particularly the large-scale numerical
simulation) on this interesting issue is highly desirable and we expect the
current study on HK model will be helpful to understand generic quantum
oscillation in correlated electron materials.

##  I Introduction

Recently, an exactly solvable many-body fermionic model with an infinite-range
interaction, i.e. the Hatsugai-Kohmoto (HK) model,Hatsugai1992 ; Baskaran1991
; Hatsugai1996  and its extension has been hotly studied.Phillips2018 ;
Yeo2019 ; Phillips2020 ; Yang2021 ; Zhu2021 ; Zhao2022 ; Setty2021 ; Mai2022 ;
Huang2022 ; Li2022 ; Setty2020 ; Setty2021b ; Zhong2022  In contrast with
celebrated and more familiar Sachdev-Ye-Kitaev, Kitaev’s toric code and
honeycomb model with either quenched disorder or local $Z_{2}$ gauge
symmetry,Sachdev ; Maldacena ; Chowdhury ; Kitaev1 ; Kitaev2 ; Prosko ;
Zhong2013 ; Smith2017  the original HK model has translation invariance with
topologically trivial nature, but surprisingly, it provides a strictly exact
playground for non-Fermi liquid (NFL) and featureless Mott insulator in any
spatial dimension, which is rare in statistical mechanics and condensed matter
physics. (See Fig. 1)

The solvability of HK model results from its locality in momentum space and
one can diagonalize HK Hamiltonian (just diagonal $4\times 4$-matrix) for each
momentum. The current studies have mainly focused on an interesting extension
of HK model, i.e. the superconducting instability from the intrinsic NFL state
in HK model,Phillips2020  which is inspired by ubiquitous NFL behaviors and
their link to unconventional superconductivity in cuprate, iron-based
superconductors (SC) and many heavy fermion compounds. Unexpected properties
such as topological $s$-wave pairing and two-stage superconductivity have been
discovered.Zhu2021 ; Zhao2022 ; Li2022

Figure 1:  (a) The Hatsugai-Kohmoto (HK) model with hopping strength $t$ and
interaction $U$ in one spatial dimension and (b) on a square lattice. (c) The
exact ground-state phase diagram for HK model exhibits a Mott insulator and a
non-Fermi-liquid-like metal. ($U_{c}=W$ with $W$ being the band-width) The
transition from metallic state to gapped Mott insulating phase belongs to the
universality of the continuous Lifshitz transition.

At the same time, it is well-known that observable (like magnetization and
resistivity) in metals show periodic oscillation under external magnetic
field.Shoenberg1984  Such phenomena is called quantum oscillation and is
generally believed to result from the oscillation of electron’s density of
state at Fermi energy when Landau level crosses Fermi energy periodically. The
underlying microscopic description is captured by the famous Lifshitz-Kosevich
(LK) formula, which provides a direct tool to extract effective mass and Fermi
surface from period and amplitude of oscillation. Furthermore, LK formula is
still valid if electron-electron interaction effect does not alter the Fermi
liquid nature of the considered system.Luttinger1961 ; Wasserman1996

However, a natural question arises when the system is not described by Fermi
liquid, and we may ask how about its quantum oscillation and the fate of LK
formula. This question is partially motivated by recent experiments of quantum
oscillation in many strongly correlated electron systems, including cuprate
and iron-based superconductor,Sebastian2015 ; Carrington2011  heavy fermion
compounds,McCollam2005  topological Kondo insulator,Tan2015  excitonic
insulator and twisted-bilayer graphene.Wang2021 ; Cao2016  In literature,
electron correlation effect in quantum oscillation has been widely studied by
mean-field approximation,Doh1998  slave-particle theory and dynamic mean-field
theory.Galitski2009 ; Acheche2017 ; Vucicevic2021  It should be emphasized
that all these mentioned treatments involve either uncontrollable
approximation or infinite-dimension limit, thus our understanding on quantum
oscillation in interacting many-electron system is obviously not complete.

In this work, inspired by the NFL state in solvable HK model, we study its
possible quantum oscillation. With the Luttinger approximation, the NFL state
indeed shows quantum oscillation and its zero-temperature behavior can be
captured by LK-like formula, which reflects the existence of two Fermi surface
structure of (non-Landau) quasiparticle in NFL. We also perform an exact
diagonolization calculation to check the validity of Luttinger approximation,
but due to the limited size of the system, although numerical results exhibit
certain oscillation behavior, it is hard to extract its oscillation period and
amplitude. Therefore, more work (particularly the large-scale numerical
simulation) on this interesting issue is highly desirable and we expect the
current study on HK model will be helpful to understand generic quantum
oscillation in correlated electron materials.

##  II The Hatsugai-Kohmoto model

###  II.1 Quick review of Hatsugai-Kohmoto model

The Hatsugai-Kohmoto model we study has the following form,

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle-\sum_{i,j,\sigma}t_{ij}\hat{c}_{i\sigma}^{{\dagger}}\hat{c}_{j\sigma}-\mu\sum_{j\sigma}\hat{c}_{j\sigma}^{{\dagger}}\hat{c}_{j\sigma}$
(1) $\displaystyle+$
$\displaystyle\frac{U}{N_{s}}\sum_{j_{1},j_{2},j_{3},j_{4}}\delta_{j_{1}+j_{3}=j_{2}+j_{4}}\hat{c}_{j_{1}\uparrow}^{{\dagger}}\hat{c}_{j_{2}\uparrow}\hat{c}_{j_{3}\downarrow}^{{\dagger}}\hat{c}_{j_{4}\downarrow}.$

Here, $\hat{c}_{j\sigma}^{{\dagger}}$ is the creation operator of conduction
electron (called $c$-electron for simplicity) at site $j$. $t_{ij}$ are
hopping integral between $i,j$ sites. Furthermore, chemical potential $\mu$
has been added to fix electron’s density. $N_{s}$ is the number of sites. The
last term of $\hat{H}$ is the HK interaction,Hatsugai1992  unlike the usual
Hubbard interaction
$U\sum_{j}\hat{c}_{j\uparrow}^{{\dagger}}\hat{c}_{j\uparrow}\hat{c}_{j\downarrow}^{{\dagger}}\hat{c}_{j\downarrow}$,
HK interaction is an infinite-range interaction between four electrons but
preserves the center of motion for $c$-electron due to the constraint of
$\delta$ function. This interaction plays a fundamental role in solving this
model as we will see later.

Importantly, Eq. 1 is local in momentum space after Fourier transformation and
the resultant Hamiltonian reads as $\hat{H}=\sum_{k}\hat{H}_{k}$,

$\displaystyle\hat{H}_{k}$ $\displaystyle=$
$\displaystyle\sum_{\sigma}(\varepsilon_{k}-\mu)\hat{c}_{k\sigma}^{{\dagger}}\hat{c}_{k\sigma}+U\hat{f}_{k\uparrow}^{{\dagger}}\hat{f}_{k\uparrow}\hat{f}_{k\downarrow}^{{\dagger}}\hat{f}_{k\downarrow},$
(2)

where $\varepsilon_{k}$ are dispersion of electrons. It is emphasized that the
locality of above Hamiltonian stems from infinite-range HK interaction
preserving center of motion. In contrast, the Hubbard interaction in momentum
space is rather nonlocal as
$U\sum_{k,k^{\prime},q}\hat{c}_{k+q\uparrow}^{{\dagger}}\hat{c}_{k\uparrow}\hat{c}_{k^{\prime}-q\downarrow}^{{\dagger}}\hat{c}_{k^{\prime}\downarrow}$,
thus it cannot lead to simple formalism in our model.

Now, if we choose Fock state

$|n_{1},n_{2}\rangle\equiv(\hat{c}_{k\uparrow}^{{\dagger}})^{n_{1}}|0\rangle(\hat{c}_{k\downarrow}^{{\dagger}})^{n_{2}}|0\rangle$
(3)

with $n_{i}=0,1$ as basis, $\hat{H}_{k}$ can be written as a diagonal $4\times
4$ matrix, whose eigen-energy is
$0,\varepsilon_{k}-\mu,\varepsilon_{k}-\mu,2(\varepsilon_{k}-\mu)+U$ and the
corresponding eigen-state is
$|0\rangle_{k}\equiv|00\rangle,|\sigma=\uparrow\rangle_{k}\equiv|10\rangle,|\sigma=\downarrow\rangle_{k}\equiv|01\rangle,|\uparrow\downarrow\rangle_{k}\equiv|11\rangle$,
which means states are empty, single occupation with spin-up and spin-down,
and double occupation.

Therefore, the many-body ground-state of $\hat{H}$ is just the direct-product
state of each $\hat{H}_{k}$’s ground-state, i.e.
$|\Psi_{g}\rangle=\prod_{k\in\Omega_{0}}|0\rangle_{k}\prod_{k\in\Omega_{1}}|\sigma\rangle_{k}\prod_{k\in\Omega_{2}}|\uparrow\downarrow\rangle_{k}$.($\Omega_{0},\Omega_{1},\Omega_{2}$
are the momentum range for different occupation) If $\Omega_{0}=\Omega_{2}=0$,
the system is a Mott insulator otherwise, we obtain a metallic state with non-
Fermi liquid properties (e.g. absence of Fermi surface, violation of Luttinger
theorem). Similarly, excited states and their energy are easy to be
constructed, so $\hat{H}$ (Eq. 1) has been solved since all eigen-states and
eigen-energy are found.

It is noted that the solvability of our model involves only locality in
momentum space, therefore Eq. 1 is solvable for arbitrary lattice geometry,
spatial dimension and electron filling. Furthermore, HK model does not rely on
disorder average and large-$N$ limit, which are crucial in classic
Sherrington-Kirkpatrick’s spin glass model and the more recent Sachdev-Ye-
Kitaev model.Sherrington1975 ; Sachdev ; Maldacena ; Chowdhury  In addition,
(mean-field-based) pairing term like
$\hat{c}_{k\uparrow}^{{\dagger}}\hat{c}_{-k\downarrow}^{{\dagger}}$ has been
studied in Ref. Phillips2020 ; Zhu2021 ; Zhao2022 .

For our purpose, it is useful to present the single-particle Green’s function
and some ground-state or thermodynamic quantities for HK model. For example,
the single-particle Green’s function can be obtained in terms of equation of
motion, which is read as (See Appendix.A)

$\displaystyle G_{\sigma}(k,\omega)$ $\displaystyle=$
$\displaystyle\frac{1+\frac{U\langle\hat{n}_{k\bar{\sigma}}\rangle}{\omega-(\varepsilon_{k}-\mu+U)}}{\omega-(\varepsilon_{k}-\mu)}$
(4) $\displaystyle=$
$\displaystyle\frac{1-\langle\hat{n}_{k\bar{\sigma}}\rangle}{\omega-(\varepsilon_{k}-\mu)}+\frac{\langle\hat{n}_{k\bar{\sigma}}\rangle}{\omega-(\varepsilon_{k}-\mu+U)}$

where $\langle\hat{n}_{k\bar{\sigma}}\rangle$ is the expectation value of
electron number operator
$\hat{n}_{k\bar{\sigma}}=\hat{c}^{{\dagger}}_{k\bar{\sigma}}\hat{c}_{k\bar{\sigma}}$
with spin $\bar{\sigma}=-\sigma$. The pole of above Green’s function tells us
that there exist two quasiparticle (it is not Landau quasiparticle since
adiabatical continuity into non-interacting limit $U=0$ does not work) energy
bands $\varepsilon_{k}-\mu,\varepsilon_{k}-\mu+U$.

Next, at finite-$T$, the thermodynamics of HK model is determined by its free
energy density $f$, which is related to partition function $\mathcal{Z}$ as

$\displaystyle
f=-\frac{T}{N_{s}}\ln\mathcal{Z},\mathcal{Z}=\mathrm{Tr}e^{-\beta\hat{H}}=\prod_{k}\mathrm{Tr}e^{-\beta\hat{H}_{k}}=\prod_{k}f_{k}$
(5)

Here, one notes that the partition function is easy to calculate since each
$k$-state contributes independently. We have defined
$f_{k}=1+2z_{k}+z_{k}^{2}e^{-\beta U}$ and
$z_{k}=e^{-\beta(\varepsilon_{k}-\mu)}$.Then, the typical thermodynamic
quantity, i.e. the heat capacity, is calculated by standard thermodynamic
relation $C_{V}=-T\frac{\partial^{2}f}{\partial T^{2}}$. In addition to
$C_{V}$, one can also calculate spin susceptibility $\chi_{s}$ if inserting
Zeeman energy term
$\hat{H}_{h}=-B(\hat{c}_{k\uparrow}^{{\dagger}}\hat{c}_{k\uparrow}-\hat{c}_{k\downarrow}^{{\dagger}}\hat{c}_{k\downarrow})$
into Hamiltonian $\hat{H}_{k}$. Then, it follows that the magnetization
$M=-\frac{\partial f}{\partial B}$ and $\chi_{s}=\frac{\partial M}{\partial
B}=-\frac{\partial^{2}f}{\partial B^{2}}$.

At zero temperature, the free energy density reduces into the ground-state
energy density, which has very simple expression,

$e_{g}=\frac{1}{N_{s}}\sum_{k}[(\varepsilon_{k}-\mu)\theta(\mu-\varepsilon_{k})+(\varepsilon_{k}-\mu+U)\theta(\mu-\varepsilon_{k}-U)],$
(6)

where $\theta(x)$ is the standard unit-step function ($\theta(x)=1$ for $x>0$
and $\theta(x)=0$ if $x<0$). Therefore, the electron density at $T=0$ is found
to be

$n=\frac{1}{N_{s}}\sum_{k}[\theta(\mu-\varepsilon_{k})+\theta(\mu-\varepsilon_{k}-U)].$
(7)

which indicates two Fermi surface located at $\varepsilon_{k}=\mu$ and
$\varepsilon_{k}=\mu-U$.

###  II.2 Adding orbital magnetic field

Because we are interested in possible quantum oscillation of HK model, orbital
magnetic field should be included. This can be realized easily via Pierls
substitution as $t_{ij}\rightarrow t_{ij}e^{ia_{ij}}$, where phase factor
$a_{ij}=\frac{e}{\hbar}\int_{r_{j}}^{r_{i}}d\vec{r}^{\prime}\cdot\vec{A}(\vec{r}^{\prime})$
and $\vec{A}$ is the vector potential of external magnetic field. Then, we can
write HK model as $\hat{H}=\hat{H}_{0}+\hat{H}_{\mu}+\hat{H}_{U}$

$\displaystyle\hat{H}_{0}=-\sum_{i,j,\sigma}t_{ij}e^{ia_{ij}}\hat{c}_{i\sigma}^{{\dagger}}\hat{c}_{j\sigma},$
$\displaystyle\hat{H}_{\mu}=-\mu\sum_{j\sigma}\hat{c}_{j\sigma}^{{\dagger}}\hat{c}_{j\sigma},~{}~{}~{}~{}$
$\displaystyle\hat{H}_{U}=\frac{U}{N_{s}}\sum_{j_{1},j_{2},j_{3},j_{4}}\delta_{j_{1}+j_{3}=j_{2}+j_{4}}\hat{c}_{j_{1}\uparrow}^{{\dagger}}\hat{c}_{j_{2}\uparrow}\hat{c}_{j_{3}\downarrow}^{{\dagger}}\hat{c}_{j_{4}\downarrow}.$
(8)

Frankly speaking, the above Hamiltonian is hard to solve since adding external
magnetic field breaks the translation invariance, which leads to the violation
of solvability of original HK model (Eq. 1).

Instead, following the original treatment of Luttinger,Luttinger1961 ;
Wasserman1996  whose approximation is able to capture quantum oscillation in
interacting electron systems such as Fermi liquid, we can still use results of
last section but only need to replace single-particle energy $\varepsilon_{k}$
with eigen-energy of $\hat{H}_{0}$ (denoted as $\\{E_{m}\\}$). In the language
of many-body physics, such replacement is a statement of that the external
magnetic field only modifies the single-particle spectrum but not the
expression of self-energy function.

Therefore, the ground-state energy density under external magnetic field
($\vec{B}=\nabla\times\vec{A}$) has the new formalism,

$e_{g}=\frac{1}{N_{s}}\sum_{m}[(E_{m}-\mu)\theta(\mu-
E_{m})+(E_{m}-\mu+U)\theta(\mu-E_{m}-U)],$ (9)

and the related electron density is

$n=\frac{1}{N_{s}}\sum_{m}[\theta(\mu-E_{m})+\theta(\mu-E_{m}-U)].$ (10)

Thus, the magnetization and susceptibility in ground-state can be found by
$M=-\frac{\partial e_{g}}{\partial B},\chi_{s}=\frac{\partial M}{\partial B}$.
For thermodynamics at finite $T$, the only modification is $f_{k}\rightarrow
f_{m}=1+2z_{m}+z_{m}^{2}e^{-\beta U}$ with
$z_{m}=e^{-\beta(\varepsilon_{m}-\mu)}$ in Eq. 5.

To be specific, let us consider the square lattice on $xy$-plane with nearest-
neighbor-hopping ($t_{ij}\rightarrow t$) and the magnetic field is along the
$z$-axis $\vec{B}=B\vec{e}_{z}$. In Landau gauge, the corresponding vector
potential is $\vec{A}=Bx\vec{e}_{y}$, so $a_{i,i+x}=0$ and
$a_{i,i+y}=\frac{e}{\hbar}Ba^{2}i_{x}$, ($a$ is the lattice constant and will
be set to unit) which gives rise to

$\displaystyle\hat{H}_{0}=-t\sum_{i,\sigma}(\hat{c}_{i\sigma}^{{\dagger}}\hat{c}_{i+x,\sigma}+e^{i\frac{e}{\hbar}Bi_{x}}\hat{c}_{i\sigma}^{{\dagger}}\hat{c}_{i+y,\sigma}).$
(11)

and phase factor part can be parameterized as
$e^{i\frac{e}{\hbar}Bi_{x}}=e^{i\frac{e}{\hbar}\Phi i_{x}}=e^{i2\pi wi_{x}}$
with flux $\Phi=2\pi\frac{\hbar}{e}w$ on each plaquette. In fact $\hat{H}_{0}$
is just the famous Hofstadter model with spin degree of freedom and has been
widely studied in recent years due to its realization in cold atom experiments
and relevance to moiré superlattice systems.Hofstadter1976 ; Aidelsburger2013
; Miyake2013 ; Kennedy2015 ; Dean2013 ; Hunt2013 ; Spanton2018

After diagonalization of $\hat{H}_{0}$, we obtain its eigen-energy
$\\{E_{m}\\}$ for given magnetic field $B$, thus interested quantities can be
calculated straightforwardly.

##  III Quantum oscillation in non-Fermi liquid: The case study on Hatsugai-
Kohmoto model

###  III.1 Lattice calculation

In this section, we study the HK model under magnetic field on square lattice
whose single-particle Hamiltonian is $\hat{H}_{0}$ (Eq. 11). For parameters
$t=e=\hbar=1$, the single-particle spectrum $\\{E_{m}\\}$ has been shown in
Fig. 2, in which we consider a $60\times 60$ system with periodic boundary
condition. It is clear to see the well-known butterfly spectrum of
$\hat{H}_{0}$ for $E_{m}$ versus flux $\Phi=2\pi w=B$.Hofstadter1976

Figure 2:  The butterfly spectrum of $\hat{H}_{0}$ for $E_{m}$ versus flux
$\Phi=2\pi w=B$.

Next, using Eq. 10, we have plotted particle density $n$ versus $\Phi$ and
$1/\Phi$ for different interaction strength $U$.

Figure 3:  The particle density $n$ versus $\Phi$ (a) and $1/\Phi$ (b) for
different $U/t$.

To have regular data for our model on square lattice, we have used periodic
boundary condition along $y$-direction while an open boundary condition is
assumed for $x$-direction. Thus, a clear quantum oscillation of particle
density appears in Fig. 3. (setting $\mu=0$) Similarly, in Fig. 4 we have
shown the data of magnetization, which also exhibits the quantum oscillation
signal though with irregular error introduced by numerical differentiation.
Therefore, we finds that the HK model indeed shows quantum oscillation
phenomena in terms of the Luttinger’s approximation.

Figure 4:  The magnetization $M$ versus $\Phi$ for $U/t=0.5,1.0,1.5,2.0$.

###  III.2 Continuity limit

After presenting the results of direct lattice calculation, in this
subsection, we try to understand the origin of quantum oscillation from the
continuity limit.

In continuity limit, one can approximate electron’s dispersion
$\varepsilon_{k}$ with free particle energy $\frac{k^{2}}{2m_{e}}$. At the
same time, $\varepsilon_{m}=\hbar\omega_{c}(m+1/2)$ if we consider usual
Landau gauge. ($\omega_{c}=eB/m_{e}$) Then, the electron’s density reads

$n=\frac{1}{N_{\Lambda}}\sum_{k_{y}}\sum_{m=0}^{\Lambda}[\theta(\mu-\hbar\omega_{c}(m+1/2))+\theta(\mu-\hbar\omega_{c}(m+1/2)-U)].$
(12)

Here, $\Lambda$ is the cutoff of Landau level index, which acts as a band-
width. The factor $N_{\Lambda}$ has been introduced to give sensible
definition of electron’s density. The summation over $k_{y}$ gives the
degeneracy of Landau level ($D$).

To proceed, we can rewrite $n$ as follows,

$\displaystyle n$ $\displaystyle=$
$\displaystyle\frac{D}{N_{\Lambda}}\lim_{\beta\rightarrow\infty}\int_{0}^{\infty}d\omega\frac{\sum_{m}\delta(\omega-\varepsilon_{m})+\sum_{m}\delta(\omega-\varepsilon_{m}-U)}{e^{\beta(\omega-\mu)}+1}$
(13) $\displaystyle=$
$\displaystyle\frac{D}{N_{\Lambda}}\lim_{\beta\rightarrow\infty}\int_{0}^{\infty}d\omega\frac{1}{e^{\beta(\omega-\mu)}+1}$
$\displaystyle\times$
$\displaystyle\mathrm{Re}\int_{0}^{\infty}d\tau\sum_{m}(e^{i(\omega-\varepsilon_{m})\tau}+e^{i(\omega-\varepsilon_{m}-U)\tau}).$

Then, using Poisson summation formula
$\sum_{m}e^{-im\hbar\omega_{c}\tau}=\frac{2\pi}{\hbar\omega_{c}}\sum_{p=0,\pm
1,\pm 2...}\delta(\tau-\frac{2\pi p}{\hbar\omega_{c}})$, and integrate over
$\tau$, one finds

$\displaystyle n$ $\displaystyle=$ $\displaystyle\frac{2\pi
D}{N_{\Lambda}\hbar\omega_{c}}\lim_{\beta\rightarrow\infty}\int_{0}^{\infty}d\omega\left(\frac{1}{e^{\beta(\omega-\mu)}+1}+\frac{1}{e^{\beta(\omega-\mu+U)}+1}\right)$
(14) $\displaystyle\times$ $\displaystyle\mathrm{Re}\sum_{p}e^{i\frac{2\pi
p}{\hbar\omega_{c}}(\omega-\frac{\hbar\omega_{c}}{2})}$ $\displaystyle\propto$
$\displaystyle\lim_{\beta\rightarrow\infty}\frac{2\pi}{\beta\hbar\omega_{c}}\mathrm{Re}\sum_{\omega_{n}>0}\sum_{p>0}(-1)^{p}\left(e^{i\frac{2\pi
p}{\hbar\omega_{c}}(\mu+i\omega_{n})}+e^{i\frac{2\pi
p}{\hbar\omega_{c}}(\mu-U+i\omega_{n})}\right)$ $\displaystyle=$
$\displaystyle\lim_{\beta\rightarrow\infty}\sum_{p>0}\frac{(-1)^{p}}{2\pi
p}\frac{X}{\sinh X}\left(\cos\frac{2\pi p\mu}{\hbar\omega_{c}}+\cos\frac{2\pi
p(\mu-U)}{\hbar\omega_{c}}\right).$

Here, $X=\frac{2\pi^{2}pT}{\hbar\omega_{c}}$ and usually one defines
$R_{T}=\frac{X}{\sinh X}$ as damping factor for temperature. It is amusing to
note that the above Eq. 14 is just the well-known LK formula with oscillation
frequency $\mu/(\hbar\omega_{c})$ and $(\mu-U)/(\hbar\omega_{c})$. When
$T\rightarrow 0$ or $\beta\rightarrow\infty$, we obtain
$n\propto\sum_{p>0}\frac{(-1)^{p}}{2\pi p}\left(\cos\frac{2\pi
p\mu}{\hbar\omega_{c}}+\cos\frac{2\pi p(\mu-U)}{\hbar\omega_{c}}\right)$.
Thus, it seems that under the approximation of Luttinger, despite the non-
Fermi liquid nature, the quantum oscillation exists in the ground-state of HK
model and it has two oscillation period determined by chemical potential $\mu$
and $\mu-U$.

##  IV Exact calculation based on exact diagonolization

Until now, our calculation is based on Luttinger’s approximation, which is in
principle not known whether it works or not.

Instead, one may use (numerical) exact diagonolization to extract reliable
information from our model (Eq.8 and 11). Since $y$-direction of our model has
been chosen to have periodic boundary condition under Landau gauge, we can use
Fourier transformation to write

$\hat{c}_{j\sigma}=\frac{1}{\sqrt{L_{y}}}\sum_{k_{y}}e^{ik_{y}j_{y}}\hat{c}_{j_{x}\sigma}(k_{y}),$
(15)

where $x-$direction remains to be intact. Then, the Hamiltonian reads as
$\hat{H}=\sum_{k_{y}}\hat{H}(k_{y})$,

$\displaystyle\hat{H}(k_{y})$ $\displaystyle=$
$\displaystyle-t\sum_{j_{x}\sigma}(\hat{c}_{j_{x}\sigma}^{{\dagger}}(k_{y})\hat{c}_{j_{x}+1\sigma}(k_{y})+\hat{c}_{j_{x}+1\sigma}^{{\dagger}}(k_{y})\hat{c}_{j_{x}\sigma}(k_{y}))$
(16) $\displaystyle+$ $\displaystyle\sum_{j_{x}\sigma}\left(-2t\cos(\Phi
j_{x}+k_{y})-\mu\right)\hat{c}_{j_{x}\sigma}^{{\dagger}}(k_{y})\hat{c}_{j_{x}\sigma}(k_{y})$
$\displaystyle+$
$\displaystyle\frac{U}{L_{x}}\sum_{j_{1x},j_{2x},j_{3x},j_{4x}}\delta_{j_{1x}+j_{3x}=j_{2x}+j_{4x}}\hat{c}_{j_{1x}\uparrow}^{{\dagger}}(k_{y})\hat{c}_{j_{2x}\uparrow}(k_{y})$
$\displaystyle\times$
$\displaystyle\hat{c}_{j_{3x}\downarrow}^{{\dagger}}(k_{y})\hat{c}_{j_{4x}\downarrow}(k_{y}).$

The above Hamiltonian means that for given momentum $k_{y}$, one can just
study the reduced object $\hat{H}(k_{y})$, which is able to be solved in terms
of exact diagonolization up to $L_{x}$ sites along $x-$direction (i.e.
$j_{x}=1,2...Lx$).

In Fig. 5, we have shown the particle density $n$ versus $\Phi$ for $L_{x}=6$
and $L_{y}=16$. However, due to the limited size, particularly the smallness
of $L_{x}$, although certain oscillation behavior appears, it is hard to
detect clear quantum oscillation and extract its oscillation period and
amplitude. Therefore, much larger size of $L_{x}$ should be used to explore
the true signature of quantum oscillation of HK model. (system with $L_{x}$ up
to $16$ may be calculated in present workstation and it will be explored in
our future work)

Figure 5:  The particle density $n$ versus $\Phi$ for $U/t=0,0.5,1.0,1.5,2.0$
from exact diagonolization.

##  V Conclusion and Future direction

In conclusion, we have taken Hatsugai-Kohmoto model as an example to study the
quantum oscillation behavior in strongly correlated system. With Luttinger
approximation, it is found that the quantum oscillation indeed appears in non-
Fermi liquid state and one can use Lifshitz-Kosevich-like formula to extract
its basic properties. We have also performed a small size exact
diagonolization calculation, its results exhibits certain oscillation behavior
but we cannot extract exact value of oscillation period and amplitude.
Therefore, we expect that future work on this interesting issue is highly
desirable, particularly the large scale numerical simulation.

_Note added_ : After completing this work, we have noticed the paper of Leeb
and Knolle,Leeb2023  whose results generally agree with ours though their
calculation is based on the projection into Landau level.

##  Appendix A Derivation of singe-particle Green’s function

Follow the treatment of Hubbard model,Hubbard1963  let us define the single-
particle Green’s function as
$G_{\sigma}(k,\omega)=\langle\langle\hat{c}_{k\sigma}|\hat{c}_{k\sigma}^{{\dagger}}\rangle\rangle_{\omega}$,
which is just the Fourier transformation of the retarded Green’s function

$G_{\sigma}(k,t)=-i\theta(t)\langle[\hat{c}_{k\sigma}(t),\hat{c}_{k\sigma}^{{\dagger}}]_{+}\rangle.$

Then, in terms of

$\displaystyle[\hat{c}_{k\sigma},\hat{H}]=(\varepsilon_{k}-\mu)\hat{c}_{k\sigma}+U\hat{c}_{k\sigma}\hat{n}_{k\bar{\sigma}},$
$\displaystyle[\hat{c}_{k\sigma}\hat{n}_{k\bar{\sigma}},\hat{H}]=(\varepsilon_{k}-\mu+U)\hat{c}_{k\sigma}\hat{n}_{k\bar{\sigma}},$

we find

$\omega\langle\langle\hat{c}_{k\sigma}|\hat{c}_{k\sigma}^{{\dagger}}\rangle\rangle_{\omega}=1+(\varepsilon_{k}-\mu)\langle\langle\hat{c}_{k\sigma}|\hat{c}_{k\sigma}^{{\dagger}}\rangle\rangle_{\omega}+U\langle\langle\hat{c}_{k\sigma}\hat{n}_{k\bar{\sigma}}|\hat{c}_{k\sigma}^{{\dagger}}\rangle\rangle_{\omega}$

and

$\omega\langle\langle\hat{c}_{k\sigma}\hat{n}_{k\bar{\sigma}}|\hat{c}_{k\sigma}^{{\dagger}}\rangle\rangle_{\omega}=\langle\hat{n}_{k\bar{\sigma}}\rangle+(\varepsilon_{k}-\mu+U)\langle\langle\hat{c}_{k\sigma}\hat{n}_{k\bar{\sigma}}|\hat{c}_{k\sigma}^{{\dagger}}\rangle\rangle_{\omega}$

Since above equations are closed, we obtain

$\langle\langle\hat{c}_{k\sigma}\hat{n}_{k\bar{\sigma}}|\hat{c}_{k\sigma}^{{\dagger}}\rangle\rangle_{\omega}=\frac{\langle\hat{n}_{k\bar{\sigma}}\rangle}{\omega-\varepsilon_{k}+\mu-U}$

and

$\displaystyle G_{\sigma}(k,\omega)$ $\displaystyle=$
$\displaystyle\frac{1+\frac{U\langle\hat{n}_{k\bar{\sigma}}\rangle}{\omega-(\varepsilon_{k}-\mu+U)}}{\omega-(\varepsilon_{k}-\mu)}$
$\displaystyle=$
$\displaystyle\frac{1-\langle\hat{n}_{k\bar{\sigma}}\rangle}{\omega-(\varepsilon_{k}-\mu)}+\frac{\langle\hat{n}_{k\bar{\sigma}}\rangle}{\omega-(\varepsilon_{k}-\mu+U)}$

which is just the wanted Eq. 4 in the main text.

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