# Influence of strong magnetic field on the structure properties of strange
quark stars

Fatemeh Kayanikhoo1 , Kazem Naficy 1 and Gholam Hossein Bordbar
2,3111Corresponding author. E-mail<EMAIL_ADDRESS>1 Department of
Physics, University of Birjand, Birjand, Iran
2 Department of Physics, Shiraz University, Shiraz 71454, Iran222Permanent
address
and
3 Department of Physics and Astronomy, University of Waterloo, 200 University
Avenue West, Waterloo, Ontario, N2L 3G1, Canada

###### Abstract

We investigate the thermodynamic properties of strange quark matter under the
strong magnetic field in the framework of the MIT bag model in two cases of
bag constants. We consider two cases of the magnetic field, the uniform
magnetic field and the density-dependent magnetic field to calculate the
equation of state of strange quark matter. For the case of density-dependent
magnetic field, we use a Gaussian equation with two free parameters $\beta$
and $\theta$ and use two different sets of the parameters for the magnetic
field changes (a slow and a fast decrease of the magnetic field from the
center to the surface). Our results show that the energy conditions based on
the limitation of the energy-momentum tensor, are satisfied in the
corresponding conditions. We also show that the equation of state of strange
quark matter becomes stiffer by increasing the magnetic field. In this paper,
we also calculate the structure parameters of a pure strange quark star using
the equation of state. We investigate the compactification factor ($2M/R$) and
the surface redshift of star in different conditions. The results show that
the strange quark star is denser than the neutron star and it is more compact
in the presence of the stronger magnetic field. As another result, the
compactification factor increases when we use a slow increase of the magnetic
field from the surface to the center. Eventually, we compare our results with
the observational results for some strange star candidates, and we find that
the structure of the strange star candidates is comparable to that of the star
in our model.

Strange quark matter, strange quark star, MIT bag model, Landau quantization
effect, density dependent bag constant, strong magnetic field

##  I Introduction

Like white dwarfs, neutron stars, and black holes, strange quark stars (SQS)
are compact objects, created at the end of the life of the massive stars. The
SQS is denser than the neutron star. In other words, they have the maximum
gravitational masses around $1.7-2M_{sun}$ as the neutron stars but the
smaller radii (around $4-8\ km$) rk1 . The SQS is considered in two cases: 1)
The pure SQS that all over is made of strange quark matter (SQM) 2) The hybrid
star that is the neutron star with the core made of SQM.

When the nuclear matter is compressed to the high density (more than
$10^{15}g/cm^{3}$), the nucleons overlap and it is expected that the nuclear
matter converts to the quark matter (consisting of up and down quarks) by
phase transition rk2 ; rk3 . The weak interaction between up and down quarks
creates strange quarks rk4 . For the first time, Gell-Mann and Zweig suggested
that hadrons are made up of the smaller particles later called quarksrk5 ; rk6
. Ivanenco explained the possibility of the existence of a quark core in some
compact objects rk7 ; rk8 . In 1970, Itoh computed the maximum gravitational
mass of the quark stars rk9 . In addition, in 1984, Witten suggested that the
basic matter of the stars could be the strange matter rk10 . There have been
candidates for quark stars, RX J185635-3754 and 3C58 observed by Chandra X-ray
Observatory in 2002 rk1 , and SWIFT J1749.42807 rk11 , etc.

The most important properties of compact objects is a strong magnetic field.
The surface magnetic field that observed for white dwarfs is about
$10^{6}-10^{8}\ G$ rk12 , and for the neutron stars and SQS’s is about
$10^{12}-10^{14}\ G$ rk13 . It is clear that the interior magnetic field of
compact objects is a few orders of magnitudes stronger. So the interior
magnetic field around $10^{12}$-$10^{14}\ G$ for the white dwarfs and
$10^{18}-10^{20}\ G$ for the neutron stars and SQS’s are expected rk13 ; rk14
; rk15 . Although the source of these strong magnetic fields is not clear, it
can be considered that existence of a large number of spin particles creates a
strong magnetic flux and as a result a strong magnetic field. In presence of
the strong magnetic field, the Landau quantization effect is considerable. In
other words, the strong magnetic field causes the cyclotron orbits of the
charged particles to be quantized rk16 ; rk17 . Several papers have already
been published on the magnetic field effect on compact objects. Mukhopadhyay
investigated the Landau quantization effect on white dwarfs rk12 . Rezaei and
Bordbar published papers in subjects of the effect of Landau quantization on
neutron matter rk18 ; rk19 . In addition, Lai and Shapiro published a paper on
the same subject rk13 . Lopes and Menezes considered the Landau quantization
effect in neutron stars rk20 .

Bordbar et. al. published some papers on the SQS’s in the various conditions
in recent years. They investigated the thermodynamic properties and structure
of cold polarized SQS rk21 ; rk22  and the magnetized SQS at the finite
temperature rk23 ; rk24 . In addition, Non-polarized SQS using MIT bag model
and NJL model have been investigated rk25 ; rk26 ; rk27 . Furthermore, papers
have been published to investigate properties of neutron stars with the quark
core rk28 ; rk29 .

In the current paper, we investigate the effect of the strong magnetic field
on SQS with Landau quantization effect using the MIT bag model in two cases of
bag constants. It should be mentioned that in our calculations, we consider an
isotropic magnetic field for the interior of SQS. In the next section, we
study the formalism of calculation of the thermodynamic properties of SQM. The
structure parameters of SQS are calculated in the third section. The
conclusion is given in the final section.

##  II Formalism of Calculation

In this section, we investigate the thermodynamic properties of a pure SQS
made by up, down and strange quarks. Since the density of electrons is low
compared to that of quarks, it is ignored rk30 ; rk31 ; rk32 . The SQM is a
relativistic Fermi system, so we should use Fermi-Dirac statistics to
calculate its thermodynamic properties rk33 . We use the MIT bag model to
calculate the energy density of the system. The MIT bag model is an approach
based on QCD model, in this model quarks are considered as the free particles
in a bag under the potential of a bag constant rk34 ; rk35 .

###  II.1 Energy density and equation of state of strange quark matter

To study the thermodynamic properties and structure of SQS, the first step is
to calculate the energy density and the equation of state (EOS). As we
mentioned, we use the Fermi-Dirac statistics and consider the Landau
quantization effect to calculate the thermodynamic properties of SQM in the
presence of a strong magnetic field. By considering the Landau quantization
effect the single particle energy of $J$th Landau levels is (see Refs. rk12 ;
rk13 ; rk16 ; rk17  for more details),

$\epsilon^{i}=[p_{i}^{2}c^{2}+m_{i}^{2}c^{4}(1+2JB_{D})]^{1/2},$ (1)

where superscript $i$ represents the particles ($i=$ up, down, strange
quarks), $p$, $c$ and $m$ are momentum, speed of light and mass of particles,
respectively, and $B_{D}=B/B_{c}$ ($B$ is the magnetic field and $B_{c}$ is
equal to $m_{i}^{2}c^{3}/q_{i}\hbar$ where $q$ is charge of particles and
$\hbar$ is Plank constant). The number density of quarks is obtained as
follows,

$\rho=\sum_{J=0}^{J_{max}}\frac{2qB}{h^{2}c}g_{J}p_{F}(J),$ (2)

where $p_{F}(J)$ is the Fermi momentum corresponding to the $J$th. Landau
level, $g_{J}$ is the degeneracy of the $J$th. Landau level and $J_{max}$ is
the upper limit of Landau level,

$J_{max}=\frac{\epsilon_{Fmax}^{2}-1}{2B_{D}}$ (3)

in the above equation, $\epsilon_{Fmax}$ is the Fermi energy density
corresponding to the Landau level.

In the frame work of MIT bag model, the total energy density of the system is,

$\varepsilon_{tot}=\sum_{j=+,-}\varepsilon_{i}^{(j)}+\varepsilon_{M}+B_{bag},$
(4)

in the above equation, we denote the bag constant as $B_{bag}$, we consider
two cases for $B_{bag}$, a fixed bag constant and a density-dependent bag
constant. For the fixed bag constant we use $B_{bag}^{fixed}=90MeV/fm^{3}$,
and for the density-dependent bag constant, we use the Gaussian equation given
by experimental data of CERN as follows,

$B_{bag}^{dep.}=B_{bag}(\rho)=B_{\infty}+(B_{0}-B_{\infty})exp{(-\beta(\frac{\rho}{\rho_{0}})^{2})},$
(5)

where $\beta$ is the numerical parameter equal to the density of normal
nuclear matter ($\rho_{0}=0.17\ fm^{-3}$), and $B_{0}=B(\rho=0)$ is equal to
$400\ MeV/fm^{3}$ rk36a ; rk36b ; rk36 ; rk37 . Also, the value of the
parameter $B_{\infty}$ depends only on the value of the parameter $B_{0}$ and
is obtained by the LOCV method in our calculations rk38 ; rk39 ; rk40 ; rk41 .
In Eq. 3 the energy density due to the interaction of the magnetic field and
the dipole moment of the quarks is denoted by $\varepsilon_{M}$
($\varepsilon_{M}=0.299\rho\xi\mu_{N}B$, $\mu_{N}$ is $5.05\times
10^{-27}$$J/T$ and $\xi$ is the polarization parameter of system
$\xi=\frac{\rho^{(+)}-\rho^{(-)}}{\rho}$) and the kinetic energy density of
quarks is denoted by $\varepsilon_{i}^{(j)}$ ($j=+,-$, for spin up and spin
down particles, respectively). The kinetic energy density of each quark
$\varepsilon_{i}^{(j)}$ is as follows rk33 ,

$\varepsilon_{i}^{(j)}=\frac{2B_{D}}{(2\pi)^{2}\lambda^{3}}m_{i}c^{2}\sum_{J=0}^{J_{max}}g_{J}(1+2JB_{D})\eta(\frac{X_{F}^{(j)}}{(1+2JB_{D})^{1/2}}),$
(6)

where

$\eta(x)=\frac{1}{2}[x\sqrt{1+x^{2}}+ln(x\sqrt{1+x^{2}})],x=\frac{X^{(j)}_{F}}{(1+2JB_{D})^{1/2}}$
(7) $X^{(j)}_{F}=(\epsilon_{F}^{(j)2}-1-2JB_{D})^{1/2}.$ (8)

The equation of state (EOS) of SQM is given by the following relation,

$P(\rho)=\rho(\frac{\partial\varepsilon_{tot}}{\partial\rho})-\varepsilon_{tot}.$
(9)

###  II.2 Magnetic energy density and magnetic pressure

As there is a strong magnetic field, the magnetic energy density and magnetic
pressure cannot be ignored. The magnetic energy density
$\varepsilon_{B}=\frac{B^{2}}{8\pi}$ and the magnetic pressure
$P_{B}=\frac{B^{2}}{8\pi}$ ($B$ is the magnetic field) are directly added to
the total energy density ($\varepsilon_{tot}$) and the EOS ($P(\rho)$) rk42 ;
rk43 ; rk44 ; rk45 .

Here we study the uniform magnetic field and the density-dependent magnetic
field. In the first case, $B$ is constant from the center to the surface of
SQS. We compare the different magnetic fields effects in the section of
calculation results. In the second case, the magnetic field is considered as a
function of the density of the system. We use a Gaussian function as follows,

$B(\rho)=B_{surf}+B_{0}[1-exp{\beta(\frac{\rho}{\rho_{0}})^{\theta}}],$ (10)

where $B_{surf}$ is the magnetic field of the surface of the SQS that we
consider $5\times 10^{13}\ G$ and $B_{0}$ is the interior magnetic field that
is expected for the star. Also, $\beta$ and $\theta$ are the parameters that
define the magnetic field changes based on the density of SQS rk42 ; rk46 ;
rk47 . In other words, these parameters select as the magnetic field decrease
fast or slow from the center to the surface of the star. There are different
sets of $\beta$ and $\theta$ investigated in the literature rk48 ; rk49 .

###  II.3 Thermodynamic calculation results

We have shown the pressure of the system versus the total energy density in
Figs. 1 and 2 by considering $B_{bag}^{fixed}$ and $B_{bag}^{dep.}$,
respectively. The figures are plotted in the case of the uniform magnetic
fields and the effect of the different magnetic fields are compared. The
figures show that pressure increases by increasing the energy density in all
cases. Furthermore, The EOS becomes stiffer by increasing the magnetic field.
Also, It can be seen that the energy density in the case of strong magnetic
fields is much greater than the case with no magnetic field ($B=0$). The
Landau quantization effect breaks the spherical symmetry and thereby increases
the energy density up to several times of the normal nuclear energy density.

Figure 1: The pressure versus the energy density of system in presence of
different magnetic fields using $B_{bag}^{fixed}=90\ MeV/fm^{3}$.

$\begin{array}[]{cc}\includegraphics[width=256.0748pt,height=179.25244pt]{peuniformdep.eps}\end{array}$

Figure 2: The pressure versus the energy density of system in presence of
different magnetic fields using $B_{bag}^{dep.}$.

In Figs. 3 and 4 the case of the density-dependent magnetic field have been
plotted by considering $B_{bag}^{fixed}$ and $B_{bag}^{dep.}$, respectively.
We considered two different sets of $\beta$ and $\theta$: The fast changes of
the magnetic field (Set A: $\beta=0.05$ and $\theta=2$) and the slow changes
of the magnetic field (Set B: $\beta=0.1$ and $\theta=1$). It can be seen from
Fig. 3 that considering the slow changes of the magnetic field leads to the
softer equation of state in comparison to the fast changes of that. The same
behavior can be seen in Fig. 4 for the case of density-dependent bag constant.

Figure 3: The pressure versus the energy density of system in presence of the
case of density-dependent magnetic field (Set A and Set B) using
$B_{bag}^{fixed}=90\ MeV/fm^{3}$.  Figure 4: The pressure versus the energy
density of system in presence of the case of density-dependent magnetic field
(Set A and Set B) using $B_{bag}^{dep.}$.

We know that any theory of physics must comply with the energy conditions that
result from the limitation in energy-momentum tensor, $T^{\mu\nu}$ rk48 ; rk49
. We can ensure the accuracy of the results obtained for the energy and the
pressure of the system by investigation the energy conditions:
a) Null energy condition (NEC) $\longrightarrow$ $P_{c}$ \+
$\rho_{c}c^{2}$$\geq$0,
b) Weak energy condition (WEC) $\longrightarrow$ $P_{c}$ \+
$\rho_{c}c^{2}$$\geq$0 and $\rho_{c}$$\geq$ 0,
c) Strong energy condition (SEC) $\longrightarrow$ $P_{c}$ \+
$\rho_{c}c^{2}$$\geq$ 0 and $3P_{c}$ \+ $\rho_{c}c^{2}$$\geq$ 0,
d) Dominate energy condition (DEC) $\longrightarrow$
$\rho_{c}c^{2}$$\geq$$\mid$ $P_{c}$ $\mid$,

where $\rho_{c}$ and $P_{c}$ are the mass density and pressure at the center
of SQS ($r=0$). We have found that the energy conditions are satisfied
regarding the equation of state for all considered cases of this paper.

##  III Structure properties of strange quark star

The structure of stars is usually determined by their mass, radius and some
other parameters. In this section, we study the structure of the SQS by
gravitational mass ($M/M_{sun}$), radius ($R$), compactification factor
($2M/R$)and surface redshift ($Z_{s}$). For the compact objects, we should use
the general relativistic equation of hydrostatic equilibrium known as Tolman-
Oppenheimer-Volkov (TOV) equations rk50 ; rk51 ; rk52 ,

$\frac{dP}{dr}=-\frac{G\left[\varepsilon(r)+\frac{P(r)}{c^{2}}\right]\left[m(r)+\frac{4\pi
r^{3}P(r)}{c^{2}}\right]}{r^{2}\left[1-\frac{2Gm(r)}{rc^{2}}\right]},$ (11)
$\frac{dm}{dr}=4\pi r^{2}\varepsilon(r).$ (12)

Using the thermodynamic results of the previous section and numerical solution
of TOV equations the gravitational mass and the radius of the star have been
computed. We consider the usual boundary conditions: $P(r=0)=P_{c}$,
$P(r=R)=0$, $m(r=0)=0$ and $m(r=R)=M_{max}$.

### Results for calculations of the structure

We have shown the gravitational mass versus the central energy density of the
SQS in Figs. 5 and 6 for the cases of $B_{bag}^{fixed}$ and $B_{bag}^{dep.}$,
respectively. In these figures, we compared the behavior of the gravitational
mass of SQS in the presence of different magnetic fields. These figures show
that for all curves the gravitational mass increases by increasing the central
energy density until it reaches a maximum value (The maximum gravitational
mass). Also, it can be seen that the maximum gravitational mass of SQS has a
larger value in the absence of the magnetic field.

Figure 5: The gravitational mass versus the central energy density of SQS in
presence of different magnetic fields using $B_{bag}^{fixed}=90\ MeV/fm^{3}$.
Figure 6: The gravitational mass versus the central energy density of SQS in
presence of different magnetic fields using $B_{bag}^{dep.}$.

In Figs. 7 and 8, the changes of the gravitational mass versus the central
energy density are plotted for two cases of bag constant ($B_{bag}^{fixed}$
and $B_{bag}^{dep.}$). In these figures, we have compared the effect of two
cases of the fast (Set A) and the slow (Set B) increasing of the magnetic
field from the surface to the center of SQS.

Figure 7: The gravitational mass versus the central energy density of SQS in
presence of two different cases for the density-dependent of the magnetic
fields (Set A and Set B) using $B_{bag}^{fixed}=90\ MeV/fm^{3}$.  Figure 8:
The gravitational mass versus the central energy density of SQS in presence of
two different cases for the density-dependent of the magnetic fields (Set A
and Set B) using $B_{bag}^{dep.}$.

The figures show that considering the slow changes of the magnetic field
(using Set B) leads to larger values of the maximum gravitational masses.

Figure 9: The gravitational mass versus the radius of SQS in presence of the
different magnetic fields using $B_{bag}^{fixed}=90\ MeV/fm^{3}$.  Figure 10:
The gravitational mass versus the radius of SQS in presence of the different
magnetic fields using $B_{bag}^{dep.}$.  Figure 11: The gravitational mass
versus the radius of SQS in presence of two different cases for the density-
dependent of the magnetic fields (Set A and Set B) $B_{bag}^{fixed}=90\
MeV/fm^{3}$.  Figure 12: The gravitational mass versus the radius of SQS in
presence of two different cases for the density-dependent of the magnetic
fields (Set A and Set B) $B_{bag}^{dep.}$.

We have plotted the gravitational mass versus the radius of SQS for the
uniform magnetic fields in Figs. 9 and 10 and for the density-dependent
magnetic fields in Figs. 11 and 12, in the both cases of $B_{bag}^{fixed}$ and
$B_{bag}^{dep.}$, respectively. The results of calculation regarding SQS shows
a self-bound object, in contrast to the neutron star which is bounded by
gravity. The mass-radius relation of SQS follows $M\propto R^{\alpha}$
($\alpha\sim 3$) rk0 . We have found that the radius increases until the
gravitational mass reaches the maximum value, then the radius decreases by
increasing the gravitational mass. In other words, increasing the
gravitational mass more than a certain value causes the SQS to collapses and
the singularity is formed. Also, it is clear from Figs. 9 and 10, the maximum
gravitational mass and the radius decrease by increasing the magnetic field in
all considered cases. Furthermore, Figs. 11 and 12 show that considering Set B
for the density-dependent magnetic field leads to larger values of the maximum
gravitational mass and the radius. In addition, the $M-R$ plot corresponding
to Set A (the fast drop of the magnetic field from the center to the surface
of SQS) increases with the more steep slope in comparison to the same plot of
Set B. We have found with the curve fitting of Fig. 11 that $\alpha$ is equal
to $3.65$ for Set A and $3.41$ for Set B. Also, using Fig. 12, $\alpha$ is
found $3.21$ and $3.20$ for Set A and Set B, respectively. These values
indicate that the slow changes of the magnetic field results in the smaller
value of $\alpha$.

The numerical results of the structure parameters of the SQS are shown in
tables 1 and 2 for two cases of bag constant, $B_{bag}^{fixed}$ and
$B_{bag}^{dep.}$, respectively. We have shown the values of compactification
factor ($2M/R$) and the corresponding surface redshift
($Z_{s}=[1-2\frac{GM}{c^{2}R}]^{-\frac{1}{2}}-1$) for the different uniform
magnetic fields and $SetA$ and $SetB$ for the density-dependent magnetic field
in the two latest columns. It can be seen that the Buchdahl condition
($2M/R\leq 8/9$) is establishedrk53  for all considered cases. Furthermore,
the value of $Z_{s}$ is in a range of $0.42-0.71$ that are within acceptable
limits for compact objects rk54 ; rk55 ; rk56 ; rk57 . Also, the maximum value
of $Z_{s}$ in our calculations (for the case of density-dependent bag constant
using Set B) is about $16.47\%$ lower than the upper bound of that for the
subluminal EOS ($Z_{s}^{CL}=0.85$) rk057 . As expected, the values of the
surface redshift of SQS is larger than that for the results of neutron stars
rk58 . The surface redshift of a neutron star is reported in a range of $0.49$
\- $0.56$ in rk59 . Also, $Z_{s}=0.51$ is calculated for a magnetic neutron
star in rk60 . The results in tables 1 and 2 show that the compactification
factor increases by increasing the magnetic field. By investigation of the
corresponding figures related to EOS for the uniform magnetic fields (Figs. 1
and 2), we have found that the stiffer EOS leads to the star with lower
gravitational mass. This behavior is different for the neutron stars where the
stiffer EOS leads to the higher gravitational mass rk60a . Interaction
behavior of quarks in the strange quark matter differs from nucleon-nucleon
interaction behavior in nuclear matter within the neutron star. In addition,
it is found that the stiffer EOS leads to the more compact SQS and as a result
the larger value for the surface redshift. Furthermore, the structure
parameters results for Set A and Set B in tables 1 and 2 show that the slower
increase of the magnetic field from the surface to the center of SQS leads to
the larger value of the compactification factor and the surface redshift. It
should be noted that in the same range of the number density, the central
magnetic field reach $\simeq 4.9\times 10^{18}G$ using Set A and $\simeq
3.01\times 10^{18}\ G$ using Set B, so using Set B the EOS is softer as it can
be seen from Figs. 3 and 4. In addition, by comparing the results of tables 1
and 2 the values of compactification factor in the case of $B_{bag}^{dep.}$ is
larger than that for $B_{bag}^{fixed}$ with the same condition of the magnetic
field. In other words, considering the density-dependent bag constant leads to
the more compact SQS. It can be seen from tables 1 and 2 that the maximum
value of compactification factor and the corresponding surface redshift are
$0.89$ and $0.71$ regarding Set B of $B_{bag}^{dep.}$.

Table 1: The structure parameters of SQS in presence of different uniform magnetic fields and two different cases for the density-dependent of the magnetic field (Set A and Set B) by considering $B_{bag}^{fixed}=90\ MeV/fm^{3}$ . $Magneticfield$ |  | $\frac{M}{M_{sun}}$ |  | $R(km)$ |  | $\frac{2M}{R}$ |  |  | $Z_{s}$ |  |
---|---|---|---|---|---|---|---|---|---|---|---
$0$ |  | 1.22 |  | 7.10 |  | $0.68$ |  |  | $0.42$ |  |
$5\times 10^{17}\ G$ |  | 1.11 |  | 6.06 |  | $0.72$ |  |  | $0.47$ |  |
$5\times 10^{18}\ G$ |  | 0.97 |  | 5.32 |  | $0.73$ |  |  | $0.47$ |  |
Set A |  | 1.21 |  | 6.40 |  | $0.75$ |  |  | $0.50$ |  |
Set B |  | 1.37 |  | 6.99 |  | $0.78$ |  |  | $0.54$ |  |
Table 2: The structure parameters of SQS in presence of different uniform magnetic fields and two different cases for the density-dependent of the magnetic field (Set A and Set B) by considering $B_{bag}^{dep.}$. $Magneticfield$ |  | $\frac{M}{M_{sun}}$ |  | $R\ (km)$ |  | $\frac{2M}{R}$ |  |  | $Z_{s}$ |  |
---|---|---|---|---|---|---|---|---|---|---|---
$0$ |  | 1.80 |  | 8.65 |  | $0.82$ |  |  | $0.61$ |  |
$5\times 10^{17}\ G$ |  | 1.65 |  | 7.43 |  | $0.88$ |  |  | $0.70$ |  |
$5\times 10^{18}\ G$ |  | 1.42 |  | 6.37 |  | $0.88$ |  |  | $0.71$ |  |
Set A |  | 1.41 |  | 6.33 |  | $0.88$ |  |  | $0.70$ |  |
Set B |  | 1.44 |  | 6.45 |  | $0.89$ |  |  | $0.71$ |  |

The structure parameters of some SQS candidates are shown in table 3. As it
can be seen the value of the maximum gravitational mass of candidates is in
the range of $1.10$ \- $1.74$ $M_{sun}$, the compactification factor is
between $0.57$ \- $0.83$ and the surface redshift changes from $0.31$ to
$0.62$. By comparing the values of surface redshift and the compactification
factors in three tables we have found that the structure parameters of our
model in all cases are in the range of those for the SQS candidates. As our
calculation results in table 2 are shown, the values of $2M/R$ and $Z_{s}$ in
the case of $B_{bag}^{dep.}$ (especially using Set A and Set B) are nearer to
that values for $SAXJ1808.4-3659$. Actually, the comparison shows that the SQS
model in the current paper is compatible with $SAXJ1808.4-3659$. In addition,
the values of $2M/R$ and $Z_{s}$ for $4U1608-52$ and $4U1820-30$ is comparable
with the results of table 1. Also, the compactification factor of
$RXJ185635-3754$ and $HerX-1$ are $0.60$ and $0.57$, respectively, and are
comparable with the results of table 1 that are in the range of $0.68$-$0.78$.

Table 3: Structure parameters of strange quark stars candidates. Observed stars |  | $\frac{M}{M_{sun}}$ |  | $R(km)$ |  | $\frac{2M}{R}$ |  |  | $Z_{s}$ |  |
---|---|---|---|---|---|---|---|---|---|---|---
$RXJ185635-3754$ rk1  |  | 1.20 |  | 8.00 |  | $0.60$ |  |  | $0.34$ |  |
$HerX-1$ rk62  |  | 1.10 |  | 7.70 |  | $0.57$ |  |  | $0.31$ |  |
$4U1608-52$ rk63  |  | 1.74 |  | 9.3 |  | $0.74$ |  |  | $0.49$ |  |
$4U1820-30$ rk64  |  | 1.58 |  | 9.1 |  | $0.69$ |  |  | $0.43$ |  |
$SAXJ1808.4-3659(SS1)$ rk65  |  | 1.44 |  | 7.07 |  | $0.80$ |  |  | $0.57$ |  |
$SAXJ1808.4-3659(SS2)$ rk65  |  | 1.32 |  | 6.35 |  | $0.83$ |  |  | $0.62$ |  |

##  IV Conclusion

In the current paper, we have investigated the thermodynamic properties of
strange quark matter in the core of strange quark stars. We have calculated
the equation of state of strange quark matter using the MIT bag model in the
presence of the strong magnetic field. We have considered uniform magnetic
fields, the density-dependent magnetic field and two cases of the bag constant
(fixed bag constant and the density-dependent bag constant). It is shown that
the equation of state of the strange quark matter becomes stiffer by
increasing the magnetic field in all considered cases. Also, we have
investigated the energy conditions based on the limitation in the energy-
momentum tensor $T^{\mu\nu}$. We have found that the energy conditions are
satisfied regarding the equation of state for all considered cases in this
paper. Next, we have calculated the structure parameters of the strange quark
star using the Tolman-Oppenheimer-Volkov equations. We have plotted the
gravitational mass versus the central energy density of the strange quark
star. It is shown that the maximum gravitational mass decreases by increasing
the magnetic field. Also, in the case of the density-dependent magnetic field,
the maximum gravitational mass has a larger value when the magnetic field
increases faster from the surface to the center of the star. We have plotted
the gravitational mass versus the radius of the star in the all considered
cases and it is shown that the maximum gravitational mass and the radius
decrease by increasing the magnetic field. In addition, we have calculated the
mass-radius relation of the strange quark star in the density-dependent
magnetic field case. We have found that $\alpha$ in the mass-radius relation
of the strange quark stars ($M\propto R^{\alpha}$) is in the range of the
expected values ($\alpha\sim 3$). Also, the $\alpha$ value is larger in the
case of the fast drop of the magnetic field from the center to the surface of
the star. Finally, the compactification factor and the surface redshift of the
star are calculated in all considered cases. As the results have shown, the
strange quark star becomes more compact by increasing the magnetic field.
Also, it becomes a more compact star when magnetic field has a slower rate of
changes interior of the star. Furthermore, by considering the density-
dependent bag constant in comparison with the fixed bag constant the star is
denser. It is found that the surface redshift values are in the ranges of that
of the strange stars candidates. furthermore, as it is expected the larger
compactification factor results in the larger redshift value. As the results
show the surface redshift and the compactification factor of the strange quark
star under the conditions in this paper is comparable with the reported values
of the strange quark stars candidates, considered in this paper.

## Acknowledgements

G. H. Bordbar wishes to thank Shiraz University Research Council.

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[figure]style=plain,subcapbesideposition=top [table]capposition=top

# Scalable telomere-to-telomere assembly for diploid and polyploid genomes
with double graph

Haoyu Cheng  Department of Data Sciences, Dana-Farber Cancer Institute,
Boston, MA, USA  Department of Biomedical Informatics, Harvard Medical School,
Boston, MA, USA  Mobin Asri  Genomics Institute, University of California,
Santa Cruz, CA, USA  Julian Lucas  Genomics Institute, University of
California, Santa Cruz, CA, USA  Sergey Koren  Genome Informatics Section,
Computational and Statistical Genomics Branch, National Human Genome Research
Institute, National Institutes of Health, Bethesda, MD, USA  Heng Li
Department of Data Sciences, Dana-Farber Cancer Institute, Boston, MA, USA
Department of Biomedical Informatics, Harvard Medical School, Boston, MA, USA
To whom correspondence should be addressed<EMAIL_ADDRESS>
###### Abstract

Despite recent advances in the length and the accuracy of long-read data,
building haplotype-resolved genome assemblies from telomere to telomere still
requires considerable computational resources. In this study, we present an
efficient _de novo_ assembly algorithm that combines multiple sequencing
technologies to scale up population-wide telomere-to-telomere assemblies. By
utilizing twenty-two human and two plant genomes, we demonstrate that our
algorithm is around an order of magnitude cheaper than existing methods, while
producing better diploid and haploid assemblies. Notably, our algorithm is the
only feasible solution to the haplotype-resolved assembly of polyploid
genomes.

The emergence of accurate PacBio High-Fidelity (HiFi) long reads has
revolutionized the assembly of large genomes, making high-quality haplotype-
resolved assembly a routine procedure [1, 2, 3]. However, HiFi reads are often
not long enough to resolve long exact repeats, resulting in fragmented
components around repeat-rich regions such as centromeres [4]. Recent advances
by Oxford Nanopore Technologies (ONT) have enabled the generation of ultra-
long reads, which are approximately 5–10 times longer than HiFi reads though
at relatively lower accuracy [5]. The Telomere-to-Telomere (T2T) consortium
has demonstrated that with careful manual curation, combining HiFi and ultra-
long reads could perfectly reconstruct the haploid CHM13 human genome [6].

Learning from the complete human genome assembly of CHM13, Verkko is a first
effort towards automated telomere-to-telomere assembly of diploid samples [7].
It can produce high-quality assembly when parental sequence data are
available. However, as we will show later, Verkko does not fully phase a
single diploid sample without parental data and thus results in incomplete
assembly. It may produce relatively fragmented assembly at lower read coverage
and is unable to produce haplotype-resolved assemblies of polyploid samples.
Verkko is also compute intensive, making it costly to deploy Verkko to a large
number of samples.

Figure 1: Hybrid assembly with PacBio HiFi and ONT ultra-long reads. (a)
Overall workflow. Hifiasm (UL) corrects HiFi reads, constructs a string graph
with HiFi reads alone and aligns ultra-long reads to the HiFi graph. Based on
the graph alignment, hifiasm (UL) encodes an ultra-long read as a sequence of
integers with each integer uniquely corresponding to a node (also known as a
unitig) in the HiFi graph. It then constructs a string graph of integer-
encoded ultra-long reads, and merges the HiFi graph and the ultra-long graph
to generate the final assembly. (b) HiFi assembly graph and ultra-long
alignment. Circles in orange and blue represent heterozygous nodes constructed
by HiFi reads from haplotype 1 and haplotype 2, respectively. Green circles
represent homozygous nodes within the HiFi string graph. The alignment paths
of ultra-long reads from haplotype 1 and haplotype 2 are represented by orange
and blue lines, respectively. (c) Ultra-long reads encoded as sequences of
integer unitig identifiers in the HiFi graph. Nucleotide sequences are ignored
at this step. (d) Ultra-long assembly graph and the resulting contigs in the
integer encoding. (e) Final assembly graph by incorporating the ultra-long
contigs into the HiFi graph. From the initial HiFi graph, hifiasm (UL) removes
unitigs that are present on the ultra-long contigs and adds the ultra-long
contigs back together with edges between remaining unitigs and unitigs on the
ultra-long contigs. Some unitigs (green circles in the example) may appear
multiple times in the final graph.

For the efficient near telomere-to-telomere assembly of diploid and polyploid
samples, we developed hifiasm (UL) that tightly integrates PacBio HiFi, ONT
ultra-long, Hi-C reads and trio data and produces high-quality assembly in one
go. Unlike Verkko that is based on the multiplex de Bruijn graph [8, 9],
hifiasm (UL) represents sequences with two string graphs [10] (Fig. 1a). The
first string graph is built from HiFi reads (Fig. 1b), the same as the
original hifiasm graph [1]. The second string graph is built from ultra-long
reads in reduced representation (Fig. 1b–d). Hifiasm (UL) then merges the two
graphs to produce the final assembly graph (Fig. 1e). The use of two assembly
graphs at different scales separates hifiasm (UL) from other assemblers.

Figure 2: Statistics of different assemblies. Hifiasm(UL)_trio and verkko_trio
assemblies were generated using HiFi and ultra-long reads, along with parental
short reads. Hifiasm(UL)_hic and verkko(UL)_gfase assemblies were constructed
using HiFi, ultra-long, and Hi-C reads obtained from the same sample.
Verkko(UL)_gfase applied the standalone Hi-C phasing algorithm, gfase [11], to
the Verkko assembly graph. (a) Assembly length of 11 human samples. (b) Contig
N50 representing the assembly contiguity of human samples. (c) Problematical
autosomal genes reported by the asmgene method [12]. The number of each
assembly is the sum of the asmgene results for haplotype 1 and 2. (d) Cloud
computing cost for assembling human data. Only three samples were assembled by
Verkko using cloud computing. (e) Assembly length of the haploid _Arabidopsis
thaliana_ sample and the autotetraploid potato sample. Hifiasm(HiFi)
represents hifiasm assemblies without the ultra-long integration. (f) Contig
N50 of Arabidopsis and potato assemblies by filtering out contigs shorter than
500kb. (g) BUSCO [13] scores of Arabidopsis and potato assemblies by filtering
out contigs shorter than 500kb.

To compare hifiasm (UL) with Verkko at a population scale, we evaluated both
approaches using 22 human samples selected from the Human Pangenome Reference
Consortium (HPRC) [14]. Eleven of these samples were chosen from the Year-1
dataset of the HPRC, while the remaining eleven samples were selected from the
Year-2 dataset (Supplementary Table 3). We carried out trio assembly for all
22 samples but only did Hi-C-based single-sample assembly for the 11 Year-1
samples. Verkko natively supports trio binning assembly. As it does not
support internal Hi-C phasing, we utilized the Hi-C phasing approach, gfase
[11], in combination with Verkko for the single-sample phased assembly. In
total, we collected a total of 132 assembled haplotypes for comprehensive
evaluation of hifiasm (UL) and Verkko.

For each sample, both hifiasm (UL) and Verkko yielded assemblies of similar
sizes (Fig. 2a) and exhibited comparable phasing accuracy (Supplementary Table
1). However, when assembling HPRC Year-1 samples at lower HiFi and ultra-long
coverage (Supplementary Table 3), hifiasm (UL) tended to produce more
contiguous assemblies (Fig. 2b). It generated contiguous contigs spanning from
telomere to telomere for multiple chromosomes, whereas Verkko did not produce
telomere-to-telomere contigs for Year-1 samples (Supplementary Fig. 1a). The
consistent improvement to assembly contiguity highlights the advantages of our
approach. Although Verkko could produce scaffolds that bridge entire
chromosomes (Supplementary Table 4), the assembly gaps in the scaffolds will
complicate downstream analysis. In addition, Verkko could not assemble
chromosome-long scaffolds for all chromosomes. We anyway need a Hi-C-based
scaffolder for reliable scaffolding.

For HPRC Year-2 datasets at higher coverage (Supplementary Table 1), Verkko
assemblies were broadly comparable to hifiasm (UL) assemblies in terms of
assembly contiguity (Fig. 2b), the number of telomere-to-telomere contigs
(Supplementary Fig. 1a), and phasing accuracy (Supplementary Table 1). A
noticeable difference between the two assemblers is that Verkko did not assign
all contigs to specific haplotypes given Hi-C data. We observed that the
majority of unassigned sequences come from unpaired sex chromosomes of male
samples, but there are also relatively larger numbers of unassigned sequences
from paired sex chromosomes and autosomes. Due to these unassigned contigs,
Verkko assemblies missed more autosomal genes in comparison to hifiasm (UL)
and were thus less complete (Fig. 2c). Meanwhile, for samples HG01099 and
HG03710, Verkko produced noticeably more duplicated genes. Close inspection of
these errors revealed that Verkko duplicated a few regions on one haplotype
but left these regions blank on the other haplotype. Hifiasm (UL) was less
affected by this issue. We assembled all Year-2 samples with hifiasm (UL) and
three samples with Verkko using cloud computing and recorded the cost. Hifiasm
(UL) is 8–15 times more cost-effective. The low computational cost of hifiasm
(UL) is particularly important for population-scale telomere-to-telomere
assembly projects.

We used all HiFi reads and ultra-long reads with a minimum length of 50kb from
the Arabidopsis thaliana (Col-0) dataset [15] to evaluate the assembly results
for non-human genomes (Fig. 2e–g). As an inbred plant strain, A. thaliana
Col-0 has five long chromosomes with a large number of ribosome DNAs (rDNAs)
on the short arms of chromosomes 2 and 4. Hifiasm (UL) produced exactly five
contigs that are 500 kb or longer. Three of them were telomere-to-telomere
contigs corresponding to chromosomes 1, 3 and 5 (Supplementary Fig. 1b). The
other two contigs represented the majority of chromosomes 2 and 4 except the
rDNA arrays on their short arms. Hifiasm (UL) assembled tens of Mb of small
contigs $<$500 kb (Fig. 2e). Almost all of them could be aligned to rDNA or
the chloroplast DNA. Also interestingly, the contig corresponding to
chromosome 2 integrated 294 kb of mitochondrial DNA towards the telomere end
of the short arm. This integration is also present in the assembly done by the
authors who produced the dataset [15] but is absent from the _A. thaliana_
reference genome or the assembly done by Naish et al [16]. For the _A.
thaliana_ dataset, Verkko only generated one telomere-to-telomere contig
corresponding to chromosome 5 (Supplementary Fig. 1b), partly due to
homozygous regions that are longer than ultra-long reads but do not span
entire chromosome arms. The Verkko assembly at present was less contiguous
(Fig. 2f) and less complete based on the BUSCO evaluation [13] (Fig. 2g). The
Verkko contig corresponding to chromosome 2 was fragmented on the short arm
and did not reveal the mitochondrion integration. Both Verkko and hifiasm (UL)
assemblies were more contiguous hifiasm HiFi-only assembly, indicating the
additional power of ultra-long reads.

To evaluate polyploid assembly, we further assembled an autotetraploid potato
genome [17]. As Verkko does not support polyploid phasing, only hifiasm (UL)
and hifiasm (HiFi) were applied with all HiFi reads and ultra-long reads with
a minimum length of 50kb. By leveraging the additional genetic map information
from progeny, both hifiasm (UL) and hifiasm (HiFi) could assemble four
haplotypes based on the polyploidy graph-binning approach (Methods). The
integration of ultra-long reads not only significantly increased assembly
contiguity (Fig. 2f and Supplementary Fig. 1b) but also improved the
completeness for all haplotypes (Fig. 2g). For the polyploid genome assembly,
the main limitation of our current algorithm is that it requires genetic map
information from progeny. In order to address this issue, we implemented an
experimental single-sample approach using Hi-C phasing, and applied it to the
autotetraploid potato dataset. This resulted in four haplotype assemblies,
which have slightly worse phasing accuracy and contiguity in comparison to the
genetic-map-based assemblies. However, the four Hi-C phased haplotype
assemblies are imbalanced, with one assembly being 20% larger than the others.
In the future, we plan to address this issue by proposing Hi-C phasing
approaches specifically designed for polyploid genomes.

The availability of ultra-long or accurate long reads has significantly
advanced the development of _de novo_ genome assemblies. Recently, the Human
Pangenome Reference Consortium (HPRC) has successfully applied our original
hifiasm algorithm to achieve high-quality haplotype-resolved assemblies in a
population-scale utilizing accurate HiFi reads, while the Telomere-to-Telomere
(T2T) consortium has demonstrated the feasibility of reconstructing a human
genome from telomere to telomere by co-assembling HiFi and ultra-long reads.
In this study, we present a new hybrid assembly algorithm, hifiasm (UL), which
provides an ultra-fast and robust solution for telomere-to-telomere genome
assemblies in a population-scale. We anticipate that hifiasm (UL) will be a
highly competitive _de novo_ assembler for numerous large-scale telomere-to-
telomere assembly projects in the coming years. In the long term, hifiasm (UL)
will facilitate a more comprehensive understanding of complex genomic regions
such as centromeres and highly repetitive segmental duplications.

## Methods

Extended Data Fig. 1: Accurate HiFi string graph combining PacBio HiFi and ONT
ultra-long reads. (a) Effect of contained reads in the string graph.
Rectangles in orange and blue represent heterozygous HiFi reads from haplotype
1 and haplotype 2, respectively. Green rectangles are HiFi reads originating
from homozygous regions, whereas red rectangles are contained reads. The
string graph is constructed using all reads, except for two contained reads.
(b) Hifiasm (UL) aligns ultra-long reads to the HiFi string graph with
contained reads to alleviate the contained read problem. The alignment paths
of ultra-long reads from haplotype 1 and haplotype 2 are represented by orange
and blue lines, respectively. Despite being a contained read, h12 is retained
as the critical read because it is covered by ultra-long reads u6 and u7. To
ensure accurate graph cleaning, hifiasm (UL) also tracks the number of ultra-
long reads that support each edge as its weight. For instance, the edge weight
between h5 and h8 is 2 because ultra-long reads u4 and u5 cover it.

Overview of hifiasm (UL). The main objective of hifiasm (UL) is to leverage
the benefits of HiFi and ultra-long reads, simplifying the assembly graph as
much as possible (Fig. 1). A complete and clean assembly graph will
substantially simplify the following steps like Hi-C phasing and phased contig
generation. Our previous phasing algorithms [1, 18] are then applied to the
graph to produce haplotype-resolved telomere-to-telomere assemblies. In
building the high-quality assembly graph, hifiasm (UL) generally follows the
traditional hybrid assembly paradigm, which uses the accurate HiFi graph as
the backbone and extends the graph by aligning ultra-long reads to it.
However, unlike existing methods, hifiasm (UL) performs an additional round of
ultra-long-to-HiFi alignments in advance. This provides extra information for
accurately constructing the HiFi graph and alleviates the contained read
problem specifically for the string graph [19]. We then create an integer
graph for the ultra-long reads and subsequently merge it with the initial HiFi
graph to produce the final assembly graph.

The advantages of hifiasm (UL) stem mainly from the novel double graph
framework for co-assembly (Fig. 1). Indeed, there are several existing hybrid
assemblers designed to combine shorter accurate reads as well as longer noisy
reads, but they all rely on the straightforward accurate-read-first assembly
strategy: that is, they first build the assembly graph with accurate reads,
and then further resolve the graph by aligning long noisy reads onto the
graph. Although this approach takes the information among accurate reads as
well as the information between accurate reads and noisy reads, it disregards
the critical information among noisy reads. To fully exploit all reads, the
double graph framework in hifiasm (UL) employs a two-stage approach. First, it
builds two string graphs individually, one for HiFi reads (Fig. 1b) and
another for ultra-long reads (Fig. 1d). Second, it merges the two graphs to
produce a final graph that combines both HiFi and ultra-long reads (Fig. 1e).
This approach ensures that the information contained in both types of reads is
fully leveraged, resulting in a more accurate and complete genome assembly.

Building an accurate string graph as the backbone. A string graph is an
assembly graph that preserves the information of complete reads, where each
node represents a read, and edges connecting the nodes correspond to overlaps
between reads. Hifiasm (UL) builds the initial backbone graph with HiFi reads,
as they are much more accurate than ultra-long reads. To further eliminate
sequencing errors, all HiFi reads are self-corrected with the haplotype-
resolved error correction algorithm described in the original hifiasm [1].
Once the graph is constructed, it is necessary to perform multiple rounds of
graph cleaning to simplify the graph by removing edges that are less likely to
be real.

Although the string graph has been widely utilized in many long-read
assemblers, the issue of the contained read remains unclear and could
potentially impact the completeness of the graph [19]. Given two reads X and
Y, if there is an overlap between X and Y that covers a part of X and the
whole Y, Y is a contained read that is totally contained in X. Extended Data
Fig. 1a gives an example. Read h11 and h12 are two contained reads covered by
read h3. Practical implementations of the string graph remove all contained
reads when building graphs, since the edges in the string graph correspond to
the prefix-to-suffix or suffix-to-prefix overlaps between reads [10]. However,
simply ignoring contained reads could introduce breakpoints in the string
graph, especially in highly repetitive regions and homologous regions between
two haplotypes. For instance, read h12 is a critical read for one haplotype
(reads in blue) but is an unnecessary contained read for another haplotype
(reads in orange), as shown in Extended Data Fig. 1a. Removing read h12 does
not affect the haplotype in orange but leads to a breakpoint for the haplotype
in blue, resulting in a fragmented assembly graph. Identifying critical
contained reads and retaining them in the string graph is the primary
challenge posed by the contained read problem. Several approaches have been
proposed to tackle it based on the simplified assumptions of read coverage or
length [19], which are not always reliable, especially in highly repetitive
regions.

Hifiasm (UL) alleviates the contained read problem within the HiFi string
graph by utilizing ultra-long-to-HiFi read alignments. A HiFi read is
considered a critical contained read only if it lacks sufficient informative
variants to distinguish it from reads originating from other repeat copies
(read h12 in Extended Data Fig. 1a). Given that ultra-long reads are
frequently ten times longer than HiFi reads (with a median length exceeding
100kb), it is less probable that an ultra-long read is a critical contained
read without any informative variant. As a result, when ultra-long reads are
aligned to HiFi reads, the HiFi contained reads that must be covered by ultra-
long read alignments are expected to be the critical reads. To this end,
hifiasm (UL) theoretically constructs a HiFi string graph that includes both
contained and uncontained reads (Extended Data Fig. 1b). It then employs the
graph alignment to align all ultra-long reads to this graph. As shown in
Extended Data Fig. 1b, the contained read h12 must be covered by the alignment
paths of ultra-long reads u6 and u7, while another contained read h11 could be
skipped by read h3. Consequently, hifiasm (UL) retains the critical read h12
for constructing a complete string graph of HiFi reads, while safely removing
read h11 to simplify the graph.

The ultra-long-to-HiFi read alignment could also be used to avoid the
incorrect graph cleaning. In an ideal scenario where all HiFi reads are longer
than any homozygous or repetitive regions, each node in the string graph
should have a maximum of one edge extending towards the left and right sides.
However, due to the limited length of HiFi reads, some nodes may have multiple
edges, making it difficult for assemblers to determine the real number of
edges to be retained. For instance, hifiasm and HiCanu [3] utilize a length-
based strategy that prioritizes the edge with the longest overlap length and
often removes other shorter edges. These heuristics graph cleaning solutions
may result in the overcutting of real edges or retaining unrelated edges. If
the initial backbone HiFi graph is either oversimplified or too complex, the
downstream steps of hifiasm (UL) may not accurately resolve difficult-to-
assemble regions. By utilizing the ultra-long-to-HiFi read alignment, hifiasm
(UL) is able to ascertain the number of ultra-long reads supported for each
edge, providing additional information to prevent incorrect graph cleaning
(Extended Data Fig. 1b).

Integer graph with ultra-long reads. To fully capture the length information
of ultra-long reads, hifiasm (UL) constructs another string graph using only
those reads. However, generating the string graph requires the computationally
intensive all-versus-all pairwise read comparison, which constitutes the
primary bottleneck in the long-read assembly workflow. Moreover, identifying
correct overlaps among ultra-long reads is particularly challenging due to
their significantly higher error rate compared to HiFi reads. Furthermore, the
high frequency of recurrent sequence errors in ONT ultra-long reads makes it
nearly impossible to accurately identify overlaps in difficult regions.

Hifiasm (UL) constructs a lightweight integer graph to entirely avoid the
expensive all-versus-all base-level read comparison and ensure the accuracy of
the ultra-long graph is comparable to that of the HiFi graph. In short, all
ultra-long reads are converted from the base pair space to a low-dimensional
integer space using the graph alignments of ultra-long reads. By working in
the integer space, the graph construction procedure is both efficient and
straightforward. The detailed steps of the integer graph construction are
listed as follows.

  1. 1.

_Mapping ultra-long reads into the integer space._ All ultra-long reads are
aligned to the HiFi graph to obtain the alignment paths (Fig. 1b). Given an
ultra-long read, hifiasm (UL) first collects its linear alignments to the
nodes of the HiFi graph using pairwise base-level alignments. Linear
alignments are then chained in the graph space using the approach described in
minigraph [20]. A graph alignment path is a sequence of the aligned node
identifiers. For each ultra-long read, hifiasm (UL) only keeps the node
identifiers and disregards all alignment details and base pairs. Node
identifiers can be represented as integers, meaning that ultra-long reads of
over tens of kilobases are transformed into ultra-long sequences consisting of
tens of integers (Fig. 1c).

  2. 2.

_Calculating overlaps among ultra-long integer sequences._ To construct a
string graph in the integer space, obtaining overlaps between integer
sequences is essential. As the base-level sequencing errors within ultra-long
reads have already been corrected through the graph alignment to the accurate
HiFi graph, hifiasm (UL) only allows exact overlaps in the integer space.
Notably, this step is considerably faster than the conventional all-versus-all
inexact pairwise alignment.

  3. 3.

_Constructing an integer graph._ An integer graph is a type of string graph
where each node is an integer sequence. Hifiasm (UL) constructs an integer
graph by utilizing ultra-long integer sequences and their overlaps (Fig. 1d).
Specifically, each node in this graph represents an ultra-long integer
sequence, and the edges connecting the nodes correspond to exact overlaps
between these sequences. However, even after this initial construction,
multiple rounds of standard graph cleaning are still necessary to further
simplify the integer graph. As ultra-long reads are typically long enough to
assemble through repetitive or homozygous regions, hifiasm (UL) employs highly
aggressive graph cleaning strategies to eliminate ambiguous edges associated
with each node.

  4. 4.

_Producing integer contigs._ A contig corresponds to a non-branching path in
the string graph. Given a contig in the integer graph, hifiasm (UL) produces
its sequence by concatenating the subsequences of nodes within the
corresponding path (Fig. 1d). After the contig generation process, each
resulting contig is an integer sequence that is significantly longer than any
individual ultra-long read. In fact, these integer contigs represent the paths
that can untangle intricate structures within the initial HiFi graph.

Building final assembly graph by graph incorporation. The integer graph
produces ultra-long integer contigs that correspond to assembly paths within
the initial HiFi graph. These integer contigs represent another HiFi string
graph that resolves the majority of tangles and homozygous regions within the
initial HiFi graph into linear sequences. By incorporating ultra-long integer
contigs into the initial HiFi graph, hifiasm (UL) can produce the final
assembly. Specifically, hifiasm (UL) first removes all nodes within the
initial HiFi graph that also appear in ultra-long integer contigs, and then
merges the remaining nodes and overlaps with ultra-long integer contigs. Fig.
1e provides an example. In the final assembly graph, all nodes except h7 come
from ultra-long integer contigs. This is because all nodes except node h7 are
present in both the initial HiFi graph (Fig. 1b) and ultra-long integer
contigs (Fig. 1d).

Constructing haplotype-resolved assemblies. The high-quality assembly graphs
combining HiFi and ultra-long reads significantly simplifies the generation of
haplotype-resolved assemblies. With the addition of Hi-C or parental short
reads, hifiasm (UL) can reuse previous Hi-C [18] or trio-binning [1]
algorithms to assign haplotype-specific markers to the nodes of the assembly
graph. The final haplotype-resolved assemblies are then produced using the
graph-binning strategy [1]. For polyploid genomes, we implemented a polyploidy
graph-binning approach that extends our previous diploid graph-binning method.
In the polyploidy graph-binning approach, when emitting the assembly of one
haplotype, all nodes with other haplotype-specific markers are discarded from
the assembly graph. This is the main difference between the polyploidy graph-
binning and the diploid graph-binning approaches.

Optimizing for cloud computing. To evaluate the computational cost of both
hifiasm (UL) and Verkko, we assembled all human samples with hifiasm (UL) and
three human samples with Verkko using the Terra platform on top of Google
Cloud Platform. We further reduced the computational costs by executing
assemblers with preemptible instances. A preemptible instance takes much lower
cost but its running times often cannot exceed 24 hours. As a result, both
hifiasm (UL) and Verkko were divided into multiple short tasks, which were
executed individually using preemptible instances (Supplementary Section 1.4).

## Acknowledgements

This study was supported by US National Institutes of Health (grant
R01HG010040, U01HG010971 and U41HG010972 to H.L., grant 1K99HG012798 to H.C.).
We thank the Human Pangenome Reference Consortium for making Year-1 and Year-2
datasets publicly available.

## Author contributions

H.C. and H.L. designed the algorithm, implemented hifiasm (UL) and drafted the
manuscript. H.C. benchmarked hifiasm (UL) and other assemblers. M.A., J.L. and
S.K. designed the evaluation of human genome assemblies.

## Competing interests

The authors declare no competing interests.

## Data availability

Human reference genome: GRCh38; HiFi reads of HPRC Year-2 samples:
https://s3-us-west-2.amazonaws.com/human-
pangenomics/index.html?prefix=submissions/1E2DD570-3B26-418B-B50F-5417F64C5679
--HIFI_DEEPCONSENSUS/; ONT ultra-long reads of HPRC Year-2 samples:
https://s3-us-west-2.amazonaws.com/human-
pangenomics/index.html?prefix=submissions/90A1F283-2752-438B-917F-53AE76C9C43E
--UCSC_HPRC_nanopore_Year2/; Hi-C reads of HPRC Year-2 samples: https://s3-us-
west-2.amazonaws.com/human-
pangenomics/index.html?prefix=submissions/4C696EB9-9AD2-47A2-8011-2F43977CC4E0
--Y2-HIC/; Parental short reads of HPRC Year-2 samples: https://s3-us-
west-2.amazonaws.com/human-
pangenomics/index.html?prefix=submissions/AD30A684-C7A8-4D24-89B2-040DFF021B0C
--Y2_1000G_DATA/; All reads of HPRC Year-1 samples: https://github.com/human-
pangenomics/HPP_Year1_Data_Freeze_v1.0; All reads of Arabidopsis:
https://ngdc.cncb.ac.cn/search/?dbId=gsa&q=CRA004538; All reads of potato:
https://ngdc.cncb.ac.cn/gsa/browse/CRA006012; Hifiasm (UL) assemblies of HPRC
Year-2 samples: “*hifiasm_v0.19.5*” from https://s3-us-
west-2.amazonaws.com/human-
pangenomics/index.html?prefix=submissions/53FEE631-4264-4627-8FB6-09D7364F4D3B
--ASM-COMP/; Verkko assemblies of HPRC Year-2 samples: “*verkko_1.3.1*” from
https://s3-us-west-2.amazonaws.com/human-
pangenomics/index.html?prefix=submissions/53FEE631-4264-4627-8FB6-09D7364F4D3B
--ASM-COMP/; All evaluated HPRC Year-1 and plant assemblies are available at
https://zenodo.org/record/7996422 and https://zenodo.org/record/7962930,
respectively.

## Code availability

Hifiasm (UL) is available at https://github.com/chhylp123/hifiasm.

## Reporting Summary

Further information on research design is available in the Nature Research
Reporting Summary linked to this article

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