[a]Yu Zhang

# Exploring the QCD phase diagram with three flavors of Möbius domain wall
fermions

Yasumichi Aoki  Shoji Hashimoto  Issaku Kanamori  Takashi Kaneko  Yoshifumi
Nakamura

###### Abstract

We present an update on the study of the QCD phase transition with 3 flavors
of Möbius domain wall fermions at zero baryon density. We performed
simulations on lattices of size $36^{3}\times 12\times 16$ and $24^{3}\times
12\times 32$ with a variety of quark masses at a fixed lattice spacing
$a=0.1361(20)$ fm, which correspond to a temperature 121(2) MeV. By analyzing
the chiral condensate, chiral susceptibilitities and Binder cumulant on
$36^{3}\times 12\times 16$ lattices together with the result obtained from our
previous study on $24^{3}\times 12\times 16$ lattices [1], we identified a
crossover occurring at quark mass around
$m_{q}^{\mathrm{\overline{MS}}}(2\,\mathrm{GeV})\sim 3-4$ MeV for this
temperature. Besides, we show the effects of residual chiral symmetry breaking
on chiral condensate and chiral susceptibilities between $L_{s}=16$ and 32.

KEK-CP-0396

##  1 Introduction

The QCD chiral phase transition for three degenerate quark flavors, on the
lower left corner of the Columbia plot, is yet to be determined. Based on the
renormalization group flow of 3D linear sigma model, Pisarski and Wilczek
predicate a first-order transition in the $N_{f}=3$ chiral limit [2]. If this
is the case, the first-order phase transition weakens moving away from the
chiral limit and finally terminates at a critical point of a second-order
transition belonging to the 3D $Z(2)$ Ising universality class. On the
contrary, a recent study of the RG flow of all couplings up to $\phi^{6}$ in
3D Ginzburg-Landau theory claims a possible second order phase transition in
the $N_{f}=3$ chiral limit [3]. Whether the first-order region exists in the
$N_{f}=3$ light quark regime at all is still an open question. If the first-
order region exists, then what is the value of the critical mass? Answering
those questions requires nonperturbative lattice QCD simulations.

It is quite difficult to pin down the location of critical mass or to
determine the order of phase transition in the $N_{f}=3$ chiral regime using
lattice QCD simulations, as the computational cost increases significantly
when simulating smaller quark masses. Previous studies using Wilson and
staggered, or their improved fermion actions, showed that the bound of
critical pion mass decreases towards finer lattices, and the critical pion
mass obtained from Wilson-type fermions differs significantly from those
obtained from staggered ones [4, 5, 6, 7, 8, 9]. This indicates strong cutoff
and discretization scheme dependence. A recent study with unimproved staggered
fermions suggests a second-order chiral phase transition for $N_{f}=3$
massless quark flavors [10]. Later, a study using the Highly Improved
Staggered Quark action also did not find evidence of a phase transiton in pion
mass range between 80 and 140 MeV [11]. This leaves little room for a first
order region to exist, but it cannot be ruled out. The current existing
results are either from staggered or Wilson type fermions which break chiral
symmetry partially or entirely. Therefore, it is important to study using a
chiral fermion formulation, which preserves chiral symmetry on the lattice, to
further investigate this problem. We use the Möbius domain wall fermion to
study the $N_{f}=3$ chiral region in this work.

##  2 Lattice Setup

Using the tree-level improved Symanzik gauge action and Möbius domain wall
fermion action [12], we conducted $N_{f}=3$ QCD simulations using the Grid
code set optimized for the Fugaku CPU A64FX [13]. We fixed the gauge coupling
at $\beta=4.0$, corresponding to a lattice spacing of $a=0.1361(20)$ fm. This
value was determined from the Wilson flow $t_{0}$, employing the result of
continuum- extrapolated $t_{0}$ at physical quark mass for $N_{f}=2+1$ QCD as
input [14]. In our previous study [1], we conducted simulations on
$24^{3}\times 12\times 16$ lattices with 26 bare quark masses ranging from
$-$0.006 to 0.1. For this work, we performed a large volume simulation on
$36^{3}\times 12\times 16$ lattices with 7 bare quark masses within the range
of $[-0.005,0.001]$, aiming to study the volume effect. Additionally, for
studying the resdiual chiral symmetry breaking effect, we performed
simulations on $24^{3}\times 12\times 32$ lattices with 5 bare quark masses
ranging from $-$0.001 to 0.003. Each parameter set involved approximately
20,000 trajectories, and measurements were taken at every 10th trajectory.

##  3 Numerical Results

###  3.1 Chiral observables

The chiral condensate is a measure of chiral symmetry breaking in QCD. It
vanishes in the chirally symmetric phase at temperature $T$ higher than the
critical termperature $T_{c}$, retaining a nonzero value at $T<T_{c}$, which
is the chiral symmetry broken phase. The chiral condensate is defined as a
derivative of the partition function $Z$ with respect to the quark mass,

$\displaystyle\langle\bar{\psi}\psi\rangle=\frac{T}{V}\frac{\partial\ln
Z}{\partial
m_{q}}=\frac{N_{f}}{N_{s}^{3}N_{t}}\left\langle\mathrm{Tr}M^{-1}\right\rangle\,,$
(1)

Where $M$ denotes the Dirac matrix, and $N_{f}$ represents the number of
degenerate quark masses. At finite quark mass, the chiral condensate diverges
both additively and multiplicatively in the continuum limit. The former arises
from the power-law divergence, which is proportional to $m_{q}a^{-2}$. For
domain wall fermions with finite $L_{s}$, the leading effect of small but
finite violation of chiral symmetry causes an additive renormalziation to the
bare input quark mass, known as the residual mass $m_{\rm{res}}$. There is an
additional power divergence induced by this residual chiral symmetry breaking
effect, which is proportional to $xm_{\mathrm{res}}a^{-2}$ with an unknown
coefficient $x$ [15]. Therefore, the chiral condensate for domain wall
fermions is expected to behave as

$\displaystyle\langle\bar{\psi}\psi\rangle|_{\mathrm{DWF}}\sim
C\frac{m_{q}+xm_{\mathrm{res}}}{a^{2}}+\langle\bar{\psi}\psi\rangle|_{\mathrm{cont}}+...\,.$
(2)

In the chiral limit $m_{q}+m_{\mathrm{res}}=0$, there still remains some UV
divergent pieces, represented as $C\frac{(x-1)m_{\mathrm{res}}}{a^{2}}$. One
way of eliminating the additive divergence is to use the finite temperature
chiral condensate $\langle\bar{\psi}\psi\rangle^{T>0}$ minus the zero
tempearture chiral condensate $\langle\bar{\psi}\psi\rangle^{T=0}$ at the same
quark mass, as this divergence term ($C\frac{m_{q}+xm_{\mathrm{res}}}{a^{2}}$
) remains temperature independent. To address the remaining multiplicative
logarithmic divergence, we use the bare chiral condensate divided by the quark
mass renormalization constant $Z_{m}$ in the $\overline{\rm{MS}}$ scheme at a
scale of $\mu=2$ GeV. $Z_{m}^{\overline{\rm{MS}}}(2\,\rm{GeV})$ is determined
using an inter/extra-polation as a function of $\beta$ with those determined
in a non-perturbative renormalization [16]. See a description in Ref.[17] for
detail. Then the renormalized chiral condensate can be defined as

$\displaystyle[\langle\bar{\psi}\psi\rangle^{\mathrm{T}>0}-\langle\bar{\psi}\psi\rangle^{\mathrm{T}=0}]^{\overline{\mathrm{MS}}}(2\,\mathrm{GeV})=\frac{\langle\bar{\psi}\psi\rangle^{\mathrm{T}>0}-\langle\bar{\psi}\psi\rangle^{\mathrm{T}=0}}{Z_{m}^{\overline{\mathrm{MS}}}(2\,\mathrm{GeV})}\,.$
(3)

For the determination of the transition point, we use the disconnected chiral
susceptibility $\chi_{\rm{disc}}$, which describes the fluctuation of the
chiral condensate. For $N_{f}$ quark flavors, it is given by

$\displaystyle\chi_{{\rm
disc}}=\frac{N_{f}^{2}}{N_{s}^{3}N_{t}}\Bigl{(}\left\langle({\rm
Tr}M^{-1})^{2}\right\rangle-\left\langle{\rm
Tr}M^{-1}\right\rangle^{2}\Bigr{)}\,.$ (4)

$\chi_{\rm{disc}}$ is a part of the total chiral susceptibility
$\chi_{\rm{\sigma}}$, which is defined as the derivative of
$\langle\bar{\psi}\psi\rangle$ with respect to the quark mass. The other part
is the connected chiral susceptibility,

$\displaystyle\chi_{\rm{\sigma}}=\frac{\partial\langle\bar{\psi}\psi\rangle}{\partial
m_{q}}=\chi_{{\rm disc}}+\chi_{\rm{con}}\,,$ (5) $\displaystyle\chi_{{\rm
con}}=-\frac{N_{f}}{N_{s}^{3}N_{t}}\left\langle{\rm Tr}M^{-2}\right\rangle\,.$
(6)

We will show the results of both disconnected chiral susceptibility and total
chiral susceptibility with only multiplicative renormalization applied.
Although the total chiral susceptibility suffers from an additive divergence,
it is constant as a function of quark mass thus has no effect for the
determination of the transition point.

Figure 1: Left: The multiplicatively renormalized chiral condensate on finite
temperature ensembles $N_{s}\times 12\times 16$ with $N_{s}=24,36$ and zero
temperature ensembles $24^{3}\times 48\times 16$ as a function of quark mass
for $\beta=4.0$. The dashed line represents the quadratic extrapolation for
zero temperature chiral condensate. Right: The renormalzied subtracted chiral
condensate as a function of quark mass for $N_{s}\times 12\times 16$ lattices
with $N_{s}=24$ and $36$.

On the left panel of Fig. 1, we show the multiplicatively renormalized chiral
condensate as a function of the renormalized quark mass for finite temperature
$N_{t}=12$ lattices alongside zero temperature lattices. The negative finite
temperature result near the chiral limit arises from the remaining UV
divergent piece $C\frac{(x-1)m_{\mathrm{res}}}{a^{2}}$. To eliminate the UV
divergence term $C\frac{m_{q}+xm_{\mathrm{res}}}{a^{2}}$, we utilize
$\left\langle\bar{\psi}\psi\rangle^{\mathrm{T}>0}-\langle\bar{\psi}\psi\rangle^{\mathrm{T}=0}\right|_{m_{q}+m_{\mathrm{res}}}$.
The zero temperature results are derived from the green dashed line
representing the quadratic extrapolation based on existing zero-temperature
data. Then the renormalized subtracted chiral condensate is depicted on the
right panel of Fig. 1. We observe a small finite volume effect, a difference
between $24^{3}$ and $36^{3}$ lattices, at
$(m_{q}+m_{\mathrm{res}})^{\overline{\mathrm{MS}}}(2\,\mathrm{GeV})\sim 4-9$
MeV and a rapid transition at
$(m_{q}+m_{\mathrm{res}})^{\overline{\mathrm{MS}}}(2\,\mathrm{GeV})\sim 1-5$
MeV. To precisely locate the inflection point, we focus on the chiral
susceptibilities, which are more sensitive observables for studying the phase
transition as they measure the response of the chiral condensate to a small
change in the quark mass.

Figure 2: The renormalized disconnected chiral susceptibility (left) and the
multiplicatively renormalized total chiral susceptibility (right) as a
function of quark mass for $N_{t}=12$ lattices at $\beta=4.0$.

On the left panel of Fig. 2, we present the disconnected chiral susceptibility
as a function of quark mass for $N_{t}=12$ lattices with two different
volumes. The observed transition mass point is around 4 MeV. We notice a
sizeable finite volume effect in the vicinity of the transition range,
approximately
$(m_{q}+m_{\mathrm{res}})^{\overline{\mathrm{MS}}}(2\,\mathrm{GeV})\sim 2-5$
MeV. On the right panel of Fig. 2, we observe a relatively small but
significant finite volume effect from the total chiral susceptibility within
the same quark mass range.

Based on finite-size scaling analysis, if it is a true phase transition, then
at the critical point, the maximun of the total chiral susceptibility would be
proportional to $N_{s}^{3}$ for a first-order phase transition and
$N_{s}^{1.966}$ for a Z(2) second-order phase transition. However, the total
chiral susceptibility contains a constant term due to UV power divergence,
which is independent of volume and will distort the volume scaling. Therefore,
we calculated the subtracted total chiral susceptibility
$\chi_{\sigma_{\mathrm{sub}}}^{\mathrm{\overline{MS}}}$ with the UV divergence
eliminated explicitly. The subtracted total chiral susceptibility times
inverse of the scale factor for these two possibilities is depicted in Fig. 3.
Here, the rescaled subtracted total chiral susceptibility at the peak location
significantly differs from what is expected in the case of a first-order
transition or a Z(2) second order phase trasnsition. This suggests that it’s
most likely a crossover transition. The transition mass point estimated from
the disconnected chiral susceptibility and total chiral susceptibility is
around 4 MeV and 3 MeV, respectively. This discrepancy is understandable for a
corrossover, where the transiton occurs over a broad range. We expect the
transition mass point obtained from these two susceptibilities will concide at
the chiral phase transition termperature.

Figure 3: Rescaled subtracted total chiral susceptibility with the scaling
factor corresponding to a first order phase transition (left) and Z(2) second
order phase transition (right) for $N_{t}=12$ lattices.

###  3.2 Binder cumulant

To investigate the nature of phase transition, we utilize the Binder cumulant
of the chiral condensate $B_{4}(\bar{\psi}\psi)$, defined by [18]

$\displaystyle
B_{4}(\bar{\psi}\psi)=\frac{\left\langle(\delta\bar{\psi}\psi)^{4}\right\rangle}{\left\langle(\delta\bar{\psi}\psi)^{2}\right\rangle^{2}},\quad\delta\bar{\psi}\psi=\bar{\psi}\psi-\langle\bar{\psi}\psi\rangle\,.$
(7)

For the chiral critical point in $N_{f}=3$ QCD, $B_{4}(\bar{\psi}\psi)$ will
approach a universal value of 1.604 [19] on finite volumes. However, for
transitions rather than critical points, $B_{4}(\bar{\psi}\psi)$ takes a
characteristic value in the infinite volume limit. Specifically, for a
crossover, $B_{4}(\bar{\psi}\psi)=3$ in the infinite volume limit. For a
first-order transition, the Binder cumulant at infinite volume is
$B_{4}(\bar{\psi}\psi)=1$.

Figure 4: Left: The histogram of chiral condensate in the vincity of the
transition mass point for $36^{3}\times 12\times 16$ lattice. Right: The
binder cumulant as a function of quark mass for $N_{t}=12$ ensembles.

Before presenting the result of Binder cumulant, we display the distribution
of the chiral condensate near the transiton point for $36^{3}\times 12\times
16$ lattice, in the left panel of Fig. 4. We observe it behaves like a
Gaussian distribution rather than a double peak structure, as expected for a
first order phase transition. This provides a further evidence for a crossover
transition. The right panel of Fig. 4 illustrates $B_{4}(\bar{\psi}\psi)$ as a
function of quark mass for $N_{t}=12$ lattice ensembles. The results for
$36^{3}\times 12$ lattice ensembles are consistent with 3 near the transition
masses of 3 and 4 MeV, as determined from the peak of total and disconnected
chiral susceptibility. For $24^{3}\times 12$ lattice ensembles, the result is
consistent with 3 near transition mass point 4 MeV. However, it is quite large
and not consistent with 3 in one standard deviation near transition point 3
MeV but it is much far from 1.604 and 1. These indicate a crossover
transition.

###  3.3 Residual chiral symmetry breaking effect for different $L_{s}$

The Möbius domain wall fermion with finite $L_{s}$ introduces a small but non-
negligible amount of chiral symmetry breaking. To the leading order in an
expansion in lattice spacing, this residual chiral symmetry breaking can be
characterized by the residual mass $m_{\rm{res}}$, which acts as an additive
shift to the bare input quark mass, resulting in the total quark mass
$m=m_{q}+m_{\rm{res}}$. One way of simulating the small value of target quark
mass is to keep the value of $m_{\rm{res}}$ small, which can be achieved by
increasing $L_{s}$.

Figure 5: Left: The residual mass as a function of bare input quark mass for
zero temperature and finite temperature ensembles with different $L_{s}$ at
$\beta=4.0$. The dashed line denotes the linear fit to the zero temperature
result. Right: chiral condensate as a function of quark mass for $24^{3}\times
12\times L_{s}$ lattices with $L_{s}=16$ and $32$.

The left panel of Fig. 5 displays the residual mass as a function of bare
input quark mass for zero temperature ensembles with $24^{3}\times 48\times
16$ lattice and finite temperature ensembles with $L_{s}=16$ and $32$ at
$\beta=4.0$. The green dashed line represents the linear fit to the zero
temperature data. The common method for defining a mass-independent
$m_{\rm{res}}$ involves an extrapolation to the zero input quark mass limit,
which is determined to be 0.00613(9) for $L_{s}=16$ and 0.00325(3) for
$L_{s}=32$. A $1/L_{s}$ dependence dominates the contribution to
$m_{\rm{res}}$. We find that the $m_{\rm{res}}$ calculated from both zero
temperature and finite temperature ensembles are consistent and nearly
independent of the volume, see also [20].

The right panel of Fig. 5 shows the chiral condensate as a function of quark
mass for $24^{3}\times 12\times L_{s}$ lattices with $L_{s}=16$ and $32$. A
negative result near the chiral limit is observed for $24^{3}\times 12\times
16$ lattices, attributed to the remaining UV divergence term
$C\frac{(x-1)m_{\mathrm{res}}}{a^{2}}$, as previously mentioned, indicating
$C(x-1)<0$. An increase of the chiral condensate is observed when increasing
$L_{s}$ from 16 to 32 while keeping the total quark mass the same. This can be
easily understood from $\langle\bar{\psi}\psi\rangle|_{\mathrm{DWF}}\sim
C\big{(}\frac{(x-1)m_{\mathrm{res}}}{a^{2}}+\frac{m_{q}+m_{\mathrm{res}}}{a^{2}}\big{)}+\langle\bar{\psi}\psi\rangle|_{\mathrm{cont}}+...\,.$,
given that $m_{\rm{res}}$ for $L_{s}=16$ is larger than for $L_{s}=32$, and
$C(x-1)<0$. If we do a simple extrapolation, we will see the
$\langle\bar{\psi}\psi\rangle$ will have a smaller chiral symmetry breaking
effect for $L_{s}=32$ compared to $L_{s}=16$ in the chiral limit, as expected.

Figure 6: The disconnected chiral susceptibility (left) and total chiral
susceptibility (right) as a function of quark mass for $24^{3}\times 12\times
L_{s}$ lattices with $L_{s}=16$ and $32$.

Fig. 6 shows the result of disconnected chiral susceptibility (left) and total
chiral susceptibility (right) as functions of quark mass for two different
$L_{s}$. We expect that the $xm_{\mathrm{res}}$ effect will not show up in the
susceptibility as it is only constant as we change the quark mass. In fact,
with only the proper quark mass shift, the results from different $L_{s}$
appear consistent with each other. This is taken as evidence that we have a
good control of the explicit chiral symmetry breaking already at smaller fifth
size $L_{s}=16$, where the position of the susceptibility peak is in the
negative input quark mass range.

##  4 Summary and Outlook

We carried out the study of $N_{f}=3$ QCD phase transition using Möbius domain
wall fermions at a fixed lattice spacing of $a=0.1361(20)$ fm on $36^{3}\times
12\times 16$ lattices, corresponding to a temperature of 121(2) MeV. We
simulated various quark masses for this temperature. Through the analysis of
the chiral condensate, chiral susceptibilities and Binder cumulant, comparing
them with the results obtained on $24^{3}\times 12\times 16$ lattices in our
previous work [1]. We find the transition mass point is around
$m_{q}^{\mathrm{\overline{MS}}}(2\,\mathrm{GeV})\sim 4$ MeV and 3 MeV, as
determined from the disconnected and total chiral susceptibility,
respectively, at a temperature of 121(2) MeV and it is a crossover transition.
Should there be a first-order region in the lower left corner of the Columbia
plot, the critical quark mass would be smaller than 4 MeV. This is consistent
with the findings using Wilson and staggered type fermions.

To reduce the residual chiral symmetry breaking effects, we conducted
simulations on $24^{3}\times 12\times 32$ lattice. We found $1/L_{s}$
dependence dominates the contribution to the residual mass. By investigating
the residual chiral symmetry breaking effect on chiral condensate and chiral
susceptibilities between $L_{s}=16$ and 32 lattices, we found that in the
chiral limit the residual chiral symmetry breaking effect for chiral
condensate is smaller for $L_{s}=32$ than for 16, as expected. The chiral
susceptibilities results are consistent at the approximates same total quark
mass between these two different $L_{s}$, which indicates we have a good
control of the explicit chiral symmetry breaking already at smaller
$L_{s}=16$.

For the crossover transition, the peak height of chiral susceptibilities will
remain constant for larger volumes. However, we observed some volume
dependence in the chiral susceptibilities for $N_{s}^{3}\times 12\times 16$
lattices between $N_{s}=36$ and 24, though not as substantial as one would
expect from a true phase transition. Presently, we are conducting simulations
on larger lattice of $48^{3}\times 12\times 16$. Additionally, to directly
probe the first order region, if it exist, we are working toward even lower
temperatures through simulations on $N_{t}=14$ lattices, corresponding to a
temperature of 104(2) MeV.

## Acknowledgments

This work used the computational resources of Supercomputer Fugaku provided by
the RIKEN Center for Computational Science through HPCI project hp210032 and
Usability Research ra000001 as well as Wisteria/BDEC-01 Supercomputer System
at Tokyo University/JCAHPC through HPCI project hp220108 and Ito supercomputer
at Kyushu University through HPCI project hp190124 and hp200050 and also the
Hokusai BigWaterfall at RIKEN. This work is also supported in part by JSPS
KAKENHI grant No 20H01907 and 21H01085. We acknowledge the Grid Lattice QCD
framework111https://github.com/paboyle/Grid and its extension for A64FX
processors [13] which is used to generate the QCD configurations for this
study. We thank N. Meyer and T. Wettig for discussions on the use of Grid for
A64FX. For the measurement, we used Hadrons code [21].

## References

  * [1] Y. Zhang, Y. Aoki, S. Hashimoto, I. Kanamori, T. Kaneko and Y. Nakamura, _Finite temperature QCD phase transition with 3 flavors of Mobius domain wall fermions_ , _PoS_ LATTICE2022 (2023) 197 [2212.10021].
  * [2] R.D. Pisarski and F. Wilczek, _Remarks on the chiral phase transition in chromodynamics_ , _Phys. Rev. D_ 29 (1984) 338.
  * [3] G. Fejős, _Second-order chiral phase transition in three-flavor quantum chromodynamics?_ , _Phys. Rev. D_ 105 (2022) L071506.
  * [4] F. Karsch, E. Laermann and C. Schmidt, _The chiral critical point in 3-flavour qcd_ , _Physics Letters B_ 520 (2001) 41.
  * [5] P. de Forcrand and O. Philipsen, _The qcd phase diagram for three degenerate flavors and small baryon density_ , _Nuclear Physics B_ 673 (2003) 170.
  * [6] Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai and T. Yoshié, _Finite temperature transitions in lattice qcd with wilson quarks: Chiral transitions and the influence of the strange quark_ , _Phys. Rev. D_ 54 (1996) 7010.
  * [7] X.-Y. Jin, Y. Kuramashi, Y. Nakamura, S. Takeda and A. Ukawa, _Critical endpoint of the finite temperature phase transition for three flavor qcd_ , _Phys. Rev. D_ 91 (2015) 014508.
  * [8] X.-Y. Jin, Y. Kuramashi, Y. Nakamura, S. Takeda and A. Ukawa, _Critical point phase transition for finite temperature 3-flavor qcd with nonperturbatively $\mathrm{O}(a)$ improved wilson fermions at ${N}_{\mathrm{t}}=10$_, _Phys. Rev. D_ 96 (2017) 034523.
  * [9] A. Bazavov, H.-T. Ding, P. Hegde, F. Karsch, E. Laermann, S. Mukherjee et al., _Chiral phase structure of three flavor qcd at vanishing baryon number density_ , _Phys. Rev. D_ 95 (2017) 074505.
  * [10] L. Dini, P. Hegde, F. Karsch, A. Lahiri, C. Schmidt and S. Sharma, _Chiral phase transition in three-flavor qcd from lattice qcd_ , _Phys. Rev. D_ 105 (2022) 034510.
  * [11] F. Cuteri, O. Philipsen and A. Sciarra, _On the order of the QCD chiral phase transition for different numbers of quark flavours_ , _JHEP_ 11 (2021) 141 [2107.12739].
  * [12] Y. Nakamura, Y. Aoki, S. Hashimoto, I. Kanamori, T. Kaneko and Y. Zhang, _Finite temperature phase transition for three flavor QCD with Möbius-domain wall fermions_ , _PoS_ LATTICE2021 (2022) 080.
  * [13] N. Meyer, D. Pleiter, S. Solbrig and T. Wettig, _Lattice QCD on upcoming Arm architectures_ , _PoS_ LATTICE2018 (2019) 316 [1904.03927].
  * [14] S. Borsanyi et al., _High-precision scale setting in lattice QCD_ , _JHEP_ 09 (2012) 010 [1203.4469].
  * [15] S.R. Sharpe, _Future of Chiral Extrapolations with Domain Wall Fermions_ , in _Workshop on Domain Wall Fermions at Ten Years_ , 6, 2007 [0706.0218].
  * [16] JLQCD collaboration, _Renormalization of domain-wall bilinear operators with short-distance current correlators_ , _Phys. Rev. D_ 94 (2016) 054504 [1604.08702].
  * [17] Y. Aoki, S. Aoki, H. Fukaya, S. Hashimoto, I. Kanamori, T. Kaneko et al., _2+1 flavor fine lattice simulation at finite temperature with domain-wall fermions_ , _PoS_ LATTICE2021 (2022) 609 [2112.11771].
  * [18] K. Binder, _Finite size scaling analysis of ising model block distribution functions_ , _Zeitschrift für Physik B Condensed Matter_ 43 (1981) 119.
  * [19] H.W.J. Blote, E. Luijten and J.R. Heringa, _Ising universality in three dimensions: a monte carlo study_ , _Journal of Physics A: Mathematical and General_ 28 (1995) 6289–6313.
  * [20] C. Jung, R.G. Edwards, X.-D. Ji and V. Gadiyak, _Residual chiral symmetry breaking in domain wall fermions_ , _Phys. Rev. D_ 63 (2001) 054509 [hep-lat/0007033].
  * [21] A. Portelli, N. Asmussen, P. Boyle, F. Erben, V. Gülpers, R. Hodgson et al., _aportelli/hadrons: Hadrons v1.2_ , Nov., 2020.  10.5281/zenodo.4293902.