2022 November 29 2023 February 28 $\langle$publication date$\rangle$

accretion, accretion disks — black hole physics — stars: black holes — line:
profiles — X-rays: binaries — galaxies: active

# On the Zeeman Effect in Magnetically-Arrested Disks

Yoshiyuki Inoue11affiliation: Department of Earth and Space Science, Graduate
School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
22affiliation: Interdisciplinary Theoretical & Mathematical Science Program
(iTHEMS), RIKEN, 2-1 Hirosawa, Saitama 351-0198, Japan 33affiliation: Kavli
Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The
University of Tokyo, Kashiwa, Chiba 277-8583, Japan<EMAIL_ADDRESS>
###### Abstract

Magnetically arrested disk (MAD) has been argued as the key accretion phase to
realize the formation of relativistic jets. However, due to the lack of
magnetic field measurements of accreting systems, MAD has not been
observationally confirmed yet. Here we propose that a strong magnetic field
accompanied by MAD would induce the Zeeman splitting of relativistically
broadened Fe K$\alpha$ fluorescence lines in X-ray binaries and active
galactic nuclei, where we consider a two-phase medium in the inner accretion
disk, magnetically dominated hot corona and cold reflector. Such a geometrical
configuration is suggested from X-ray observations and recently confirmed by
numerical simulations. Although turbulence in accretion flows would broaden
the split lines, future X-ray high-energy resolution satellites, XRISM and
Athena, would be capable of seeing the Zeeman effect on the Fe lines in X-ray
binaries in the case with the MAD configuration. The signature of the Zeeman
split lines would provide observational evidence for MAD.

##  1 Introduction

It has been more than a century since astrophysical jets were first observed
(Curtis, 1918). Astrophysical jets are collimated relativistic magnetized
plasma outflows launched from compact accreting objects. They are found in
stellar-mass black holes (BH; Fender et al. (2004)) and supermassive BHs
(Blandford et al., 2019). Powers of some relativistic jets can exceed the
Eddington limit of BHs, requiring highly efficient energy conversion processes
from accretion to outflows. However, the formation mechanism of powerful
relativistic jets has yet to be answered.

Theoretically, the Blandford-Znajek (BZ) mechanism (Blandford & Znajek, 1977)
is believed as the plausible explanation for the jet launch. In the BZ
mechanism, the jet power is extracted by the rotation of BHs with the support
of the magnetic fields threading the central BH. General relativistic
magnetohydrodynamic (GRMHD) simulations confirm this process as a plausible
and efficient jet power extraction mechanism (see, e.g., Komissarov et al.
(2007); Tchekhovskoy et al. (2010, 2011); McKinney et al. (2012); Takahashi et
al. (2016); Avara et al. (2016); Liska et al. (2022)).

The BZ mechanism has two key parameters: the spin parameter of the BH and the
large-scale magnetic field threading in the BH. Numerical simulations suggest
that strong magnetic fields are required to realize observed powerful jets
(e.g., McKinney et al. (2012)). Such large magnetic flux accumulation is
expected to appear in the magnetically arrested disk (MAD) scenario (Narayan
et al. (2003); Igumenshchev et al. (2003), see also Bisnovatyi-Kogan &
Ruzmaikin (1974, 1976)), where magnetic field dominates the dynamics of the
inner disk. Although the event horizon telescope (EHT) has resolved inner
accretion disks of M 87 and Sgr A* with unprecedented spatial resolutions, it
is still not conclusive whether their accretion processes are dominated by MAD
or other processes (Event Horizon Telescope Collaboration et al., 2021, 2022;
Blandford & Globus, 2022). Therefore, observational evidence of MAD in
accretion systems is still lacking.

Here, the presence of a magnetic field induces spectral line splitting by the
Zeeman effect due to the interaction of the magnetic dipole moment of an
electron with the magnetic field. The Zeeman effect has been applied to
measure magnetic fields of various astrophysical systems such as sunspots
(Hale, 1908), active stars (Donati et al., 1997), molecular clouds (Nakamura
et al., 2019), and an outer maser disk of an AGN (Modjaz et al., 2005).
Detectability of the Zeeman effect in the X-raying accreting neutron stars has
also been argued in the literature (Sarazin & Bahcall, 1977; Loeb, 2003).

BH-powered X-ray binaries (XRBs) and active galactic nuclei (AGNs)
ubiquitously have the Fe K$\alpha$ fluorescence line in their X-ray spectra.
Broaden Fe K$\alpha$ lines imply the location of the cold reflecting medium
near the BHs and have been used for the BH spin measurements (Reynolds, 2021).
Here, hot plasma, namely coronae, should also exist in the vicinity of BHs to
reproduce X-ray continuum spectra of accreting BHs. Geometrical configurations
of hot coronae and cold reflectors have been debated in literature (see e.g.,
Done et al. (2007); Meyer-Hofmeister & Meyer (2011)). Recently, by performing
two temperature GR-radiation-MHD simulations, Liska et al. (2022) demonstrated
that a geometrically thin accretion disk transitions into a two-phase medium
of cold gas clumps and a hot, magnetically dominated corona, when the thin
disk is threaded by large-scale poloidal magnetic fields. This numerical
result can naturally explain the coexistence of broaden Fe K$\alpha$ line and
hard X-ray continuum emission in XRBs and AGNs.

In this Letter, we consider the Zeeman effect on the Fe K$\alpha$ lines of
XRBs and AGNs in the MAD state assuming the two-pahse medium in the inner
accretion disk. Future X-ray satellite missions such as X-Ray Imaging and
Spectroscopy Mission (XRISM; Tashiro et al. (2020)) and Advanced Telescope for
High ENergy Astrophysics (Athena; Nandra et al. (2013)) will carry an X-ray
microcalorimeter with an energy resolution down to several eV at 6 keV. We
also discuss whether future X-ray missions can probe the MAD via the Zeeman
effect.

##  2 Zeeman Effect on MAD

Magnetic flux $\Phi_{\mathrm{BH}}$ threading a BH is described as

$\Phi_{\mathrm{BH}}\equiv\phi_{\mathrm{BH}}(\dot{M}_{\mathrm{BH}}R_{g}^{2}c)^{1/2},$
(1)

where $\phi_{\mathrm{BH}}$ is the dimensionless magnetic flux,
$\dot{M}_{\mathrm{BH}}$ is the accretion rate onto the BH, and
$R_{g}=GM_{\mathrm{BH}}/c^{2}$ is the gravitational radius. $M_{\mathrm{BH}}$
is the BH mass, $G$ is the gravitational constant, and $c$ is the speed of the
light. $\Phi_{\mathrm{BH}}$ in accretion flows are always nonzero since the
magnetic flux is transported inward via accretion. $\phi_{\rm BH}$ is
typically 20–50 depending on accretion rates based on GRMHD simulations (e.g.,
McKinney et al. (2012); Avara et al. (2016); Liska et al. (2022)). We set
$\phi_{\rm BH}=30$ as a fiducial value, which is based on the recent GR-
radiation-MHD simulation accounting for both hot corona and cold medium at
$\sim 35$% of the Eddington luminosity (Liska et al., 2022).

The MAD magnetic field at a distance $R\equiv rR_{g}$ from the BH is, assuming
$BR^{p}=\mathrm{const.}$,

$\displaystyle B(R)$ $\displaystyle=$
$\displaystyle\frac{B(R_{g})R_{g}^{p}}{R^{p}}$ (2) $\displaystyle=$
$\displaystyle\frac{\Phi_{\mathrm{BH}}}{\pi R_{g}^{2}r^{p}}$ (3)
$\displaystyle\simeq$ $\displaystyle 2.3\times
10^{9}\mathrm{G}\biggl{(}\frac{\phi}{30}\biggr{)}\biggl{(}\frac{m}{10}\biggr{)}^{-1/2}\biggl{(}\frac{\dot{m}}{0.3}\biggr{)}^{1/2}\biggl{(}\frac{r}{1}\biggr{)}^{-p},$
(4)

where $m\equiv M_{\mathrm{B}H}/M_{\odot}$ and
$\dot{m}\equiv\dot{M}_{\mathrm{BH}}/\dot{M}_{\mathrm{Edd}}$ with a 10%
radiative efficiency. $\dot{M}_{\mathrm{Edd}}$ is the Eddington accretion rate
for the mass of $M_{\mathrm{BH}}$.

The energy separation by the Zeeman effect is given by

$\Delta
E_{\mathrm{split}}\approx\frac{e\hbar}{2m_{e}c}\left(M_{L}+2M_{S}\right)B\simeq
11.6~{}\mathrm{eV}~{}\biggl{(}\frac{B}{10^{9}~{}\mathrm{G}}\biggr{)}$ (5)

where $M_{L}$ and $M_{S}$ are the quantum numbers of the orbital angular
momentum and the spin angular momentum. We consider the split transition lines
associated with changes of $\Delta M_{L}=\pm 1$ and $\Delta M_{S}=0$.

Figure 1: Fe K$\alpha$ line splitting by the Zeeman Effect for various masses
and accretion rates as indicated in the figure. We set
$\phi_{\mathrm{BH}}=30$, $p=2$, and $r=2$. The horizontal dashed and dotted
line represents the energy resolution of XRISM and Athena, respectively.

In the accreting system, Eq. 5 can be replaced as

$\Delta E_{\mathrm{split}}\simeq
27~{}\mathrm{eV}~{}\biggl{(}\frac{\phi}{30}\biggr{)}\biggl{(}\frac{m}{10}\biggr{)}^{-1/2}\biggl{(}\frac{\dot{m}}{0.3}\biggr{)}^{1/2}\biggl{(}\frac{r}{1}\biggr{)}^{-p}.$
(6)

Therefore, we can expect the line splitting of Fe K$\alpha$ lines at the level
of several tens of eV for XRBs and $10^{-3}$ eV for AGNs with
$\phi_{\mathrm{BH}}=30$, $\dot{m}=0.3$, and $p=2$.

The energy resolution of current X-ray CCDs at the Fe K$\alpha$ line is $\sim
120$ eV (Ezoe et al., 2021), larger than the typically expected split. With
this resolution, we may infer $B<10^{10}$ G for any available observations.
Next-generation X-ray telescopes such as XRISM and Athena will have X-ray
micro-calorimeter instruments. The planned energy resolution is 7.0 eV and 2.5
eV at 6 keV for XRISM and Athena, respectively. Figure 1 shows the expected Fe
K$\alpha$ line split for various BH masses and accretion rates with
$\phi_{\mathrm{BH}}=30$, $p=2$, and $r=2$. Figure 1 also shows the energy
resolutions of XRISM and Athena. Next-generation X-ray telescopes will enable
us to see the Zeeman line splitting in XRBs with the presence of MAD.

About 20 XRBs have dynamically confirmed BHs (Corral-Santana et al., 2016).
Among them, low-mass black holes such as GX 339-4 and GRO J1655-40, whose BH
mass is $5.8\pm 0.5~{}M_{\odot}$ (Hynes et al., 2003) and $5.4\pm
0.3~{}M_{\odot}$ (Beer & Podsiadlowski, 2002), respectively, would be possible
candidates to see the Zeeman effect on MAD. Another candidate is a persistent
X-ray binary Cyg X-1 having the BH mass of $14.8\pm 1.0~{}M_{\odot}$ (Orosz et
al., 2011)111Recent radio astrometric observation suggests its mass of
$m=21.2\pm 2.2$ (Miller-Jones et al., 2021). A broad iron K$\alpha$ line has
also been reported for Cyg X-1 (Duro et al., 2011, 2016), with the inner edge
of the disk of $\sim 1.6R_{g}$ (Fabian et al., 2012). Although the accretion
rate depends on the states, it is about 1% even in the low/hard state (Yamada
et al., 2013). The expected $\Delta E_{\mathrm{split}}$ is $2.7$ and $8.5$ eV
for $\dot{m}=0.03$ and $0.3$, respectively. Therefore, XRISM and Athena would
see the Zeeman effect on MAD even for a relatively massive stellar BH object
Cyg X-1.

##  3 Discussions

Figure 2: Simulated broad Fe K$\alpha$ line spectrum accounting for the Zeeman
Effect based on the kerrdisk model. Detailed parameters are described in the
text. We set $\Delta E_{\mathrm{split}}=24$ eV. The inset in the figure shows
an enlarged view of the $6.50$ to $6.56$ keV region to clarify the split
feature.

The Fe K$\alpha$ lines appear broadened and skewed by the Doppler effect and
gravitational redshift (Fabian et al., 1989; Laor, 1991). Those relativistic
effects would blur the Zeeman split lines. Figure 2 shows a simulated Fe
K$\alpha$ line in the MAD. We apply the kerrdisk code (Brenneman & Reynolds,
2006) with emissivity indices of $\alpha_{1}=\alpha_{2}=0$, inclination of
$i=30^{\circ}$, dimensionless spin parameter of $a=0.9$, inner and outer radii
of $r_{\mathrm{min}}=r_{\mathrm{ms}}$ and
$r_{\mathrm{max}}=100~{}r_{\mathrm{ms}}$, and redshift $z=0$.
$r_{\mathrm{ms}}$ is the marginally stable orbit radius. We include three
lines split by the Zeeman effect assuming $\Delta
E_{\mathrm{split}}=24~{}\mathrm{eV}$, corresponding to $B\simeq 2\times
10^{9}~{}\mathrm{G}$, we also set $p=0$. As the figure shows, three split
lines would be distinguishable within $\Delta E_{\mathrm{split}}=24$ eV.

There also exists Doppler broadening due to turbulent motions of accreting
flows. The turbulence speed is characterized by the sound speed. The
broadening by turbulence is expected at the level of $\sim 10$ eV at the
temperature of $\sim 10^{7}$ K, which can be comparable to the expected Zeeman
split (Eq. 6). Three split lines would be more broaden by turbulence than
shown in Figure 2. In addition, the effect of the continuum is not included in
Figure 2 and the energy split would depend on radius if $p\neq 0$. Further
detailed spectral simulations, including instrument response functions of
XRISM and Athena, will be needed.

Broad Fe K$\alpha$ lines are commonly reported in various XRBs and AGNs
(Reynolds, 2021). However, line broadening is known to depend on the spectral
modeling (see, e.g., Done & Zycki (1999); Makishima et al. (2008)) and disk
winds further would disturb the ionization states (see e.g., Tomaru et al.
(2019)). If the location of reflecting iron atoms is further away from the BH,
the expected line split drops with $r^{-p}$. Determination of the reflecting
medium is necessary to probe the MAD through X-ray Zeeman effect measurements.
Therefore, the Zeeman effect on the Fe K$\alpha$ line is realized only with
the MAD state and the near BH reflector case. In other words, if we do not see
the Zeeman effect even with sufficient energy resolutions, it will imply that
MAD is absent or the reflector is distant.

With a strong magnetic field like MAD, we would also have the quadratic Zeeman
effect producing displacements of lines toward shorter wavelengths (Jenkins &
Segrè, 1939; Schiff & Snyder, 1939; Preston, 1970; Loeb, 2003). The energy
shift is given as

$\displaystyle\Delta
E_{\mathrm{shift}}=\frac{e^{2}a_{0}^{2}}{8Z^{2}m_{e}c^{2}}n^{4}(1+M_{L}^{2})B^{2}$
(7) $\displaystyle\simeq 9.6\times
10^{-3}~{}\mathrm{eV}~{}\biggl{(}\frac{\phi}{30}\biggr{)}^{2}\biggl{(}\frac{m}{10}\biggr{)}^{-1}\biggl{(}\frac{\dot{m}}{0.3}\biggr{)}\biggl{(}\frac{r}{1}\biggr{)}^{-2p},$
(8)

where $n$ is the principal quantum number, $a_{0}$ is the Bohr radius, and $Z$
is the nuclear charge. We set $n=1$, $M_{L}=1$ and $Z=26$ in Eq. 8. Thus,
several orders of magnitude better energy resolution would be necessary to see
the quadratic energy shift.

##  4 Summary

In this Letter, we consider the Zeeman effect on the MAD state. MAD is
expected to be associated with the jet production (Tchekhovskoy et al., 2010,
2011; McKinney et al., 2012). In the black hole accretion systems, broad Fe
K$\alpha$ fluorescence lines have been often reported. A strong magnetic field
environment by MAD would induce line splitting of the Fe K$\alpha$ line by the
Zeeman effect (Eq. 6). Next-generation X-ray telescopes such as XRISM and
Athena will be able to see the Zeeman splitting of Fe lines in XRBs, if
reflectors exist near BHs. The detection of the Zeeman effect would be clear
evidence of the MAD in the BH accretion systems. If the Zeeman effect does not
appear even with sufficient energy resolutions, it would imply that MAD is
absent or the iron reflector is distant.

We would like to thank the anonymous referee for thoughtful and helpful
comments. We would also like to thank Roger Blandford, Chris Done, Norita
Kawanaka, Katsunori Kusakabe, Shin Mineshige, Hirokazu Odaka,and Shinsuke
Takasao for useful discussions and comments. Y.I. is supported by JSPS KAKENHI
Grant Number JP18H05458, JP19K14772, and JP22K18277. This work was supported
by World Premier International Research Center Initiative (WPI), MEXT, Japan.

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OU-HET-1221

# A lattice formulation of Weyl fermions on a single curved surface

Shoto Aoki  Hidenori Fukaya  Naoto Kan  Department of Physics, Osaka
University, Toyonaka 560-0043, Japan

###### Abstract

In the standard lattice domain-wall fermion formulation, one needs two flat
domain-walls where both of the left- and right-handed massless modes appear.
In this work we investigate a single domain-wall system with a nontrivial
curved background. Specifically we consider a massive fermion on a three-
dimensional square lattice, whose domain-wall is a two-dimensional sphere. In
the free theory, we find that a single Weyl fermion is localized at the wall
and it feels gravity through the induced spin connection. With a topologically
nontrivial $U(1)$ link gauge field, however, we find a zero mode with the
opposite chirality localized at the center where the gauge field is singular.
In the latter case, the low-energy effective theory is not chiral but
vectorlike. We discuss how to circumvent this obstacle in formulating lattice
chiral gauge theory in the single domain-wall fermion system.

##  1 Introduction

Lattice gauge theory has been successful in understanding and computing
nonperturbative dynamics of particle physics. Numerical simulation of QCD on a
flat Euclidean lattice has been an essential tool for examining the standard
model in various hadronic processes. Domain-wall fermions, in particular,
defined on a five-dimensional square lattice, achieve a good control of chiral
symmetry, where the unwanted mixing between the left- and right-handed quarks
is well suppressed.

In the standard domain-wall fermion formulation KAPLAN1992342AMethod ;
Shamir1993Chiral ; Furman1995Axial  on a flat five-dimensional lattice, we
need at least two domain-walls where the left- and right-handed massless
fermions are localized. If one of them is decoupled from the gauge fields, the
system may be able to formulate a chiral gauge theory on a lattice
Golterman:1992ub ; Jansen:1994ym ; Kaplan:1995pe ; Kikukawa:2000kd 111 Instead
of directly using the domain-wall fermions, the effective four-dimensional
Dirac operator or known as the overlap Dirac operator Neuberger:1997fp  was
used to formulate some successful examples of nonperturbative formulation of
lattice chiral gauge theory. See Refs. Luscher:1998du ; Luscher:1999un ;
Kikukawa:2000kd  and the references therein. . In order to decouple the
unwanted modes, a smooth elimination of gauge fields has been considered in
Grabowska:2015qpk ; Grabowska:2016bis  but it turned out that this task is not
that easy. For instance, the cobordism invariance of the instanton background
forces the two domain-walls to have the same number of zero modes
Okumura:2016dsr  (see also related works Hamada:2017tny ; Makino:2016auf ;
Makino:2017pbq ; Ago:2019yzj ). This indicates that the low energy theory has
both of the left- and right-handed modes. Another approach is the so-called
symmetric mass generation Wen:2013ppa ; Wang:2013yta ; Kikukawa:2017ngf ;
Kikukawa:2017gvk ; Wang:2018jkc ; you2018symmetric ; Tong:2021phe ;
Zeng:2022grc ; Wang:2022ucy ; Lu:2023emm ; Guo:2023rnz ; Lu:2023cev ;
Liu:2023msa ; Golterman:2023zqf , which gaps out the unwanted chiral modes by
generating mass without breaking the chiral symmetry. This, however, requires
a highly nonperturbative dynamics.

In our recent works Aoki:2022cwg ; Aoki:2022aez  we have been studying a
massive fermion system with a nonflat single domain-wall. In the case with an
$S^{1}$ domain-wall embedded into a flat two dimensional square lattice, we
observed that massless Dirac fermion appears on the wall, having a definite
eigenvalue of the gamma matrix perpendicular to the domain-wall. Similarly, we
observed that a two-flavor massive fermion, which respects the time-reversal
symmetry, yields a massless Dirac fermion on the $S^{2}$ domain-wall embedded
in flat three dimensions. Also in this case, the edge-localized modes are the
eigenstates of the gamma matrix accompanied by an operator exchanging the
flavor index.

Moreover, it was found through the Dirac operator spectrum that the edge-
localized modes feel gravity via the induced spin connection on the curved
domain-wall. This is a realization of the Einstein equivalence principle,
which tells that any acceleration is indistinguishable from gravity. We have
numerically confirmed that the rotational symmetry is monotonically recovered
towards the classical continuum limit of the higher-dimensional square
lattice. The limit of the Dirac operator eigenvalue spectrum is consistent
with that in the continuum theory, which has a manifest gauge symmetry under
the general coordinate transformation.

In this work, we extend our study on the curved domain-wall fermions in two
new directions. One is to employ Shamir’s formulation Shamir1993Chiral ;
Furman1995Axial . We consider a two-dimensional spherical domain-wall embedded
into a three-dimensional square lattice, but we ignore the lattice points
outside of the domain-wall, which is equivalent to taking the fermion mass to
$+\infty$. Second is to consider a single flavor fermion, which loses the
time-reversal symmetry in three dimensions. Then the edge-localized modes
should represent not Dirac but Weyl fermions. From the Dirac operator
spectrum, we investigate the gravity induced on the wall, and examine its
continuum limit. A preliminary result of this work was presented in Ref.
Aoki:2024pad .

In this setup, a single massless fermion with a definite chirality is
localized on a single domain-wall. Namely, it looks as if a Weyl fermion on an
$S^{2}$ surface is nonperturbatively regularized on a lattice. In recent
publications Kaplan:2023pxd ; Kaplan:2023pvd  this single-domain-wall system
was proposed as a lattice regularization of chiral gauge theories.

Contrary to the expectation, when it is coupled to the $U(1)$ gauge field, we
observe a nontrivial dynamics where the low-energy theory becomes not chiral
but rather vectorlike. This happens when the gauge field on the $S^{2}$ has a
nontrivial topological charge. The Gauss law in three dimensions requires a
singular source of the gauge flux inside the spherical domain-wall. This
configuration is an analog of a magnetic monopole in three spacial dimensions,
whose magnetic flux penetrates the $S^{2}$ surface. The additive mass
renormalization through the Wilson term due to the singular gauge field
becomes strong enough to flip the sign of the effective mass of the fermion,
and another domain-wall is created around the singularity. The dynamically
created domain-wall captures a single fermion zero mode with the opposite
chirality to the original edge modes on the $S^{2}$. This dynamical creation
of domain-walls was already reported in our previous work of Dirac fermion
system Aoki:2023lqp ; Kan:2023latticepro  which gives a microscopic
description of the Witten effect Witten:1979Dyons : the magnetic monopole
gains an electric charge because the electron zero mode is bound to the small
but finite spherical domain-wall around the monopole. We discuss how to
circumvent this obstacle in formulating chiral gauge theory.

The rest of this paper is organized as follows. In Sec. 2 we show that the
edge-localized modes generally exist and they feel the induced gravity. In
Sec. 3 we analyze the lattice domain-wall fermion system numerically. Our
result shows that a Weyl fermion appears on the $S^{2}$ domain-wall, as is
expected. In Sec. 4, however, we show that an oppositely chiral zero mode
appears when the gauge field has a point-like singularity. We discuss in Sec.
5 how to avoid the unwanted zero modes with the opposite chirality. Summary is
given in Sec. 6.

##  2 Continuum analysis of three-dimensional Shamir domain-wall fermion with
a single $S^{2}$ boundary

We first consider a Dirac fermion action in a three-dimensional flat continuum
spacetime $\mathbb{R}^{3}$,

$S=\int d^{3}x\;\bar{\psi}D\psi(x),$ (1)

where the Dirac operator with a position-dependent mass $m(x)$ is given by

$D=\sum_{i=1}^{3}\sigma^{i}\quantity(\partialderivative{x^{i}}-iA_{i}(x))+m(x).$
(2)

Here we have introduced a background $U(1)$ gauge field $A_{i}(x)$ and
$\sigma^{i}$ denote the Pauli matrices representing the gamma matrices in
three dimensions.

We assume that the mass function $m(x)$ has a spherical domain-wall structure:

$\displaystyle m(x)=\left\\{\begin{array}[]{cc}-m&\text{for $|x|\leq
r_{0}$}\\\ +M&\text{otherwise}\end{array}\right.,$ (5)

where we assume $m>0,M>0$. In order to obtain the Shamir domain-wall fermion
Dirac operator, we take the $M\to+\infty$ limit, which is equivalent to
imposing a boundary condition

$\sigma_{r}\psi(x)=+\psi(x)\;\;\;\text{at $|x|=r_{0}$}.$ (6)

Here the chirality operator $\sigma_{r}$ is given by

$\sigma_{r}=\frac{\sum_{i}\sigma^{i}x_{i}}{|x|},$ (7)

which is well-defined at any point on the $S^{2}$ surface $|x|=r_{0}$. In the
same way, the conjugate operator $D^{\dagger}$ requires the boundary condition
with the opposite chirality $\sigma_{r}=-1$.

We show below that the chiral edge-localized modes generally exist on the
curved spherical domain-wall, at least approximately in Sec. 2.1. We also
derive in Sec. 2.2 an exact eigenfunction when the gauge field configuration
respects the rotational symmetry.

###  2.1 Edge-localized modes in general background gauge field

First, let us confirm the existence of the edge-localized modes in a general
$U(1)$ gauge field background. Here we assume that the gauge field is smooth
everywhere without any singularity.

In the polar coordinates and taking the radial gauge condition $A_{r}=0$, the
Dirac operator inside the domain-wall $|x|\leq r_{0}$ is expressed by

$\displaystyle D=$
$\displaystyle\sigma_{r}\quantity(\partialderivative{r}+\frac{1}{r}-\frac{r_{0}}{r}iD^{S^{2}})-m,$
(8)

where the Hermitian operator $iD^{S^{2}}$ is defined by

$iD^{S^{2}}:=\frac{1}{r_{0}}\left(1-i\sigma_{2}\exp(i\phi\sigma_{3})\left({\partial\over\partial\theta}-i\hat{A}_{\theta}\right)-{1\over\sin\theta}\sigma_{r}\sigma_{2}\exp(i\phi\sigma_{3})\left({\partial\over\partial\phi}-i\hat{A}_{\phi}\right)\right).$
(9)

Here we have introduced the dimensionless vector potentials:
$\hat{A}_{\theta}:=rA_{\theta}$ and $\hat{A}_{\phi}:=r\sin\theta A_{\phi}$.

In fact, we can identify $D^{S^{2}}$ as the effective Dirac operator on
$S^{2}$ for the low-energy edge-localized modes. Taking a gauge transformation
or the Euclidean version of the local Lorentz transformation,

$\displaystyle
R(\theta,\phi)=\exp(i\theta\sigma^{2}/2)\exp(i\phi\sigma^{3}/2),$ (10)

as well as a $U(1)$ gauge transformation, we obtain the two-dimensional
massless Dirac operator:

$\displaystyle
e^{-i\frac{\phi}{2}}R(\theta,\phi)iD^{S^{2}}e^{i\frac{\phi}{2}}R(\theta,\phi)^{-1}=-\frac{1}{r_{0}}\sigma_{3}\left[\sigma_{1}\left(\partialderivative{\theta}-i\hat{A}_{\theta}\right)+\frac{\sigma_{2}}{\sin\theta}\quantity(\partialderivative{\phi}-i\hat{A}_{\phi}+\frac{i}{2}-\frac{i\cos\theta}{2}\sigma_{3})\right].$
(11)

We can see that the spin connection $\sigma_{3}\cos\theta/2$ is induced222 To
be precise, the induced connection is spin-c with an additional $U(1)$ part
$\frac{i}{2}$, which, however, can be eliminated by the gauge transformation.
We keep this unphysical $U(1)$ part nonzero to make the analysis simpler,
since otherwise, the gauge transformation requires multiple-patch covering of
$\mathbb{R}^{3}$ to avoid multivaluedness of the gauge transformation with
respect to $\theta$.. It is also important to note that the rotated two-
dimensional Dirac operator anticommutes with the chirality operator
$R(\theta,\phi)\sigma_{r}R(\theta,\phi)^{-1}=\sigma_{3}$.

Let $\chi(\theta,\phi)$ be a function in the domain of $iD^{S^{2}}$ at
$r=r_{0}$. Then we can construct an edge-localized mode

$\displaystyle\psi_{+}^{e}=\frac{1}{r}\exp[-m(r_{0}-r)]P_{+}\chi(\theta,\phi),$
(12)

to which the Dirac operator effectively acts as

$\displaystyle D=P_{-}(iD^{S^{2}})P_{+},$ (13)

with the chiral projection operators $P_{\pm}:=(1+\sigma_{r})/2$. Here we have
used an approximation in the large $m$ limit,

$\displaystyle\frac{1}{r}iD^{S^{2}}\exp[-m(r_{0}-r)]\sim\frac{1}{r_{0}}iD^{S^{2}}\exp[-m(r_{0}-r)].$
(14)

Thus, the edge-localized mode $\psi_{+}^{e}$ represents one massless fermion
with positive chirality on the single $S^{2}$ surface with radius $r_{0}$.

When we take $M$ in Eq. (5) to negative infinity keeping the sign of $m$
unchanged, the boundary condition requires the negative chirality at
$r=r_{0}$, which prohibits the edge-localized modes. If we take both $m$ and
$M$ negative, the negatively chiral edge-mode

$\displaystyle\psi_{-}^{e}=\frac{1}{r}\exp[m(r_{0}-r)]P_{-}\chi(\theta,\phi),$
(15)

appears. The case with $mM<0$ is analogous to a normal insulator where no
edge-localized mode appears, and $mM>0$ corresponds to a topological insulator
where the existence of the edge excitation is topologically protected. Thus,
our curved domain-wall fermion system with $mM>0$ has the edge-localized
modes, which describe a single Weyl fermion on a single curved surface.

The edge-localized Weyl fermion feels gravity through the induced spin
connection as explicitly shown in Eq. (11). This is consistent with the
Einstein’s equivalence principle, which indicates that any constraint force
can be identified as gravity. Since the edge modes are constrained on the
curved $S^{2}$ surface, gravity is naturally induced on them. Interestingly
the induced gravity reflects only intrinsic geometry and the extrinsic
information is invisible. This is not special for $S^{2}$ but true on general
curved domain-walls as discussed in Ref. Aoki:PhD2024 . The extrinsic
curvature contribution is suppressed exponentially in the large $m$ limit.

###  2.2 Exact edge-localized modes with rotational symmetry

Next we exactly solve the eigenproblem of the curved domain-wall fermion Dirac
operator in the case where the system has a rotational symmetry. Let us take
the radial gauge $A_{r}=0$ and assume that the field strength $F_{\mu\nu}$
perpendicular to the domain-wall is uniformly distributed. Such $U(1)$
backgrounds are classified by the topological charge,

$\displaystyle n$
$\displaystyle=\frac{1}{4\pi}\int_{S^{2}}d^{2}x\epsilon^{\mu\nu}F_{\mu\nu},\;\;\;F_{12}=-F_{21}=\frac{n}{2r_{0}^{2}}.$
(16)

Our target gauge field background in the whole three-dimensional space is
achieved by putting a “monopole” at the origin333 It may not be adequate to
name this configuration a “monopole” since our gauge potential is considered
not in three-dimensional space but in $2+1$ dimensional spacetime. But the
computation is completely parallel to the case with the monopole in three
spacial dimensions we discussed in Ref. Aoki:2023lqp . . The vector potential
generated by the monopole is

$\displaystyle
A_{1}=\frac{-q_{m}y}{r(r+z)},~{}A_{2}=\frac{q_{m}x}{r(r+z)},~{}A_{3}=0,$ (17)

where $q_{m}$ is the magnetic charge of the monopole, which satisfies the
Dirac’s quantization condition $q_{m}=\frac{n}{2}~{}(n\in\mathbb{Z})$. To be
precise, this expression has a singularity at $x=y=0$ and $z<0$, which is the
so-called Dirac string. In order to eliminate the Dirac string, we need
another set of the gauge potentials in the $z\leq 0$ region but here we simply
ignore the neighborhood of the string. In fact, the Dirac string automatically
disappears in the lattice regularization we will discuss in the next section.
The field strength we have obtained is

$\displaystyle
F_{ij}=\partial_{i}A_{j}-\partial_{j}A_{i}=q_{m}\epsilon_{ijk}\frac{x_{k}}{r^{3}}.$
(18)

With the above spherically symmetric gauge field background, the effective
two-dimensional Dirac operator $iD^{S^{2}}$ can be written as

$D^{S^{2}}=\frac{1}{r_{0}}\left[\sigma^{i}\left(L_{i}+n\frac{x_{i}}{2r}\right)+1\right],$
(19)

where $L_{i}$ is the orbital angular momentum operator in the presence of the
monopole,

$L_{i}=-i\epsilon_{ijk}x^{j}\left(\frac{\partial}{\partial
x^{k}}-iA_{k}\right)-n\frac{x_{i}}{2r}.$ (20)

Instead of $D$ itself, let us solve the eigenproblem of the Hermitian operator
$D^{\dagger}D$. Note that $D^{\dagger}D$ commutes with the total angular
momentum operators $J^{i}=L^{i}+\sigma^{i}/2$ as well as $D^{S^{2}}$ so that
it is convenient to consider the simultaneous eigenstates of
$\bm{J}^{2}=\sum_{i}(J^{i})^{2}$, $J^{3}$ and $D^{S^{2}}$ . Let $j$ be the
highest weight of $\bm{J}$ where takes discrete values

$\displaystyle
j=\frac{\absolutevalue{n}-1}{2},~{}\frac{\absolutevalue{n}-1}{2}+1,\frac{\absolutevalue{n}-1}{2}+2\cdots,$
(21)

with a constraint $j\geq 0$. The eigenfunction denoted by $\chi_{j,j_{3},\pm}$
satisfies

$\displaystyle\bm{J}^{2}\chi_{j,j_{3},\pm}$
$\displaystyle=j(j+1)\chi_{j,j_{3},\pm},$ (22) $\displaystyle
J_{3}\chi_{j,j_{3},\pm}$ $\displaystyle=j_{3}\chi_{j,j_{3},\pm},$ (23)
$\displaystyle iD^{S^{2}}\chi_{j,j_{3},\pm}$
$\displaystyle=\pm\frac{\nu}{r_{0}}\chi_{j,j_{3},\pm},\;\;\;\nu=\sqrt{\quantity(j+\frac{1}{2})^{2}-\frac{n^{2}}{4}},$
(24) $\displaystyle\sigma_{r}\chi_{j,j_{3},\pm}$
$\displaystyle=\chi_{j,j_{3},\mp}.$ (25)

As will be shown below, the edge-localized eigenvalue of $D^{\dagger}D$
depends only on $j$. Therefore the degeneracy of the eigenvalue is $2(2j+1)$.

Since the chirality operator $\sigma_{r}$, which determines the boundary
condition at $r=r_{0}$, anti-commutes with $D^{S^{2}}$, the eigenstate $\psi$
of $D^{\dagger}D$ is written by a linear combination of $\chi_{j,j_{3},\pm}$:

$\displaystyle\psi=\frac{1}{\sqrt{r}}\left[g_{+}(r)\chi_{j,j_{3},+}+g_{-}(r)\chi_{j,j_{3},-}\right].$
(26)

Substituting this to the equation $D^{\dagger}D\psi=E^{2}\psi$, where we
assume $m^{2}>E^{2}$, we find that the functions $g_{\pm}(r)$ satisfy the
modified Bessel equations with a rescaled variable $z=\kappa r$ where
$\kappa=\sqrt{m^{2}-E^{2}}$,

$\displaystyle\left[\frac{\partial^{2}}{\partial
z^{2}}+\frac{1}{z}\frac{\partial}{\partial z}-\frac{(\nu\mp
1/2)^{2}}{z^{2}}-1\right]g_{\pm}=0.$ (27)

From the regularity condition at $r=0$, we find the solutions in terms of the
modified Bessel function of the first kind: $g_{+}(r)=AI_{\nu-1/2}(\kappa r)$
and $g_{-}(r)=BI_{\nu+1/2}(\kappa r)$ with numerical constants $A$ and $B$,
respectively. Since $I_{\mu}(z)$ are exponentially increasing function of $z$,
the solutions are exponentially localized at the edge, $r=r_{0}$.

The boundary condition at $r=r_{0}$: $\sigma_{r}\psi=+\psi$ and
$\sigma_{r}D\psi=-D\psi$ leads to two equations

$\displaystyle AI_{\nu-1/2}(\kappa r_{0})$ $\displaystyle=BI_{\nu+1/2}(\kappa
r_{0}),$ (28) $\displaystyle(\kappa B-mA)I_{\nu-1/2}(\kappa r_{0})$
$\displaystyle=-(\kappa A-mB)I_{\nu+1/2}(\kappa r_{0}),$ (29)

which determine the coefficients $A,B$ up to normalization, and the eigenvalue
$E^{2}$. Note in the large $\kappa r_{0}$ limit, the eigenvalue converges to

$E^{2}={\nu^{2}\over r_{0}^{2}}$ (30)

which is independent of the mass parameter $m$. Moreover, the eigenvalue is
shared with that of the massless two-dimensional Dirac operator
$(iD^{S^{2}})^{2}$ shown in Eq. (24). It is important to remark in the free
case with $n=0$ that the eigenvalue is gaped from zero, which is a
gravitational effect aoki2021chiral ; Aoki:2022cwg .

The case of $j=\frac{\absolutevalue{n}-1}{2}$ with $n\neq 0$, is qualitatively
different from the above result. Let $\chi_{j,j_{3},0}$ be an eigenstate
satisfying

$\displaystyle\bm{J}^{2}\chi_{j,j_{3},0}$
$\displaystyle=j(j+1)\chi_{j,j_{3},0}$ (31) $\displaystyle
J_{3}\chi_{j,j_{3},0}$ $\displaystyle=j_{3}\chi_{j,j_{3},0}$ (32)
$\displaystyle iD^{S^{2}}\chi_{j,j_{3},0}$ $\displaystyle=0$ (33)
$\displaystyle\sigma_{r}\chi_{j,j_{3},0}$
$\displaystyle=\text{sign}(n)\chi_{j,j_{3},0}.$ (34)

Assuming the edge-localized mode has the following form,

$\displaystyle\psi_{0}=\frac{1}{\sqrt{r}}f(r)\chi_{j,j_{3},0},$ (35)

we find that $f(r)$ satisfies the same equation as Eq. (27), but with $\nu=0$
and thus we obtain

$\displaystyle f(r)=CI_{1/2}(\kappa r)=C\sqrt{\frac{2}{\pi\kappa
r}}\sinh(\kappa r),$ (36)

with a normalization constant $C$.

It is interesting to note that the chiral boundary condition
$\sigma_{r}\psi_{0}(r=r_{0})=+\psi_{0}(r=r_{0})$ is never satisfied with
$n<0$. Also, another boundary condition
$\sigma_{r}D\psi_{0}(r=r_{0})=-D\psi_{0}(r=r_{0})$ is only satisfied by
$D\psi_{0}(r=r_{0})=0$ when we take the $m\to\infty$ limit. This reflects the
Atiyah-Singer index theorem Atiyah1963TheIndexOfEllipticOperator  on the
sphere $S^{2}$,

${\rm
Ind}D^{S^{2}}=\frac{1}{4\pi}\int_{S^{2}}d^{2}x\epsilon^{\mu\nu}F_{\mu\nu}=n.$
(37)

The Weyl fermion with positive chirality cannot give negative contribution to
the index when $n<0$. Only in the $m\to\infty$ limit, we have an exactly
chiral zero mode of both $D^{\dagger}D$ and $iD^{S^{2}}$.

Thus, we have obtained the exact massless edge-localized modes with the
positive chirality, describing a Weyl fermion on the sphere with radius
$r_{0}$. Their existence is guaranteed even in the presence of nontrivial
gauge field background. This single domain-wall fermion system looks a good
candidate for nonperturbatively formulating chiral gauge theory on a lattice.
Note, however, that the chirality is not perfect unless we take the
$mr_{0}\to\infty$ limit. In particular, it is not obvious in the above
analysis what will happen to the chiral zero mode when $mr_{0}$ is finite. So
far we have ignored the short-distance singularity at the location of the
monopole. As will be seen below, a highly nonperturbative and careful short-
distance analysis is needed to understand this problem.

##  3 Lattice analysis in free theory

In this section we discuss a lattice regularization of the curved domain-wall
fermion system. In the free fermion theory with trivial link variables we show
that a single Weyl fermion appears on a single spherical domain-wall. We also
discuss its continuum and finite volume systematics.

For discretization of the Dirac operator, we employ the Wilson Dirac operator

$\displaystyle D_{W}=$
$\displaystyle\sum_{i=1}^{3}\sigma^{i}\frac{\nabla_{i}-\nabla_{i}^{\dagger}}{2a}+\sum\frac{w}{2a}\nabla_{i}\nabla_{i}^{\dagger}-m,$
(38)

where the lattice spacing is denoted by $a$, and we assume that $m>0$. The
difference operator in the $i$-direction for a spinor field located at the
lattice site $x$ is defined by

$\displaystyle(\nabla_{i}\psi)_{x}=\psi_{x+\hat{i}}-\psi_{x},$ (39)

with the unit vector $\hat{i}$ in the corresponding direction. As the standard
choice, we take the coefficient of the Wilson term $w=1$. With this choice,
the Dirac operator can be written as

$\displaystyle
D_{W}=-\frac{1}{a}\sum_{i=1}^{3}\left(P^{i}_{-}\nabla_{i}+P^{i}_{+}\nabla_{i}^{\dagger}\right)-m,\;\;\;P_{\pm}^{i}=\frac{1\pm\sigma_{i}}{2}.$
(40)

It was shown by Shamir Shamir1993Chiral  in the standard flat domain-wall
fermion that there is no need to impose a chiral boundary condition on the
lattice but it is enough to simply set $\psi_{x}=0$ outside the domain-wall.
The same is true for our curved case. As shown in Fig. 1 where a two-
dimensional slice at $z=1/2$ is drawn, we impose

$\psi_{x}=0\;\;\;\mbox{if $|x-x_{0}|\geq r_{0}$},$ (41)

where the center of the domain-wall is put at the origin $x_{0}=(0,0,0)$. Note
that our lattice sites are put at half-integral points:
$x=((2n_{1}-1)/2,(2n_{2}-1)/2,(2n_{3}-1)/2)$ with integer values $n_{i}$. Put
$x$ be one of the inner nearest-neighbor sites of the domain-wall, where
$x+\hat{j}$ is put outside of the wall. Then the Dirac operator acts as

$\displaystyle(D_{W}\psi)_{x}=-\frac{1}{a}\left[P^{j}_{-}\left(-\psi_{x}\right)-P^{j}_{+}\left(\psi_{x}-\psi_{x-\hat{j}}\right)\right]+\cdots$
(42)

which imposes the boundary condition $P^{j}_{-}\psi_{x}=0$ in the continuum
limit $a\to 0$, while $P^{j}_{+}\psi_{x}$ can be taken arbitrarily, as far as
$\psi_{x}-\psi_{x-\hat{j}}\sim O(a)$. With this simple boundary condition, the
Hermitian conjugate of the Wilson Dirac operator $D^{\dagger}_{W}$ is
canonically defined.

To be precise, our domain-wall is not a smooth sphere but a rather digitized
zig-zag surface like a block toy as depicted in Fig. 1. It is, therefore,
important to check if the rotational symmetry is recovered in the continuum
limit. In our previous work on Dirac fermions aoki2021chiral , we already
obtained positive numerical evidence. The rotational symmetry of the lowest
eigen modes was almost linearly recovered in the $a\to 0$ limit. We will
address this issue for the Weyl fermion at the end of this section.

Figure 1: A two-dimensional slice at $z=1/2$ of the spherical domain-wall in
our lattice space. The hopping to the outside of the sphere $r=r_{0}$ is cut
off.

We numerically solve the eigenproblem of $D_{W}^{\dagger}D_{W}$ as well as
$D_{W}D_{W}^{\dagger}$ which are both Hermitian. With the obtained normalized
eigenfunction $\phi_{k}(x)$, where $k$ is the label for the low-lying modes,
we measure the expectation value

$\displaystyle\sum_{x}\phi_{k}^{\dagger}(x)\sigma_{r}(x)\phi_{k}(x),$ (43)

where the chirality operator $\sigma_{r}(x)=\sum_{i}\sigma_{i}x^{i}/|x|$ is
unambiguously defined at all our half-integral lattice sites. We also scan the
local chirality, defined by

$\displaystyle\phi_{k}^{\dagger}(x)\sigma_{r}(x)\phi_{k}(x)/\phi_{k}^{\dagger}(x)\phi_{k}(x),$
(44)

for each lattice site $x$ in order to study its spacial distribution. This
turns out to be useful in analyzing the case with a nontrivial $U(1)$ gauge
field background in the next section.

In Fig. 2, we plot the eigenvalue spectrum of $D_{W}^{\dagger}D_{W}$ at
$r_{0}=24a$ and $ma=0.35$. Solving
$D_{W}^{\dagger}D_{W}\phi_{k}=E_{k}^{2}\phi_{k}$, its square root normalized
by $r_{0}$: $r_{0}|E|$ is plotted by filled circle symbols. The color
gradation shows the chirality expectation value. We can see that the low-lying
modes below $mr_{0}$, shown by the dotted horizontal line, have the positive
chirality. The obtained eigenvalues agree with the continuum prediction in
cross symbols in which the finite volume correction to the infinite volume
value $r_{0}|E|=\nu=1,2,3,\cdots$ in Eq. (30) is taken into account. The
$2(j+1/2)$ (or $2\nu$ in this case) degeneracy with $j=1/2,3/2,\cdots$ looks
also consistent. The gravitational effect through the induced spin connection,
or the gap from zero is clearly seen.

We also plot the eigenvalue spectrum of $D_{W}D_{W}^{\dagger}$ on the same
lattice at Fig. 3. The spectrum is the same as that of $D_{W}^{\dagger}D_{W}$
but the chirality is opposite.

Figure 2: The spectrum of $D_{W}^{\dagger}D_{W}$ at $r_{0}=24a$ and $ma=0.35$.
The square rooted value $r_{0}|E|$ is plotted. The color gradation shows the
chirality expectation value. The dotted horizontal line indicates the value of
mass $mr_{0}$, above which the eigenmodes are extended to the bulk. Our low-
lying eigenvalues below $mr_{0}$ of the edge-localized modes agree well with
the continuum prediction marked by cross symbols. Figure 3: The spectrum of
$D_{W}D_{W}^{\dagger}$. The lattice setup is the same as Fig. 2.

Figure 4 presents the amplitude of the lowest eigenmode at the $z=1/2$ slice
of $D_{W}^{\dagger}D_{W}$ (left panel) and that of $D_{W}D_{W}^{\dagger}$
(right). The color gradation indicates the local chirality defined by Eq.
(44). As is expected, these eigenmodes are localized at the domain-wall
$r=r_{0}$ and have a uniform distribution of the chirality.

Figure 4: The amplitude of the lowest eigenstate of $D_{W}^{\dagger}D_{W}$ at
$r_{0}=24a$ and $ma=0.35$ (left panel) and that of $D_{W}D_{W}^{\dagger}$
(right) The color gradation shows the local chirality.

In order to estimate the systematic error due to the lattice discretization,
we vary the lattice spacing keeping the dimensionless quantity $mr_{0}=8.4$
fixed. In Fig. 5, we plot the relative ratio of the first eigenvalue,

$\Delta\epsilon_{j}:=\frac{|E_{j}|-|E_{j}^{\rm con.}|}{|E_{j}^{\rm con.}|},$
(45)

where $E_{j}^{\rm con.}$ is the corresponding continuum prediction obtained by
numerically solving Eq. (28). We can see that our lattice values are linearly
approaching the continuum limit with the lattice spacing $a$.

Figure 5: The relative deviation of the eigenvalue $E_{1/2}$ as a function of
the lattice spacing $a$ with $mr_{0}=8.4$.

In Fig. 6 , we plot $r_{0}|E_{1/2}|$ with a fixed lattice spacing $ma=0.35$,
as a function of $r_{0}/a$. The lattice results in lower triangle symbols are
consistent with the continuum prediction given by the solid curve, which is

$1+\frac{1}{2mr_{0}}+O(1/(mr_{0})^{2}),$

in the large $mr_{0}$ expansion. Note that the finite volume correction is not
an exponential but a power function of $r_{0}$, since the propagating fermion
at the edge is massless.

Figure 6: The finite volume effect on the eigenvalue of the lowest mode as a
function of the radius of the sphere at $ma=0.35$. The solid line represents
the continuum prediction $Er_{0}=1+1/{2mr_{0}}$.

Finally let us examine the rotational symmetry in the continuum limit. The
three plots in Fig. 7 show the same eigenfunction with $j=1/2$ with three
different lattice spacings $a/r_{0}=1/16,1/24$ and $1/32$. At the finer
lattice spacing, the better the rotational symmetry looks. The symmetry can be
quantified by taking the standard deviation $\sigma$ of the amplitude peaks at
the nearest-neighbor points to the domain-wall $r=r_{0}$ normalized by their
average $\mu$ as presented in Fig. 8. The result is not monotonic but still
shows a reasonable convergence to zero in the $a\to 0$ limit. This supports
that our digitized surface is a good regularization of the two-dimensional
sphere.

Figure 7: Top panel: The amplitude of the lowest eigenstate of
$D_{W}^{\dagger}D_{W}$ at $mr_{0}=8.4$ and $a/r_{0}=1/16$. Middle: That at
$a/r_{0}=1/24$. Bottom: That at $a/r_{0}=1/32$.

The above numerical evidence in the free case shows that we can formulate a
single Weyl fermion by this single curved domain-wall fermion on a square
lattice. In the large $mr_{0}$ limit, the finite volume effect is suppressed
by $1/mr_{0}$ and the chirality becomes exact. But below we will see that the
system shows a nontrivial behavior when a gauge field is turned on.

Figure 8: The standard deviation $\sigma$ divided by the average $\mu$ of the
peaks is plotted as a function of the lattice spacing $a$ with $mr_{0}=8.4$.

##  4 With nontrivial $U(1)$ gauge fields

Next we set nontrivial $U(1)$ gauge link variables,

$\displaystyle U_{j}(x)=\exp(i\int_{x}^{x+\hat{j}}A_{j}(x^{\prime})dx^{\prime
j}),$ (46)

where the vector potential $A_{j}(x)$ is defined in Eq. (17). The covariant
difference operator in the Wilson Dirac operator acts as

$\displaystyle(\nabla_{j}\psi)_{x}=U_{j}(x)\psi_{x+\hat{j}}-\psi_{x}.$ (47)

With these link variables, the Dirac string has no physical effect. In other
words, the singularity of the continuum gauge field is automatically canceled
by the multi-valuedness of the $U(1)$ link variables. To confirm this, let us
compare the values of the two plaquettes $p_{+}$ and $p_{-}$ in the $x$-$y$
plane whose centers are located at $x_{+}=(0,0,1/2)$ and $x_{-}=(0,0,-1/2)$,
respectively. If the Dirac string is physical, the plaquette values should be
asymmetric since the string intersects the plaquette at $x_{-}$ but does not
for the one at $x_{+}$. But this is not true: by a direct computation, we
obtain the symmetric result:

$\displaystyle p_{\pm}$
$\displaystyle=\exp\left[i\int_{-1/2}^{1/2}dxA_{1}(x,-1/2,\pm
1/2)+i\int_{-1/2}^{1/2}dyA_{2}(1/2,y,\pm 1/2)\right.$
$\displaystyle\left.-i\int_{-1/2}^{1/2}dxA_{1}(x,1/2,\pm
1/2)-i\int_{-1/2}^{1/2}dyA_{2}(-1/2,y,\pm 1/2)\right]$
$\displaystyle=\exp(\pm\pi in/3),$ (48)

which indicates that the magnetic flux from the monopole penetrating the two
plaquettes at $x_{\pm}$ is the same. On this lattice, the total magnetic flux
from the monopole is nonzero, which allows violation of the Bianchi identity
in the continuum limit.

In Fig. 9, we plot the eigenvalue spectrum of $D_{W}^{\dagger}D_{W}$ with the
monopole charge $n=1$ (top panel) and that with $n=-1$ (bottom). The lattice
set up is the same as Fig. 2. For $n=+1$, the lattice result is consistent
with the continuum prediction in cross symbols. The low lying modes with
$r_{0}|E|<mr_{0}$ have $\sim+1$ chirality as the color gradation indicates.
For $n=-1$, however, we find one near-zero mode with the opposite chirality
$\sigma_{r}=-1$, which does not exist in the continuum analysis.

Figure 9: Top: the eigenvalue spectrum of $D_{W}^{\dagger}D_{W}$ with the
monopole charge $n=1$. Other lattice setup is the same as Fig. 2. Bottom : the
same as top but with $n=-1$. We can see a near-zero mode, which does not exist
in the continuum prediction.

The amplitude of the oppositely chiral near zero mode at the $z=1/2$ slice is
plotted in Fig. 10. This mode is not located at the domain-wall but localized
at the center where the monopole sits.

Figure 10: The amplitude of the zero mode of $D_{W}^{\dagger}D_{W}$ at
$r_{0}=16a,~{}ma=0.35$ and $n=-1$.

What is the origin of the new center-localized zero mode with the opposite
chirality? In fact, in our previous work Aoki:2023lqp , we observed a similar
phenomenon in the four-spinor system where a zero Dirac spinor mode appears in
the vicinity of the monopole. Our microscopic analysis revealed that the
additive mass renormalization from the Wilson term becomes strong enough at
the singularity of the gauge field to flip the sign of the effective mass,
which creates another small but finite domain-wall near the origin. This
dynamically created domain-wall captures one edge-localized zero mode (of
electron), which explains why the monopole becomes dyon in topological
insulators Witten:1979Dyons .

In the top panel of Fig. 11, we present the distribution in the $z=1/2$ slice
of the effective mass defined by

$\displaystyle M_{\rm
eff}(x)=\frac{\phi_{0}^{\dagger}(x)\quantity[\sum_{i=1}^{3}\frac{1}{2a}\nabla_{i}\nabla_{i}^{\dagger}-m]\phi_{0}(x)}{\phi_{0}^{\dagger}(x)\phi_{0}(x)},$
(49)

where $\phi_{0}$ denotes the center-localized zero mode with $n=-1$. We can
see that an island of the positive mass region whose edge is another domain-
wall, is created. For comparison, we also plot the same quantity but with the
lowest eigen mode with $n=0$ in the bottom panel of Fig. 11, where the mass is
negative everywhere except for the edge $r=r_{0}$. Thus we can identify the
center localized zero mode as the edge mode of the new domain-wall near the
monopole.

Figure 12 shows the eigenvalue distribution of $D_{W}D_{W}^{\dagger}$. A
similar center-localized mode with positive chirality, which is opposite to
that localized at $r=r_{0}$ appears when $n=+1$.

The appearance of another domain-wall and the opposite chiral zero mode on it
may indicate that the low-energy theory is not a simple chiral theory with a
single Weyl fermion on a single domain-wall but a nontrivial vectorlike theory
on the two domain-walls.

Figure 11: Top panel: The effective mass of the zero mode at $n=-1$. Bottom:
That of the lowest mode at $n=0$. The red-white color gradation expresses the
sign of the effective mass.

Figure 12: Top: the eigenvalue spectrum of $D_{W}D_{W}^{\dagger}$ with the
monopole charge $n=1$. Other lattice setup is the same as Fig. 2. Bottom : the
same as top but with $n=-1$.

##  5 How to avoid opposite chiral modes

In this section we show in continuum analysis that the center-localized mode
is stable against perturbation of background gauge and gravity field since it
is topologically protected. It is, therefore, crucial for constructing a
chiral gauge theory in this set up, to find a formulation which eliminates
such oppositely chiral zero modes.

In order to make the problem explicit, let us consider a Hermitian operator in
continuum

$\hat{D}:=\left(\begin{array}[]{cc}0&D^{\dagger}\\\ D&0\end{array}\right),$
(50)

where the domain-wall Dirac operator $D$ is defined as Eq. (2) but with two
domain-walls:

$\displaystyle m(x)=\left\\{\begin{array}[]{cc}-m&\text{for $r_{1}\leq|x|\leq
r_{0}$}\\\ +M&\text{otherwise}\end{array}\right.,$ (53)

where we assume that $r_{1}$ is put in the vicinity of the monopole at the
inverse of the cutoff scale.

In the $M\to+\infty$ limit, the edge-localized states are the eigenmode of

$\gamma:=\sigma_{3}\otimes\sigma_{r}.$ (54)

The positive $\gamma$ modes are located at the outer domain-wall at $r=r_{0}$,
while the negative $\gamma$ modes are at $r=r_{1}$. At the outer domain-wall,
the Dirac operator acts as the two-dimensional massless operator Eq. (19) and
the same is true for $r=r_{1}$ but with different scale

$\bar{D}^{S^{2}}=\frac{1}{r_{1}}\left[\sigma^{i}\left(L_{i}+n\frac{x_{i}}{2r}\right)+1\right].$
(55)

It is not difficult to see that the two massless Dirac operator $D^{S^{2}}$ at
$r=r_{0}$ and $\bar{D}^{S^{2}}$ at $r=r_{1}$ share the same index:

$\displaystyle{\rm Ind}D^{S^{2}}={\rm Ind}\bar{D}^{S^{2}}$
$\displaystyle=\frac{1}{4\pi}\int_{S^{2}}d^{2}x\epsilon^{\mu\nu}F_{\mu\nu}=n.$
(56)

Namely, the existence of the zero modes at the two spherical domain-walls are
topologically guaranteed by the Atiyah-Singer index on them. For $n>0$, the
two zero modes with $\sigma_{r}={\rm sgn}(n)$ appear for $D$ on the $r=r_{0}$
domain-wall and another for $D^{\dagger}$ is localized at $r=r_{1}$. For
$n<0$, there are also two zero modes but the roles of the $r_{0}$ and $r_{1}$
are flipped.

The above equivalence ${\rm Ind}D^{S^{2}}={\rm Ind}\bar{D}^{S^{2}}$ is a
consequence of cobordism invariance of the AS index. Since the two spheres at
$r=r_{0}$ and $r=r_{1}$ share the same bulk region in $r_{1}<r<r_{0}$, they
are cobordant and must have the same index.

Therefore, the opposite chiral zero modes are as stable as those localized at
the edge $r=r_{0}$ against any perturbative deformation of the configuration.
Their existence in this continuum analysis looks a serious obstacle in
constructing a chiral gauge theory. The situation is apparently the same as
the flat two domain-wall system where it is difficult to decouple the unwanted
opposite chiral modes from the theory. Sometime ago, a promising proposal
using the Yang-Mills gradient flow to turn off the gauge interaction for one
of the chiral modes in the two domain-wall fermion system, was made
Grabowska:2015qpk ; Grabowska:2016bis . However it was pointed out in Ref.
Okumura:2016dsr  (see also related works Hamada:2017tny ; Makino:2016auf ;
Makino:2017pbq ; Ago:2019yzj ) that topologically nontrivial gauge background
cannot eliminate the zero modes originated from the AS index. This is also a
consequence of the cobordism invariance of the index.

Note, however, that the situation on the lattice is not exactly the same as
the continuum analysis above. First, the inner domain-wall is not fixed but
can appear only with singular gauge field configuration. Second, the center-
localized mode is effectively a zero-dimensional object, having no $\theta$
and $\phi$ dependence on the small sphere, which is isolated from the bulk
modes. Taking these differences from the flat two-domain-wall case into
account, we would like to propose two ideas which may be useful to avoid
appearance of the opposite chiral modes.

One is to impose the admissibility condition Luscher:1981zq  or a smoothness
condition for every plaquette on the lattice. Under this condition, the only
admissible link gauge fields, satisfying

$\displaystyle||1-P_{ij}(x)||<\epsilon,$ (57)

with some small real number $\epsilon$ are taken to construct the theory.
Then, singular configurations are not allowed in the theory. In the continuum
limit, the condition is equivalent to imposing the Bianchi identity.

Under the admissibility condition, we cannot put a monopole on the lattice. If
$\epsilon$ is small enough, we will be able to limit the additive mass
renormalization through the Wilson term and avoid dynamical creation of the
domain-walls. However, this would restrict the surface theory to be fixed in a
topologically trivial sector where the instanton number is forced to be zero.
By enlarging the gauge symmetry, which is a well-known prescription by ’t
Hooft and Polyakov t_Hooft1974-yf ; Polyakov1974-up , this fixing topology
problem may be circumvented but the discussion is beyond the scope of this
work.

The second proposal is to resort to the so-called symmetric mass generation.
Recently many studies Wen:2013ppa ; Wang:2013yta ; Kikukawa:2017ngf ;
Kikukawa:2017gvk ; Wang:2018jkc ; you2018symmetric ; Tong:2021phe ;
Zeng:2022grc ; Wang:2022ucy ; Lu:2023emm ; Guo:2023rnz ; Lu:2023cev ;
Liu:2023msa ; Golterman:2023zqf  have discussed possibility of gaping out the
edge-localized modes for special combinations of Weyl fermions by nontrivial
interactions such as four-Fermi vertices, without breaking the chiral
symmetry. This is possible only when their both perturbative and
nonperturbative gauge anomalies are canceled. Since the unwanted center-
localized modes in our system are essentially zero-dimensional, the analysis
may be a lot easier than those in general dimensions. Once the center-
localized zero modes were gaped out, they would have an eigenvalue which
scales as $1/r_{1}$, and we will be able to show that they are decoupled from
the low-energy dynamics and the total system is essentially chiral.

##  6 Summary and discussion

In this work, we have investigated a Shamir lattice domain-wall fermion system
with a single curved surface. We have embedded a two-dimensional spherical
domain-wall into a three-dimensional flat square lattice ignoring fermion
hopping to the outside of the domain-wall. Solving the Dirac equation of the
three-dimensional negatively massive Wilson fermion in this system, we have
examined existence of massless chiral edge-localized modes and the gravity
they feel through the induced spin connections.

In the free fermion theory with trivial gauge link variables, we have verified
the existence of the edge-localized modes on the surface. Moreover, we have
shown that these edge modes are almost chiral and massless. We have also
identified the effect of the induced gravity as a gap in the Dirac eigenvalue
spectrum. The continuum extrapolation and large volume limit are well under
control, including recovery of the rotational symmetry. Our single domain-wall
fermion system thus describes a single Weyl fermion in the low energy limit.

With nontrivial $U(1)$ gauge link variables, however, it has turned out that
the system shows a dramatic change. With the magnetic monopole-like
configuration, we have observed that a center-localized zero mode appears in
the vicinity of the monopole, having the opposite chirality to that of the
edge modes. This was not expected in the continuum analysis, under a condition
where the eigenfunctions are smooth everywhere. We have found that the
singularity of the gauge field creates another small but finite domain-wall
via additive mass renormalization near the origin, which makes the massless
zero mode localized at the wall.

We have discussed stability of the dynamically created domain-wall and the
center-localized zero mode on it and showed that they are topologically
protected by the AS index theorem on the sphere. We have then made two
proposals to avoid appearance of these opposite chiral zero modes. One is to
impose the admissibility condition, and another is the use of symmetric mass
generation. Smallness of the domain-wall radius may help in the analysis of
the center-localized mode as it is essentially a zero-dimensional object,
which is far isolated from the massive bulk modes.

We thank S. Aoki, M. Furuta, S. Iso, D.B. Kaplan, Y. Kikukawa, M. Koshino, Y.
Matsuki, S. Matsuo, T. Onogi, S. Yamaguchi, M. Yamashita and R. Yokokura for
useful discussions. In particular, we thank Y. Kikukawa for his instruction to
technical details on the symmetric mass generation. The work of SA was
supported by JSPS KAKENHI Grant Number JP23KJ1459. The work of HF and NK was
supported by JSPS KAKENHI Grant Number JP22H01219.

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