OU-HET-1164

# Curved domain-wall fermion and its anomaly inflow

Shoto Aoki  Hidenori Fukaya<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, Osaka University,
Toyonaka, Osaka 560-0043, Japan

###### Abstract

We investigate the effect of $U(1)$ gauge field on lattice fermion systems
with a curved domain-wall mass term. In the same way as the conventional flat
domain-wall fermion, the chiral edge modes appear localized at the wall, whose
Dirac operator contains the induced gravitational potential as well as the
$U(1)$ vector potential. In the case of $S^{1}$ domain-wall fermion on a two-
dimensional flat lattice, we find a competition between the Aharonov-Bohm(AB)
effect and gravitational gap in the Dirac eigenvalue spectrum, which leads to
anomaly of the time-reversal ($T$) symmetry. Our numerical result shows a good
consistency with the Atiyah-Patodi-Singer index theorem on a disk inside the
$S^{1}$ domain-wall, which describes the cancellation of the $T$ anomaly
between the bulk and edge. When the $U(1)$ flux is squeezed inside one
plaquette, and the AB phase takes a quantized value $\pi$ mod
$2\pi\mathbb{Z}$, the anomaly inflow drastically changes: the strong flux
creates another domain-wall around the flux to make the two zero modes
coexist. This phenomenon is also observed in the $S^{2}$ domain-wall fermion
in the presence of a magnetic monopole. We find that the domain-wall creation
around the monopole microscopically explains the Witten effect.

xxxx, xxx

##  1 Introduction

Lattice gauge theory has played an essential role in theoretical computations
of hadronic processes. It is usually formulated on a square lattice with
periodic or anti-periodic boundary conditions, to regularize the quantum field
theory to a well-defined integral of finite degrees of freedom. The
systematics due to finite lattice spacing and finite volume size is well
understood and the continuum limit can be taken in a controlled manner.

With a gravitational background, on the other hand, lattice field theory is
less developed, compared to the case in a flat space time. In order to
describe the nontrivial curved metric, one needs to modify the structure of
the lattice itself, which has been often studied by triangular lattices
Hamber2009Quantum ; Regge1961general ; brower2016quantum ;
AMBJORN2001347Dynamicallytriangulating ; Brower2017LatticeDirac ;
Catterall2018Topological ; ambjorn2022topology ; Brower:2019kyh ;
Brunekreef:2021iay ; Asaduzzaman:2022kxz ; Ambjorn:2022naa . The continuum
limit is, however, nontrivial as it is difficult to control the number of
sites, links, and their angles and lengths at the same time. The recovery of
the symmetry that the target continuum theory has is not obvious, either
Brower2015Quantum .

In our previous work Aoki:2022cwgCurved , we proposed a novel formulation of
curved space field theory using a fermion system on a square lattice. As a
mathematical base, we rely on Nash’s embedding theorem Nash1956TheImbedding ;
Gromov1970Embeddingsand  which states that any curved Riemannian manifold can
be isometrically embedded into a finite-dimensional flat Euclidean space and
their metric and vielbein are uniquely determined up to gauge transformations.
In our formulation a square lattice is used to regularize the flat higher
dimensional space, and the embedding is realized by a curved domain-wall in
the fermion mass term. In the same way as in the flat domain-wall fermion
KAPLAN1992342AMethod ; Shamir1993Chiral ; Furman1995Axial  the massless edge-
localized modes appear on the domain-wall, and they feel “gravity” through the
Einstein’s equivalence principle. Note that we do not assign any link or site
variables to describe gravitational degrees of freedom. They appear as
effective low-energy fields.

In the analysis of the circle $S^{1}$ domain-wall fermion on a two-dimensional
square lattice as well as the $S^{2}$-shaped curved domain-wall fermion in
three dimensions, we found that 1) chiral edge-localized modes appear at the
domain-wall, 2) they feel gravity through the induced spin connection, and 3)
the continuum extrapolation including the rotational symmetry is well
controlled, tuning the lattice spacing of the higher dimensional square
lattice.

A similar approach to our study was already discussed in condensed matter
physics. In Imura2012Spherical ; Takane2013UnifiedDescription , they found in
the continuum effective theory of a topological insulator that the edge-
localized modes on a spherical curved surface are described by the induced
spin connections. Our lattice result below agrees well with their analytic
observation. In Hotta:2013qta ; Hotta:2022aiv , it was proposed how to
demonstrate the inflation mechanism using the expanding edge of the quantum
Hall systems. In fact, our study has triggered a collaboration AFKKMinpre
with condensed matter physicists for microscopically understanding the Witten
effect Witten:1979Dyons , which is a theoretical prediction that a magnetic
monopole becomes a dyon inside topological insulators.

In this work, in constant to our previous work Aoki:2022cwgCurved  where we
studied a free fermion system only, we introduce nontrivial $U(1)$ link
variables to investigate the anomaly inflow between the bulk and edge. In
continuum theory, it is known that the time-reversal symmetry anomaly Alvarez-
Gaume:1984zst  of the bulk and edge modes is canceled through the Atiyah-
Patodi-Singer index theorem Atiyah1975spectral ; Witten:2015Fermion , to which
a mathematical relation with the domain-wall fermion Dirac operator was proved
Fukaya_2017Atiyah-Patodi-Singer ; fukayaFuruta2020physicistfriendly .

With a weak and smooth $U(1)$ flux in the $S^{1}$ domain-wall fermion, we will
show that the $\eta$ invariant of the massive Dirac operator is consistent
with the APS index on a disk. As far as we know, this is the first example of
a nontrivial APS index realized on a lattice except for the flat two or four-
dimensional tori FukayaKawai2020TheAPS .

We will also study the cases where the gauge field has a singular defect: a
vortex in two dimensions Lee:2019rfbFractionalchargebound ; Khalilov:2014rka
and a magnetic monopole Dirac:1931kp  in three dimensions Yamagishi:1982wpTHE
; Yamagishi1983Fermion-monopole ; Yamagishi1984Magnetic ; WU1976365 . For
these strong field cases, the Wilson term in the fermion action plays a key
role: it dynamically creates another domain-wall near the defects. The
creation/annihilation of the domain-walls corresponds to topology changes of
the total system, but the anomaly inflow is still consistent with the
cobordism. If we consider the Dirac operator as a Hamiltonian in one dimension
higher system Imura2012Spherical ; Takane2013UnifiedDescription ;
Rosenberg:2010iaWitteneffect ; Qi:2008ewTopologicalFieldTheory , the creation
of the domain-wall naturally explains Witten effect Witten:1979Dyons ;
Yamamoto2020Magneticmonopoles ; Rosenberg:2010iaWitteneffect ;
Qi:2008ewTopologicalFieldTheory  which indicates that the defects are
electrically charged inside topological insulators.

The rest of the paper is organized as follows. In Sec. 2, we explain our
lattice setup for curved domain-wall systems. In Sec. 3, we embed $S^{1}$
domain-wall into a square lattice and assign a weak and uniform $U(1)$
connection inside the wall. We analyze how the connection affects the dynamics
of edge modes and how the anomaly inflow is detected. In Sec. 4, we squeeze
the $U(1)$ flux in a single plaquette. We show an extra domain-wall is created
by a Wilson term and localized mode appears at the plaquette. In Sec. 5, we
confirm a similar bound state around a magnetic monopole in the $S^{2}$
domain-wall fermion system.

##  2 Curved domain-wall fermion on a lattice

In our previous work Aoki:2022cwgCurved , we investigated free fermion systems
on two and three-dimensional square lattices having a curved domain-wall in
their mass term. We found that the edge-localized modes appear at the
spherical ($S^{1}$ and $S^{2}$) domain-walls Imura2012Spherical ;
Takane2013UnifiedDescription , the effect of gravity or induced Spin and Spinc
connections encoded in the Dirac spectrum, and a monotonic scaling behavior
towards the continuum limits including recovery of the rotational symmetry.

The key mathematical ideas in our previous work are the Nash’s embedding
theorem Nash1956TheImbedding  and Einstein’s equivalence principle. The Nash’s
theorem assures that any curved Riemannian manifold can be isometrically
embedded into a finite-dimensional flat Euclidean space and its metric and
vielbein are uniquely induced by the embedding function. When the motion of
some particles or fields are constrained to the embedded manifold, they feel
gravity, by the equivalence principle, through the induced metric or Spin and
Spinc connections.

In our previous work Aoki:2022cwgCurved , we regularized the higher-
dimensional Euclidean space on a square lattice, and realized the embedding
function by the domain-wall mass term in the fermion. It this work, we
investigate the effect of electro-magnetic gauge fields focusing on the
anomaly. On a higher-dimensional square lattices, it is straightforward to
introduce the gauge fields by the standard link variables and associated
covariant difference operators.

We denote a $(n+1)$-dimensional periodic square lattice by
$(\mathbb{Z}/N\mathbb{Z})^{n+1}$. The physical lattice size is $L=N/a$ with
the lattice spacing $a$. The lattice site coordinate is given by
$x_{i}=a\hat{x}_{i}$, where $\hat{x}_{i}$ takes a half-integer value in the
range $-(N-1)/2,-(N-1)/2+1,\cdots,(N-1)/2$111Here we set the origin at the
center of $n+1$-dimensional hypercube.. The covariant difference operators in
the $i$-th direction $\nabla_{i}$ and its adjoint $\nabla_{i}^{\dagger}$ are
given by

$\displaystyle(\nabla_{\mu}\psi)_{x}$
$\displaystyle=U_{\mu}(x)\psi_{x+a\hat{\mu}}-\psi_{x}$ (1)
$\displaystyle(\nabla_{\mu}^{\dagger}\psi)_{x}$
$\displaystyle=U_{\mu}^{\dagger}(x-a\hat{\mu})\psi_{x-a\hat{\mu}}-\psi_{x},$
(2)

where $U_{\mu}(x)$ is a link variable and defined by

$\displaystyle U_{\mu}(x)=P\exp(i\int_{x+a\hat{\mu}}^{x}A).$ (3)

$A$ is a gauge field. Then the hermitian domain-wall fermion Dirac operator is
given by

$\displaystyle H$
$\displaystyle=\frac{\bar{\gamma}}{a}\quantity(\sum_{i=1}^{n+1}\quantity[\gamma^{i}\frac{\nabla_{i}-\nabla^{\dagger}_{i}}{2}+\frac{1}{2}\nabla_{i}\nabla^{\dagger}_{i}]+\text{sign}(f)ma),$
(4)

where $f:\mathbb{R}^{n+1}\to\mathbb{R}$ is a real-valued smooth function and
generates a domain-wall $Y=\set{x\in\mathbb{R}^{n+1}}{f(x)=0}$. Here we take
$m$ to be positive and the Wilson term to be unity. We denote the Dirac’s
gamma matrices by

$\displaystyle\gamma^{a}=-\sigma_{2}\otimes\tilde{\gamma}^{a},\
\gamma^{n+1}=\sigma_{1}\otimes 1,\ \bar{\gamma}=\sigma_{3}\otimes 1,$ (5)

where $\tilde{\gamma}^{a}~{}(a=1,\cdots n)$ are the $2^{[n/2]}\times
2^{[n/2]}$ gamma matrices ($[\alpha]$ denotes the Gauss symbol or the integer
part of $\alpha$) which satisfy the Clifford algebra in $n$ dimensions.

Note that the Pauli matrices $\sigma_{1,2,3}$ operate as flavor matrices
rather than spinors when $n$ is even. The doubling of the flavor space is
introduced to make the massive Dirac operator Hermitian with the
$\mathbb{Z}_{2}$ grading operator $\bar{\gamma}$.

In the continuum limit $a\to 0$, this operator converges to

$\displaystyle
H=\bar{\gamma}\quantity(\sum_{I=1}^{n+1}\gamma^{I}\quantity(\partialderivative{x^{I}}-iA_{I})+m\text{sign}(f)).$
(6)

In our previous work, we compared our numerical result for the lattice Dirac
eigenvalues and those analytically obtained in continuum to find the footprint
of gravity in the spectrum of the edge states localized at $S^{1}$ or $S^{2}$
domain-walls. In this work, we extend the analysis to the system with
nontrivial $U(1)$ link gauge fields and investigate the anomaly inflow between
bulk and edge fermion modes.

##  3 Anomaly of the $S^{1}$ domain-wall fermion

In this section, we consider an $S^{1}$ domain-wall fermion coupled to $U(1)$
gauge field on a two-dimensional square lattice. The gauge field changes the
eigenvalues of the edge-localized modes, which yields the anomaly of the time
reversal ($T$) symmetry Alvarez-Gaume:1984zst . As the whole two-dimensional
fermion system respects the $T$ symmetry, this $T$ anomaly of the edge modes
must be canceled by the bulk modes. We show how this $T$ anomaly
correspondence between the bulk and edge is protected in the curved domain-
wall system, and its consistency with the Atiyah-Patodi-Singer index on the
disk whose boundary is located at the $S^{1}$ domain-wall Witten:2015Fermion .

First let us consider a weak and uniform $U(1)$ $1$-form gauge field in
continuum theory in flat two dimensions, given by the vector potential

$\displaystyle
A=\left\\{\begin{array}[]{ll}\alpha\quantity(-\frac{y}{r_{1}^{2}}dx+\frac{x}{r_{1}^{2}}dy)=\alpha\frac{r^{2}}{r_{1}^{2}}d\theta&(r<r_{1})\\\
\alpha\quantity(-\frac{y}{r^{2}}dx+\frac{x}{r^{2}}dy)=\alpha
d\theta&(r>r_{1})\end{array}\right.,$ (9)

whose field strength is $F_{12}=2\alpha/r_{1}^{2}$ for $r<r_{1}$ and zero for
$r>r_{1}$. $\alpha$ represents the total flux (divided by $2\pi$) of the
entire system.

It is straightforward to translate this continuum vector potential into the
link variables on the lattice. The corresponding covariant difference operator
in the $x$-direction is given by (denoting $x=a\hat{x}$ and $y=a\hat{y}$)

$\displaystyle\nabla_{1}\psi_{(\hat{x},\hat{y})}=\exp(i\int_{(\hat{x}+1,\hat{y})}^{(\hat{x},\hat{y})}A)\psi_{(\hat{x}+1,\hat{y})}-\psi_{(\hat{x},\hat{y})},$
(10)

and that in $y$-direction is defined by the same manner. It is obvious that
this operator converges to $\partial_{i}-iA_{i}$ in the continuum limit. In
order to achieve the periodic boundary condition on the fermion, we set the
link variables connecting $x=N/2$ and $-N/2$ for all $y$ and those connecting
$y=N/2$ and $-N/2$ for all $x$ to unity. The field strength around the
boundary gives little effect on the edge-mode spectrum.

With these link variables, we numerically solve the eigenproblem of the
hermitian Dirac operator

$\displaystyle
H=\frac{1}{a}\sigma_{3}\quantity(\sum_{i=1,2}\quantity[\sigma_{i}\frac{\nabla_{i}-\nabla^{\dagger}_{i}}{2}+\frac{1}{2}\nabla_{i}\nabla^{\dagger}_{i}]+a\epsilon
m),$ (11)

where $\sigma_{i}$ are the Pauli matrices, and we assign the $S^{1}$ domain-
wall with the radius $r_{0}>r_{1}$ by the step function

$\displaystyle\epsilon(\hat{x},\hat{y})=\left\\{\begin{array}[]{cc}-1&(a\hat{r}<r_{0})\\\
1&(a\hat{r}>r_{0})\end{array}\right..$ (14)

We illustrate the lattice system in Fig. 1.

Figure 1: Our two-dimensional square lattice set up. The outer circle
represents the $S^{1}$ domain-wall with radius $r_{0}$ and inside the inner
circle with radius $r_{1}$ we put the uniform $U(1)$ gauge field strength.

If we regard this two-dimensional system as a layer or sheet in three space
dimensions, the $U(1)$ flux corresponds to a magnetic field penetrating the
sheet in the range $r<r_{1}$. For the fermion modes localized at $r=r_{0}$,
the magnetic field does not directly interact, but the Aharonov-Bohm effect
Aharonov:1959fk  changes the spectrum of $H$.

In order to test the essential property of the edge-localized modes, we also
numerically evaluate the expectation value of the gamma matrix in the normal
direction of the $S^{1}$ domain-wall

$\displaystyle\gamma_{\text{normal}}=\frac{x}{r}\sigma_{1}+\frac{y}{r}\sigma_{2},$
(15)

which is well-defined when $N$ is even. We call $\gamma_{\text{normal}}$ the
“chirality” operator by analogy from the standard flat domain-wall fermion in
five dimensions, although in our two-dimensional case there is already a more
legitimate operator $\bar{\gamma}=\sigma_{3}$.

In Fig. 2, we plot the lattice results for the eigenvalues of $H$ as a
function of $\alpha$. Here we set $m=14/L$, $L=40a$, $r_{0}=\frac{L}{4}$ and
$r_{1}=\frac{r_{0}}{2}$. We can clearly see that the spectrum is monotonically
decreasing linearly with $\alpha$.

Figure 2: The eigenvalue spectrum of $H$ at $m=14/L$, $r_{0}/a=10$,
$r_{1}/a=5$ and $L/a=40$. The filled circles show the lattice data and the
solid lines are the continuum prediction in Eq. (19). The color gradation of
the data points from deep blue through light yellow to deep red represents the
expectation value of the “chirality” operator $\gamma_{\text{normal}}$.

The linear dependence of the spectrum is due to the Aharonov-Bohm effect. In
the large $m$ limit in the continuum theory, we can analytically solve the
eigenproblem of $H$ (see Appendix A). The effective Dirac operator on the
edge-localized modes reads

$\displaystyle
ir_{0}{\ooalign{\hfil/\hfil\crcr$D$}}^{S^{1}}=-i\partialderivative{\theta}+\frac{1}{2}-\alpha,$
(18)

where $\theta$ denotes the coordinate of the $S^{1}$ in units of the radius
$r_{0}$. $1/2$ in the second term is the induced $Spin^{c}$ connection and
$\alpha$ originates from the Aharonov-Bohm phase of the wave function. The
eigenvalues are then obtained as

$\displaystyle E_{n}r_{0}=n+\frac{1}{2}-\alpha$
$\displaystyle~{}(n\in\mathbb{Z}).$ (19)

Note that the effect of the $Spin^{c}$ connection $1/2$ is canceled by the
Aharonov-Bohm effect at $\alpha=1/2$. At $\alpha=1$ the spectrum comes back to
the original distribution at $\alpha=0$ but each eigenvalue shifts from the
level $n$ to the next lower level $n-1$. This behavior is similar to the
Thouless charge pump Thouless1983Quantization . This continuum prediction
presented in Fig. 2 by the solid lines agrees well with the lattice data.

The cut-off dependence of the relative eigenvalue difference for
$E_{i}(i=0,1,2,3)$ at $\alpha=0.25$ is plotted in Fig. 3, which shows a
monotonous decrease towards the continuum limit.

Figure 3: The relative deviation of the eigenvalue
$\quantity(E_{n}-E_{n}^{\text{cont.}})/E_{n}^{\text{cont.}}$, where
$E_{n}^{\text{cont.}}$ denotes the continuum prediction.

As the color gradation of the data points from the dark red ($\gamma=-1$) to
dark blue ($\gamma=+1$) shows, we find that the eigenmodes with
$\absolutevalue{E}<m$ are almost “chiral” or $\gamma_{\text{normal}}\sim+1$.
We also find that these modes are well localized at $r=r_{0}$ as Fig. 4
indicates.

Figure 4: The amplitude of the edge modes with $E_{0\mathchar 45\relax 3}$ at
$\alpha=0.5$, $m=14/L$, $r_{0}/a=10$, $r_{1}/a=5$ and $L/a=40$.

The linear shift of the Dirac eigenvalues due to $\alpha$ reflects the fact
that the edge massless Dirac fermion suffers from the anomaly of the $T$
symmetry. The asymmetry of the spectrum is measured by the Atiyah-Patodi-
Singer $\eta$ invariant Atiyah1975spectral

$\displaystyle\eta(i{\ooalign{\hfil/\hfil\crcr$D$}}^{S^{1}})=$
$\displaystyle\lim_{\epsilon\to 0}\sum_{\lambda\neq
0}\frac{\lambda}{\absolutevalue{\lambda}^{1+\epsilon}}+\dim\text{Ker}(i{\ooalign{\hfil/\hfil\crcr$D$}}^{S^{1}})\sum_{\lambda=0}1,$
(24)

where $\lambda$ are the eigenvalues of
$i{\ooalign{\hfil/\hfil\crcr$D$}}^{S^{1}}$. We can directly confirm with the
Pauli-Villars regularization that the phase of the fermion determinant is the
$\eta$ invariant:

$\displaystyle\lim_{\mu\to\infty}\det\frac{{\ooalign{\hfil/\hfil\crcr$D$}}^{S^{1}}}{{\ooalign{\hfil/\hfil\crcr$D$}}^{S^{1}}+\mu}\sim\prod_{\lambda}\frac{-i\lambda}{\mu}\propto\exp(-i\frac{\pi}{2}\eta(i{\ooalign{\hfil/\hfil\crcr$D$}}^{S^{1}})).$
(31)

Since the sign of each eigenvalue flips under $T$ transformation, the edge
mode partition function breaks the $T$ symmetry when
$\eta(i{\ooalign{\hfil/\hfil\crcr$D$}}^{S^{1}})/2$ is a non-integer, even when
the fermion action is $T$ invariant.

However, for the total two-dimensional lattice Dirac operator, $\eta(H)$ is
always an even integer, and $\det H$ is real. This indicates that the above
$T$ anomaly of the edge modes is canceled by the bulk contribution so that the
$T$ symmetry of the total system is protected. In Refs. Fukaya_2017Atiyah-
Patodi-Singer ; fukayaFuruta2020physicistfriendly  it was mathematically
proved that for any domain-wall fermion Dirac operator $H$ on a closed
Riemannian manifold $X$, $-\eta(H)/2$ is guaranteed to be an integer since it
is equal to the Atiyah-Patodi-Singer index on a submanifold $X_{-}$, or the
negative region of the fermion mass. In FukayaKawai2020TheAPS , it was shown
on a square lattice with a flat domain-wall, $-\eta(H)/2$ is consistent with
the APS index, assuming that the link variables are sufficiently smooth.

In our two-dimensional curved domain-wall fermion system, we can analytically
estimate the edge mode contribution as

$\displaystyle-\frac{1}{2}\eta(i{\ooalign{\hfil/\hfil\crcr$D$}}^{S^{1}})=$
$\displaystyle-\frac{1}{2}\quantity(\lim_{\epsilon\to
0}\sum_{n\in\mathbb{Z}}\frac{n+\frac{1}{2}-\alpha}{\absolutevalue{n+\frac{1}{2}-\alpha}^{1+\epsilon}}+\\#\set{\text{zero
modes}})$ (34) $\displaystyle=$ $\displaystyle[\alpha+\frac{1}{2}]-\alpha,$
(35)

where $[A]$ denotes the Gauss symbol taking the integer value of $A$. Since it
is obvious that the total $\eta$ invariant is
$-\eta(H)/2=[\alpha+\frac{1}{2}]$, the bulk contribution must be $+\alpha$,
which agrees well with the perturbative evaluation FukayaKawai2020TheAPS

$\displaystyle-\frac{1}{2}\eta(H)^{\text{b}ulk}=\frac{1}{2\pi}\int_{r<r_{1}}F=\frac{1}{2\pi}\int_{S^{1}}A=\alpha.$
(36)

This observation indicates that the $T$ anomaly matching between bulk and edge
is well described through the $\eta$ invariant of $H$ or the APS index on our
square lattice even when the domain-wall is curved.

As far as we know, this is the first example of a nontrivial APS index on a
curved manifold realized in lattice gauge theory setup.

##  4 Topology change of the domain-wall and Witten effect

In the previous section, a uniform weak gauge field background has been
considered. We have confirmed the $T$ anomaly inflow on a square lattice,
where the asymmetry of the edge localized mode’s spectrum is compensated by
the bulk contribution, and the $T$ symmetry is protected as a consequence of
the APS index theorem on a two-dimensional disk. In this section, we consider
a strong gauge field concentrated in a short range, which yields a more
dynamical change to the fermion system.

###  4.1 Topology changes: creation and annhilation of the domain-wall

Let us take a very small value of $r_{1}$, while the total $U(1)$ flux is
unchanged. Then, at $r_{1}<a/2$, only one plaquette at the origin gives $-1$,
representing a strong gauge field. With such a high energy configuration in a
short range, it is nontrivial if the anomaly inflow persists via the APS index
theorem. As is shown below, the anomaly matching is still valid, but in a
drastically different way from the weak field case in the previous section.

Figure 5 shows the $\alpha$ dependence of the Dirac eigenvalue spectrum. The
uniform linear shift with $\alpha$ of the edge-localized modes with positive
chirality is seen as in the weak field case. However, a remarkable difference
is that one negative chirality mode appears when $\alpha=0.2$ at energy level
$\sim-m$, crosses zero at $\alpha=1/2$ and disappears at the level $\sim m$.

Figure 5: The eigenvalue spectrum of (11) $m=14/L$, $\hat{r}_{0}=r_{0}/a=10$,
$\hat{r}_{1}=0.001$ and $L=40a$. The filled circles show the lattice data and
the solid lines are the continuum prediction.

In fact, as presented in Fig. 7 this mode is localized at the origin, where we
do not assign any domain-wall structure in the mass term but the strong $U(1)$
gauge field is located. Since there was no such mode in the weak field case in
the previous section, it is natural to assume that this mode is somehow
excited by the strong gauge field potential.

Figure 6: The amplitude of the eigenstate localized at the origin when
$\alpha=0.4$, $m=14/L$, $r_{0}/a=10$, $r_{1}<a/2$ and $L/a=40$.

Figure 7: The ratio of the effective mass $M_{eff}$ and $m=\absolutevalue{M}$
when $m=14/L$, $\hat{r}_{0}=r_{0}/a=10$, $\hat{r}_{1}=0.001$ and $L=40a$.

We numerically investigate this issue and find that the Wilson term plays a
key rule. As is well known, the Wilson term gives additive mass
renormalization due to the violation of the chiral symmetry. In lattice QCD it
is known that the stronger the coupling is, the larger the additive mass shift
becomes in the positive direction. We confirm that the same mass shift happens
in our system but only locally near the origin where the strong background is
given. In Fig. 7, we plot the local expectation value of the “effective” mass

$\displaystyle
M_{eff}=\frac{1}{\psi(x)^{\dagger}\psi(x)}\psi(x)^{\dagger}\quantity(\epsilon
m+\sum_{i=1,2}\frac{1}{2a}\nabla_{i}\nabla^{\dagger}_{i})\psi(x),$ (37)

where the second term contains the dependence on the link variables. The
renormalized mass near the origin becomes positive, creating another domain-
wall around it. We can identify this as dynamical creation of the domain-wall
and the center-localized mode appears as the new edge mode on it.

In terms of the APS index expressed by the $\eta$ invariant of $H$, the
topology of the target spacetime defined by the negative mass region, is
changed from a disk with one $S^{1}$ boundary to a cylinder with two $S^{1}$
boudaries at $r=r_{0}$ and $r=r_{1}$. Thus, the $U(1)$ gauge field can make
creation/annihilation of the domain-walls and topology changes of the bulk
manifold on which the APS index describes the anomaly inflow.

In particular, it is interesting to see the case $\alpha=0.5$, where two zero
modes appear: one is on the original edge at $r=r_{0}$ and another center-
localized mode (to be precise, they are split due to the interference, which
will be discussed later in detail). In this case, the Dirac operator becomes
real and the number of zero modes is known to be the mod-two index
AtiyahSinger1971TheIndex5 , exhibiting a global anomaly Witten:1982fp . On
$S^{1}$, the mod-two index, being a cobordism invariant, must be zero when it
is boundary of a disk. Existence of the two zero modes can be understood as a
new type of anomaly inflow: changing topology from disk to cylinder by the
dynamical creation of the domain-wall, another zero-mode at the center appears
in order to make the total mod-two index zero (mod 2) and the system free from
global anomaly.

In the above discussion on the dynamical creation of the domain-wall, the
Wilson term, which represents a higher derivative term in the continuum limit,
is essential. It is interesting to ask if we can understand the center-
localized mode in continuum theory, too. We find a positive answer to this
question but the Dirac equation requires a second derivative term with $M=-m$
which can be understood as a Pauli-Villars contribution.

The details of the computation are given in Appendix B. We confirm that for
$\absolutevalue{E}<m$ only one solution exists and the obtained $E$ as a
function of $\alpha$ is plotted in Fig. 5, where agrees well with our lattice
data.

###  4.2 Microscopic description of Witten effect

The appearance of the center-localized fermion can be viewed as an electric
“charge” of the electro-magnetic defect. It was shown by Witten
Witten:1979Dyons  that a monopole in a topological insulator, or equivalently
in the $\theta=0$ vacuum, obtains a half electric charge. Although our object
is not a monopole but a vortex, let us call the phenomenon that some defect in
the electro-magnetic field obtains a fractional charge
Lee:2019rfbFractionalchargebound ; Khalilov:2014rkaBoundstates ;
Rosenberg:2010iaWitteneffect ; Qi:2008ewTopologicalFieldTheory ;
Zhao2012Amagneticmonopole  the “Witten effect” since we believe that their
microscopic mechanism is essentially the same, as explained below222We will
discuss the monopole case in the next section..

To this end, let us switch the view from two-dimensional Euclidean spacetime
to two-dimensional space with the Hamiltonian given by (11). The negative mass
region with $r<r_{0}$ is regarded to be a topological insulator, while outside
is normal insulator. We can regard the $U(1)$ flux as a strong magnetic field
penetrating the sheet.

Let us then set the Fermi energy to zero so that all negative states are
occupied by the valence electrons. This is the so-called half-filled state.
Then Fig. 5 indicates that one conduction band localized at the $S^{1}$
domain-wall meets zero at $\alpha=0.5$ and one valence electron is pumped up
to the zero energy at the center. By quantum tunneling effect, these two zero-
modes mix and the energy levels are split, as Fig. 8 shows. In Fig. 9, we plot
the distribution of $\psi^{\dagger}\psi(r)\times 2\pi r$ (left panel), and its
integral up to $r$ (right panel). We can see that the $50$% of the electron
wave function is concentrated at the origin, where the point-like magnetic
field is located.

Figure 8: Enlarged view of the $\alpha=0.5$ of Fig. 2

Figure 9: Left panel shows the amplitude of two zero-modes in $r$ direction
and right panel indicates the accumulation of that when $m=14/L$,
$\hat{r}_{0}=r_{0}/a=10$, $\hat{r}_{1}=0.001$, $L=40a$ and $\alpha=0.5$. The
two zero-modes are localized at the wall and the origin by the tunneling
effect. So their energy are slightly deviated from zero.

We can interpret the above discussion as a microscopic description of the
Witten effect. In the effective theory approach, the Witten effect is
described by the Chern-Simons action, which induces an electric field to the
monopole, becoming a dyon with charge $1/2$. In our lattice set up, we have
shown that the point-like magnetic field (by the vortex) creates a domain-wall
around it to capture the electron wave function clinging to the defect. At
$\alpha=1/2$ where the time-reversal symmetry is recovered, the electron is
captured without any energy cost. Thus every vortex is charged.

It is important to note that another half of the electron wave function exists
at the surface of the topological insulator, and the total electron charge is
kept integral. It is also remarkable that the $U(1)$ gauge field is singular
at $x=-L/2,~{}L/2$ and $y=-L/2,~{}L/2$ due to the periodic boundary conditions
on the lattice, but no localized modes appear near the singularity where the
mass term is positive.

Even if we remove the domain-wall, and set the mass negative everywhere, we
can not find a zero mode localized at $x=-L/2,~{}L/2$ and $y=-L/2,~{}L/2$
since the flux at the singular edge is too weak to constrain a fermion. The
singular edge breaks the $T$-symmetry so mod $2$ index can’t classify this
system.

On the other hand, when we set a gauge field

$\displaystyle U_{1}(x,y)$ $\displaystyle=1$ (38) $\displaystyle U_{2}(x,y)$
$\displaystyle=\left\\{\begin{array}[]{cc}-1&(x>0,~{}y=-\frac{a}{2})\\\
+1&(\text{otherwize})\end{array}\right..$ (41)

We can maintain the $T$ symmetry. In this case, an anti-vortex appears at
$(x,y)=(0,\frac{L}{2})$ in addition to the original vortex. Each vortex binds
one fermion with $E=0$ (see Fig. 10). Even number of zero-modes appear on the
lattice space to cancel the anomaly.

Figure 10: Two zero-modes appear at the vortex and anti-vortex.

##  5 Anomaly of the $S^{2}$ domain-wall and Witten effect

In this section, we consider a two-flavor $S^{2}$ domain-wall fermion on a
three-dimensional square lattice. Because of the two-flavor structure having
both of left- and right-handed chiralities, the edge modes represent a Dirac
fermion field, rather than a Weyl fermion. The only possible anomaly they have
is the standard axial $U(1)$ anomaly Adler1969Axial-Vector ;
Bell1969APCACpuzzle , which gives a nontrivial AS index atiyah1963index ;
AtiyahSinger1986TheIndex1 ; AtiyahSegal1968TheIndex2 ;
AtiyahSinger1968TheIndex3 ; AtiyahSinger1971TheIndex4  of the Dirac operator
on $S^{2}$.

The AS index is, however, a cobordism invariant, which cannot be nonzero when
the $S^{2}$ is a boundary of a three-dimensional bulk ball $B^{3}$. Therefore,
when we force the domain-wall fermion to have a zero mode, we expect a
dynamical domain-wall creation at the singularity of the gauge field to change
the topology of the negative region of the fermion mass, and have another zero
mode to cancel the axial $U(1)$ anomaly333The region where the mass is
negative is topologically equivalent to $I\times S^{2}$, and there are two
boundaries whose orientations are opposite to each other. Since the chirality
is defined with the opposite sign for the two $S^{2}$, the $U(1)$ anomaly is
canceled between them. .

The gauge field configuration which gives a nonzero index of the massless
Dirac operator on $S^{2}$ is equivalent to putting a magnetic monopole
Dirac:1931kp  inside the domain-wall. Therefore, if the field strength is
strong enough to change the sign of the mass term from negative to positive
near the monopole, creating another domain-wall, the appearance of the edge
modes makes the monopole charged. This is exactly the same microscopic
description as the previous section where a vortex is changed.

The Dirac monopole on the three-dimensional square lattice can be put by the
$U(1)$ link variables assigned as

$\displaystyle
U_{\mu}(p)=\exp(i\int_{p+a\hat{\mu}}^{p}A)~{}(A=n\frac{1-\cos\theta}{2}d\phi),$
(42)

where $n$ is a magnetic charge. In the same way as the two-dimensional
lattice, we impose the periodic boundary conditions on the fermions, setting
the link variables from connecting $x=N/2$ and $-N/2$ for all $y,z$, those
connecting $y=N/2$ and $-N/2$ for all $x,z$ and those at $z=N/2$ for all $x,y$
to unity. The field strength produced around the boundary gives little effect
on the edge-mode spectrum. Note in our formulation that the Dirac string
located between $(0,0,0)$ and $(0,0,-L/2)$ is invisible as the plaquette
becomes $\exp(2\pi n)=1$. This reflects the fact that the AB effect disappears
when the magnetic flux is quantized.

With the above link configuration, let us consider the lattice Dirac operator

$\displaystyle
H=\frac{1}{a}\gamma^{5}\quantity(\quantity[\sum_{i=1}^{3}\gamma^{i}\frac{\nabla_{i}-\nabla^{\dagger}_{i}}{2}+\frac{1}{2}\nabla_{i}\nabla^{\dagger}_{i}]+\epsilon_{A}am),$
(43)

where we set $\gamma^{i}=\sigma_{1}\otimes\sigma_{i}$ and
$\gamma^{5}=\sigma_{3}\otimes 1$. The chirality operator is defined by

$\displaystyle\gamma_{\text{normal}}=\frac{x}{r}\gamma^{1}+\frac{y}{r}\gamma^{2}+\frac{z}{r}\gamma^{3}=\sigma_{1}\otimes\frac{x^{a}\sigma_{a}}{r}.$
(44)

In Fig. 11, we plot the lattice data for the eigenvalues of $H$, where we set
$m=14/L$, $L=20a$, $r_{0}=\frac{L}{4}$ and the monopole charge $n=1$. Similar
to the $S^{1}$ domain-wall fermion case, the edge modes with
$\gamma_{\text{normal}}=+1$ appear between $\pm m$.

Figure 11: The spectrum of (43) near $E=0$. We set $m=14/L$, $L=20a$,
$r_{0}=\frac{L}{4}$ and $n=1$.

Let us compare the result with the continuum theory in the large $m$ limit.
The effective Dirac operator on $S^{2}$ on the edge modes is

$\displaystyle
i{\ooalign{\hfil/\hfil\crcr$D$}}^{S^{2}}=i\quantity(\sigma_{1}\partialderivative{\theta}+\sigma_{2}\frac{1}{\sin\theta}\quantity(\partialderivative{\phi}+\frac{i}{2}-\frac{\cos\theta}{2}\sigma_{1}\sigma_{2}-in\frac{1-\cos\theta}{2})),$
(47)

where $\theta$ and $\phi$ represent polar angle and azimuthal angle,
respectively Tamm1931monopole ; WU1976365 . In the innermost parenthesis, the
second and third terms are the induced gravitational $Spin^{(c)}$ connection.
The fourth term is the $U(1)$ connection given by the monopole with magnetic
charge $n$.

Since the angular momentum is conserved in the continuum theory, the
eigenvalues labelled by its quantized numbers $j$ (of the angular momentum
squared) and $j_{3}$ (in the $z$ direction), are obtained as Tamm1931monopole

$\displaystyle
E_{j,j_{3}}r_{0}=\pm\sqrt{\quantity(j+\frac{1}{2})^{2}-\frac{n^{2}}{4}},~{}j=\frac{\absolutevalue{n}-1}{2},~{}\frac{\absolutevalue{n}-1}{2}+1,\cdots,~{}j_{3}=-j,-j+1,\cdots+j,$
(48)

where we have $2j+1$ degeneracies as usual. Note when $n=1$ and $j=0$, we have
a zero mode. However, we can see two would-be zero modes in Fig. 11, which
cannot be explained by the continuum Dirac operator on $S^{2}$.

As shown in Fig. 12, these two near zero modes have peaks at $r=0$ and
$r=r_{0}$. which gives a strong evidence for the topology change of our
domain-wall fermion system and the two zero modes around $r=0$ and $r=r_{0}$
are mixed through the tunneling effect. The charge the monopole dresses is
thus, $\sim 1/2$.

Figure 12: The amplitude of two zero-modes in $r$ direction when $m=14/L$,
$L=20a$, $r_{0}=L/4$ and $n=1$.

Let us also examine the effective mass shift due to the strong magnetic field.
Fig. 13 represents the position dependence of the effective mass,

$\displaystyle
M_{eff}=\frac{1}{\psi(x)^{\dagger}\psi(x)}\psi(x)^{\dagger}\quantity(\epsilon
m+\sum_{i=1}^{3}\frac{1}{2a}\nabla_{i}\nabla^{\dagger}_{i})\psi(x)$ (49)

which clearly indicates that another domain-wall is created near the origin,
where the mass is shifted from negative to positive. Nothing special happens
to the anti-monopole located in the normal insulator region.

Figure 13: The ratio of the effective mass $M_{eff}$ and $m=\absolutevalue{M}$
when $m=14/L$, $\hat{r}_{0}=r_{0}/a=5$ and $L=20a$.

Our numerical data above supports our microscopic scenario for the Witten
effect that the $U(1)$ gauge field from the monopole inside the topological
insulator yields a mass shift, creating another domain-wall and the edge-
localized mode is the origin of the electric charge. The electric charge of
the dyon is 1/2 since another half is distributed at the surface of the
topological insulator by the tunneling effect.

##  6 Summary

We have analyzed lattice fermion systems with the spherical $S^{1}$ and
$S^{2}$ domain-wall mass terms embedded into two- and three-dimensional square
lattices, respectively. Putting nontrivial $U(1)$ link variables, we have
investigated the Dirac eigenvalue spectrum as well as the profile of the
corresponding eigenfunctions to understand the interplay between induced
gravity and gauge field on the edge-localized modes.

In sec. 3, we have considered the $S^{1}$ domain-wall fermion on a two-
dimensional square lattice. When we put a weak $U(1)$ gauge field inside the
domain-wall, the gravitational effect encoded in the gap of the Dirac spectrum
competes with the AB effect caused by the $U(1)$ flux. The uniform eigenvalue
shift in the Dirac spectrum is consistent with the nonzero $\eta$ invariant of
the edge-localized mode, or its time-reversal symmetry anomaly. Then the
$\eta$ invariant of the total two-dimensional domain-wall fermion Dirac
operator is consistent with the APS index theorem in continuum theory on the
two-dimensional disk, with which the $T$ symmetry of the whole system is
protected.

When we shrink the $U(1)$ flux to form a vortex, the AB effect on the edge-
localized modes is unchanged but the intense gauge field inside makes a
drastic change in the bulk. We have revealed an exponentially localized mode
at the flux, which has an opposite chirality to the edge modes on the $S^{1}$
domain-wall. In our microscopical analysis, this mode can be interpreted as an
edge mode sitting at another small domain-wall dynamically created by the
additive mass renormalization via the Wilson term around the vortex.

In particular, the two Dirac eigenvalues, one is located at the $S^{1}$
domain-wall, and the other localized at the new domain-wall around the vortex,
approach to zero when the AB phase is $\pi$, which is the $T$-symmetric point.
Pairing of the two zero modes reflects the fact that the mod two index is a
cobordism invariant: they must appear in pairs, to make the total index
trivial, on a manifold that is a boundary of a higher dimensional manifold.
When we regard the Dirac operator as a Hamiltonian in $d=2$ space dimensions,
rather than in $1+1$ space-time dimensions, the appearance of the two zero
modes and their small split due to tunneling between two domain-walls gives a
microscopic description how the vortex gains a fractional electric charge.

We have also analyzed the $S^{2}$ domain-wall fermion system in the presence
of a magnetic monopole at the origin. The Atiyah-Singer index theorem on the
two-dimensional sphere indicates a zero edge localized mode on it. The axial
$U(1)$ anomaly cancellation of the total three-dimensional system is then
required as the same cobordism argument as the $S^{1}$ domain-wall, and
suggests a dynamical creation of another domain-wall around the monopole.

We have numerically confirmed the sign flip of the effective fermion mass
around the monopole, and thus creation of the domain-wall, on which the edge-
localized zero-mode appears. We have also found that 50% of the zero mode’s
amplitude is localized at the monopole, while the other 50% is at the $S^{2}$
domain-wall. We believe that the analysis gives a microscopic description of
the Witten effect that the monopole gains a 1/2 charge of an electron. We are
now trying to analytically confirm the results exactly solving the
“negatively” massive Dirac equation with a Wilson term AFKKMinpre .

## Acknowledgment

We thank M. Furuta, K. Hashimoto, S. Iso, M. Kawahira, N. Kan, M. Koshino, Y.
Matsuki, S. Matsuo, T. Onogi, S. Yamaguchi, M. Yamashita and R. Yokokura for
useful discussions. This work was supported by JST SPRING, Grant Number
JPMJSP2138, and in part by JSPS KAKENHI Grant Number JP18H01216, JP18H04484
and JP22H01219.

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##  Appendix A Edge mode at $S^{1}$ domain-wall with the $U(1)$ flux

In this section, we analytically solve the eigenvalue of edge modes of

$\displaystyle
H=\sigma_{3}\quantity(\sum_{i=1,2}\sigma_{j}(\partial_{j}-iA_{j})+M),$ (50)

where $M=m\text{sign}(r-r_{0})$ and $A$ is given by (9). This operator is the
continuum limit of (11) and acts on a periodic spinor field. Setting $A=\alpha
d\theta$ on the whole of the two-dimensional Euclidean space, the operator
becomes

This operator commutes with the total angular momentum
$J=-i\partialderivative{\theta}+\frac{1}{2}\sigma_{3}$, whose eigenvalue takes
$j\in\frac{1}{2}+\mathbb{Z}$. The eigenfunction with $J=j$ localized at the
wall is given by

$\displaystyle\psi^{E,j}$
$\displaystyle=\left\\{\begin{array}[]{ll}A\matrixquantity(\sqrt{m^{2}-E^{2}}I_{\absolutevalue{j-\frac{1}{2}-\alpha}}(\sqrt{m^{2}-E^{2}}r)e^{i(j-\frac{1}{2})\theta}\\\
(m+E)I_{\absolutevalue{j+\frac{1}{2}-\alpha}}(\sqrt{m^{2}-E^{2}}r)e^{i(j+\frac{1}{2})\theta})&(r<r_{0})\\\
B\matrixquantity((m+E)K_{j-\frac{1}{2}-\alpha}(\sqrt{m^{2}-E^{2}}r)e^{i(j-\frac{1}{2})\theta}\\\
\sqrt{m^{2}-E^{2}}K_{j+\frac{1}{2}-\alpha}(\sqrt{m^{2}-E^{2}}r)e^{i(j+\frac{1}{2})\theta})&(r>r_{0}).\end{array}\right.$
(53)

The connection condition $r=r_{0}$

$\displaystyle\frac{I_{\absolutevalue{j-\frac{1}{2}-\alpha}}}{I_{\absolutevalue{j+\frac{1}{2}-\alpha}}}\frac{K_{j+\frac{1}{2}-\alpha}}{K_{j-\frac{1}{2}-\alpha}}(\sqrt{m^{2}-E^{2}}r_{0})=\frac{m+E}{m-E},$
(54)

determines the eigenvalue $E$. In the large mass limit or $m\gg E$, the
eigenvalue converges to

$\displaystyle E\simeq\frac{j-\alpha}{r_{0}}\
\quantity(j=\pm\frac{1}{2},\pm\frac{3}{2},\cdots).$ (55)

##  Appendix B Eigenstate localized at the $U(1)$ flux

In this appendix, we analytically show that a domain-wall is dynamically
created near the vortex, which captures a bound state of an electron. Here, we
ignore the outside of the $S^{1}$ domain-wall or take the $r_{0}\to\infty$
limit and consider the operator

$\displaystyle
H=\sigma_{3}\quantity(\sigma_{j}D_{j}-\frac{1}{2M_{PV}}D^{2}+M),$ (56)

where $M=-m<0$, $M_{PV}=1/a$ and $D_{j}=\partial_{j}-iA_{j}$ is a covariant
derivative in $j$-th direction. Note that the Wilson term is put by hand, in
order to make the sign of fermion mass well-defined Zhao2012Amagneticmonopole
. This operator also commutes with
$J=-i\partialderivative{\theta}+\frac{1}{2}\sigma_{3}$. The eigenfunction can
be written as $\psi=\matrixquantity(f(r)e^{i(j-\frac{1}{2})\theta}\\\
g(r)e^{i(j+\frac{1}{2})\theta})$.

###  B.1 Exterior $(r>r_{1})$

In the region of $r>r_{1}$, the operator (56) is given by

where
$D^{2}=\partialderivative[2]{r}+\frac{1}{r}\partialderivative{r}-\frac{1}{r^{2}}\quantity(i\partialderivative{\theta}+\alpha)^{2}$.
Then the equation in the $r$-direction is

where $E$ is an eigenvalue of the localized mode. $a_{j}$ and
$a_{j}^{\dagger}$ are a differential operator defined by

$\displaystyle
a_{j}=-\partialderivative{r}+\frac{j-\frac{1}{2}-\alpha}{r},~{}a_{j}^{\dagger}=\partialderivative{r}+\frac{j+\frac{1}{2}-\alpha}{r}.$
(57)

Assuming $a_{j}f\propto g$ and $a_{j}^{\dagger}g\propto f$, we find a set of
solutions in the term $f=aK_{j-\frac{1}{2}-\alpha}(\kappa r)$ and
$g=bK_{j+\frac{1}{2}-\alpha}(\kappa r)$ for a complex number $a$, $b$ and
$\kappa$. The coefficients $a$ and $b$ satisfy

There are two $\kappa$’s that satisfy the above equation:

$\displaystyle\frac{\kappa^{2}}{M_{PV}^{2}}=\frac{\kappa_{\pm}^{2}}{M_{PV}^{2}}=\quantity(\pm
1+\sqrt{1+2\frac{M}{M_{PV}}+\frac{E^{2}}{M_{PV}^{2}}})^{2}-\frac{E^{2}}{M_{PV}^{2}}.$
(58)

Thus two eigenstates are

$\displaystyle\psi_{o\pm}=\matrixquantity(\quantity(M-\frac{\kappa_{\pm}^{2}}{2M_{PV}}+E)K_{j-\frac{1}{2}-\alpha}(\kappa_{\pm}r)e^{i(j-\frac{1}{2})\theta}\\\
\kappa_{\pm}K_{j+\frac{1}{2}-\alpha}(\kappa_{\pm}r)e^{i(j+\frac{1}{2})\theta}).$
(59)

###  B.2 Interior $(r<r_{1})$

For $r<r_{1}$ the Dirac operator is given by

where
$D^{2}=\partialderivative[2]{r}+\frac{1}{r}\partialderivative{r}-\frac{1}{r^{2}}\quantity(i\partialderivative{\theta}+\alpha\frac{r^{2}}{r_{1}^{2}})^{2}$.
$f$ and $g$ satisfy

where $a_{j}$ and $a_{j}^{\dagger}$ are defined by

$\displaystyle
a_{j}=-\partialderivative{r}+\frac{1}{r}\quantity(j-\frac{1}{2}-\alpha\frac{r^{2}}{r_{1}^{2}}),~{}a_{j}^{\dagger}=\partialderivative{r}+\frac{1}{r}\quantity(j+\frac{1}{2}-\alpha\frac{r^{2}}{r_{1}^{2}}).$
(60)

Assuming that $f=aF$ is a eigenfunction of $a_{j}^{\dagger}a_{j}F=LF$. We find

$\displaystyle F=\left\\{\begin{array}[]{ll}r^{j-\frac{1}{2}}e^{-\frac{\alpha
r^{2}}{2r_{1}^{2}}}{}_{1}F_{1}(-\frac{r_{1}^{2}}{4\alpha}L,j+\frac{1}{2};\alpha\frac{r^{2}}{r_{1}^{2}})&(j=\frac{1}{2},\frac{3}{2},\cdots)\\\
r^{-j+\frac{1}{2}}e^{-\frac{\alpha
r^{2}}{2r_{1}^{2}}}{}_{1}F_{1}(-j+\frac{1}{2}-\frac{r_{1}^{2}}{4\alpha}L,-j+\frac{3}{2};\alpha\frac{r^{2}}{r_{1}^{2}})&(j=-\frac{1}{2},-\frac{3}{2},\cdots)\end{array}\right.$
(63)

an overall constant.
${}_{1}F_{1}(a,b;z)=\sum_{n=0}^{\infty}\frac{\Gamma(a+n)}{\Gamma(a)}\frac{\Gamma(b)}{\Gamma(b+n)}\frac{z^{n}}{n!}$
is a confluent hypergeometric function which satisfies

$\displaystyle\quantity(z\derivative[2]{z}+(b-z)\derivative{z}-a){}_{1}F_{1}(a,b;z)=0.$
(64)

Denoting $g$ by $ba_{j}F$, we obtain an equation for $a$ and $b$,

Similarly to the case $r>r_{1}$, we have two solution for $L$:

$\displaystyle\frac{L}{M_{PV}^{2}}=\frac{L_{\pm}}{M_{PV}^{2}}=-\quantity(\pm
1+\sqrt{1+2\frac{M}{M_{PV}}+\frac{\quantity(E-\frac{\alpha}{M_{PV}r_{1}^{2}})^{2}}{M_{PV}^{2}}})^{2}+\frac{\quantity(E-\frac{\alpha}{M_{PV}r_{1}^{2}})^{2}}{M_{PV}^{2}}$
(65)

and

$\displaystyle\psi_{i\pm}=\matrixquantity(\quantity(M+\frac{L_{\pm}}{2M_{PV}}+E-\frac{\alpha}{M_{PV}r_{1}^{2}})F(r)e^{i(j-\frac{1}{2})\theta}\\\
a_{j}F(r)e^{i(j+\frac{1}{2})\theta}).$ (66)

###  B.3 Continuity at $r=r_{1}$

The total eigenfunction $\psi$ is a liner combination of $\psi_{i\pm}$ when
$r<r_{1}$ and $\psi_{o\pm}$ when $r>r_{1}$:

$\displaystyle\psi=\left\\{\begin{array}[]{cc}c_{1}\psi_{i+}+c_{2}\psi_{i-}&(r<r_{1})\\\
d_{1}\psi_{o+}+d_{2}\psi_{o-}&(r>r_{1})\end{array}\right..$ (69)

Since $\psi$ and $\partialderivative{r}\psi$ are a continuous at $r=r_{1}$ and
$\psi$ is normalized, we can determine $c_{1},~{}c_{2},~{}d_{1},~{}d_{2}$ and
the eigenvalue $E$.

The connection of the outer/inner solutions requires $\psi$ and its derivative
$\partial_{r}\psi$ are continuous at $r=r_{1}$. Knowing the fact that the
Hamiltonian relates the eigenfunction $\psi$ and its derivative
$\partial_{r}\psi$ to the operation of the Laplacian $D_{\mu}D^{\mu}\psi$, we
replace the continuity condition for $\partial_{r}\psi$ by that for
$D_{\mu}D^{\mu}\psi$.

We furthermore assume $1/r_{1}\gg M_{PV}\gg\absolutevalue{M}$ and $j>0$, then
the functions are approximated as

$\displaystyle\psi_{i+}$
$\displaystyle\simeq\matrixquantity(-\frac{2\alpha}{M_{PV}r_{1}^{2}}{}_{1}F_{1}(+\frac{1}{2},j+\frac{1}{2};\alpha\frac{r^{2}}{r_{1}^{2}})e^{i(j-\frac{1}{2})\theta}\\\
-\frac{\alpha}{r_{1}(j+1/2)}{}_{1}F_{1}(+\frac{3}{2},j+\frac{1}{2};\alpha\frac{r^{2}}{r_{1}^{2}})e^{i(j+\frac{1}{2})\theta})r^{j-\frac{1}{2}}e^{-\frac{\alpha
r^{2}}{2r_{1}^{2}}},$ (70) $\displaystyle D^{2}\psi_{i+}$
$\displaystyle\simeq\matrixquantity(\frac{2\alpha}{M_{PV}r_{1}^{2}}M_{PV}^{2}{}_{1}F_{1}(+\frac{1}{2},j+\frac{1}{2};\alpha\frac{r^{2}}{r_{1}^{2}})e^{i(j-\frac{1}{2})\theta}\\\
-\frac{\alpha}{r_{1}(j+1/2)}\frac{4\alpha}{r_{1}^{2}}{}_{1}F_{1}(+\frac{3}{2},j+\frac{1}{2};\alpha\frac{r^{2}}{r_{1}^{2}})e^{i(j+\frac{1}{2})\theta})r^{j-\frac{1}{2}}e^{-\frac{\alpha
r^{2}}{2r_{1}^{2}}},$ (71)

$\displaystyle\psi_{i-}$
$\displaystyle\simeq\matrixquantity(-\frac{M_{PV}}{2}{}_{1}F_{1}(-\frac{1}{2},j+\frac{1}{2};\alpha\frac{r^{2}}{r_{1}^{2}})e^{i(j-\frac{1}{2})\theta}\\\
\frac{\alpha}{r_{1}(j+1/2)}{}_{1}F_{1}(+\frac{1}{2},j+\frac{1}{2};\alpha\frac{r^{2}}{r_{1}^{2}})e^{i(j+\frac{1}{2})\theta})r^{j-\frac{1}{2}}e^{-\frac{\alpha
r^{2}}{2r_{1}^{2}}},$ (72) $\displaystyle D^{2}\psi_{i-}$
$\displaystyle\simeq\matrixquantity(-\frac{M_{PV}}{2}\frac{4\alpha}{r_{1}^{2}}{}_{1}F_{1}(-\frac{1}{2},j+\frac{1}{2};\alpha\frac{r^{2}}{r_{1}^{2}})e^{i(j-\frac{1}{2})\theta}\\\
-\frac{\alpha}{r_{1}(j+1/2)}M_{PV}^{2}{}_{1}F_{1}(+\frac{1}{2},j+\frac{1}{2};\alpha\frac{r^{2}}{r_{1}^{2}})e^{i(j+\frac{1}{2})\theta})r^{j-\frac{1}{2}}e^{-\frac{\alpha
r^{2}}{2r_{1}^{2}}},$ (73)

$\displaystyle\psi_{o+}$
$\displaystyle\simeq\matrixquantity(-2M_{PV}K_{j-\frac{1}{2}-\alpha}(2M_{PV}r)e^{i(j-\frac{1}{2})\theta}\\\
2M_{PV}K_{j+\frac{1}{2}-\alpha}(2M_{PV}r)e^{i(j+\frac{1}{2})\theta}),$ (74)
$\displaystyle D^{2}\psi_{o+}$
$\displaystyle\simeq\matrixquantity((2M_{PV})^{3}K_{j-\frac{1}{2}-\alpha}(2M_{PV}r)e^{i(j-\frac{1}{2})\theta}\\\
-(2M_{PV})^{3}K_{j+\frac{1}{2}-\alpha}(2M_{PV}r)e^{i(j+\frac{1}{2})\theta}),$
(75)

$\displaystyle\psi_{o-}$
$\displaystyle\simeq\matrixquantity((M+E)K_{j-\frac{1}{2}-\alpha}(\sqrt{M^{2}-E^{2}}r)e^{i(j-\frac{1}{2})\theta}\\\
\sqrt{M^{2}-E^{2}}K_{j+\frac{1}{2}-\alpha}(\sqrt{M^{2}-E^{2}}r)e^{i(j+\frac{1}{2})\theta}),$
(76) $\displaystyle D^{2}\psi_{o-}$
$\displaystyle\simeq\matrixquantity(-(M+E)\sqrt{M^{2}-E^{2}}^{2}K_{j-\frac{1}{2}-\alpha}(\sqrt{M^{2}-E^{2}}r)e^{i(j-\frac{1}{2})\theta}\\\
-\sqrt{M^{2}-E^{2}}^{3}K_{j+\frac{1}{2}-\alpha}(\sqrt{M^{2}-E^{2}}r)e^{i(j+\frac{1}{2})\theta}).$
(77)

From the continuity, $E$ must satisfy

where
$u_{\frac{1}{2}+n}={}_{1}F_{1}(+\frac{1}{2}+n,j+\frac{1}{2}+n;\alpha)e^{-\frac{\alpha}{2}}r_{1}^{j-\frac{1}{2}}\simeq
r_{1}^{j-\frac{1}{2}},~{}v_{-\frac{1}{2}+n}={}_{1}F_{1}(-\frac{1}{2}+n,j+\frac{1}{2}+n;\alpha)e^{-\frac{\alpha}{2}}r_{1}^{j-\frac{1}{2}}\simeq
r_{1}^{j-\frac{1}{2}}$ and
$k_{n}(z)=K_{j-\frac{1}{2}+n-\alpha}(z)\simeq\frac{\Gamma(\absolutevalue{j-\frac{1}{2}+n-\alpha})}{2}\quantity(\frac{2}{z})^{\absolutevalue{j-\frac{1}{2}+n-\alpha}}~{}(n=0,1)$.
In the limit $\absolutevalue{M}\ll M_{PV}\ll 1/r_{1}$, the ４-1 component
$\sim\frac{1}{r_{1}^{3}}$ in the determinant is the most dominant, and the 3-2
component $\sim\frac{1}{r_{1}^{2}}$ is the secondary dominant. Then the
determinant reduces to

Thus the two solution on $r>r_{1}$ must be parallel at $r=r_{1}$ to obtain a
continuum solution. Since $M_{PV}r_{1}$ and $\sqrt{M^{2}-E^{2}}r_{1}$ are
small, $E$ is determined by

$\displaystyle\frac{M+E}{\sqrt{M^{2}-E^{2}}}\quantity(\frac{\sqrt{M^{2}-E^{2}}}{2M_{PV}})^{\absolutevalue{j+\frac{1}{2}-\alpha}-\absolutevalue{j-\frac{1}{2}-\alpha}}+1=0.$
(78)

There exists a solution if and only if $M<0$ since $E^{2}<M^{2}$. We obtain
the same condition for $j<0$, where the eigenfunction is approximated by

$\displaystyle\psi_{i+}$
$\displaystyle\simeq\matrixquantity(-\frac{2\alpha}{M_{PV}r_{1}^{2}}{}_{1}F_{1}(-j+\frac{1}{2}+\frac{1}{2},-j+\frac{3}{2};\alpha\frac{r^{2}}{r_{1}^{2}})e^{i(j-\frac{1}{2})\theta}\\\
-2\frac{-j+\frac{1}{2}}{r}{}_{1}F_{1}(-j+\frac{1}{2}+\frac{1}{2},-j+\frac{1}{2};\alpha\frac{r^{2}}{r_{1}^{2}})e^{i(j+\frac{1}{2})\theta})r^{-j+\frac{1}{2}}e^{-\frac{\alpha
r^{2}}{2r_{1}^{2}}}$ (79) $\displaystyle D^{2}\psi_{i+}$
$\displaystyle\simeq\matrixquantity(\frac{2\alpha}{M_{PV}r_{1}^{2}}M_{PV}^{2}{}_{1}F_{1}(-j+\frac{1}{2}+\frac{1}{2},-j+\frac{3}{2};\alpha\frac{r^{2}}{r_{1}^{2}})e^{i(j-\frac{1}{2})\theta}\\\
-2\frac{-j+\frac{1}{2}}{r}\frac{4\alpha}{r_{1}^{2}}{}_{1}F_{1}(-j+\frac{1}{2}+\frac{1}{2},-j+\frac{1}{2};\alpha\frac{r^{2}}{r_{1}^{2}})e^{i(j+\frac{1}{2})\theta})r^{-j+\frac{1}{2}}e^{-\frac{\alpha
r^{2}}{2r_{1}^{2}}}$ (80)

and the dependence of $r_{1}$ is the same as $j>0$.

$\absolutevalue{j+\frac{1}{2}-\alpha}-\absolutevalue{j-\frac{1}{2}-\alpha}$
takes three values

$\displaystyle\absolutevalue{j+\frac{1}{2}-\alpha}-\absolutevalue{j-\frac{1}{2}-\alpha}=\left\\{\begin{array}[]{cc}+1&(j-\frac{1}{2}>\alpha)\\\
2(j-\alpha)&(j+\frac{1}{2}>\alpha>j-\frac{1}{2})\\\
-1&(j+\frac{1}{2}<\alpha)\end{array}\right..$ (84)

When $j-1/2>\alpha$ or $j+\frac{1}{2}<\alpha$, $E\sim M_{PV}$ so it is
contradict with $E^{2}<M^{2}$. A solution of (78) exists only when
$j+\frac{1}{2}>\alpha>j-\frac{1}{2}$, i.e. $j=[\alpha]+\frac{1}{2}$. $E$ is an
odd function of $\alpha-[\alpha]-1/2$ and described as the inverse function of

$\displaystyle\alpha-[\alpha]=\frac{1}{2}\frac{\log(-\frac{M+E}{2M_{PV}})}{\log(\frac{\sqrt{M^{2}-E^{2}}}{2M_{PV}})}.$
(85)

Thus $E$ is approximated as

$\displaystyle
E\simeq\left\\{\begin{array}[]{cc}-\absolutevalue{M}&(\alpha-[\alpha]\sim
0)\\\
\absolutevalue{2M\log(\frac{\absolutevalue{M}}{2M_{PV}})}\quantity(\alpha-[\alpha]-\frac{1}{2})&(\alpha-[\alpha]\sim\frac{1}{2})\\\
\absolutevalue{M}&(\alpha-[\alpha]\sim 1)\end{array}\right..$ (89)