# Generalized Hall current on a finite lattice

Srimoyee Sen  Semeon Valgushev
Department of Physics and Astronomy, Iowa State University, Ames, IA, 50011

###### Abstract

Gapped fermion theories with gapless boundary fermions can exist in any number
of dimensions. When the boundary has even space-time dimensions and hosts
chiral fermions, a quantum Hall current flows from the bulk to the boundary in
a background electric field. This current compensate for the boundary chiral
anomaly. Such a current inflow picture is absent when the boundary theory is
odd dimensional. However, in recent work, the idea of quantum Hall current has
been generalized to describe odd dimensional boundary theories in continuous
Euclidean space-time dimension of infinite volume. In this paper we extend
this idea to a lattice regulated finite volume theory of 1+1 dimensional
Wilson-Dirac fermions. This fermion theory with a domain wall in fermion mass
can host gapless modes on the wall. The number of gapless fermions is equal to
the integral of the divergence of the lattice generalized Hall current.

##  1 Introduction

Odd dimensional Dirac fermion field theories are interesting when there is a
domain wall in fermion mass. In that case, the domain wall defect is even
dimensional and hosts massless chiral fermions [Callan:1984sa]. When this
theory is coupled to electromagnetic fields, the boundary suffers from chiral
anomaly leading to non-conservation of vector current in the presence of
background electromagnetic fields. However, as Callan-Harvey showed
[Callan:1984sa], a vector current flows from the bulk to the boundary
restoring current conservation in the higher dimensional theory. In order to
compute this current one integrates out the fermion away from the domain wall
which leaves behind a Chern-Simons theory for the electromagnetic field. This
explains the inflowing current from the bulk to the boundary. As is well
known, the odd dimensional gapped bulk theory of free Dirac fermion describes
the physics of quantum Hall effect. The inflowing current is analogous to the
quantum Hall current whereas the massless chiral fermions on the domain wall
are analogs of the quantum Hall edge states.

More generally, gapped fermion field theories can host massless fermions on
domain walls irrespective of whether the wall is even or odd dimensional. They
describe the physics of topological insulators and superconductors with
corresponding edge states in various dimensions [Kitaev:2009mg, Ryu_2010,
PhysRevB.85.085103, PhysRevB.78.195125, Ludwig_2015]. When the boundary is odd
dimensional, in contrast to quantum Hall effect, the boundary theory does not
have a chiral anomaly. Therefore, we don’t expect an inflowing current from
the bulk to the boundary as in the case of quantum Hall effect. Although, the
boundary theory can have discrete anomalies which connects the existence of
the edge states to the gapped bulk theory[Witten:2015aba]. In a recent paper
[kaplan2021index, Kaplan:2022uoo] the authors showed that the idea of the Hall
current can be generalized to odd dimensional boundaries. The idea was
inspired by index calculation of fermion vortex system in [PhysRevD.24.2669].
This generalization of the Hall current relies on the following step: the
Minkowski space domain wall fermion theory with massless boundary fermion is
first connected to another Euclidean fermion theory where the Euclidean
fermion operator has a nonzero index. This index equals the number of massless
fermions in the original Minkowski theory. From there, it was shown
[kaplan2021index, Kaplan:2022uoo] that one can construct a generalized Hall
current: the space-time integral of its divergence equaling the index of the
fermion operator. The construction outlined in [kaplan2021index,
Kaplan:2022uoo] holds for non-interacting fermions in infinite volume and
continuum space-time. The goal of this paper is to extend that analysis to a
discrete space-time lattice of infinite and finite volume. The analysis in
[kaplan2021index, Kaplan:2022uoo] included several different fermion theories
in various space-time dimensions. In this paper, we choose to work with the
simplest example: $1+1$ dimensional Dirac fermion with a domain wall in its
mass [PhysRevD.13.3398]. The domain wall hosts a massless fermion which may
suffer from discrete anomalies ([Witten:2015aba, PhysRevB.83.075103]), but
does not suffer from chiral anomaly. As a result, one doesn’t expect a Hall
current flowing from bulk to the boundary. However, the generalized Hall
current exists for this system in infinite volume and continuum space-time. We
explore how the generalized Hall current for this system can be constructed on
an infinite and finite lattice.

A crucial observation which makes the continuum construction of the
generalized Hall current possible is the following. Whenever the index of a
Euclidean elliptic fermion operator is nonzero, there is a current in the
system: the space-time integral of the divergence of this current equals the
index. We call this current the generalized Hall current. Note that the index
of a Euclidean elliptic operator is the difference between the number of zero
modes of that operator and that of its Hermitian conjugate [PhysRevD.24.2669].
Therefore, no generalized Hall current exists if the index of the operator is
zero. This observation is not meant to be self-evident and its proof is
outlined in [kaplan2021index, Kaplan:2022uoo]. We will discuss the proof
briefly in the next section of this paper. This observation can then be used
to construct the generalized Hall current for massless fermion edge states of
any Minkowski fermion theory as follows. The first step is to use the
Minkowski fermion operator to construct its Euclidean counterpart. Since the
Minkowski fermion operator has massless states living on the defect, the
corresponding Euclidean operator has unnormalizable zero eigenvalue
eigenstates living on the same defect. These states are not zero modes since
they are not normalizable. As a result the index of the Euclidean operator at
this stage is zero. Ref. [kaplan2021index, Kaplan:2022uoo] then introduces a
slight deformation to this Euclidean operator through the introduction of a
background diagnostic field in such a way that this unnormalizable zero
eigenvalue state becomes localized and normalizable. i.e. the deformed
Euclidean operator has a zero mode iff the original Minkowski fermion operator
had a massless fermion in its spectrum. The introduction of this diagnostic
field also creates an imbalance between the number of zero modes of the
fermion operator and its Hermitian conjugate resulting in a nonzero index for
the deformed theory. Additionally, the construction carefully ensures that the
index survives in the limit of the diagnostic field being taken to zero. We
expect a generalized Hall current to flow as long as the index is nonzero. In
the continuum analysis, one can obtain this Hall current by simply perturbing
in the diagnostic field and integrating out the fermions in a one loop
diagram. This is analogous to the Goldstone-Wilczek calculation
[PhysRevLett.47.986].

As we embark on generalizing the above construction on the lattice, both
infinite and finite, we explore which elements of the continuum construction
can be carried over to the lattice without significant modification and which
elements need to be reformulated. Since we will work with the $1+1$
dimensional fermion theory, from this point onward we exclusively focus on it.
The organization of the paper is as follows. We will begin with a brief
overview of the generalized Hall current construction in the continuum
specializing to the case in $1+1$ dimensions. We will then discuss how this
construction is generalized to an infinite lattice analytically. The following
section will describe the numerical analysis of this construction and
demonstrate that a generalized Hall current exists on a finite lattice.

##  2 Infinite volume continuum analysis

The procedure for constructing the generalized Hall current in the continuum
in infinite volume is described in detail in [kaplan2021index,
Kaplan:2022uoo]. We briefly review this construction here. Consider a
Minkowski fermion operator $D_{M}$ with a mass defect which causes it to have
a massless fermion in the spectrum that is stuck to the defect. To construct
the generalized Hall current

  1. 1.

We analytically continue this fermion operator to Euclidean space-time,
denoting it by $\mathcal{D}$.

  2. 2.

Introduce background diagnostic field to deform the fermion operator
$\mathcal{D}$ to have an index of one (equal to the number of massless
fermions in the original Minkowski theory).

  3. 3.

Obtain the generalized Hall current following a Goldstone-Wilczek
[PhysRevLett.47.986] inspired calculation using one loop Feynman diagram.

  4. 4.

Take the background diagnostic field to zero at the end of calculation and
confirm that the generalized Hall current and the index survives taking this
limit.

Before we apply this construction to $1+1$ dimensional example, let’s first
attempt to understand how the index of a fermion operator gives rise to an
inflowing current in infinite volume continuous space-time of Euclidean
signature. Note that the index of the fermion operator $\mathcal{D}$ is given
by

$\displaystyle
I=\text{Dim}(\text{ker}\,D)-\text{Dim}(\text{ker}\,D^{\dagger}).$ (1)

In the example we will consider, the number of zero modes of either the
operator $\mathcal{D}$ or the operator $\mathcal{D}^{\dagger}$ is zero. As a
result the magnitude of the index ends up being equal to the number of zero
modes of one operator or the other. Furthermore, the number of zero modes of
the operator $\mathcal{D}$ coincides with the number of zero modes for the
operator $\mathcal{D}^{\dagger}\mathcal{D}$ and the number of zero modes of
$\mathcal{D}^{\dagger}$ coincides with that of
$\mathcal{D}\mathcal{D}^{\dagger}$. Therefore the formula for the index can be
re-expressed by defining

$\displaystyle\mathcal{I}(M)=\frac{M^{2}}{M^{2}+\mathcal{D}^{\dagger}\mathcal{D}}-\frac{M^{2}}{\mathcal{D}\mathcal{D}^{\dagger}+M^{2}}$
(2)

and noting that

$\displaystyle I=\lim_{M\rightarrow 0}\mathcal{I}(M).$ (3)

Interestingly, the quantity $\mathcal{I}(M)$ can now be recast as the matrix
element $\mathcal{I}(M)=-\int
d^{d+1}x\langle\bar{\Psi}\Gamma_{\chi}\Psi\rangle$ in a fermion theory with
the following action

$\displaystyle\mathcal{S}=\int d^{d+1}x\,\bar{\Psi}(K+M)\Psi$ (4)

where

$\displaystyle K=\begin{pmatrix}0&&-\mathcal{D}^{\dagger}\\\
\mathcal{D}&&0\end{pmatrix}$ (5)

and

$\displaystyle\Gamma_{\chi}=\begin{pmatrix}1&&0\\\ 0&&-1\end{pmatrix}.$ (6)

Note that, $d+1$ is the number of space-time dimensions in which the original
fermion operator $\mathcal{D}$ is defined. The spinor $\Psi$ has twice the
dimension of the spinors of the original theory. The gamma matrices for this
theory can be easily read off using

$\displaystyle\Gamma_{\mu}=i\partial\tilde{K}(p)/\partial p_{\mu}$ (7)

where $\tilde{K}$ is the Fourier transform of $K$. The theory of Eq. 4 has its
own fermion number symmetry which works as $\Psi\rightarrow e^{i\theta}\Psi$.
In the $M\rightarrow 0$ limit, it also has an axial symmetry $\Psi\rightarrow
e^{i\Gamma_{\chi}\alpha}\Psi$ where this new axial symmetry has nothing to do
with the symmetries of the original theory. We can now construct an axial
current $\mathcal{J}_{\mu}^{\chi}=\bar{\Psi}\Gamma_{\mu}\Gamma_{\chi}\Psi$ and
write down the Ward identity for it

$\displaystyle\partial_{\mu}\mathcal{J}_{\mu}^{\chi}=2M\bar{\Psi}\Gamma_{\chi}\Psi-\mathcal{A}$
(8)

where $\mathcal{A}$ is the “anomaly contribution"

$\displaystyle\mathcal{A}=-2\lim_{\Lambda\rightarrow\infty}\text{Tr}(\Gamma_{\chi}e^{K^{2}/\Lambda^{2}})=-2\mathcal{I}(\infty).$
(9)

This anomaly contribution can be computed using the methods outlined in
Fujikawa [PhysRevLett.42.1195]. It is found to vanish for the theory under
consideration Eq. 4 and was elaborated in [kaplan2021index, Kaplan:2022uoo].
At this point we can take the limit $M\rightarrow 0$ in Eq. 8 to write

$\displaystyle I=\mathcal{I}(0)=-\lim_{M\rightarrow 0}M\int
d^{d+1}x\langle\bar{\Psi}\Gamma_{\chi}\Psi\rangle=-\lim_{M\rightarrow
0}\frac{1}{2}\int
d^{d+1}x\langle\partial_{\mu}\mathcal{J}_{\mu}^{\chi}\rangle.$ (10)

We have now expressed the index of the fermion operator in terms of the
“axial" current of the theory in Eq. 4. We call this current the generalized
Hall current. This generalized Hall current
$\bar{\Psi}\Gamma_{\mu}\Gamma_{\chi}\Psi$ can now be computed using one loop
Feynman diagrams by perturbing in the mass defect as well as the other
background fields. We will review how this is done for $1+1$ dimensional Dirac
fermion with a domain wall in its mass.

###  2.1 $1+1$ dimensional Dirac fermion in continuum

Let’s consider the Lagrangian of a Dirac fermion in Minkowski space-time with
Dirac mass denoted as $\phi_{1}$. It has the Lagrangian

$\displaystyle\mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-\phi_{1})\psi$
(11)

where $\mu$ takes values $0$ and $1$, $x_{0}$ is the temporal and $x_{1}$ is
the spatial coordinate. We can take the $\gamma$ matrices as

$\displaystyle\gamma^{0}=\sigma_{2},\gamma^{1}=-i\sigma_{1},\gamma^{\chi}=\sigma_{3}$
(12)

where $\gamma_{\chi}$ is the chirality operator. If we introduce a domain wall
in $\phi_{1}$ along the spatial coordinate $x_{1}$,
$\phi_{1}=m_{0}\epsilon(x_{1})$ with $m_{0}>0$ and

$\displaystyle\epsilon(x)=\begin{cases}+1,&x\geq 0\\\ -1,&x<0\end{cases},$
(13)

then we will have a massless fermion mode living on the domain wall at
$x_{1}=0$ as seen from the Dirac equation in the domain wall background

$\displaystyle
i\gamma^{0}\partial_{0}\psi+i\gamma^{1}\partial_{1}\psi-\phi_{1}\psi=0.$ (14)

To look for massless state, we can set $\partial_{0}\psi=0$ and find that the
Dirac equation is solved by

$\displaystyle\psi=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
-1\end{pmatrix}e^{-m_{0}|x_{1}|}.$ (15)

In order to construct the generalized Hall current we first have to
analytically continue to Euclidean space-time where the Lagrangian is now

$\displaystyle\mathcal{L}_{\text{E}}=\bar{\psi}(\gamma_{\mu}\partial_{\mu}+\phi_{1})\psi$
(16)

with Euclidean gamma matrices defined as

$\displaystyle\gamma_{0}=\sigma_{2},\gamma_{1}=-\sigma_{1},\gamma_{\chi}=\sigma_{3}.$
(17)

We also denote two dimensional identity matrix as $\sigma_{0}$. The
corresponding fermion operator $\gamma_{\mu}\partial_{\mu}+\phi_{1}$ has an
unnormalizable zero eigenvalue eigenstate. However this state doesn’t count as
zero mode which should be normalizable. In order to engineer a zero mode we
turn on a background pseudo-scalar field with a domain wall profile in the
Euclidean time direction. We also refer to this field as a diagnostic field.
The corresponding Lagrangian is of the form

$\displaystyle\mathcal{L}_{\text{E}}=\bar{\psi}(\gamma_{\mu}\partial_{\mu}+\phi_{1}+i\phi_{2}\gamma_{\chi})\psi$
(18)

where $\phi_{2}=\mu_{0}\epsilon(x_{0})$ with $\mu_{0}>0$. Let’s denote this
fermion operator as $\mathcal{D}$ with

$\displaystyle\mathcal{D}=(\gamma_{\mu}\partial_{\mu}+\phi_{1}+i\phi_{2}\gamma_{\chi}).$
(19)

We find that the operator $\mathcal{D}$ has one zero mode of the form

$\displaystyle\psi=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
-1\end{pmatrix}e^{-m_{0}|x_{1}|-\mu_{0}|x_{0}|}.$ (20)

We can also look for zero modes for the operator $\mathcal{D}^{\dagger}$ and
find that there are none for this specific choice of domain wall profile
($m_{0}>0,\mu_{0}>0$). More generally, for other choices of the domain wall
profile, e.g. with $m_{0}>0,\mu_{0}<0$ or $m_{0}<0,\mu_{0}>0$ we find a zero
mode for the operator $\mathcal{D}^{\dagger}$ and the operator $\mathcal{D}$
has no zero modes. Similarly, the choice of $m_{0}<0,\mu_{0}<0$ yields a
zeromode for $\mathcal{D}$ and none for $\mathcal{D}^{\dagger}$. In other
words, the magnitude of the index of the fermion operator remains $1$ as long
as there is a domain wall in both $\phi_{1}$ and $\phi_{2}$. However, whether
the index is positive or negative depends on the profile of choice.

There is a simple way to relate the domain wall profile with the index of the
fermion operator. To see this, we can first express $\phi_{1}+i\phi_{2}$ as
$\phi_{1}+i\phi_{2}=ve^{i\theta}$. It is easy to see that for a crossed domain
wall profile in $\phi_{1}$ and $\phi_{2}$, if one considers a polar coordinate
system centered at $x_{0}=x_{1}=0$, then the phase variable $\theta$ completes
a winding of $2\pi$ or $-2\pi$ as one travels along a contour encircling the
center over a polar angle of $2\pi$. The crossed domain wall defect can
therefore be thought of as a vortex in $\phi_{1}+i\phi_{2}$. We have now
constructed the intended fermion operator whose index is equal to the winding
in the crossed domain wall configuration. Note that the index and the winding
survives in the limit $\mu_{0}\rightarrow 0$.

###  2.2 Generalized Hall current (GHC) in the continuum

We now review the one loop Feynman diagram calculation to compute the
generalized Hall current and then verify that the space-time integral of its
divergence equals the index.

Following the prescription outlined in Eq. 4,5,6 we construct the $K$ matrix
which we can re-express in momentum space as

$\displaystyle K=\Gamma_{\mu}k_{\mu}+i\phi_{2}\Gamma_{2}+i\phi_{1}\Gamma_{3}$
(21)

where we have defined

$\displaystyle\Gamma_{i}=\sigma_{1}\otimes\gamma_{i},\Gamma_{2}=\sigma_{1}\otimes\gamma_{\chi},$
$\displaystyle\Gamma_{3}=-\sigma_{2}\otimes\sigma_{0},\Gamma_{\chi}=\sigma_{3}\otimes\sigma_{0}.$
(22)

To compute the “axial" current, we rewrite the mass terms as
$\phi_{1}+i\phi_{2}=(v+\rho(x))e^{i\theta(x)}$ and expand the $K$ matrix in
$\theta$ with

$\displaystyle K=K_{0}+\delta K$ (23)

where $K_{0}=\gamma_{\mu}k_{\mu}+\rho$ and $\delta
K=iv\theta\Gamma_{2}+i\rho\Gamma_{3}$. Up to linear order in $\theta$ we get

$\displaystyle\mathcal{J}_{\mu}^{\chi}$ $\displaystyle=$
$\displaystyle+v\frac{\partial\theta}{\partial
x_{\mu}}\int\frac{d^{2}q}{(2\pi)^{2}}\text{Tr}\left(\Gamma_{\mu}\Gamma_{\chi}\frac{dK_{0}^{-1}}{dq_{\nu}}\Gamma_{2}K_{0}^{-1}\right)$
(24) $\displaystyle=$
$\displaystyle\epsilon_{\mu\nu}\partial_{\nu}\theta\int\frac{d^{2}q}{(2\pi)^{2}}\frac{4v^{2}}{(q^{2}+v^{2})^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\epsilon_{\mu\nu}\partial_{\nu}\theta$

We can now compute the space-time integral of the divergence of this current
and relate it to the index with

$\displaystyle\mathcal{I}(0)=-\frac{1}{2}\int
d^{2}x\partial_{\mu}\mathcal{J}_{\mu}^{\chi}=-\nu_{\theta}$ (25)

where $\nu_{\theta}$ is the winding of the crossed domain wall or vortex
configuration. For the specific domain wall profile we have chosen this
winding is $-1$. Therefore we get an index of $1$ which is consistent with the
index we obtained for the fermion operator in the previous subsection. This
demonstrates that whenever the Minkowski theory specified by Eq. 11 has a
domain wall in fermion mass hosting massless edge state, one can construct a
corresponding Euclidean fermion operator with the following properties:

  1. 1.

The Euclidean fermion operator has an index of $\pm 1$ in the presence of a
background diagnostic field.

  2. 2.

In the limit of diagnostic field going to zero this Euclidean operator
coincides with the Euclidean analytic continuation of the Minkowski operator
in Eq. 11.

  3. 3.

The index of this Euclidean operator persists in the limit of the diagnostic
field being taken to zero and is equal to the space-time integral of the
divergence of the GHC.

In the next section, we will to extend our Euclidean fermion operator
construction to discrete space-time. In order to mimic the continuum
construction sufficiently closely we will have to maintain the following

  1. 1.

The lattice fermion operator or its Hermitian conjugate should not have more
than one zeromode.

  2. 2.

We will exclude regions in parameter space where the number of zeromodes for
the fermion operator and its Hermitian conjugate are the same.

The second condition ensures that the index of the fermion operator is
nonzero.

##  3 1+1 case on the lattice in infinite volume

We begin with the fermion operator in Eq. 19 and discretize spacetime setting
the lattice spacing to $1$. If we first set $\phi_{2}=0$ and naively
discretize space-time, we observe an important difference from the spectrum in
the continuum : i.e. we see fermion doubling. This is to say, in the continuum
we had a single solution to the equation $\mathcal{D}|_{\phi_{2}=0}\psi=0$
with $\psi$ being localized in the $x_{1}$ direction and constant in the
$x_{0}$ direction. On the lattice, there are more than one solution of this
form. In order to remove fermion doubling so as to retain only one solution
will require us to introduce higher dimensional operators to the Lagrangian
similar to the Wilson terms used in domain wall fermions [Kaplan:1992bt,
Jansen:1992yj, Golterman:1992ub, Jansen:1992tw, Sen:2022dkl]. Since our end
goal is to construct a Euclidean fermion operator with a single zeromode we
have two simple choices for this higher dimensional term:

  1. 1.

Wilson-like operator: Inspired by the Wilson term in lattice field theory, we
introduce the following higher-derivative operators to the Lagrangian which we
call Wilson-like terms [Sen:2022dkl]:

$\displaystyle\mathcal{D}_{1}=\sum_{\mu}\gamma_{\mu}\nabla_{\mu}+\frac{R}{2}\nabla_{1}^{2}+i\gamma_{\chi}\frac{R}{2}\nabla_{0}^{2}.$
(26)

We set parameter $R=1$.

  2. 2.

Fermion operator with Wilson term: We introduce in the Lagrangian the standard
Wilson term:

$\displaystyle\mathcal{D}_{2}=\sum_{\mu}\gamma_{\mu}\nabla_{\mu}+\frac{R}{2}\sum_{\mu}\nabla_{\mu}^{2}$
(27)

We again set Wilson parameter to $R=1$.

We now look for the zeromodes of these operators by varying the paramaters of
our theory.

###  3.1 Zeromodes

In this subsection we aim to obtain zeromode solutions, by varying the
parameters like the domain wall heights for the two types of lattice fermion
operator introduced in the previous section. We first present an analytic
calculation for the zeromode of the Wilson-like operator in infinite and
finite volume. The corresponding expressions for the zeromode profile are
simple and illuminating. An analogous analytic calculation for the Wilson
fermion case is more difficult and not particularly illuminating. Therefore we
defer the discussion of the Wilson fermion operator to the subsection 3.1.3
where we present numerical analysis of both the Wilson fermion case and the
Wilson-like cases.

####  3.1.1 Analytic solution for the zeromode in infinite volume

We begin with Wilson-like operator given by Eq. 26:

$\displaystyle\mathcal{D}_{1}=\begin{pmatrix}\tilde{\phi}_{1}+i\tilde{\phi}_{2}&&-i\nabla_{0}-\nabla_{1}\\\
i\nabla_{0}-\nabla_{1}&&\tilde{\phi}_{1}-i\tilde{\phi}_{2}\end{pmatrix}$ (28)

where $\tilde{\phi}_{1}=\phi_{1}+\frac{1}{2}\nabla_{1}^{2}$ and
$\tilde{\phi}_{2}=\phi_{2}+\frac{1}{2}\nabla_{0}^{2}$. With an ansatz of
$\psi_{+}=\begin{pmatrix}1\\\ -1\end{pmatrix}\varphi_{+}$ of $\gamma_{1}$
eigenvalue $+1$ we get two equations for $\varphi_{+}$,

$\displaystyle\nabla_{1}\varphi+\tilde{\phi}_{1}\varphi_{+}=0,$ (29)
$\displaystyle\nabla_{0}\varphi+\tilde{\phi}_{2}\varphi_{+}=0.$ (30)

Then using an ansatz of $\varphi_{+}=z_{0}^{x_{0}}z_{1}^{x_{1}}$ we see that
there exists normalizable solution with

$\displaystyle z_{0}=(1-\phi_{2})$ $\displaystyle z_{1}=(1-\phi_{1})$ (31)

when $0<m_{0}<2$ and $0<\mu_{0}<2$. Let’s fix $m_{0}=1$ and $\mu_{0}=1$. Now
we consider the ansatz of $\psi_{-}=\begin{pmatrix}1\\\
1\end{pmatrix}\varphi_{-}$. The EOMs for $\varphi_{-}$ are

$\displaystyle\nabla_{1}\varphi_{-}-\tilde{\phi}_{1}\varphi_{-}=0,$ (32)
$\displaystyle\nabla_{0}\varphi_{-}-\tilde{\phi}_{2}\varphi_{-}=0.$ (33)

These are solved by the ansatz

$\displaystyle\varphi_{-}=z_{0}^{x_{0}}z_{1}^{x_{1}}\,\,\,\,\text{with}\,\,\,\,z_{0}=\frac{1}{(1-\phi_{2})},\,\,\,z_{1}=\frac{1}{(1-\phi_{1})}.$
(34)

The solution is not normalizable for our choice of $m_{0}=1$ and $\mu_{0}=1$.
Therefore $\psi_{-}$ is not a zeromode of $\mathcal{D}_{1}$; thus
$\mathcal{D}_{1}$ has a single zeromode specified by the expression for
$\psi_{+}$ in Eq. 3.1.1. Now, let’s look at the zero modes for
$\mathcal{D}^{\dagger}$. With an ansatz of $\xi_{-}=\begin{pmatrix}1\\\
1\end{pmatrix}\chi_{-}$ and $\xi_{+}=\begin{pmatrix}1\\\
-1\end{pmatrix}\chi_{+}$ we get the following EOMs for $\chi_{-}$ and
$\chi_{+}$,

$\displaystyle\nabla_{1}\chi_{-}+\tilde{\phi}_{1}\chi_{-}=0$
$\displaystyle\nabla_{0}\chi_{-}-\tilde{\phi}_{2}\chi_{-}=0$ (35)

and

$\displaystyle\nabla_{1}\chi_{+}-\tilde{\phi}_{1}\chi_{+}=0$
$\displaystyle\nabla_{0}\chi_{+}+\tilde{\phi}_{2}\chi_{+}=0$ (36)

Using an ansatz of the form $z_{0}^{x_{0}}z_{1}^{x_{1}}$ for $\chi_{-}$ and
$\chi_{+}$ we see that there are no normalizable solutions for either. Thus we
have accomplished what we set out do, i.e. engineer a Euclidean fermion
operator on the lattice with an index of $+1$ using the Wilson-like terms.

Note that, if we vary parameters the pattern of zeromodes change. E.g. for
$-2<m_{0}<0,-2<\mu_{0}<0$ we find a zeromode solution with $\gamma_{1}$
eigenvalue $-1$. Similarly, with $2>m_{0}>0,0>\mu_{0}>-2$ and
$0>m_{0}>-2,2>\mu_{0}>0$ we find no normalizable zeromode for the operator
$\mathcal{D}_{1}$. However, we find a zeromode for the operator
$\mathcal{D}_{1}^{\dagger}$: $\gamma_{1}$ eigenvalue $-1$ for
$2>m_{0}>0,0>\mu_{0}>-2$ and $\gamma_{1}$ eigenvalue $1$ for
$2>\mu_{0}>0,0>m_{0}>-2$.

####  3.1.2 Finite volume

Our next goal is to generalize the infinite volume construction to finite
volume, i.e. on $S^{1}\times S^{1}$. At this point we will have to resort to
numerical techniques. We will take the lattice size to be $L\times L$ where
the domain wall in $\phi_{1}$ is located at $x_{1}=0$ and the anti-wall is
located at $x_{1}=L/2$. Similarly, the domain wall in $\phi_{2}$ is located at
$x_{0}=0$ with anti-wall at $x_{0}=L/2$. Therefore in effect we have four
vortex-like defects at $(x_{0}=0,x_{1}=0)$, $(x_{0}=0,x_{1}=L/2)$,
$(x_{0}=L/2,x_{1}=0)$ and $(x_{0}=L/2,x_{1}=L/2)$. There are several
subtleties with this finite volume analysis which we describe below.

##### Exact zeromode and tuning:

The two types of lattice fermion operators, which we call the Wilson-like or
Wilson fermion operator, will in general not exhibit exact zeromodes in finite
volume for arbitrary choice of domain wall heights. To understand why this is
the case, consider the Wilson-like fermion operator. Since we are considering
$S^{1}\times S^{1}$ with periodic boundary condition, any solution to the
equation of motion including the zeromode should satisfy:

$\displaystyle\phi_{+}(x_{\mu}=-L/2)=\phi_{+}(x_{\mu}=L/2)$ (37)

for $\mu=0,1$. The solution obtained in Eq. 3.1.1 for an infinite lattice with
equal magnitude of domain wall height on the two sides of the wall will not
satisfy this periodic boundary condition (PBC) in finite volume. In order to
obtain an exact zeromode solution which satisfies PBC, we will need to assume
more general domain wall configuration:

$\displaystyle\phi_{1}(x_{1})$ $\displaystyle=$
$\displaystyle\begin{cases}m_{+}&x_{1}\geq 0\\\ m_{-}&x_{1}<0\\\ \end{cases},$
(38) $\displaystyle\phi_{2}(x_{0})$ $\displaystyle=$
$\displaystyle\begin{cases}\mu_{+}&x_{0}\geq 0\\\ \mu_{-}&x_{0}<0\\\
\end{cases}.$

Then we find an exact zeromode for the choice

$\displaystyle\frac{1}{1-m_{-}}=(1-m_{+}),$ (39)
$\displaystyle\frac{1}{1-\mu_{-}}=(1-\mu_{+}).$

Note that these equations do not depend on the lattice size, thus if they are
satisfied then the exact zeromode of $\mathcal{D}_{1}$ will exist in any
volume. A similar analysis is much more complicated for the Wilson case and is
not particularly interesting.

It’s important consider however, that the Minkowski space domain wall theory
in continuous space-time and in infinite volume hosts massless edge states
without requiring any tuning of the domain wall height. Therefore, on the
finite lattice too, we seek a formulation which does not rely on tuning of the
domain wall heights. Since, a finite volume lattice fermion operator
$\mathcal{G}$ does not have an exact zeromode in general, we shift our
attention to the operator $\mathcal{G}^{\dagger}\mathcal{G}$. This is also
motivated by the observation that the index formula for the fermion operator
involves the kernel of the operator $\mathcal{G}^{\dagger}\mathcal{G}$ and
$\mathcal{G}\mathcal{G}^{\dagger}$. However, the operator
$\mathcal{G}^{\dagger}\mathcal{G}$ (or $\mathcal{G}\mathcal{G}^{\dagger}$)
doesn’t have exact zeromodes in finite volume either. In order to recover them
one has to take an infinite volume limit. Interestingly, this limit is smooth
for $\mathcal{G}^{\dagger}\mathcal{G}$ (or $\mathcal{G}\mathcal{G}^{\dagger}$)
but not necessarily for $\mathcal{G}$ (or $\mathcal{G}^{\dagger}$) itself. We
will use this observation to enable the GHC construction. The index formula in
infinite volume is related to the the difference of the zeromodes of the
operators $\mathcal{D}_{1/2}\mathcal{D}_{1/2}^{\dagger}$ and
$\mathcal{D}_{1/2}^{\dagger}\mathcal{D}_{1/2}$. We will work with the same
definition for the “index” in finite volume. As we will see, in finite volume,
the operators, $\mathcal{D}_{1/2}\mathcal{D}_{1/2}^{\dagger}$ and
$\mathcal{D}_{1/2}^{\dagger}\mathcal{D}_{1/2}$ will exhibit smooth convergence
towards infinite volume zeromodes without any fine tuning for the domain wall
heights, whereas $\mathcal{D}_{1/2}$ will not. This will enable us to
construct a tuning independent lattice GHC. Although we don’t need fine tuning
of domain wall heights, the domain wall heights must satisfy the following
constraints to host a zeromode in the infinite volume limit. E.g. for a
crossed domain wall configuration of the form $\phi_{1}=m_{0}\epsilon(x_{1})$
and $\phi_{2}=\mu_{0}\epsilon(x_{0})$ we must have $0<m_{0},\mu_{0}<2$ in
order for there to be a zeromode. Therefore, in the rest of the paper we will
choose parameters that satisfy this condition. Finally, even though our goal
is to construct a GHC formulation which does not rely on tuning of the domain
wall height, we will present the results for the tuned case of the Wilson-like
fermion operator to illustrate a GHC in the presence of an exact lattice
zeromode.

##### Index in finite volume:

In a finite volume, a domain wall setup will appear accompanied by an anti-
wall. As a result, with a domain wall in mass and the diagnostic field, we
will have four vortex, two vortex and two anti-vortex defects in finite volume
as described in the beginning of this subsection. Clearly the net winding of
this system is zero. Therefore the net “index" in this finite volume lattice
theory is also zero. However, locally in a region near each of the vortex
defect we should be able to define an "index" which we can then attempt to
connect to a lattice version of the generalized Hall current. In other words,
in finite volume, the operators $\mathcal{D}_{1/2}\mathcal{D}_{1/2}^{\dagger}$
and $\mathcal{D}_{1/2}^{\dagger}\mathcal{D}_{1/2}$ have the same number of
zeromodes. This implies that the difference between the number of zeromodes
for the two is zero, or the net “index" is zero. However, the zeromodes for
these two operators will be localized on different vortex defects. E.g.
$\mathcal{D}_{1}^{\dagger}\mathcal{D}_{1}$ will have a zeromode on the defect
at $(x_{0}=0,x_{1}=0)$ and $(x_{0}=L/2,x_{1}=L/2)$. Similarly,
$\mathcal{D}_{1}\mathcal{D}_{1}^{\dagger}$ will have zeromodes at
$(x_{0}=L/2,x_{1}=0)$ and $(x_{0}=0,x_{1}=L/2)$. As a result, e.g., near the
vortex at $(x_{0}=0,x_{1}=0)$ we expect the index to be $1$. Our goal is to
show that the integral of the divergence of the lattice GHC in a region around
the vortex equals the index.

| Wilson-like operator $\mathcal{D}_{1}$ | Wilson operator $\mathcal{D}_{2}$
---|---|---
Tuned domain walls? | Yes | No | No
Exact zeromode?
Singular values | — | Similar to $\mathcal{D}_{2}$ | Fig. 3 and Fig. 6 for $\phi_{2}(x_{0})\to 0$
Singular vectors | Fig. 4 and Fig. 5
Generalized Hall Current | Fig. 7(a) and Fig. 8(a) | Fig. 7(b) and Fig. 8(b)
“Index” | Fig. 9 | Fig. 10 and Fig. 11
Table 1: List of figures for different numerical setups presented in the
paper.

####  3.1.3 Zeromode numerics and singular value decomposition (SVD)

In this subsection we study the eigenvalues of the finite volume lattice
operators numerically. Our goal is to map the lowest eigenstate of the
suitable finite volume lattice operator to the zeromode of the infinite volume
continuum fermion operator. As stated earlier, this mapping cannot be
performed smoothly in the infinite volume limit by directly considering the
eigenvalues of $\mathcal{D}_{1/2}$ and $\mathcal{D}_{1/2}^{\dagger}$. Instead,
we need to consider the eigenvalues of
$\mathcal{D}_{1/2}^{\dagger}\mathcal{D}_{1/2}$ and
$\mathcal{D}_{1/2}\mathcal{D}_{1/2}^{\dagger}$. Our goal therefore, is to find
the lowest eigenvalues of $\mathcal{D}_{1/2}^{\dagger}\mathcal{D}_{1/2}$ and
$\mathcal{D}_{1/2}\mathcal{D}_{1/2}^{\dagger}$ and confirm that they go to
zero in the infinite volume limit. This discussion is organized as follows:
first, we present numerical methods for finding the zeromodes of
$\mathcal{D}_{1/2}\mathcal{D}_{1/2}^{\dagger}$ and
$\mathcal{D}_{1/2}^{\dagger}\mathcal{D}_{1/2}$. We first apply this method to
study a $0+1$ dimensional Wilson fermion operator with domain wall. We then
apply it to the lattice fermion operators we wish to study in $1+1$
dimensions, i.e. $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$.

To describe the numerical technique, we use a fermion operator $\mathcal{D}$
which would serve as a proxy for both $\mathcal{D}_{1/2}$. We can now consider
the spectrum of the operators $\mathcal{D}\mathcal{D}^{\dagger}$ and
$\mathcal{D}^{\dagger}\mathcal{D}$ using the eigenvalue equation

$\displaystyle\mathcal{D}\mathcal{D}^{\dagger}u_{i}=\sigma_{i}^{2}u_{i},$ (40)
$\displaystyle\mathcal{D}^{\dagger}\mathcal{D}v_{i}=\sigma_{i}^{2}v_{i},$ (41)

where $\sigma_{i}^{2}$ is an eigenvalue. The eigenvectors $u_{i}$ and $v_{i}$
are called left and right singular vectors and corresponding $\sigma_{i}\geq
0$ is called a singular value of $\mathcal{D}$. Note that the vectors $u_{i}$
and $v_{i}$ are linearly independent since the fermion operator is not normal,
i.e. $[\mathcal{D}^{\dagger},\mathcal{D}]\neq 0$.

Another possible way to arrive at the same is to look for a vector
$v^{\prime}$ which will minimize the norm $|\mathcal{D}v^{\prime}|$. The
square of this norm is positive-definite quadratic form given by
$\mathcal{D}^{\dagger}\mathcal{D}$, therefore the minimum is delivered by
eigenvector $v_{min}$ corresponding to the smallest eigenvalue
$\sigma^{2}_{min}$. Analogously, $u_{min}$ will deliver the minimum of
$|\mathcal{D}^{\dagger}u^{\prime}|$.

Interestingly, there is a simple relationship between $u_{i}$ and $v_{i}$,
since they together with $\sigma_{i}\geq 0$ define a singular values
decomposition (SVD) of the operator $\mathcal{D}$:

$\displaystyle\mathcal{D}^{\dagger}u_{i}=\sigma_{i}v_{i},$ (42)
$\displaystyle\mathcal{D}v_{i}=\sigma_{i}u_{i}.$

The SVD can be written in a compact matrix form as follows:

$\displaystyle\mathcal{D}=U\Sigma V^{\dagger},$ (43)

where unitary matrix $U$ is composed of (column) singular vectors $u_{i}$,
unitary matrix $V$ – of singular vectors $v_{i}$ and $\Sigma$ is a diagonal
matrix of corresponding singular values $\sigma_{i}$. It is clear from SVD
that neither $u_{i}$ nor $v_{i}$ are straightforwardly related to eigenvectors
of $\mathcal{D}$ if the operator is not normal. However, the singular values
of the operator $\mathcal{D}$ and $\mathcal{D}^{\dagger}$ map one to one to
the eigenvalues of the operators $\mathcal{D}^{\dagger}\mathcal{D}$ and
$\mathcal{D}\mathcal{D}^{\dagger}$. Therefore, the SVD of
$\mathcal{D}/\mathcal{D}^{\dagger}$ is equivalent to eigen-decomposition of
$\mathcal{D}^{\dagger}\mathcal{D}/\mathcal{D}\mathcal{D}^{\dagger}$ etc. In
the rest of the paper we will refer to the lowest eigenmode of
$\mathcal{D}^{\dagger}\mathcal{D}/\mathcal{D}\mathcal{D}^{\dagger}$ as near-
zeromode of the operator $\mathcal{D}/\mathcal{D}^{\dagger}$ and the vectors
$u_{i},v_{i}$ as singular vectors.

##### Wilson fermion operator in $0+1$ dimension:

We first demonstrate the utility of our approach in the simple case of $0+1$
dimensional Wilson fermion operator $\mathcal{D}_{1d}$ in the presence of a
domain wall. We use periodic boundary conditions on $S^{1}$ and a domain wall
in the fermion mass:

$\displaystyle m(x)$ $\displaystyle=$
$\displaystyle\begin{cases}m_{+}&L/2>x\geq 0\\\ m_{-}&-L/2\leq x<0\\\
\end{cases}.$ (44)

The equation of motion is given by:

$\displaystyle\mathcal{D}_{1d}\psi(x)=\frac{1}{2}\left(\psi(x+1)-\psi(x-1)\right)+m(x)\psi(x)+\frac{R}{2}\left(\psi(x+1)+\psi(x-1)-2\psi(x)\right)=0,$
(45)

which in the case of $R=1$ can be simplified to:

$\displaystyle\mathcal{D}_{1d}\psi(x)=(m(x)-1)\psi(x)+\psi(x+1)=0.$ (46)

We numerically find singular vectors $u_{i}$ and $v_{i}$ together with
singular values $\sigma_{i}$ of this operator and study their dependence on
the lattice size $L$.

Let us first consider singular values $\sigma(L)$ which we depicted on the
Fig. 1. We observe that the smallest singular value $\sigma_{0}$ approaches
zero exponentially fast: $\sigma_{0}\sim O(e^{-L})$, whereas other singular
values remain finite. This indicates that in the infinite volume there exists
a zero mode of $\mathcal{D}_{1d}$ given by infinte volume limit of
corresponding singular vector $v_{min}$.

We show the near-zero mode $v_{0}$ on the Fig. 2(a). We compare it to exact
solution of equation $\mathcal{D}_{1d}\psi_{inf}(x)=0$ in the infinite volume
which is given by:

$\displaystyle\psi_{0}^{inf}(x)=\begin{cases}\left({1-m_{+}}\right)^{x},&\qquad
x\geq 0,\\\ \left({1-m_{-}}\right)^{x},&\qquad x<0,\end{cases}$ (47)

where $m_{\pm}$ are bulk fermion masses on either sides of the domain wall.
Here we work with $m_{-}=3/4,m_{+}=-1$. We find an excellent agreement between
$v_{0}$ and $\psi_{inf}$ already for lattice sizes $L>20$. We also show on the
Fig. 2(b) how near-zeromodes of $\mathcal{D}_{1d}$ and
$\mathcal{D}_{1d}^{\dagger}$ are related by SVD in Eq. 42.

Figure 1: The flow of $10$ smallest singular values $\sigma_{i}$ of the
$0+1$-dimensional Wilson fermion operator $\mathcal{D}_{1d}$ (Eq. 46) as a
function of the lattice size $L$. One can clearly see that the smallest
singular value follows exponential law $\sigma_{0}\sim e^{-\alpha L}$.
Corresponding singular vector is localized on one of the the domain walls. The
domain wall profile is given by Eq. 44 with $m_{+}=-1$ and $m_{-}=3/4$. Note
logarithmic scale.

(a) Volume dependence

(b) Near-zeromodes of $\mathcal{D}_{1d}$ and $\mathcal{D}^{\dagger}_{1d}$. We
show how acting on $v_{0}$ with $\mathcal{D}=\mathcal{D}_{1d}$ takes $v_{0}$
to $u_{0}$ and acting on $u_{0}$ with $\mathcal{D}_{1d}^{\dagger}$ takes
$u_{0}$ to $v_{0}$.

Figure 2: Near-zero modes of $\mathcal{D}_{1d}$ (Eq. 46) in $0+1$ dimensions
with $m_{+}=-1,m_{-}=3/4$. Figure 3: The flow of $40$ smallest singular values
$\sigma_{i}$ of the Wilson fermion operator $\mathcal{D}_{2}$ as a function of
the lattice size $L\times L$. One can clearly see two degenerate singular
values which follow exponential law $\sigma_{i=0,1}\sim e^{-\alpha L}$.
Corresponding singular vectors are localized on two different vortices with
winding numbers $\nu_{\theta}=-1$. The domain wall profile is given by Eq. 38
with $m_{+}=\mu_{+}=1/2$ and $m_{-}=\mu_{-}=-1/2$. Note logarithmic scale.
Figure 4: Density plot of absolute values of near-zeromodes $v_{0}$ and
$v_{1}$ of the Wilson fermion operator $\mathcal{D}_{2}$ and $u_{0}$ and
$u_{1}$ of $\mathcal{D}_{2}^{\dagger}$ corresponding to two smallest singular
values $\sigma_{i=0,1}$ (which are degenerate) for the lattice size $48\times
48$. Arrows illustrate how $u_{i=0,1}$ and $v_{i=0,1}$ are related by the SVD
Eq. 42. The domain wall profile is given by Eq. 38 with $m_{+}=\mu_{+}=1/2$
and $m_{-}=\mu_{-}=-1/2$. Gray dashed line marks the position of domain walls.
Note logarithmic scale. Figure 5: Absolute value of the near-zeromode $v_{0}$
of the Wilson fermion operator $\mathcal{D}_{2}$ along several
$x_{0}=\mathrm{const}$ and $x_{1}=\mathrm{const}$ slices for the lattice size
$48\times 48$. The domain wall profile is given by Eq. 38 with
$m_{+}=\mu_{+}=1/2$ and $m_{-}=\mu_{-}=-1/2$. Note logarithmic scale. Figure
6: The flow of $40$ smallest singular values $\sigma_{i}$ of the Wilson
fermion operator $\mathcal{D}_{2}$ as a function of the lattice size $L\times
L$ for very small diagnostic field $\phi_{2}(x_{0})$ and anti-periodic
boundary conditions in the direction $\mu=0$. One can clearly see that lowest
singular values follow the law $\sigma_{i}\sim 2\pi\left(i+1/2\right)/L$
characteristic to plane waves in anti-periodic box. The domain wall profile is
given by Eq. 38 with $-m_{-}=m_{+}=1/2$ and $-\mu_{-}=\mu_{+}=5*10^{-5}$. Note
log-log scale.

##### Fermion operators in $1+1$ dimension:

Let us now consider $1+1$ dimensional fermion operators we proposed in section
3 and analyze the corresponding zeromodes and near-zeromodes. As mentioned
before, in $1+1$ D, it is possible to obtain an exact zeromode for the Wilson-
like operator in finite volume by tuning the domain wall heights. However, we
didn’t find such a solution for the Wilson fermion operator. Here we will use
SVD to instead find near-zeromodes for the Wilson fermion $\mathcal{D}_{2}$
and Wilson-like fermion operators $\mathcal{D}_{1}$. The results for the
Wilson-like case are very similar to the Wilson fermion case. Therefore, we
only present results for the Wilson fermion case here.

In order to study the singular values of the Wilson fermion operator we use
two-dimensional lattice of the size $L\times L$ and impose periodic boundary
conditions. We also use the domain wall configuration Eq. 38 with
$0>-m_{-}=m_{+}>0$ and $0>-\mu_{-}=\mu_{+}>0$.

By performing SVD numerically for different lattice sizes $L$ we find a
complete set of singular values $\sigma_{i}(L)$ and corresponding singular
vectors $v_{i}(L)$ and $u_{i}(L)$. Let us first consider few lowest singular
values $\sigma_{i}(L)$ which are presented on the Fig. 3. We observe that the
smallest two of them (take them to be $i=0,1$) are degenerate and exhibit
clear exponential decay as $L\rightarrow\infty$. Thus, we find the first
evidence for the emergence of two degenerate zero modes of the Wilson fermion
operator in the infinite volume.

Let us now study corresponding singular vectors $v_{i}(L)$ and $u_{i}(L)$.
Note that there are two degenerate singular vectors $v_{i=0,1}(L)$
corresponding to the lowest $\sigma_{0}=\sigma_{1}$. The same is true for
$u_{i}$. These degenerate vectors are some superposition of two near-zero
modes localized on appropriate vortex defects, i.e. $v_{i=0,1}$ are
superpositions of near-zeromodes on defects with winding $-1$. These two
defects are localized at $(x_{0}=0,x_{1}=0)$ and $(x_{0}=L/2,x_{1}=L/2)$.
Similarly, $u_{i=0,1}$ are superpositions of near-zeromodes located on defects
with winding $1$, $(x_{0}=0,x_{1}=L/2)$ and $(x_{0}=L/2,x_{1}=0)$.

At this point we can change basis by writing
$v^{\prime}_{i}=\alpha_{i}v_{0}+\beta_{i}v_{1}$ with
$|\alpha_{i}|^{2}+|\beta_{i}|^{2}=1$ with $i=0,1$, in order to find near-
zeromodes which are completely localized on the vortices. One can achieve this
by minimizing Inverse Participation Ratio (IPR) which can serve as a measure
of the localization:

$\displaystyle\mathrm{IPR}=\frac{1}{\sum_{x_{0},x_{1}}|v^{\prime}(x_{0},x_{1})|^{2}}.$
(48)

Intuitively, if a mode is uniformly distributed over entire lattice of volume
$V$ then one would find that $\mathrm{IPR}=V$. On the other hand, if the mode
is localized at a single point then $\mathrm{IPR}=1$.

Using this method we find two vectors $v^{\prime}_{i=0,1}(L)$ which are
exponentially localized on two vortices of the same winding number
$\nu_{\theta}=-1$, as shown on the Fig. 4 and Fig. 5. Thus, we have identified
two near-zermodes of the Wilson fermion operator $\mathcal{D}_{2}$. For
convenience, we will refer to these vectors $v_{i=0,1}$ and forego the
superscript prime, as in $v^{\prime}\rightarrow v$. We do the same for the
vectors $u_{0/1}$. The same procedure yields two vectors $u_{i}(L)$
corresponding to the same two singular values localized on the other two
vortices of winding number $\nu_{\theta}=+1$ (at $x_{0}=0,x_{1}=L/2$ and
$x_{0}=L/2,x_{1}=0$).

Finally, let us describe how near-zeromodes behave if one switches the
diagnostic field off, i.e. $\phi_{2}\rightarrow 0$. If the lattice volume is
kept fixed, then at sufficiently small $\phi_{2}$ the near-zeromodes
completely delocalize in the direction $\mu=0$, and the SVD spectrum become
consistent with that of $\phi_{2}=0$ case. Namely, we find that near-zeromodes
transform into plane wave excitations living on the two remaining domain
walls. This can be seen by direct inspection of $|v_{i}(x_{0},x_{1})|$ and
from the behavior of singular values $\sigma_{i}(L)\sim 2\pi n/L$
characteristic to the spectrum of plane waves in the finite box. Furthermore,
lowest singular values are $4$ times degenerate accounting for $2$ remaning
domain walls and $2$ possible spinor polarizations. Additionally, by imposing
anti-periodic boundary condition in the $\mu=0$ direction we again observe
that the flow of singular values $\sigma_{i}(L)\sim 2\pi(n+1/2)/L$ is
characteristic to that of plane waves in the anti-periodic box, see Fig. 6.
The true near-zeromode should not, in general, be sensitive to such change of
boundary conditions. This reorganization happens because for sufficiently
small $\phi_{2}$ the localization width of the near-zero modes become
comparable or bigger than the lattice size, thus it completely delocalizes. If
$\phi_{2}$ is kept fixed then one should recover the near-zeromodes by
increasing the volume. Therefore we find that limits $\phi_{2}\rightarrow 0$
and $L\rightarrow\infty$ do not commute. In order to correctly define the
“index" from the finite volume analysis one has to take infinite volume limit
first and only then switch the diagnostic field off.

##  4 Generalized Hall Current in the finite volume

In this part we will study the realization of the Generalized Hall Current
(GHC) for the Wilson-like and the Wilson fermions and corresponding “indices".
Before we proceed to the computations, let us outline the plan of this
section. First, we will present how we’ve computed the GHC on the lattice.
Next, we will study GHC for the Wilson-like operator $\mathcal{D}_{1}$, taking
the domain wall heights to satisfy the tuning condition (Eq. 39). This will
illustrate how the GHC reproduces the index of the fermion operator in finite
volume for the case when there is an exact zeromode. This will give us an
opportunity to study GHC and its relation to the index without complications
of the finite volume effects.

Next, we will proceed to study of Wilson fermion operator and see how near-
zeromodes and finite volume effects influence the realization of the GHC.
Results for the Wilson-like operator in the same setup (when exact zeromodes
are absent) are essentially the same, therefore we will not present them.

###  4.1 Computation of the Generalized Hall Current on the lattice

The lattice generalized Hall current $J^{H}_{\mu}(x)$ can be defined as
follows:

$\displaystyle
J^{H}_{\mu}(x)=\bar{\Psi}\tilde{\Gamma}_{\mu}(x)\Gamma_{\chi}\Psi$ (49)

where $\tilde{\Gamma}_{\mu}(x)$ is given by:

$\displaystyle\tilde{\Gamma}_{\mu}(x)=-i\left.\frac{\delta
K(A_{\mu}(x))}{\delta A_{\mu}(x)}\right|_{A_{\mu}(x)=0}.$ (50)

Here $A_{\mu}(x)$ is a $U(1)$ gauge field and $K(A_{\mu}(x))$ is a gauged
lattice Dirac operator of the double theory obtained via standard Peierls
substitution
$\delta_{x+a_{\mu},y}\rightarrow\delta_{x+a_{\mu},y}\,\exp(iA_{\mu}(x))$.

The expectation value of $J^{H}_{\mu}(x)$ is evaluated numerically by
straightforward computation of the matrix $(K+M)^{-1}$ and taking a trace. The
divergence is computed as usual with the help of lattice backward difference
$\nabla^{B}_{\mu}$:

$\displaystyle\nabla^{B}_{\mu}J^{H}_{\mu}(x)=\sum_{\mu=0,1}\left(J^{H}_{\mu}(x-a_{\mu})-J^{H}_{\mu}(x)\right).$
(51)

We posit that the space-time integral of the divergence should produce the
“index" of interest. We compute the “index" $I_{lat}$ according to the lattice
version of the Eq. 10:

$\displaystyle I_{lat}=-\frac{1}{2}\sum_{x\in
S}\nabla^{B}_{\mu}J^{H}_{\mu}(x)$ (52)

where $S$ is the area over which the divergence of the lattice GHC current
$J^{H}_{\mu}(x)$ is integrated. The area $S$ can be an entire lattice, however
in that case the total index has to vanish. Thus we will integrate only over
some portion of the lattice adjacent to the defect (vortex) of interest. To
implement this, we divide the lattice into $4$ equal squares centered around
each of the $4$ vortices created by the domain walls and then integrate the
divergence of lattice GHC on these four squares separately to compute the
corresponding index.

###  4.2 GHC for Wilson-like lattice operator and exact zeromodes

Let us first present results for GHC for Wilson-like operator
$\mathcal{D}_{1}$ when domain wall configuration satisfies the tuning
condition Eq. 39. In this case there is an exact zeromode for the fermion
operator in finite volume.

(a) Wilson-like operator $\mathcal{D}_{1}$, domain wall is $m_{-}=\mu_{-}=-1$
and $m_{+}=\mu_{+}=1/2$ which satisfies Eq. 39.

(b) Wilson operator $\mathcal{D}_{2}$, domain wall is $m_{-}=\mu_{-}=-1/2$ and
$m_{+}=\mu_{+}=1/2$.

Figure 7: The lattice GHC $J^{H}_{\mu}(x)$ and its divergence on the lattice
$32\times 32$ for (a) Wilson-like operator $\mathcal{D}_{1}$ with $M=10^{-5}$
and (b) Wilson operator $\mathcal{D}_{2}$ with $M=0.21$. Arrows represent the
current $J^{H}_{\mu}(x)$ with their size being proportional to the current
magnitude. Gray dashed lines show the location of domain walls.

(a) Wilson-like operator $\mathcal{D}_{1}$.

(b) Wilson operator $\mathcal{D}_{2}$.

Figure 8: Several slices of the (log of) divergence of $J^{H}_{\mu}(x)$ for
(a) Wilson-like operator $\mathcal{D}_{1}$ with $M=10^{-5}$ and (b) Wilson
operator $\mathcal{D}_{2}$ with $M=0.21$.

We’ve computed the GHC $J^{H}_{\mu}(M)$ and the “index" $I_{lat}(M)$ for
several values of the regulator mass from $M=10^{-5}$ to $2$ on the lattice
$L\times L=32$. We present the current $J^{H}_{\mu}(x)$ and its divergence on
the Fig. 7(a) for the smallest value of $M$, with $M=10^{-5}$. We observe that
the divergence is localized around the vortices. It has maximal value at the
vortex center. The sign is consistent with the winding number of the defect.
The current $J^{H}_{\mu}(M)$ flows preferably along the edges of the domains
from one vortex to another. The divergence exhibits an exponential decay
around the vortex as shown on Fig. 8(a).

Now we want to verify that the space-time integral of the divergence of the
lattice GHC produces the correct “index". As discussed previously, we divided
the lattice into 4 equal squares centered around each vortex and performed
integration of the divergence of GHC over them. Due to the exponential decay
of the GHC away from the defect, we expect that that the integral would
approach infinite volume value quickly. The resulting “index" $I_{lat}(M)$ is
shown in the Fig. 9(a) as function of $M$. We observe that it clearly goes
towards $\pm 1$ as $M\rightarrow 0$. The sign of the index depends on the
vortex defect in consideration. Also, as expected, for very large $M$ the
“index" approaches zero with increasing $M$.

(a) Lattice “index" as a function of $M$ for several values of $L$.

(b) Convergence of the lattice “index" to the infinite volume value $\pm 1$.

Figure 9: Dependence of the lattice “index" $I_{lat}(M)$ on $M$ and lattice
size $L$ for the Wilson-like operator $\mathcal{D}_{1}$ in the presence of
exact zeromode. The domain wall configuration is given by Eq. 38 with
$m_{-}=\mu_{-}=-1$ and $m_{+}=\mu_{+}=1/2$ which satisfies Eq. 39. Compare
this figure to Fig. 10.

In order to quantify finite volume effects we have computed the deviation:

$\displaystyle\epsilon(L)=|\pm 1-I_{lat}(M\rightarrow 0)|$ (53)

where the plus or minus sign is chosen according to the winding of the vortex
and $I_{lat}$ is the corresponding “index" computed by integrating the
$\nabla_{\mu}^{B}J_{\mu}^{H}$. This function is shown in the Fig. 9(b) where
one can see that the error is indeed exponentially small: $\epsilon(L)\sim
e^{-L}$. Therefore, after performing infinite volume extrapolation our
computations show that the lattice GHC correctly reproduces the index of the
Euclidean fermion operator. Finally, we find that generalized hall current and
divergence vanish when $\phi_{2}\rightarrow 0$ for fixed $L$ and $M$. This
shows that we have to take the infinite volume limit first and then take
$\phi_{2}$ to zero in order to retain a nonzero index in the limit of
$\phi_{2}\rightarrow 0$.

###  4.3 GHC for Wilson fermion operator and near-zero modes

We now present results for GHC and the index for the Wilson fermion operator
$\mathcal{D}_{2}$. The results for the untuned Wilson-like operator are very
similar.

Figure 10: The “index" $I_{lat}(M)$ of the Wilson fermion operator
$\mathcal{D}_{2}$ for several lattice sizes $L=8\dots 32$. The domain wall
configuration is given by Eq. 38 with $m_{-}=\mu_{-}=-1/2$ and
$m_{+}=\mu_{+}=1/2$. Here $\sigma_{0}$ is the smallest singular value and the
inset shows the same data but in the linear scale.

We use the same strategy in order to compute the “index" which is presented in
the Fig. 10 for several values of $M$ and lattice sizes $L=8\dots 32$. First
of all, we observe that the “index" vanishes when we naively take
$M\rightarrow 0$. This is an expected behaviour since the spectrum of
$\mathcal{D}_{2}$ is strictly speaking gapped: $\sigma_{0}\sim\exp(-L)\neq 0$.
In order to understand it better one can expand contribution of
$\mathrm{Dim}(\mathrm{ker}\,\mathcal{D}_{2})$ in powers of $M/\sigma_{0}\ll
1$:

$\displaystyle\frac{M^{2}}{D_{2}^{\dagger}D_{2}+M^{2}}=\frac{M^{2}}{\sigma_{0}^{2}}+O\left(\frac{M^{4}}{\sigma_{0}^{4}}\right).$
(54)

We indeed find this dependence as shown on the Fig. 10. The “index" exhibits a
pronounced maximum at some $M_{0}>\sigma_{0}$ and then decays exponentially
fast as $M\rightarrow\infty$. We find that the maximum tends to $\pm 1$ as
lattice size gets bigger, also exponentially fast, as illustrated on the Fig.
11(a). Moreover, the position of the maximum $M_{0}$ tends to zero as
$L\rightarrow\infty$ exponentially as well, see Fig. 11(b). Therefore we find
in order to reproduce the index of the fermion operator one has to take
infinite volume limit first and only then $M=M_{0}\rightarrow 0$.

(a) Convergence of the maximal value to $\pm 1$.

(b) The position of the maximum.

Figure 11: Dependence of the value and the position of the maximum of the
lattice “index" $I_{lat}(M)$ on lattice size $L$ for the Wilson fermion
operator $\mathcal{D}_{2}$. The domain wall configuration is given by Eq. 38
with $m_{-}=\mu_{-}=-1/2$ and $m_{+}=\mu_{+}=1/2$.

##  5 Conclusions

In this paper we extended the idea of generalized Hall current proposed in
[kaplan2021index, Kaplan:2022uoo] to discrete space-time in finite volume. Our
construction is focused on one of the several examples presented in
[kaplan2021index, Kaplan:2022uoo]: $1+1$ dimensional Dirac fermion with a
domain wall in its mass. It is well known that the domain wall hosts massless
fermion in the continuum. The continuum GHC construction connects the
existence of this massless fermion to a Euclidean fermion operator with an
index of $1$ by turning on some diagnostic field in the theory. We extend this
construction to discrete Euclidean space-time in finite volume ($S^{1}\times
S^{1}$) by introducing higher dimensional operators which we call Wilson-like
and Wilson terms. We tackle several nontrivial features associated with a
finite volume analysis which includes the net vorticity of the defects on
$S^{1}\times S^{1}$ being zero. We have four defects on the lattice, two
vortices and two anti-vortices. In order to mimic the GHC construction of the
continuous infinite volume space-time, we focus on the region of space-time
around only one of these vortices. We were successful in engineering a nonzero
index for the fermion operator on each of these vortices. We then computed the
lattice GHC to show that the space-time integral of its divergence computed
locally reproduced the “index" correctly.

Future research directions involve extending this lattice finite volume
construction to higher dimensional theories. Ref. [kaplan2021index,
Kaplan:2022uoo] constructed the continuum GHC for several examples, including
the $1+1$ dimensional example we focus on here. The other examples included
domain wall fermions in higher dimensions. The GHC construction in these
higher dimensional examples involved diagnostic background gauge fields as
well as diagnostic scalar and pseudo-scalar fields. Our plan is to extend
these continuum constructions to the lattice. Also, the continuum construction
of GHC in [kaplan2021index, Kaplan:2022uoo] applies to free fermion theories.
In particular, the GHC is computed using a one-loop Feynman diagram in
perturbation theory. It is however well known that in a multiflavor theory,
introducing interactions can sometimes gap out massless fermions through
nonperturbative effects. This is even more interesting when the interaction in
question do not break any anomalous symmetries of the non-interacting theory.
E.g. see symmetric mass generation [Wang:2022ucy, Wang:2013yta, You:2014vea,
Wang:2018ugf, Zeng:2022grc, Tong:2021phe, Razamat:2020kyf, PhysRevX.8.011026,
xu2021greens]. The non-perturbative effects of interactions on the GHC may not
be captured using a one loop Feynman diagram as described in [kaplan2021index,
Kaplan:2022uoo]. One may need to resort to a numerical analysis to uncover
these effects. Even though our lattice GHC construction was formulated for
non-interacting $1+1$ Dimensional fermions, it can be easily modified to take
into account interactions. This will enable us to compute the generalized Hall
current taking into account non-perturbative effects.

##  6 Acknowledgement

We acknowledges support from the U.S. Department of Energy, Nuclear Physics
Quantum Horizons program through the Early Career Award DE-SC0021892.