11institutetext: Institute for Nuclear Theory, University of Washington,
Seattle WA 9819522institutetext: Department of Physics and Astronomy, Iowa
State University, Ames IA 50011

# Generalized Hall currents in topological insulators and superconductors

David B. Kaplan<EMAIL_ADDRESS>2 Srimoyee Sen<EMAIL_ADDRESS>
###### Abstract

We generalize the idea of the quantized Hall current to count gapless edge
states in topological materials in any number of dimensions. This construction
applies equally well to theories without any continuous symmetries in the bulk
or chiral anomalies on the boundaries. The generalized Hall current is related
to the phase space index of the Euclidean fermion operator and can be
calculated via one-loop Feynman diagrams. The relevant momentum-space topology
is shown to be governed in each case by the homotopy group
$\pi_{n}(S^{n})={\mathbb{Z}}$, where in $n$ is the number of spacetime
dimensions. This holds even for topological matter which is characterized by a
${\mathbb{Z}}_{2}$ invariant, since the generalized Hall current is governed
by topology in phase space, not just momentum space. We elucidate these ideas
by explicit examples in free relativistic field theories in various spacetime
dimensions with various symmetries, and argue that this approach may work in
interacting theories as well, including the interesting cases where the
interactions gap the edge states.

††preprint: INT-PUB-22-013

subsection

##  1 Introduction

Topological insulators and superconductors, often well described by free
fermion theories Ludwig_2015 ; PhysRevB.85.085103 ; Ryu_2010 ; Kitaev:2009mg ;
PhysRevB.78.195125 , generally have gapless fermion excitations confined to
their boundary, or more generally to any boundary such as a domain wall or
vortex which breaks translation invariance. For $d+1$ dimensional defects,
where $d$ is odd, these massless fermions can be Weyl fermions. In that case
the boundary theory has a chiral anomaly, and the currents for classical axial
symmetries can have nonzero divergence in the presence of background gauge
fields. These currents are conserved in the higher dimensional bulk theory,
and the violation of axial charge in the boundary theory can be understood as
the result of a fermion number current flowing in from the bulk, as in the
Integer Quantum Hall Effect Callan:1984sa . The argument can be turned around:
the inflow of current from the bulk indicates a boundary theory with a chiral
anomaly, which in turn requires massless particles to exist at the boundary.

It is appealing that one can deduce the existence of gapless edge states from
the inflow of current from infinity, but it is not general. For example,
gapless edge states exist for superconducting systems for which there is no
conserved fermion number current, as well as for boundaries with even spatial
dimension $d$ where there are no chiral anomalies. In a recent paper we
showed, however, that in all of these cases one can generalize the idea of
Hall currents and detect the existence gapless edge states by this current’s
divergence kaplan2021index . That theory makes use of the Callias index
theorem Callias:1977kg  applied to the index of the fermion operator in the
Euclidian action and in the presence of external “diagnostic fields”. A very
simple procedure for calculating the current flow in terms of a one-loop
Feynman diagram allows one to detect the existence of gapless edge states.
Furthermore, this construction makes the topological protection of the gapless
modes manifest. In the present paper we work through the examples given in
Ref. kaplan2021index  in greater detail, including the construction of the
generalized Hall current, the topological origins of its nonzero divergence,
and the role of regularization. We also extend the previous work with a
discussion of the effect of interactions. Our discussion throughout is in the
context of relativistic quantum field theory.

##  2 General framework

Our starting point is a relativistic quantum field theory of free fermions in
infinite $d+1$ dimensional Minkowski spacetime. The action is
$S_{\text{M}}=\int d^{d+1}x\,\,\bar{\psi}{\cal D}_{\text{M}}\psi$, where the
fermion operator ${\cal D}_{\text{M}}$ can include a position dependent mass
term, e.g. one that exhibits defects like a domain wall or a vortex
singularity, which will serve as proxies for a boundary. If the fermion
operator ${\cal D}_{\text{M}}$ hosts massless fermion states trapped in these
defects, and the defects are space and time translation invariant in the co-
dimensions, then the spectrum will include a state with zero momentum that is
constant in the space-time coordinates of the defect, and is localized on the
defect in the transverse direction(s). We wish to have a simple way to detect
the existence of such states which makes manifest the underlying topology. The
approach we use is to note that when ${\cal D}_{\text{M}}$ is analytically
continued to Euclidian time, the Euclidian fermion operator ${\cal D}$ is an
elliptic operator and the propagator $1/{\cal D}$ will have poles
corresponding to gapless Minkowski edge states with zero momentum. We detect
these states by computing the index of ${\cal D}$ — the number of zeromodes of
${\cal D}$ minus that of ${\cal D}^{\dagger}$. One obstacle to this program is
that the index only counts normalizable zeromodes, while the states we are
discussing are constant in the noncompact boundary dimensions; another is that
${\cal D}^{\dagger}$ also exhibits poles corresponding to these states. Both
can be circumvented by adding “diagnostic” background fields that localize
these states within the boundary without introducing a gap. This can be easily
done when the diagnostic fields possess nontrivial spatial topology; an
interesting feature of this approach is that the diagnostic fields are
typically not fields that could have been introduced into the original
Minkowski theory, as we will explain below. In the presence of these
diagnostic fields, we find that possible solutions to ${\cal
D}^{\dagger}\psi=0$ are $delocalized$ and so the operator ${\cal D}$ acquires
a nonzero index. This effect is seen with the toy operator ${\cal
D}=\partial_{x}+\epsilon(x)$ which has a localized solution to ${\cal
D}\psi=0$ while the conjugate operator ${\cal
D}^{\dagger}=-\partial_{x}+\epsilon(x)$ does not. The index then indicates the
presence of a massless edge state in the Minkowski theory, provided that it
persists in the limit that the energy density in the diagnostic fields tends
to zero.

Computing the index then simply requires computing the divergence of an in-
flowing current by means of a 1-loop diagram, which generalizes the in-flow
picture from the quantum Hall system in any dimensions and whether the system
has a conserved current or not. Furthermore, the index is shown to be the
product of the winding number of the diagnostic fields in coordinate space
times the winding number of the fermion dispersion relation in momentum space,
making manifest the topological nature of the gapless modes. We will see that
the Euclidian momentum space topology is simpler than the topological
invariants of the Minkowski systems would suggest: in each case we examine it
is governed by the homotopy group $\pi_{n}(S^{n})$, where $n$ is the number of
spacetime dimensions. The index we compute reflects the correct invariant of
the Minkowski system, ${\mathbb{Z}}_{2}$ for example, due to the interplay of
momentum space and coordinate space topology in its definition.

Our procedure is (i) start with a Minkowski theory of interest; (ii)
analytically continue to Euclidian space time; (iii) introduce diagnostic
fields to localize edge states; (iv) compute the index of the Euclidian
fermion operator in the limit of vanishing diagnostic fields, which involves
computing the inflow of a generalized Hall current. In the cases where fermion
number is not conserved, relativistic systems analogous to topological
superconductors, the Euclidian action takes the form
$S_{\text{E}}=\frac{1}{2}\int\psi^{T}C{\cal D}\psi$, where $C$ is the charge
conjugation matrix. In these cases at step (ii) we consider instead the Dirac
action $S_{E}=\int\bar{\psi}{\cal D}\psi$ and proceed from there, since
solutions to ${\cal D}\psi=0$ are also solutions to $C{\cal D}\psi=0$. The
Dirac theory allows for the addition of diagnostic fields not possible in the
original theory, such as a $U(1)$ gauge field coupled to fermion number, or
fields in ${\cal D}$ which appear symmetrically in $C{\cal D}$, and hence
would not couple to the fermions in $\psi^{T}C{\cal D}\psi$.

The procedure for computing the index of an elliptic operator ${\cal D}$ is to
first define

$\displaystyle{\cal I}(M)={\rm Tr\,}\left(\frac{M^{2}}{{\cal D}^{\dagger}{\cal
D}+M^{2}}-\frac{M^{2}}{{\cal D}{\cal D}^{\dagger}+M^{2}}\right)$ (1)

with

$\displaystyle{\rm ind}({\cal D})={\cal I}(0)\equiv\lim_{M\to 0}{\cal I}(M)\
.$ (2)

The function ${\cal I}(M)$ can be computed by means of Feynman diagrams by
first defining

$\displaystyle K=\begin{pmatrix}0&&-{\cal D}^{\dagger}\\\ {\cal
D}&&0\end{pmatrix},\,\qquad\Gamma_{\chi}=\begin{pmatrix}1&&0\\\
0&&-1\end{pmatrix}\ ,$ (3)

in terms of which ${\cal I}(M)$ is expressed as

$\displaystyle{\cal I}(M)={\rm Tr\,}\left(\Gamma_{\chi}\frac{M}{K+M}\right).$
(4)

The matrix $K$ and $\Gamma$ have twice the dimension of ${\cal D}$. The
operator $1/(K+M)$ looks like a fermion propagator in a theory whose action is

$\displaystyle S_{K}=\int d^{d+1}x\,\kern
0.6pt\overline{\kern-0.6pt\Psi\kern-0.6pt}\kern 0.6pt(K+M)\Psi,$ (5)

where $\Psi$ has twice the number of components as the original $\psi$ field
off our Minkowski theory, but the integration is still over $d+1$ spacetime
coordinates. In this extended theory, the quantity ${\cal I}(M)$ in eq. (4)
can be expressed as the matrix element of the pseudoscalar density,

$\displaystyle{\cal I}(M)=-M\int
d^{d+1}x\,\langle\bar{\Psi}\Gamma_{\chi}\Psi\rangle$ (6)

where $\langle\bar{\Psi}\Gamma_{\chi}\Psi\rangle$ is computed using a path
integral with weight $e^{-S_{K}}$. We can also define an axial current

$\displaystyle\mathcal{J}_{\mu}^{\chi}=\bar{\Psi}\Gamma_{\mu}\Gamma_{\chi}\Psi$
(7)

where $\Gamma_{\mu}=i\partial\widetilde{K}(p)/\partial{p^{\mu}}$, where we use
the tilde to indicate the operator has been Fourier transformed to momentum
space. Assuming that ${\cal D}=\not{\partial}+\ldots$, where the ellipsis
denotes nonderivative terms, then
$\Gamma_{\mu}=\sigma_{1}\otimes\gamma_{\mu}$, in the notation where
$\Gamma_{\chi}=\sigma_{3}\otimes 1$. Applying the axial transformation
$\Psi\rightarrow e^{i\Gamma_{\chi}\theta(x)}\Psi$, we can derive the Ward-
Takahashi identity for this axial current as

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

where the first term on the right is the classical divergence due to the mass
$M$, and the second term is the anomaly, which can be computed using the
method of Ref. PhysRevLett.42.1195 ,

$\displaystyle{\cal A}=-2\,\lim_{\Lambda\rightarrow\infty}{\rm
Tr\,}\left(\Gamma_{\chi}e^{K^{2}/\Lambda^{2}}\right)=-2{\cal I}(\infty)\ .$
(9)

As we show below the anomaly vanishes in the cases we consider. With ${\cal
A}=0$, eq. (8) can then be used to express the index $\text{ind}({\cal D})$ as

$\displaystyle\text{ind}({\cal D})=-\tfrac{1}{2}\lim_{M\to 0}\int
d^{d+1}x\,\partial_{\mu}\langle{\cal J}_{\mu}^{\chi}\rangle\ .$ (10)

The current ${\cal J}_{\mu}^{\chi}$ is what we refer to as the generalized
Hall current. With the anomaly ${\cal A}$ vanishing, the nonzero divergence of
this current in the $M\to 0$ limit arises from infrared divergences in the
theory, and its inflow counts gapless edge states in the original Minkowski
theory.

The generalized Hall current in eq. (10) in the $M\to 0$ limit can be computed
from a one-loop Feynman diagram ${\rm Tr\,}\Gamma_{\mu}\Gamma_{\chi}K^{-1}$,
which may need to regulated. The UV regulator will in general contribute to
the current in odd spacetime dimensions, but not to its divergence, which
arises from infrared physics. However, in these cases the topological meaning
of the current is obscured if regulator contributions are neglected, as the
Feynman diagram can be interpreted as being proportional to the winding number
of a map from momentum space to an $n$-sphere – but only if the momentum space
of the fermion is compact. This makes the winding number sensitive to the
ultraviolet behavior of the fermion propagator. Here we use Pauli-Villars
regularization when required, computing the index of ${\cal
D}_{\text{reg}}=\frac{{\cal D}(m)}{{\cal D}(\Lambda)}$, and then sending the
regulator mass $\Lambda$ to infinity. The regulated current then is given by
${\rm Tr\,}\Gamma_{\mu}\Gamma_{\chi}K_{\text{reg}}^{-1}$, where
$K_{\text{reg}}$ is given by eq. (3) with ${\cal D}$ replaced by ${\cal
D}_{\text{reg}}$. We do this in any spacetime dimension $d+1$, but find that
for even $d+1$, the regulator does not contribute to the index.

In the next sections we work out in detail the examples of Ref.
kaplan2021index , starting with the continuum example of a single Majorana
fermion chain in $1+1$ dimensions with $0+1$ dimensional defect hosting a
Majorana zero mode. This is followed by the example of multiple flavors of
Majorana fermions in $1+1$ dimension with time reversal symmetry violation. In
both cases, the generalized Hall current correctly produces the index of the
fermion operator and counts the number of massless fermions in Minkowski
space. We then discuss Dirac fermion in $2+1$ dimensions, which exhibits the
Integer Quantum Hall Effect, with the in-flow anomaly current described in
Ref. Callan:1984sa . We then add a fermion number violating Majorana mass term
this theory, modeling a topological superconductor in $2+1$ dimensions. Again
we show how the divergence of the generalized Hall current correctly detects
edge states, even though there is no conventional Hall current in such a
system, and we discuss the topological origins of the index. After discussing
the 2+1 dimensional case, we work through an analogous example in $3+1$
dimensions: that of Dirac fermion with a domain wall in its mass which
describes three dimensional topological insulator and its edge states. We
again compute the generalized Hall current, show how it reproduces the index
and display it topological origins. The final section provides a qualitative
argument in the context of $1+1$ dimensional interacting Majorana chains that
the divergence of the generalized Hall current detects zeromodes even in the
presence of interactions, even though the index of the free fermion propagator
is no longer relevant.

##  3 Majorana fermion $1+1$ dimensions

###  3.1 One flavor of Majorana fermion

For our first example we consider a single Majorana fermion in $1+1$ dimension
with domain wall profile for the Majorana mass, with the $0+1$ dimensional
defect hosting a 1-component real massless fermion. In this theory there is no
continuous symmetry, and hence no conserved current in the bulk, and no
chirality or chiral anomaly on the $0+1$ dimensional defect. Therefore this
system cannot exhibit the conventional Hall response. As we will show,
however, we can compute generalized Hall currents which converge on the defect
when it hosts gapless states.

The Minkowski Lagrangian in this case is

$\displaystyle{\cal
L}_{M}=\tfrac{1}{2}\psi^{T}C\left(i\not{\partial}-m\right)\psi,$ (11)

which is the continuum version of the 1-flavor theory considered in Ref.
Fidkowski:2009dba . Here $\psi$ is a real, two-component Grassmann spinor. The
$\gamma$ matrices we take are

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

where $\sigma_{i}$ are the Pauli matrices. This model of Majorana fermions has
a rich phase structure when interactions are included Fidkowski:2009dba , but
in this section we restrict ourselves to analyzing the free theory. To
construct the generalized Hall current for this model we first define the
Euclidian theory111We use the mostly minus convention for the Minkowski
metric, and in $d+1$ dimensions we denote both Minkowski and Euclidean time by
$x^{0}$. The relation between Euclidian and Minkowski $\gamma$-matrices is
$\gamma^{0}_{M}=\gamma^{0}_{E}$, $\gamma^{i}_{M}=i\gamma^{i}_{E}$ and
$\gamma_{\chi}$ is the same in both Minkowski and Euclidean spacetime.

$\displaystyle{\cal L}_{E}=\tfrac{1}{2}\psi^{T}C{\cal D}\psi\ ,$ (13)

where in Euclidean spacetime

$\displaystyle{\cal D}=\not{\partial}+m\ ,\qquad\gamma_{0}=C=\sigma_{2}\
,\quad\gamma_{1}=-\sigma_{1}\ ,\quad\gamma_{\chi}=\sigma_{3}\ .$ (14)

We now wish to study the index of ${\cal D}$, somewhat modified (there being a
one-to-one correspondence between zeromodes of ${\cal D}$ and $C{\cal D}$).
The modifications involve replacing the step function mass with a general
scalar field $\phi_{1}(x)$, and adding a pseudoscalar field $\phi_{2}(x)$ as
our diagnostic field, so that the more general operator we consider is

$\displaystyle{\cal D}$
$\displaystyle=\not{\partial}+\phi_{1}+i\phi_{2}\gamma_{\chi}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}\phi_{1}+i\phi_{2}&-i\partial_{0}-\partial_{1}\\\
i\partial_{0}-\partial_{1}&\phi_{1}-i\phi_{2}\end{pmatrix}\ ,$ (15)
$\displaystyle{\cal D}^{\dagger}$
$\displaystyle=-\not{\partial}+\phi_{1}-i\phi_{2}\gamma_{\chi}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}\phi_{1}-i\phi_{2}&i\partial_{0}+\partial_{1}\\\
-i\partial_{0}+\partial_{1}&\phi_{1}+i\phi_{2}\end{pmatrix}\ .$

To understand the logic of this construction, first consider the operator we
are ultimately interested in, where $\phi_{2}=0$ and
$\phi_{1}(x)=m_{0}\epsilon(x_{1})$ with $m_{0}>0$. In this case we find two
solutions each to the equations ${\cal D}\psi=0$ and ${\cal
D}^{\dagger}\chi=0$, namely

$\displaystyle\psi_{-}(x)$ $\displaystyle=\begin{pmatrix}1\\\
1\end{pmatrix}e^{m_{0}|x_{1}|}\ ,$ $\displaystyle\qquad\psi_{+}(x)$
$\displaystyle=\begin{pmatrix}\phantom{-}1\\\
-1\end{pmatrix}e^{-m_{0}|x_{1}|}\ ,$ (16) $\displaystyle\chi_{-}(x)$
$\displaystyle=\begin{pmatrix}1\\\ 1\end{pmatrix}e^{-m_{0}|x_{1}|}\ ,$
$\displaystyle\qquad\chi_{+}(x)$ $\displaystyle=\begin{pmatrix}\phantom{-}1\\\
-1\end{pmatrix}e^{m_{0}|x_{1}|}\ ,$

where the $\pm$ subscript refers to the eigenvalue of $\gamma_{1}$. None are
normalizable since they are constant in $x_{0}$, whether or not they are
localized at the mass defect at $x_{1}=0$, and therefore the index calculation
will give us $\text{ind}({\cal D})=0$. However, if we now add
$\phi_{2}=\mu_{0}\epsilon(x_{0})\ $ with $\mu_{0}>0$, a domain wall in
Euclidian time, we find the four solutions to ${\cal D}\psi=0$ and ${\cal
D}^{\dagger}\chi=0$ are

$\displaystyle\psi_{-}(x)$ $\displaystyle=\begin{pmatrix}1\\\
1\end{pmatrix}e^{m_{0}|x_{1}|+\mu_{0}|x_{0}|}\ ,$
$\displaystyle\qquad\psi_{+}(x)$ $\displaystyle=\begin{pmatrix}\phantom{-}1\\\
-1\end{pmatrix}e^{-m_{0}|x_{1}|-\mu_{0}|x_{0}|}\ ,$ (17)
$\displaystyle\chi_{-}(x)$ $\displaystyle=\begin{pmatrix}1\\\
1\end{pmatrix}e^{-m_{0}|x_{1}|+\mu_{0}|x_{0}|}\ ,$
$\displaystyle\qquad\chi_{+}(x)$ $\displaystyle=\begin{pmatrix}\phantom{-}1\\\
-1\end{pmatrix}e^{m_{0}|x_{1}|-\mu_{0}|x_{0}|}\ ,\ (x_{2}\to x_{0})$

${\cal D}$ now has a single normalizable zeromode $\psi_{+}$ localized at
$x_{0}=x_{1}=0$ while ${\cal D}^{\dagger}$ has none. Therefore the index is
given by $\text{ind}({\cal D})=1$ in this background scalar field. By
considering the other signs for $m_{0}$ and $\mu_{0}$ one sees that more
generally we get $\text{ind}({\cal D})=-\nu_{\phi}$, where $\nu_{\phi}$ is the
winding number of $\phi=\phi_{1}+i\phi_{2}$ in the $x_{0}-x_{1}$ plane.222This
is easier to see if one replaces the singular step functions with something
smoother, such as a $\tanh$ function. In fact, it has to be smoothed out to
justify the derivative expansion we perform next. As we shall see, this
expectation is born out by the explicit calculation we give below.
Furthermore, our results persist even no matter how small we take $|\mu_{0}|$.
We also see how the zeromodes we are counting would not have existed if there
hadn’t been a solution to ${\cal D}\psi=0$ for the case of interest,
$\phi_{2}=0$ and $\phi_{1}=m_{0}\epsilon(x_{1})$ (for either sign of $m_{0}$),
and so the nonzero value for $\text{ind}({\cal D})$ informs us that the
original Minkowski theory has massless edge states.

Following the procedure laid out in the previous section, we compute the index
of ${\cal D}$ by constructing the fermion operator $K$ which serves as the
fermion operator in a theory with twice the number of fermion degrees of
freedom,

$\displaystyle K=\begin{pmatrix}0&&-{\cal D}^{\dagger}\\\ {\cal
D}&&0\end{pmatrix}=\Gamma_{\mu}\partial_{\mu}+i(\phi_{2}\Gamma_{2}+\phi_{1}\Gamma_{3})\
.$ (18)

where we have defined the five $4\times 4$ matrices

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

with $i=0,1$.

Our task is to compute the part of the chiral current ${\cal
J}_{\mu}=\bar{\Psi}\Gamma_{\mu}\Gamma_{\chi}\Psi$ that contributes to the
index, where $\mu=0,1$ and $\Psi$ is a fermion with action
$S=\bar{\Psi}K\Psi$. To do this we first write the scalars as

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\phi_{1}+i\phi_{2}=(v+\rho(x))e^{i\theta(x)}$ (20)

assuming constant $v$ and slowly varying $\rho$ and $\theta$ where
$\langle\rho\rangle=\langle\theta\rangle=0$ and the gradients of $\rho$,
$\theta$ fall off at infinity. Assuming $\langle\theta\rangle=0$ is general
since we can perform a constant chiral rotation of ${\cal D}$ which leaves the
index invariant and shifts $\theta$. To compute the part of the current that
contributes to the index we need the leading term in a $1/v$ expansion, since
higher order terms will drop off too fast at infinity to contribute to the
integral $\int\partial_{\mu}{\cal J}_{\mu}$. Thus we can write

$\displaystyle K$ $\displaystyle=$ $\displaystyle K_{0}+\delta K\ ,$ (21)

with

$\displaystyle K_{0}=\partial_{\mu}\Gamma_{\mu}+iv\,\Gamma_{3}\ ,\qquad\delta
K=iv\,\theta(x)\Gamma_{2}+i\rho(x)\Gamma_{3}\ \ .$ (22)

Then $K_{0}^{-1}$ will be our free fermion propagator, while we perturb in
$\delta K$.

When expanding ${\cal J}_{\mu}$ in $\delta K$, note that because of the
insertion of $\Gamma_{\chi}$ in the fermion loop we require that the rest of
the graph supplies one each of the other four $\Gamma$ matrices in order to
get a nonzero contribution from the trace. The matrices $\Gamma_{0,1,3}$ can
be supplied by the fermion propagators while $\Gamma_{2}$ must arise from an
insertion of $\theta$, and by the shift invariance, it must be proportional
the derivative of $\theta(x)$. Thus the leading contribution is given by the
graph in Fig. 1 expanded to linear order in the momentum carried by $\theta$.

Figure 1: The loop diagram for computing the generalized Hall current for the
$1+1$-dimension Dirac fermion. The black dot is an insertion of the chiral
current $\Gamma_{\mu}\Gamma_{\chi}$ with incoming momentum $p_{\nu}$. The
outgoing field $\alpha$ is the spatially varying part of the phase of the
complex field $\phi_{1}+i\phi_{2}$, and the fermion propagator is given by
$K_{0}^{-1}$.

Using $\tilde{K}_{0}$ to denote the Fourier transform of $K_{0}$ in momentum
space, the diagram can be computed as

$\displaystyle{\cal J}_{\mu}$ $\displaystyle=$ $\displaystyle
v\,\frac{\partial\theta}{\partial x_{\nu}}\int\frac{d^{2}q}{(2\pi)^{2}}\,{\rm
Tr\,}\Biggl{[}\Gamma_{\mu}\Gamma_{\chi}\left(\frac{\partial\tilde{K}_{0}^{-1}}{\partial
q_{\nu}}\right)\Gamma_{2}\tilde{K}_{0}^{-1}\Biggl{]}$ (23) $\displaystyle=$
$\displaystyle\epsilon_{\mu\nu}\partial_{\nu}\theta\int\frac{d^{2}q}{(2\pi)^{2}}\frac{4v^{2}}{(q^{2}+v^{2})^{2}}$
(24) $\displaystyle=$
$\displaystyle\frac{1}{\pi}\epsilon_{\mu\nu}\partial_{\nu}\theta\ .$ (25)

In this expression, the derivative $\partial_{\nu}$ acting on $\theta(x)$ is
with respect to $x_{\nu}$, while the derivative $\partial_{\nu}$ acting on
$\tilde{K}_{0}$ is with respect to $q_{\nu}$. The positive sign in the first
line above arises from $(-1)$ from the fermion loop, a $(-i)$ from Fourier
transforming $p_{\nu}\to-i\partial/(\partial x_{\nu})$, and another $(-i)$
from the $\theta$ vertex, which gives $-iv\Gamma_{2}$.

From eq. (10) it follows that the index of ${\cal D}$ is given by

$\displaystyle\text{ind}({\cal D})=-\tfrac{1}{2}\int d^{2}x\partial_{\mu}{\cal
J}_{\mu}=-\frac{1}{2\pi}\oint\frac{\partial\theta}{\partial
x_{\mu}}d\ell_{\mu}=-\nu_{\phi}\ ,$ (26)

where $\nu_{\phi}$ is the winding number of $\phi$. This result agrees with
what was predicted from our earlier heuristic argument with a configuration of
crossed domain walls.

Before turning to the question of topology we address two issues about the way
we handled the Ward-Takahashi identity for the generalized Hall current, given
in eq. (8). First of all, we set $M=0$ from the outset, rather than performing
the computation at nonzero $M$ and then taking the limit $M\to 0$ at the end.
This is discussed further in § 6, but in brief: the role of $M$ was to serve
as an IR regulator for the calculation, but instead we used our background
field $\phi$ to serve as the IR regulator. Note that the generalized Hall
current we found in eq. (25) is proportional to
$\partial_{\mu}\theta=i\phi^{*}\overleftrightarrow{\partial_{\mu}}\phi/2|\phi|^{2}$,
and that the inverse dependence on $|\phi|$ indicates its role as an IR
regulator. Secondly, we ignored the anomaly term, ${\cal A}$. This quantity
can be computed using the methods of Fujikawa PhysRevLett.42.1195 , where

$\displaystyle{\cal A}=\lim_{\Lambda\to\infty}{\rm
Tr\,}\Gamma_{\chi}e^{K^{2}/\Lambda^{2}}\ .$ (27)

(Note that $K$, defined in eq. (22), is antihermitean.) The two-derivative
term in $K^{2}$ gives rise to the Gaussian integral

$\displaystyle\int\frac{d^{2}k}{(2\pi)^{2}}\,e^{-k^{2}/\Lambda^{2}}\propto\Lambda^{2}\
.$ (28)

On the other hand, since the $\Gamma$-matrices obey the Clifford algebra for
$SO(4)$, the $\Gamma$-matrix trace with $\Gamma_{\chi}$ requires that the
expansion of $e^{K^{2}/\Lambda^{2}}$ supplies at least four different
$\Gamma_{\mu}$ to the trace before one obtains a nonzero value. As $K/\Lambda$
is linear in the $\Gamma$ matrices, these four $\Gamma$ matrices will be
accompanied by a factor of $1/\Lambda^{4}$ which overwhelms the factor of
$\Lambda^{2}$ from the momentum integral and causes the anomaly term ${\cal
A}$ to vanish as $\Lambda\to\infty$. This will occur in all of our examples,
since the doubled theory always has the same $d+1$ spacetime dimension as the
original theory, while the $\Gamma$-matrices belong to the Clifford algebra
for $SO(d+3)$.

The calculation we have performed does not make explicit the topological
quantization of the index, since it seems like an accident of the momentum
integral that the integral the divergence of the generalized Hall current came
out to be an integer times $\nu_{\phi}$. To display the topology underpinning
of the calculation explicitly we define $\tilde{\cal D}_{0}$ to equal
$\tilde{\cal D}$ with $\phi_{1}\to v$ and $\phi_{2}\to 0$ and use the
identities $\Gamma_{\mu}=-i\partial_{\mu}\tilde{K}_{0}$ and
$\\{\gamma_{\chi},\tilde{\cal D}_{0}\\}=2v\gamma_{\chi}$ to rewrite eq. (25)
as

$\displaystyle{\cal J}_{\mu}$ $\displaystyle=$ $\displaystyle-
iv\partial_{\nu}\theta\,\int\frac{d^{2}q}{(2\pi)^{2}}\,{\rm
Tr\,}\Biggl{[}\Gamma_{\chi}\left(\partial_{\nu}\tilde{K}_{0}^{-1}\right)\Gamma_{2}\tilde{K}_{0}^{-1}\partial_{\mu}\tilde{K}_{0}\Biggr{]}$
(29) $\displaystyle=$ $\displaystyle-
iv\partial_{\nu}\theta\,\int\frac{d^{2}q}{(2\pi)^{2}}\,{\rm
Tr\,}\Biggl{[}\gamma_{\chi}\left(\tilde{D}_{0}^{-1}\partial_{\mu}\tilde{D}_{0}\partial_{\nu}\tilde{D}^{-1}_{0}+\tilde{D}_{0}\to\tilde{D}^{\dagger}_{0}\right)\Biggr{]}$
(31) $\displaystyle=$
$\displaystyle\frac{i}{4}\epsilon_{\mu\nu}\partial_{\nu}\theta\epsilon_{\sigma\tau}\,\int\frac{d^{2}q}{(2\pi)^{2}}\,{\rm
Tr\,}\Biggl{[}\gamma_{\chi}\left(\tilde{D}_{0}^{-1}\partial_{\sigma}\tilde{D}_{0}\tilde{D}_{0}^{-1}\partial_{\tau}\tilde{D}_{0}+\tilde{D}_{0}\partial_{\sigma}\tilde{D}^{-1}_{0}\tilde{D}_{0}\partial_{\tau}\tilde{D}^{-1}_{0}+\tilde{D}_{0}\to\tilde{D}^{\dagger}_{0}\right)\Biggr{]}$
(33)

We can then define the special unitary matrix

$\displaystyle\xi=\frac{\tilde{\cal D}_{0}}{\sqrt{\det\tilde{\cal
D}_{0}}}=\frac{v}{\sqrt{q^{2}+v^{2}}}+i\hat{q}_{\mu}\gamma_{\mu}\frac{q}{\sqrt{q^{2}+v^{2}}}\
,$ (35)

in terms of which eq. (LABEL:eq:D3) can be rewritten as

$\displaystyle{\cal
J}_{\mu}=\frac{i}{2}\epsilon_{\mu\nu}\partial_{\nu}\theta\,\epsilon_{\sigma\tau}\int\frac{d^{2}q}{(2\pi)^{2}}{\rm
Tr\,}\gamma_{\chi}\left[\left(\xi^{\dagger}\partial_{\sigma}\xi\right)\left(\xi^{\dagger}\partial_{\tau}\xi\right)+\left(\xi\partial_{\sigma}\xi^{\dagger}\right)\left(\xi\partial_{\tau}\xi^{\dagger}\right)\right]\
,$ (36)

unaffected by the normalizing determinant factor (which can’t vanish for
nonzero $v$). As a final manipulation, we can define the “axial current”,

$\displaystyle
A_{j}=\frac{i}{2}\left(\xi^{\dagger}\partial_{j}\xi-\xi\partial_{j}\xi^{\dagger}\right)\
,$ (37)

and using the identity $\gamma_{\chi}\xi\gamma_{\chi}=\xi^{\dagger}$ we can
write the generalized Hall current as

$\displaystyle{\cal
J}_{\mu}=-i\epsilon_{\mu\nu}\partial_{\nu}\theta\,\epsilon_{\sigma\tau}\int\frac{d^{2}q}{(2\pi)^{2}}{\rm
Tr\,}\left[\gamma_{\chi}A_{\sigma}A_{\tau}\right]\ .$ (38)

In Appendix A we show that the above integral is proportional to the winding
number of the map provided by the Dirac propagator from momentum space
compactified to $S^{2}$, to “Dirac space” (the analog of the Bloch sphere),
also $S^{2}$ in two spacetime dimensions. That winding number $\nu_{q}$ is an
element of $\pi_{2}(S^{2})={\mathbb{Z}}$, and for this particular map we have
$\nu_{q}=1$. Making use of the normalization given in eq. (152) we arrive at

$\displaystyle{\cal
J}_{\mu}=\frac{\epsilon_{\mu\nu}\partial_{\nu}\theta}{\pi}\,\nu_{q}\
,\qquad\nu_{q}=1\ .$ (39)

The topological quantization in momentum space is analogous to the topological
origin of the quantization of the Integer Quantum Hall Effect discovered in
the celebrated TKNN paper and related work thouless1982quantized ;
niu1984quantised ; PhysRevB.31.3372 .

Our final result for the index is then given by

$\displaystyle\text{ind}({\cal D})=-\nu_{\phi}\nu_{q}\ ,$ (40)

where $\nu_{\phi}$ is the winding number of our diagnostic field in coordinate
space, obtained by integrating $\epsilon_{\mu\nu}\partial_{\nu}\theta$ and
determined by the homotopy group $\pi_{1}[U(1)]=\mathbb{Z}$. We see that the
index, which counts edge states, is manifestly quantized by simultaneous
nontrivial topology in both momentum and coordinate space. This result
confirms our heuristic argument about localizing the zeromode with crossed
domain walls, and agrees with eq. (26).

Relativistic quantum field theories can be assigned to the same ten
topological classes as used for condensed matter systems; for example see Ref.
PhysRevB.78.195125 . The translation is simple, with what are called time
reversal symmetry, particle hole symmetry, and sublattice symmetry replaced by
the conventional $T$, $C$ and $P$ symmetries respectively in the relativistic
theory. The ten categories are constructed from (i) time reversal symmetry
with $T^{T}=T$, or $T$ symmetry with $T^{T}=-T$, or $T$ violation; (ii) charge
conjugation symmetry with $C^{T}=C$, or $C$ symmetry with $C^{T}=-C$, or $C$
violation; (iii) both $C$ and $T$ violation, but good parity symmetry. Due to
the $CPT$ theorem, no other possibilities exist. The only caveat in comparing
with the tables from condensed matter papers is that the symmetry of the $C$
matrix is opposite that of the conventional particle-hole symmetry matrix,
since in the relativistic case charge conjugation is that the symmetry of $C$
is opposite that of the conventional particle-hole symmetry, since in the
relativistic case charge conjugation is the transformation $\psi\to
C\bar{\psi}^{T}$ for a Dirac fermion, while particle hole symmetry is $\psi\to
PHS\psi^{*}$, without the extra $\gamma^{0}$ matrix. In the present example we
have both $T$ and $C$ with symmetric $T$ and antisymmetric $C$, which puts the
system in the BDI class, with topological invariant ${\mathbb{Z}}$. If we had
considered a $N_{f}$ -flavor version of the theory considered in this section
without special global flavor symmetries, we would have trivially found
$\text{ind}({\cal D})=-N_{f}\nu_{\phi}\nu_{q}$, which can take on any value in
${\mathbb{Z}}$ as one would expect for a BDI topological class in one spatial
dimension.

###  3.2 Multiple flavors of $d=1+1$ Majorana fermions with time reversal
symmetry violation

With $N_{f}$ flavors of free fermions in our $1+1$ dimensional model, it is
possible to include both scalar and pseudoscalar mass terms:

$\displaystyle{\cal
L}_{M}=\tfrac{1}{2}\psi_{i}^{T}C\left(i\not{\partial}\delta_{ij}-m_{ij}-i\gamma_{\chi}\mu_{ij}\right)\psi_{j},$
(41)

where $m$ must be a real and symmetric matrix, while $\mu$ is imaginary and
antisymmetric. Without loss of generality it is possible to take $m_{ij}$ to
be diagonal.

The theory with $\mu_{ij}=0$ is invariant under the antiunitary time-reversal
symmetry, $\psi_{i}(x,t)\to\sigma_{1}\psi_{i}(x,-t)$. When the masses have
domain wall profiles, such as $m_{i}=m_{0}\epsilon(x)$ with $m_{0}>0$, there
will be $N_{f}$ massless Majorana modes localized at the mass defect with wave
function

$\displaystyle\eta_{i}(t)\begin{pmatrix}\phantom{-}1\\\ -1\end{pmatrix}\qquad$
(42)

in the $0+1$ dimension theory at the defect. In order to gap these modes in
the free theory, one might add a term $\mu_{ij}\eta_{i}\sigma_{1}\eta_{j}$
with $\mu$ imaginary and antisymmetric, and it is easy to see that this term
lifts into the bulk theory as the $\mu$ term in eq. (41). However, the
pseudoscalar $\mu$ term is odd under the time-reversal symmetry we identified
(since $\mu$ is imaginary), and therefore time reversal symmetry requires
$\mu=0$ and ensures that the edge states remain gapless. Such a system is in
the BDI topological class, whose gapless states are characterized by the group
$\mathbb{Z}$ wiki:periodic .

For $N_{f}\geq 3$ it is possible to show that one can choose $m$ and $\mu$
such that time reversal symmetry is broken, and generically the edge states
will be gapped pairwise, so that there will be one massless edge state for
$N_{f}$ odd, and none for $N_{f}$ even. Such a system is in the D class, with
topology characterized by the group ${\mathbb{Z}}_{2}$ wiki:periodic .

For $N_{f}=2$, time reversal is not actually broken when $\mu\neq 0$, since
the theory is invariant under the simultaneous antiunitary transformations
$\psi_{1}(x,t)\to+\sigma_{1}\psi_{1}(x,-t)$ and
$\psi_{2}(x,t)\to-\sigma_{1}\psi_{2}(x,-t)$. In this case edge states can
still be gapped because the fermions transform with opposite signs under time
reversal and the system is topologically trivial. So the system is
indistinguishable from the D class as far as edge states are concerned.

We focus on this simplest $N_{f}=2$ case to show how the index calculation
procedure we have developed here gets the correct answer, that there are no
gapless edge states. The Minkowski theory we consider is

$\displaystyle{\cal
L}_{M}=\tfrac{1}{2}\psi_{i}^{T}C\left(i\not{\partial}-m_{0}\epsilon(x)-i\gamma_{\chi}\mu\tau_{2}\right)\psi\
,$ (43)

where we have suppressed the two flavor indices. The
$(i\not{\partial}-m_{0}\epsilon(x))$ operator is diagonal in flavor, while
$\tau_{2}$ is the $y$-Pauli matrix acting in flavor space, and $\mu$ is real
and constant in spacetime.

Our Euclidean operator ${\cal D}$ with a diagnostic field $\phi(x)$ in this
case is given by

$\displaystyle{\cal
D}=\not{\partial}+\phi_{1}(x)+i\left(\phi_{2}(x)+\mu\tau_{2}\right)\gamma_{\chi}\
.$ (44)

This can be diagonalized in flavor to give two 1-flavor Dirac operators

$\displaystyle{\cal
D}_{\pm}=\not{\partial}+\phi_{1}(x)+i\left(\phi_{2}(x)\pm\mu\right)\gamma_{\chi}\
.$ (45)

Now we expect the sum of two contributions to ${\cal J}_{\mu}$ proportional to
$\epsilon_{\mu\nu}\partial_{\nu}\left(\theta_{+}+\theta_{-}\right)$ where

$\displaystyle\theta_{\pm}=\arctan\frac{\phi_{2}(x)\pm\mu}{\phi_{1}(x)}\ .$
(46)

Now we see that even if $\phi(x)=\phi_{1}(x)+i\phi_{2}(x)$ has winding number,
as we take the limit $\phi_{2}\to 0$, that winding number vanishes. For
example, if we take $\phi_{1}=m\epsilon(x)$ and
$\phi_{2}(x)=m^{\prime}\epsilon(\tau)$, the two contributions to the index
will both equal one for $|m^{\prime}|>|\mu|$, but will jump to zero for
$|m^{\prime}|<|\mu|$ as we remove our diagnostic field. So the index in this
case would give the correct answer of zero, as it will for any even number of
flavors.

If we have an odd number of flavors with a constant $\mu$ matrix for
$N_{f}=2n+1$, we will find $n$ pairs of fermions coupling to $\phi_{2}(x)$
with $\pm\mu_{i}$ shifts, each contributing zero to the index; however there
would be one flavor with no shift, and we would find an index of one then.
Thus we find that $\text{ind}({\cal D})$ takes values in ${\mathbb{Z}}_{2}$.
That is the correct answer since having no $T$ symmetry while having $C$
symmetry with antisymmetric $C$ puts the model in the $D$ topological class,
whose topological invariant in one spatial dimension is ${\mathbb{Z}}_{2}$.

We emphasize though that in our calculation the reduction of the topological
invariant from ${\mathbb{Z}}$ in the time reversal symmetric case to
${\mathbb{Z}}_{2}$ in the case with time reversal symmetry breaking manifests
itself in the spacetime coordinate topology that the fermion sees, and not due
to a change in the topology in momentum space. The index we compute sees the
topology in phase space, and is sensitive to both.

##  4 Dirac fermion and Majorana fermions in 2+1 dimensions

###  4.1 A $d=2+1$ Dirac fermion with $U(1)$ fermion number symmetry

Our next example is a massive Dirac fermion in 2+1 (Minkowski) dimensions,
which is directly analogous to the Integer Quantum Hall Effect and is quite
familiar. The coordinates are $\\{x^{0},x^{1},x^{2}\\}=\\{t,x,y\\}$ and the
fermion mass $m(y)$ which has a “domain wall” structure – a monotonically
increasing function of $y$ with $m(0)=0$ Jackiw:1975fn . For convenience we
choose the particular basis for the $\gamma$ matrices

$\displaystyle\gamma^{0}=\sigma_{2}\ ,\quad\gamma^{1}=-i\sigma_{1}\
,\quad\gamma^{2}=i\sigma_{3}\ ,$ (47)

in which case the Dirac equation $\left[i\not{\partial}-m(y)\right]\psi=0$ has
the two special solutions,

$\displaystyle\psi_{\pm}=e^{-i\omega(t\pm
x)}e^{\mp\int^{y}_{0}ds\,m(s)}\,\chi_{\pm}\ ,$ (48)

where $\chi_{\pm}$ are constant 2-component spinors satisfying
$\sigma_{3}\chi_{\pm}=\pm\chi_{\pm}$. The solution $\psi_{+}$ is localized on
the 1+1 dimensional domain wall and corresponds to a massless Weyl fermion
that travels at the speed of light in the $-x$ direction. However, since
$e^{+\int^{y}_{0}ds\,m(s)}$ is not normalizable, the $\psi_{-}$ solution does
not correspond to a state in the Hilbert space. Therefore the spectrum on the
domain wall is chiral, and if fermion number is gauged, this $1+1$ dimensional
theory on the domain wall is anomalous.

All other eigenstates of the Dirac operator are gapped and are not localized.
One might therefore expect that these heavy states could be integrated out,
leaving an effective $1+1$ dimensional theory of a Weyl fermion at $y=0$,
along with irrelevant operators. However, when fermion number symmetry is
gauged the heavy fermions do not decouple entirely, giving rise to a marginal
Chern-Simons operator after being integrated out of the theory, as pointed out
by a number of authors PhysRevLett.52.18 ; PhysRevLett.53.1615 ; Callan:1984sa
. For a constant mass $m$, this contribution is 333Our notation is that the
covariant derivative is $D_{\mu}=(\partial_{\mu}-iA_{\mu})$, where $A_{\mu}$
has mass dimension $1$, setting the electric charge to $e=1$.

$\displaystyle{\cal
L}_{\text{CS}}=\frac{1}{4\pi}\frac{m}{|m|}\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}\
.$ (49)

The coefficient of this operator depends on the sign of $m$ but not its
magnitude; the dependence on the sign is required because $m$ changes sign
under $T$ or $P$ transformations (time reversal and space reflection), as does
the Chern-Simons operator. Variation of ${\cal L}_{CS}$ with respect to the
gauge field gives rise to a current

$\displaystyle
J^{\mu}_{\text{CS}}=\frac{1}{2\pi}\frac{m}{|m|}\epsilon^{\mu\nu\rho}\partial_{\nu}A_{\rho}$
(50)

Callan and Harvey Callan:1984sa  discussed the effects of this operator in the
presence of a domain wall profile, substituting the step function
$\epsilon(y)$ for $m/|m|$ in the above expression, so that
$J^{\mu}_{\text{CS}}$ corresponds to a current flowing on or off of the domain
wall from both sides, with a divergence proportional to $\delta(y)$ that
exactly cancels the anomalous divergence of the chiral current on the domain
wall. There are a few problems with their analysis. For one, the substitution
of spatially varying mass for a constant mass in the coefficient of the Chern
Simons term is not correct where the fermion mass is varying appreciably, i.e.
near the domain wall; however to demonstrate overall charge conservation, one
need only look at the spatial integral of the divergence of the current, which
involves the Chern Simons coefficient at infinity, where the substitution is
valid if the mass is not rapidly changing there. A second problem is that
while the 1-loop integral for deriving ${\cal L}_{CS}$ is finite, the full
theory requires a regulator which contributes to the Chern-Simons operator as
well. For example, with a Pauli-Villars regulator of mass $\Lambda$, the
coefficient in eq. (49) and eq. (50) becomes proportional to
$(m/|m|-\Lambda/|\Lambda|)$. With the Pauli-Villars mass $\Lambda$ being
independent of $y$, the effect is to double the inflowing current on one side
of the domain wall, and cancel it on the other. This has no effect on the
divergence of the current, but corrects the coefficient of the Chern-Simons
term and makes a topological interpretation possible. Other regularization
schemes, such as the lattice, can have richer topological phase structure.

Even with the correct expression for the current, the connection with the
Integer Quantum Hall Effect is obscure, beyond the fact that an electric field
in the $x$ direction gives rise to a current in the $y$ direction; missing is
an analog of the famous plot of the resistivity $\rho_{xy}$ versus magnetic
field in the condensed matter system, with its characteristic steps. In the
Dirac fermion case, the domain wall plays the role of one of the boundaries of
the Quantum Hall system, and the Dirac mass — which is time reversal violating
in $2+1$ dimensions — plays the role of the magnetic field, while the
resistivity is the ratio of the applied electric field to the Chern Simons
current. However, one does not see a stepwise increase in the current as a
function of the Dirac mass. Once again, that is a result of how the short-
distance physics of the theory is being regulated. For example, when one uses
a lattice and Wilson terms to regulate the UV, one does see quantized jumps in
the Chern-Simons current as a function of the fermion mass Kaplan:1992bt ;
Jansen:1992tw ; Golterman:1992ub .

The Euclidean fermion operator for this fermion is

$\displaystyle{\cal D}=\left[\not{D}+m(y)\right]\
,\qquad\gamma_{0}=\sigma_{2}\ ,\quad\gamma_{1}=-\sigma_{1}\
,\quad\gamma_{2}=\sigma_{3}\ ,$ (51)

with $D_{\mu}=\partial_{\mu}+iA_{\mu}$. From eq. (3) we have

$\displaystyle K=\begin{pmatrix}0&&-{\cal D}^{\dagger}\\\ {\cal
D}&&0\end{pmatrix}=D_{\mu}\Gamma_{\mu}-im(y)\Gamma_{3}\ ,$ (52)

where we define the matrices (satisfying the $SO(5)$ Clifford algebra)

$\displaystyle\Gamma_{\mu}=\sigma_{1}\otimes\gamma_{\mu}\
,\qquad\Gamma_{3}=\sigma_{2}\otimes 1\ ,\qquad\Gamma_{\chi}=\sigma_{3}\otimes
1$ (53)

for $\mu=0,1,2$. Our task now is to compute the generalized Hall current
${\cal J}_{\mu}$ in eq. (7). Since the bulk mass $|m|$ serves to regulate the
IR divergences of the theory, we can take the limit $M\to 0$ in eq. (10) from
the start. Another simplification is that since we only need the current in-
flow from infinity in the limit that the gauge field strength is weak, we can
compute ${\cal J}_{\mu}$ to leading order in a $1/m$ expansion which is given
by the Feynman diagram in Fig. 2 expanded to first order in the incoming
momentum444The term we keep will give a finite contribution to the index,
while higher order operators fall off too fast at infinity to contribute to
the integral $\int\partial_{\mu}{\cal J}_{\mu}$..

Figure 2: Loop diagram for computing the generalized Hall current for the
$2+1$-dimension Dirac fermion. The black dot is an insertion of the chiral
current $\Gamma_{\alpha}\Gamma_{\chi}$ with incoming momentum $p$, and the
propagators are given by $K^{-1}$.

We first perform the calculation naively, without a Pauli-Villars regulator,
with the result

$\displaystyle{\cal
J}_{\alpha}=\partial_{\gamma}A_{\beta}\int\frac{d^{3}q}{(2\pi)^{3}}\,{\rm
Tr\,}\Gamma_{\alpha}\Gamma_{\chi}(\partial_{\gamma}\tilde{K}_{0}^{-1}(q))\Gamma_{\beta}\tilde{K}_{0}^{-1}(q)\
,$ (54)

where as before, the tilde indicates a Fourier transform to momentum space,
and $\tilde{K}_{0}=\tilde{K}|_{A_{\mu}=0}$. The factor of
$\Gamma_{\alpha}\Gamma_{\chi}$ in the trace comes from the insertion of the
generalized Hall current, while the photon vertex gives $i\Gamma_{\beta}$; the
two factors of $\tilde{K}_{0}^{-1}$ are the two fermion propagators, and the
derivative $\partial_{\gamma}=\partial/\partial q_{\gamma}$ arises from
expanding the graph to first order in the external momentum $p$. With
$\Gamma_{\alpha}=-i\partial_{\alpha}\tilde{K}_{0}$, we can rewrite this as

$\displaystyle{\cal J}_{\alpha}(p)$ $\displaystyle=$
$\displaystyle\partial_{\gamma}A_{\beta}\int\frac{d^{3}q}{(2\pi)^{3}}\,{\rm
Tr\,}\left[\Gamma_{\chi}\left(\tilde{K}_{0}^{-1}\partial_{\gamma}\tilde{K}_{0}\right)\left(\tilde{K}_{0}^{-1}\partial_{\beta}\tilde{K}_{0}\right)\left(\tilde{K}_{0}^{-1}\partial_{\alpha}\tilde{K}_{0}\right)\right]$
(55) $\displaystyle=$
$\displaystyle-\epsilon_{\alpha\beta\gamma}\partial_{\gamma}A_{\beta}\int\frac{d^{3}q}{(2\pi)^{3}}\,\frac{4m}{(m^{2}+q^{2})^{2}}$
(56) $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\frac{m}{|m|}\epsilon_{\alpha\beta\gamma}\partial_{\gamma}A_{\beta}\
.$ (57)

With a domain wall profile for $m(y)$, the above expression exhibits
generalized Hall current flow converging on (or diverging from) the wall
similar to the electromagnetic current found by Callan and Harvey
Callan:1984sa .

To better understand the topology behind the loop integral, it is convenient
to rewrite eq. (LABEL:eq:Kloop3) in terms of the fermion operator $\tilde{\cal
D}_{0}$ of the undoubled theory,

$\displaystyle{\cal J}_{\alpha}$ $\displaystyle=$
$\displaystyle-\epsilon_{\alpha\beta\gamma}\partial_{\gamma}A_{\beta}\left(\frac{1}{3}\epsilon_{ijk}\int\frac{d^{3}q}{(2\pi)^{3}}{\rm
Tr\,}\left[\left(\tilde{\cal D}_{0}^{-1}\partial_{i}\tilde{\cal
D}_{0}\right)\left(\tilde{\cal D}_{0}^{-1}\partial_{j}\tilde{\cal
D}_{0}\right)\left(\tilde{\cal D}_{0}^{-1}\partial_{k}\tilde{\cal
D}_{0}\right)\right]\right)$ (59)

As in the example in the previous section, for constant $m$ we can define the
$SU(2)$ matrix

$\displaystyle U({\mathbf{q}})=\frac{\tilde{\cal
D}_{0}({\mathbf{q}})}{\sqrt{\det\tilde{\cal
D}_{0}({\mathbf{q}})}}\equiv\cos\frac{\theta}{2}+i\hat{\boldsymbol{\theta}}\cdot{\boldsymbol{\gamma}}\sin\frac{\theta}{2}$
(62)

where ${\boldsymbol{\theta}}$ is a real 3-vector with
$\theta=|{\boldsymbol{\theta}}|$ and

$\displaystyle\cos\frac{\theta}{2}=\frac{m}{\sqrt{m^{2}+q^{2}}}\
,\qquad\sin\frac{\theta}{2}=\frac{q}{\sqrt{m^{2}+q^{2}}}\
,\qquad\hat{\boldsymbol{\theta}}=\hat{\bf q}\ .$ (63)

The coordinates $\theta_{i}$ parametrize $SU(2)\cong S^{3}$ as a three-
dimensional ball of radius $2\pi$, where we can identify $U=1$ and
$|{\boldsymbol{\theta}}|=0$ as the “north pole” of $S^{3}$, while $U=-1$ and
$|{\boldsymbol{\theta}}|=2\pi$ corresponds to the “south pole”. However, as
the magnitude of the momentum $q$ ranges from $q=0$ to $q=\infty$, we see that
$U$ only visits half of $S^{3}$. In particular, for $m>0$ we see that
$0\leq|{\boldsymbol{\theta}}|\leq\pi$, the two limits corresponding to
$q=\infty$ and $q=0$ respectively; while for $m<0$ the corresponding limits
are $2\pi\geq|{\boldsymbol{\theta}}|\geq\pi$. We can rewrite our expression
eq. (LABEL:eq:Dloop) in terms of the $\theta_{i}$ variables as

$\displaystyle{\cal J}_{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\epsilon_{\alpha\beta\gamma}\partial_{\gamma}A_{\beta}\,\epsilon_{ijk}\int\frac{d^{3}q}{(2\pi)^{3}}\,{\rm
Tr\,}\left[\left(U^{\dagger}\frac{\partial U}{\partial
q_{i}}\right)\left(U^{\dagger}\frac{\partial U}{\partial\
q_{j}}\right)\left(U^{\dagger}\frac{\partial U}{\partial q_{k}}\right)\right]$
(64) $\displaystyle=$
$\displaystyle-\frac{1}{\pi}\epsilon_{\alpha\beta\gamma}\partial_{\gamma}A_{\beta}\left(\frac{1}{24\pi^{2}}\epsilon_{ijk}\int_{V_{1/2}}d^{3}\theta\,{\rm
Tr\,}\left[\left(U^{\dagger}\frac{\partial
U}{\partial\theta_{i}}\right)\left(U^{\dagger}\frac{\partial
U}{\partial\theta_{j}}\right)\left(U^{\dagger}\frac{\partial
U}{\partial\theta_{k}}\right)\right]\right)\ ,$ (66)

where as discussed above, the volume of integration $V_{1/2}$ only covers half
of the ball, corresponding to the northern hemisphere of $S^{3}$ for $m>0$, or
the southern hemisphere for $m<0$. As such, the integral cannot be considered
to be the winding number of a map from momentum space to $S^{3}$. The
situation is remedied when a regulator is included. Here we consider a Pauli-
Villars regulator, which corresponds to replacing ${\cal D}\to{\cal
D}_{\text{reg}}={\cal D}/{\cal D}_{\text{PV}}={\cal D}(m)/{\cal D}(\Lambda)$
everywhere, where in ${\cal D}_{\text{PV}}$ we have simply replaced $m$ by the
Pauli-Villars mass $\Lambda$, which we will take to $+\infty$ after the
calculation. The calculation proceeds as before, but now the $SU(2)$ matrix in
eq. (62) gets replaced by

$\displaystyle U_{\text{reg}}({\mathbf{q}})$ $\displaystyle=$
$\displaystyle\frac{\tilde{\cal
D}_{\text{reg}}({\mathbf{q}})}{\sqrt{\det\tilde{\cal
D}_{\text{reg}}({\mathbf{q}})}}\equiv\cos\frac{\theta_{\text{reg}}}{2}+i\hat{\boldsymbol{\theta}}_{\text{reg}}\cdot{\boldsymbol{\gamma}}\sin\frac{\theta_{\text{reg}}}{2}$
(67)

where $\hat{\boldsymbol{\theta}}_{\text{reg}}=\hat{\bf q}$ as before, and

$\displaystyle\cos\frac{\theta_{\text{reg}}}{2}=\frac{\Lambda
m+q^{2}}{\sqrt{\left(m^{2}+q^{2}\right)\left(\Lambda^{2}+q^{2}\right)}}\
,\qquad\sin\frac{\theta_{\text{reg}}}{2}=\frac{q(\Lambda-m)}{\sqrt{\left(m^{2}+q^{2}\right)\left(\Lambda^{2}+q^{2}\right)}}\
.$ (68)

We note that at $q\to\infty$ we have $U_{\text{reg}}({\mathbf{q}})\to 1$,
corresponding to compactifying momentum space to $S^{3}$ and mapping the point
at $q=\infty$ to the north pole of the $S^{3}$ parametrized by
${\mathbb{\theta}}$. (This is in contrast to $U$ in eq. (62) which maps the
2-sphere at $q\to\infty$ onto the equator of $S^{3}$). At $q=0$, however, we
find that $U_{\text{reg}}(0)=\pm 1$ depending on the relative sign of $m$ and
$\Lambda$. When $\Lambda$ and $m$ have the same sign,
$\cos\frac{\theta_{\text{reg}}}{2}\geq 0$ for all values of $q$, with
$\theta_{\text{reg}}=0$ for both $q=0$ and $q=\infty$. In this case
$U_{\text{reg}}(q)$ describes a topologically trivial map from our compact
momentum space to the northern hemisphere of $S^{3}$
($0\leq\theta_{\text{reg}}\leq\pi/2$). However, when $m$ and $\Lambda$ have
opposite signs $U_{\text{reg}}(q)$ is a nontrivial map from momentum space to
$S^{3}$ with winding number equal to one. It is no surprise then that we find
that for the regulated theory

$\displaystyle{\cal J}_{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{1}{\pi}\epsilon_{\alpha\beta\gamma}\partial_{\gamma}A_{\beta}(p)\left(\frac{1}{24\pi^{2}}\epsilon_{ijk}\int_{V}d^{3}\theta_{\text{reg}}\,{\rm
Tr\,}\left[\left(U_{\text{reg}}^{\dagger}\frac{\partial
U_{\text{reg}}}{\partial\theta_{i}}\right)\left(U_{\text{reg}}^{\dagger}\frac{\partial
U_{\text{reg}}}{\partial\theta_{j}}\right)\left(U_{\text{reg}}^{\dagger}\frac{\partial
U_{\text{reg}}}{\partial\theta_{k}}\right)\right]\right)$ (70)
$\displaystyle=$
$\displaystyle\frac{\nu_{q}}{\pi}\epsilon_{\alpha\beta\gamma}\partial_{\beta}A_{\gamma}\
,$ (71)

where

$\displaystyle\nu_{q}=\tfrac{1}{2}\left(\frac{\Lambda}{|\Lambda|}-\frac{m}{|m|}\right)$
(72)

is the winding number of the map from compact momentum space to $S^{3}$, where
$\nu_{q}=0$ when $\Lambda$ and $m$ have the same signs, and $\nu_{q}=\pm 1$
when $\Lambda$ and $m$ have the opposite signs. This relation between the
topology in momentum space of the fermion dispersion relation and the
quantization of the Hall current in 2+1 dimensions has been remarked on
previously in connection with the Ward-Takahashi identity in Refs.
ishikawa1987microscopic ; Golterman:1992ub .

We arrive at the index of ${\cal D}$ as the surface integral at infinity in
Euclidian 3-space

$\displaystyle\text{ind}({\cal D})$ $\displaystyle=$
$\displaystyle-\tfrac{1}{2}\int_{S}\,{\cal
J}_{\alpha}dS_{\alpha}=-\frac{\nu_{q}}{2\pi}\int_{S}\,\epsilon_{\alpha\beta\gamma}\partial_{\beta}A_{\gamma}\,dS_{\alpha}\,$
(73)

and it remains for us to show that what is multiplying $\nu_{q}$ is an integer
winding number in coordinate space. For our diagnostic gauge field, we choose
$A_{0,1}$ to be independent of $y$, while $A_{2}=0$, and for our integration
region we take the volume to be a cylinder with its axis perpendicular to the
domain wall, which we then take to infinite size in every direction. The
surface integral only gets contributions from the end caps of the cylinder,
$\nu_{q}$ being different at the two ends when we assume that $\Lambda$ (with
$\Lambda>0$) and $m$ have the same sign for $y>0$ and the opposite signs for
$y<0$,. The expression for the index then becomes

$\displaystyle\text{ind}({\cal D})$ $\displaystyle=$
$\displaystyle-\left(\oint\vec{A}\cdot
d\vec{\ell}\right)\,\nu_{q}(y)\biggl{|}^{y=\infty}_{y=-\infty}=-\nu_{A}\nu_{q}(y)\biggl{|}^{y=\infty}_{y=-\infty}=-\frac{\nu_{A}}{2}\left[\frac{\Lambda}{|\Lambda|}-\frac{m(y)}{|m(y)|}\right]^{y=\infty}_{y=-\infty}=\nu_{A}\
,$ (74)

where $\nu_{A}$ is the winding number of our Abelian gauge field integrated
over the circle at infinity in the $\tau-x$ plane (where $\tau=x_{0}$ and
$x=x_{1}$), evaluated at $y=x_{2}=\pm\infty$. We see that a diagnostic gauge
field with winding number $\nu_{A}=1$ yields $\text{ind}({\cal D})=1$,
revealing the existence of the gapless edge state in the corresponding
Minkowski spacetime theory through a combined topological configuration in
both coordinate and momentum space. Again, with more flavors we could get any
integer value for the index, consistent with the fact that this model is in
the $D$ class (antisymmetric $C$ and broken $T$ symmetry) for which the
topological invariant is known to be ${\mathbb{Z}}$ in two spatial dimensions.

###  4.2 A $d=2+1$ Majorana fermion with only ${\mathbb{Z}}_{2}$ fermion
number symmetry

The model becomes more interesting when we explicitly break fermion number
symmetry down to ${\mathbb{Z}}_{2}$ by adding a Majorana mass (with the domain
wall profile still in the Dirac mass). This system is analogous to a
topological superconductor with the Majorana mass playing the role of the
condensation of Cooper pairs. As it has no conserve fermion number, there is
no conventional Hall current in this model, even though, as we shall see, for
some parameters there exist gapless edge states. The Minkowski theory is

$\displaystyle{\cal L}_{\text{M}}$ $\displaystyle=$
$\displaystyle\bar{\psi}\left(i\not{\partial}-m\right)\psi+\frac{i\mu}{2}\psi^{T}C\psi+\frac{i\mu}{2}\bar{\psi}C\bar{\psi}^{T}\
,\text{ sign of }\mu$ (76)

where $m$ is real, we can take $\mu$ to be real and positive, and $C$ is the
charge conjugation matrix satisfying

$\displaystyle C^{\dagger}=C^{-1}=C\ ,\qquad
C\gamma^{\mu}C^{-1}=-(\gamma^{\mu})^{T}\ .$ (77)

Once again we will assume a domain wall profile $m(y)$, while keeping the
Majorana mass $\mu$ constant. Because of the lack of continuous symmetry in
this model, there are no conserved currents and hence no anomaly current in-
flow picture at nonzero $\mu$. Nevertheless, we will show that one can still
detect massless edge states in this model by computing the generalized Hall
current, which does have in-flow onto the defect when massless edge states
exist, and we show that we can use this inflow to count such states.

Since Dirac notation is cumbersome when fermion number is violated, our first
step is to rewrite ${\cal L}_{\text{M}}$ in terms of two real spinor fields
$\chi_{1,2}$, where $\psi=(\chi_{1}+i\chi_{2})/\sqrt{2}$. The Lagrangian can
then be expressed in terms of a 4-component spinor

$\displaystyle\chi=\begin{pmatrix}\chi_{1}\\\ \chi_{2}\end{pmatrix}$ (78)

as

$\displaystyle{\cal L}_{\text{M}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\chi^{T}\left[\begin{pmatrix}[r]1&\phantom{-}i\\\
-i&1\end{pmatrix}\otimes\gamma_{M}^{0}\left(i\not{\partial}-m\right)-\mu\begin{pmatrix}[r]0&1\\\
1&0\end{pmatrix}\otimes C\right]_{\cal A}\chi$ (79)

where the subscript “${\cal A}$” means “antisymmetric part”, derivatives being
antisymmetric. We can now Wick rotate to Euclidean space and write the
Euclidean Lagrangian as

$\displaystyle{\cal L}_{\text{E}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\chi^{T}\left[\begin{pmatrix}[r]1&\phantom{-}i\\\
-i&1\end{pmatrix}\otimes\gamma_{0}\left(\not{\partial}+m\right)+\mu\begin{pmatrix}[r]0&1\\\
1&0\end{pmatrix}\otimes C\right]_{\cal A}\chi$ (81)

In the particular $\gamma$-matrix basis of eq. (51)

$\displaystyle\gamma_{0}=C=\sigma_{2}\ ,\qquad\gamma_{1}=-\sigma_{1}\
,\qquad\gamma_{2}=\sigma_{3}\ ,$ (83)

we can write ${\cal L}_{\text{E}}$ in terms of the fermion fields

$\displaystyle\zeta=\frac{1}{\sqrt{2}}\begin{pmatrix}\chi_{1}+\chi_{2}\\\
\chi_{1}-\chi_{2}\end{pmatrix}\equiv\begin{pmatrix}\zeta_{+}\\\
\zeta_{-}\end{pmatrix}$ (84)

as

$\displaystyle{\cal L}_{E}=\tfrac{1}{2}\left[\zeta_{+}^{T}C{\cal
D}_{+}\zeta_{+}+\zeta_{-}^{T}C{\cal D}_{-}\zeta_{-}\right]\ ,\qquad{\cal
D}_{\pm}=\not{\partial}+(m\pm\mu)\ .$ (85)

As in the $1+1$ dimensional case we can drop the $C$ matrix and use
$[\text{ind}({\cal D}_{+})+\text{ind}({\cal D}_{-})]$ to detect edge states in
this case, with $m(y)$ having a domain wall form and $\mu$ being constant. We
can immediately adapt the result eq. (LABEL:eq:india) in the previous section
and to evaluate this index as

$\displaystyle\text{ind}({\cal D})\equiv[\text{ind}({\cal
D}_{+})+\text{ind}({\cal D}_{-})]$ $\displaystyle=$
$\displaystyle-\left(\oint\vec{A}\cdot
d\vec{\ell}\right)\,\left(\nu^{+}_{q}(y)+\nu^{-}_{q}(y)\right)\biggl{|}^{y=\infty}_{y=-\infty}$
(86) $\displaystyle=$
$\displaystyle-\nu_{A}\left(\nu^{+}_{q}(y)+\nu^{-}_{q}(y)\right)\biggl{|}^{y=\infty}_{y=-\infty}\
,$ (87)

where

$\displaystyle\nu_{q}^{(\pm)}=\tfrac{1}{2}\left(\frac{\Lambda}{|\Lambda|}-\frac{m\pm\mu}{|m\pm\mu|}\right)\
.$ (89)

Assuming positive $\Lambda$, we have

$\displaystyle\left(\nu_{q}^{(+)}+\nu_{q}^{(-)}\right)=\begin{cases}2&m<-|\mu|\
,\\\ 1&-|\mu|<m<|\mu|\ ,\\\ 0&|\mu|<m\ .\end{cases}$ (90)

The analog of eq. (LABEL:eq:india) then follows for the index in this theory,

$\displaystyle\text{ind}({\cal D})$ $\displaystyle=$
$\displaystyle-\nu_{A}\,\left[\nu^{(+)}_{q}(y)+\nu^{(-)}_{q}(y)\right]\bigl{|}^{y=\infty}_{y=-\infty}\
,$ (91)

where $\nu_{A}$ is the winding number in the diagnostic gauge field. For
$\nu_{A}=1$ it follows that the index takes on one of the values $0,\pm 1,\pm
2$ depending on which the cases in eq. (90) pertains for the asymptotic values
of $m$ at the two sides of the domain wall. If we denote the asymptotic values
$m(\pm\infty)=m_{\pm}$, then our result for the index for various cases is
given in Table 1.

| $\ m_{+}<-|\mu|\ $ | $\ -|\mu|<m_{+}<|\mu|\ $ | $\ |\mu|<m_{+}\ $
---|---|---|---
$m_{-}<-|\mu|\ $ | $\phantom{-}0$ | $\phantom{-}1$ | $\phantom{-}2$
$-|\mu|<m_{-}<|\mu|\ $ | $-1$ | $\phantom{-}0$ | $\phantom{-}1$
$|\mu|<m_{-}\ $ | $-2$ | $-1$ | $\phantom{-}0$
Table 1: The index $\text{ind}({\cal D})$ for spatial topology $\nu_{A}=1$ as
a function of $m_{\pm}$, the asymptotic values of the fermion mass on the two
sides of the domain wall, relative to the constant Majorana mass $\mu$.

We now show that the above results agree with what one finds when constructing
explicit domain wall solutions for this model. We return the Minkowski
Lagrangian

$\displaystyle{\cal
L}_{M}=\bar{\psi}\left(i\not{\partial}-m\right)\psi+i\frac{\mu}{2}\psi^{T}C\psi+i\frac{\mu}{2}\bar{\psi}C\bar{\psi}^{T}\
,\text{sign of }\mu$ (92)

and consider the equation of motion in our basis

$\displaystyle\gamma^{0}=\sigma_{2}\ ,\quad\gamma^{1}=-i\sigma_{1}\
,\quad\gamma^{2}=i\sigma_{3}\ ,\quad C=\sigma_{2}\ .$ (93)

To obtain the solutions to the equations of motion we redefine the fermion
field $\psi=e^{i\frac{\pi}{4}}\varphi$ in which case we can write the
solutions as

$\displaystyle\varphi(x)=\begin{pmatrix}\alpha_{1}e^{(-m-\mu)y}+i\beta_{1}e^{(-m+\mu)y}\\\
\alpha_{2}e^{(m+\mu)y}+i\beta_{2}e^{(m-\mu)y}\end{pmatrix}$ (94)

We take $\mu$ to be spatially constant, and without a loss of generality we
assume $\mu>0$, while the mass $m$ depends on the coordinate $y$ and takes the
asymptotic values $m\to m_{\pm}$ as $y\to\pm\infty$. For $y>0$, a localized
solution requires that the coefficient of $y$ in the exponent in eq. (94) be
negative when $m$ is replaced by $m_{+}$. Out of the four exponentials in eq.
(94), there are always two that meet this criterion. Labeling them by their
coefficients, they are

$\displaystyle m_{+}>|\mu|:$ $\displaystyle\alpha_{1},\,\beta_{1}\ ,$ (95)
$\displaystyle-|\mu|<m_{+}<|\mu|:$ $\displaystyle\alpha_{1},\,\beta_{2}\ ,$
$\displaystyle m_{+}<-|\mu|:$ $\displaystyle\alpha_{2},\,\beta_{2}\ .$

We can do the same thing for solutions at $y<0$ and $m\to m_{-}$; in this case
the coefficient of $y$ must be positive, and again there are always two
solutions, the same as the above but with subscripts $1,2$ reversed:

$\displaystyle m_{-}>|\mu|:$ $\displaystyle\alpha_{2},\,\beta_{2}\ ,$ (96)
$\displaystyle-|\mu|<m_{-}<|\mu|:$ $\displaystyle\alpha_{2},\,\beta_{1}\ ,$
$\displaystyle m_{-}<-|\mu|:$ $\displaystyle\alpha_{1},\,\beta_{1}\ .$

When we match solutions at $y=0$ there must localized solutions of the same
chirality on both sides, and they must be both real or both imaginary. That
means that the must be the same $\alpha_{i}$ or $\beta_{i}$ solution for
positive and negative $y$. It is evident then that depending on the two values
$m_{\pm}$ relative to $|\mu|$ there can be 2, 1, or 0 solutions. For example,
if $m_{+}>|\mu|$ while $-|\mu|<m_{-}<|\mu|$, then there is one localized
positive chirality $\beta_{1}$ solution that can be matched across $y=0$. The
number and chirality of edge state solutions are given in Table 2, where the
$R,L$ entry tells us whether we have upper or lower component solutions
respectively.

| $\ m_{+}<-|\mu|\ $ | $\ -|\mu|<m_{+}<|\mu|\ $ | $\ |\mu|<m_{+}\ $
---|---|---|---
$m_{-}<-|\mu|\ $ | $0$ | R | 2R
$-|\mu|<m_{-}<|\mu|\ $ | L | $0$ | R
$|\mu|<m_{-}\ $ | 2L | L | $0$
Table 2:  The number and chirality of edge state solutions to ${\cal D}\psi=0$
before introducing a gauge field, where the R,L indicates chirality. The table
for solutions to ${\cal D}^{\dagger}\psi=0$ would be the same with
substitution $\text{L}\leftrightarrow\text{R}$.

Note that the index in Table 1 is giving us the number of positive chirality
massless edge states minus the number of negative chirality ones. The reason
for that is that the equation ${\cal D}^{\dagger}\psi=0$ has the same
solutions as ${\cal D}\psi=0$ except for a parity flip, exchanging
$R\leftrightarrow L$. When the gauge field is turned on with $\nu_{A}=1$, then
it localizes the $R$ solutions of ${\cal D}$ while delocalizing the $L$
solutions. Therefore what the index is counting is the number of $R$ solutions
for ${\cal D}\psi=0$, minus the number of $R$ solutions for ${\cal
D}^{\dagger}\psi=0$, which is equivalent to the number of positive chirality
massless edge states minus the number of negative chirality ones for the
operator ${\cal D}$.

The index we computed still takes values in ${\mathbb{Z}}$ which is
appropriate, since the Majorana mass term does not break $T$ symmetry, so the
system is still in the $D$ topological class.

##  5 Dirac fermion in $3+1$ dimensions

Our last example of a Dirac fermion $3+1$ dimensions shares many features with
the $1+1$ dimension example. When the mass $m$ has a domain wall profile, the
Dirac equation in Minkowski spacetime has exact solutions corresponding to
2-component massless edge states localized on the $2+1$ dimensional wall. Such
a domain wall describes the physics of a topological insulator, where the
region with $m<0$ is considered the interior of the topological insulator and
the region with $m>0$ is considered to be the exterior. There is no analog of
chirality in $2+1$ dimensions, and hence no charge violation on the wall in
the presence of background gauge fields and no inflowing current from the bulk
to the wall maintaining current conservation. This makes the theory of three
dimensional topological insulator another interesting example to which one
could imagine applying our construction for which to compute the generalized
Hall current and index.

The Euclidean Dirac operator is simply $\not{\partial}+m(x_{3})$ which has a
static and unnormalizable solution localized at $x_{3}=0$ and constant in all
the other coordinates. The edge state has two nonzero spinor components and is
an eigenstate of $\gamma_{3}$. To fully localize this state we add as
diagnostic fields a four dimensional gauge field and a pseudoscalar, and again
we consider the mass to be an arbitrary scalar field for now. Thus the
operator we consider is

$\displaystyle{\cal D}=D_{\mu}\gamma_{\mu}+\phi_{1}+i\phi_{2}\gamma_{\chi}\ ,$
(97)

where $D_{\mu}=(\partial_{\mu}+iA_{\mu})$ is the $d=4$ gauge covariant
derivative and $\gamma_{\mu}$, $\gamma_{\chi}$ are our 4d Dirac matrices.

To compute the index of ${\cal D}$ we once again construct the $K$ operator,

$\displaystyle K=\begin{pmatrix}0&-{\cal D}^{\dagger}\\\ {\cal
D}&0\end{pmatrix}=D_{\mu}\Gamma_{\mu}+i\phi_{2}\Gamma_{4}+i\phi_{1}\Gamma_{5}\
,$ (98)

where $\mu=0,\ldots 3$ and the $\Gamma_{a}$ are the 8-dimensional matrices

$\displaystyle\Gamma_{\mu}$ $\displaystyle=$
$\displaystyle\sigma_{1}\otimes\gamma_{\mu}\ ,\qquad\mu=0,\ldots,3\ ,$ (99)
$\displaystyle\Gamma_{4}$ $\displaystyle=$
$\displaystyle\sigma_{1}\otimes\gamma_{\chi}\ ,$ (100)
$\displaystyle\Gamma_{5}$ $\displaystyle=$ $\displaystyle-\sigma_{2}\otimes 1\
,$ (101) $\displaystyle\Gamma_{\chi}$ $\displaystyle=$
$\displaystyle\sigma_{3}\otimes 1\ .\ \text{index starts at 0}$ (102)

Our task is to compute the part of the chiral current ${\cal
J}_{a}=\bar{\Psi}\Gamma_{a}\Gamma_{\chi}\Psi$ that contributes to the index,
where $\Psi$ is a fermion with action $S=\bar{\Psi}K\Psi$. As in the $1+1$
dimensional example in eq. (20), we first write the scalars as

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\phi_{1}+i\phi_{2}=(v+\rho(x))e^{i\theta(x)}$ (103)

assuming constant $v$ and slowly varying $\rho$ and $\theta$ where
$\langle\rho\rangle=\langle\theta\rangle=0$. Assuming $\langle\theta\rangle=0$
is general since we can perform a constant chiral rotation of ${\cal D}$ which
leaves the index invariant and shifts $\theta$. We compute the leading
contribution to the chiral current in a $1/v$ expansion, since higher order
terms will drop off too fast at infinity to contribute to the integral
$\int\partial_{\mu}{\cal J}_{\mu}$. To this end we write

$\displaystyle K$ $\displaystyle=$ $\displaystyle K_{0}+\delta K\ ,$ (104)

with

$\displaystyle K_{0}=\partial_{\mu}\Gamma_{\mu}+iv\Gamma_{5}\ ,\qquad\delta
K=iA_{\mu}\Gamma_{\mu}+i\theta v\Gamma_{4}+i\rho(x)\Gamma_{5}\ .$ (105)

Then $K_{0}^{-1}$ will be the free fermion propagator, while we perturb in
$\delta K$. To compute the part of the current that contributes to the index
we need the leading term in a $1/v$ expansion, since higher order terms will
drop off too fast at infinity to contribute to the integral
$\int\partial_{\mu}{\cal J}_{\mu}$.

When expanding ${\cal J}_{\mu}$ in $\delta K$, the $\Gamma_{\chi}$ insertion
in the fermion loop requires that the rest of the graph supplies one each of
the other six $\Gamma_{a}$ matrices in order to get a nonzero contribution
from the trace. First consider the source of $\Gamma_{4,5}$ in the graph. We
see the $\Gamma_{5}$ can come from one of the fermion propagators
$K_{0}^{-1}$, but to obtain $\Gamma_{4}$ we require an insertion of $\theta$
in the graph, while to lowest order the $\rho$ contribution will vanish. To
obtain the other four $\Gamma_{\mu}$ we note that the result for ${\cal
J}_{\mu}$ will be proportional to an epsilon tensor
$\epsilon_{\mu\alpha\beta\gamma}$ and that we can only contract the
$\alpha,\beta,\gamma$ indices with a gauge field and two derivatives – one
acting on the gauge field, the other on $\theta$. Thus we must expand the
graphs in Fig. 3 to linear order in the momenta carried by the gauge field and
by $\theta$.

Figure 3: Loop diagrams for computing the generalized Hall current for the
$3+1$-dimension Dirac fermion. The black dot is an insertion of the chiral
current $\Gamma_{\alpha}\Gamma_{\chi}$ with incoming momentum $p+k$. The
outgoing fields are the gauge field and the phase $\theta$ of the complex
field $\phi_{1}+i\phi_{2}$, and the fermion propagator is given by
$K_{0}^{-1}$.

It is straightforward to evaluate this loop integral, with the result

$\displaystyle{\cal J}_{\mu}$ $\displaystyle=$ $\displaystyle
v\partial_{\gamma}\theta\partial_{\alpha}A_{\beta}\int\frac{d^{4}q}{(2\pi)^{4}}\,{\rm
Tr\,}\Bigl{(}\Gamma_{\mu}\Gamma_{\chi}\left[(\partial_{\alpha}\tilde{K}_{0}^{-1})\Gamma_{\beta}\tilde{K}_{0}^{-1}\Gamma_{4}\left(\partial_{\gamma}\tilde{K}_{0}^{-1}\right)\right.$
(107)
$\displaystyle+\left.(\partial_{\gamma}\tilde{K}_{0}^{-1})\Gamma_{4}\tilde{K}_{0}^{-1}\Gamma_{\beta}\left(\partial_{\alpha}\tilde{K}_{0}^{-1}\right)\right]\Bigl{)}$
$\displaystyle=$
$\displaystyle-\epsilon_{\mu\alpha\beta\gamma}\partial_{\gamma}\theta\partial_{\alpha}A_{\beta}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{16v^{2}}{(q^{2}+v^{2})^{3}}$
(108) $\displaystyle=$
$\displaystyle-\frac{1}{2\pi^{2}}\epsilon_{\mu\alpha\beta\gamma}\partial_{\gamma}\theta\partial_{\alpha}A_{\beta}\
,$ (109)

where the derivatives inside the integral are with respect to $q$. The
prefactor in the first line includes a $(-1)$ for the fermion loop, $(-1)$
from the Fourier transform $p^{2}\to-\partial_{x}^{2}$, $(-1)$ from the two
derivatives with respect to momentum with opposite signs due to momentum flow,
and factors $(-iv)$ and $-i\Gamma_{\beta}$ for $\theta$ and $A_{\beta}$
vertices respectively.

To highlight the momentum space topology underlying the above calculation, we
follow similar steps as in the $1+1$ dimensional example of § 3 to rewrite the
result as the analog of eq. (36)

$\displaystyle{\cal J}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\epsilon_{\mu\alpha\beta\gamma}\partial_{\gamma}\theta\partial_{\alpha}A_{\beta}$
(111)
$\displaystyle\times\epsilon_{\rho\sigma\tau\omega}\int\frac{d^{4}q}{(2\pi)^{4}}\,{\rm
Tr\,}\gamma_{\chi}\left[\left(\xi^{\dagger}\partial_{\rho}\xi\right)\left(\xi^{\dagger}\partial_{\sigma}\xi\right)\left(\xi^{\dagger}\partial_{\tau}\xi\right)\left(\xi^{\dagger}\partial_{\omega}\xi\right)+\left(\xi\partial_{\rho}\xi^{\dagger}\right)\left(\xi\partial_{\sigma}\xi^{\dagger}\right)\left(\xi\partial_{\tau}\xi^{\dagger}\right)\left(\xi\partial_{\omega}\xi^{\dagger}\right)\right]$

where

$\displaystyle\xi=\frac{\tilde{\cal D}_{0}}{(\det\tilde{\cal
D}_{0})^{\frac{1}{4}}}=\frac{v}{\sqrt{q^{2}+v^{2}}}+i\frac{q_{\mu}\gamma_{\mu}}{\sqrt{q^{2}+v^{2}}}\
.$ (113)

Making use of the currents $A_{\mu}$ and $V_{\mu}$ defined as

$\displaystyle
A_{\mu}=\frac{i}{2}\left(\xi^{\dagger}\partial_{\mu}\xi-\xi\partial_{\mu}\xi^{\dagger}\right)\
,\qquad
V_{\mu}=\frac{1}{2}\left(\xi^{\dagger}\partial_{\mu}\xi+\xi\partial_{\mu}\xi^{\dagger}\right)\
,$ (114)

the integrand in eq. (LABEL:eq:D4new) can be related to the volume form
derived for four spacetime dimensions in eq. (152) using the relation that one
can derive

$\displaystyle-\frac{\epsilon_{\rho\sigma\tau\omega}}{96\pi^{2}}\left[\left(\xi^{\dagger}\partial_{\rho}\xi\right)\left(\xi^{\dagger}\partial_{\sigma}\xi\right)\left(\xi^{\dagger}\partial_{\tau}\xi\right)\left(\xi^{\dagger}\partial_{\omega}\xi\right)+\left(\xi\partial_{\rho}\xi^{\dagger}\right)\left(\xi\partial_{\sigma}\xi^{\dagger}\right)\left(\xi\partial_{\tau}\xi^{\dagger}\right)\left(\xi\partial_{\omega}\xi^{\dagger}\right)\right]$
(115) $\displaystyle=$
$\displaystyle\epsilon_{\rho\sigma\tau\omega}\left[-\frac{1}{16\pi^{2}}A_{\rho}A_{\sigma}A_{\tau}A_{\omega}+\partial_{\sigma}\left(\frac{1}{24\pi^{2}}A_{\sigma}A_{\tau}V_{\omega}\right)\right]\
.$ (116)

On taking the trace of the above equation multiplied by $\gamma_{\chi}$, the
first term on the right is recognized from eq. (152) as the volume measure for
$S^{4}$, while the second term is the divergence of a current which is well
defined without singularities on $S^{4}$, and hence integrates to zero.
Therefore the generalized Hall current in eq. (LABEL:eq:D4new) can be written
as

$\displaystyle{\cal J}_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{\nu_{q}}{2\pi^{2}}\epsilon_{\mu\alpha\beta\gamma}\partial_{\gamma}\theta\partial_{\alpha}A_{\beta}\
,$ (117)

where $\nu_{q}$ is the element of $\pi_{4}(S^{4})$:

$\displaystyle\nu_{q}=-\frac{1}{16\pi^{2}}\epsilon_{abcd}\int d^{4}q\,{\rm
Tr\,}\gamma_{\chi}A_{a}A_{b}A_{c}A_{d}=1\ .$ (118)

The index is given by

$\displaystyle\text{ind}({\cal D})=-\tfrac{1}{2}\int d^{4}x\partial_{\mu}{\cal
J}_{\mu}=\frac{\nu_{q}}{4\pi^{2}}\int_{S}\epsilon_{\mu\alpha\beta\gamma}\partial_{\gamma}\theta\partial_{\alpha}A_{\beta}\,dS_{\mu}\
.$ (119)

To evaluate the spatial integral we first must choose a topologically
nontrivial set of diagnostic fields to localize the solutions to ${\cal
D}\psi=0$. That can be done by making use of the BPS monopole field
configuration discussed in Ref. cheng2013fermion  which considers a massless
Dirac fermion in three Euclidean dimensions interacting with a scalar field
and a gauge field,

$\displaystyle\Phi=\frac{g}{2\pi}\left(a-\frac{1}{2r}\right)\ ,\qquad{\bf
A}=-\frac{g(1+\cos\theta)}{4\pi r\sin\theta}\,\hat{\bf e}_{\varphi}\ ,$ (120)

and discuss how when the couplings obey the minimal Dirac quantization
relation, $eg=2\pi$, the $d=3$ conjugate Dirac operator ${\cal
D}_{3}^{\dagger}$ has a zeromode proportional to $\exp(-ar)$ while ${\cal
D}_{3}$ has none. This configuration can be lifted into our four-dimensional
Euclidean theory by taking ${\bf A}$ to be the first three components of our
four-component gauge field, independent of the fourth coordinate, while

$\displaystyle\phi=m(x_{4})+i\Phi({\mathbf{x}})\ ,\qquad A_{4}=0\ ,\qquad
F_{ij}=\epsilon_{ijk4}\frac{g\hat{r}_{k}}{4\pi r^{2}}\ ,$ (121)

where for notational convenience we have relabeled our coordinates so that the
domain wall mass is a function of $x_{4}$ and coordinates on the mass defect
are labeled by $x_{1,2,3}$. Since we have set the electric charge $e=1$ in our
covariant derivative, we take $g=2\pi$ for the magnetic charge. With these
background fields we then expect that the existence of a massless edge state
in the original $d=3+1$ Minkowski theory implies one zeromode for ${\cal
D}^{\dagger}$ only if $a>0$ and no zeromode for ${\cal D}$, so that
$\text{ind}({\cal D})=-\theta(a)$.

This is indeed what we find. Plugging in these fields into eq. (117), we get
the generalized Hall current

$\displaystyle{\cal J}_{i}$ $\displaystyle=$
$\displaystyle\frac{\hat{\mathbf{r}}_{i}}{2\pi^{2}r}\frac{(2ar-1)\frac{{\rm
d}m(x_{3})}{{\rm d}x_{3}}}{(2ar-1)^{2}+4r^{2}m(x_{3})^{2}}\ ,\quad{\cal
J}_{3}=\frac{m(x_{3})}{2\pi^{2}r^{2}\left((2ar-1)^{2}+4r^{2}m(x_{3})^{2}\right)}\
,$ (122)

for $i=0,1,2$. We then compute the integral of its divergence,

$\displaystyle\int d^{4}x\,\partial_{\mu}{\cal
J}_{\mu}=\lim_{L_{3},R\to\infty}\left[\int^{R}d^{3}r\,{\cal
J}_{4}\biggl{|}^{x_{3}=L_{3}}_{x_{4}=-L_{3}}+\int_{-L_{3}}^{L_{3}}dx_{3}\int
d\Omega\,R^{2}\,\hat{R}\cdot\vec{\cal J}\right]\ .$ (124)

Specializing to the mass profile $m(\pm\infty)=\pm m_{0}$, and using the
expression for current given in eq. (LABEL:eq:J31e) we find

$\displaystyle\lim_{L_{3},R\to\infty}\int_{-L_{4}}^{L_{4}}dx_{4}\int
d\Omega\,R^{2}\,\hat{r}\cdot\vec{\cal J}$ $\displaystyle=$
$\displaystyle\frac{2\tan^{-1}\left(\frac{m_{0}}{a}\right)}{\pi}\ ,$ (125)
$\displaystyle\lim_{L_{3},R\to\infty}\int^{R}d^{3}r\,{\cal
J}_{3}\biggl{|}^{x_{3}=L_{3}}_{x_{3}=-L_{3}}$ $\displaystyle=$
$\displaystyle\left(1+\frac{2\tan^{-1}\frac{a}{m_{0}}}{\pi}\right)$ (127)

Summing these two terms gives us

$\displaystyle\int d^{4}x\,\partial_{\mu}{\cal J}_{\mu}=2\theta(a)$ (128)

and yields the anticipated result for the index,

$\displaystyle\text{ind}({\cal D})=-\tfrac{1}{2}\int
d^{4}x\,\partial_{\mu}{\cal J}_{\mu}=-\theta(a)\ .$ (129)

So we see again that with diagnostic fields to localize massless edge states,
the index of the Euclidean fermion operator counts these edge states and is
quantized because of both the spacetime topology of the background fields, and
the momentum space topology of the fermion dispersion relation. As in the
$d=1+1$ case and unlike the $d=2+1$ example, we find that the $d=3+1$ result
is not affected by regulator fields, which decouple.

This model is in the DIII class, symmetric under both $C$ and $T$ with both
$C$ and $T$ matrices antisymmetric PhysRevB.78.195125 . The topological
invariant in this case for three spatial dimensions is ${\mathbb{Z}}$,
consistent with what we would find if we generalized this theory to more
flavors without any flavor symmetry.

##  6 Interactions

Our interest in understanding edge states via the index of the Euclidean
fermion operator led us to computing the incoming flux at infinity of the
generalized Hall current. This has all been for free fermions, but interacting
systems are more interesting PhysRevB.85.245132 ; PhysRevB.88.064507 ;
Gu:2013azn ; PhysRevB.92.085114 ; Gu:2015wli ; PhysRevB.90.245120 ;
PhysRevB.95.195108 ; PhysRevLett.117.206405 ; PhysRevB.92.085114 ;
Wang:2022ucy , and we know that interactions can change the topological
classification Fidkowski:2009dba . The ramifications go beyond condensed
matter systems and have been applied to lattice models for chiral gauge
theories, where vector-like fermions appear as chiral edge states until
interactions are turned on, gapping some of them and leaving behind a chiral
representation Wang:2013yta ; You:2014vea ; Wang:2018ugf ; Zeng:2022grc . For
an analogous discussion in continuum chiral gauge theories, see Tong:2021phe ;
PhysRevX.11.011063 . Clearly the index of the free fermion operator cannot
capture this physics. The generalized Hall current, on the other hand, is
well-defined even in the presence of interactions, and so it is reasonable to
ask whether it divergence still tells us about massless edge states in an
interacting theory. In this section we will give qualitative arguments for why
that is plausible, in the context of the same $1+1$ model described in Ref.
Fidkowski:2009dba .

First we examine more closely how the calculations for free fermions were
done. Our method followed the work of Callan and Harvey Callan:1984sa , which
in turn used the methods developed by Goldstone and Wilczek Goldstone:1981kk .
We computed the generalized Hall current in a derivative expansion,
integrating out the fermions in a background field. In the $d=1+1$ example,
this background field was a complex scalar field, and we obtained a
contribution to the current proportional to
$\partial_{\mu}\theta=i\phi^{*}\overleftrightarrow{\partial_{\mu}}\phi/|\phi|^{2}$.
Recall that the index was defined in eq. (10) as the integral of the
divergence of the generalized Hall current in the limit that the doubled
fermion’s mass $M$ tended to zero. As seen in eq. (1), the mass $M$ was
introduced as an infrared cutoff, and in the Ward-Takahashi identity for the
current in eq. (8),

$\displaystyle\partial^{\mu}{\cal
J}_{\mu}^{\chi}=2M\bar{\Psi}\Gamma_{\chi}\Psi-{\cal A}\ ,$ (130)

where ${\cal A}=0$. The nonzero divergence indicating the existence of the
mass less edge state comes from the $2M\bar{\Psi}\Gamma_{\chi}\Psi$ term on
the right hand side, in the limit that $M\to 0$, where $M$ serves as an IR
regulator. In our calculation, the inverse dependence on $|\phi|^{2}$ is the
sign of an infrared divergence regulated by $\langle\phi\rangle$ – it was
because the background field served as an IR regulator that we could set the
parameter $M$ in eq. (10) to zero before computing the Feynman diagrams. We
rejected $M$ as the IR regulator, since the fermion operator ${\cal D}$ did
not have normalizable zeromodes and hence had vanishing index without
background fields with nontrivial spatial topology. Thus the calculation was
performed as if the fermions were fully gapped by the background field. This
was clearly false, since we were studying systems with an exact zeromode. The
current we computed cannot be valid near the zeromode since the fermion is not
gapped there and the derivative expansion in the background field breaks down.
However, since the index is proportional to $\int\partial_{\mu}{\cal
J}_{\mu}$, it only depends on the current asymptotically far away from the
localized zeromode, For that reason we were able to treat the fermion as
gapped, and justify the derivative expansion. This is the same justification
as for the Callan-Harvey calculation Callan:1984sa .

We will argue that in theories where interactions fully gap the fermions, this
spatially trivial gap will serve as the dominant infrared regulator in
calculating the generalized Hall current, and since it cannot localize the
fermions, it will lead to a vanishing divergence of the current. We envision a
mechanism similar to what we saw in § 3 where we discussed the example of $2N$
flavors of free, $d=1+1$ Majorana fermions in the presence of time reversal
violation. There we found that if we explicitly broke time reversal invariance
via a spatially constant $i\mu\psi^{T}C\gamma_{\chi}\psi$ term, the index
vanished. The reason was that in this case, as we removed the diagnostic field
$\phi_{2}(x)\to 0$, the spatial topology experienced by the fermion abruptly
became trivial at the point where $\mu$ dominated over $\phi_{2}$ as the
infrared regulator. The question then is whether the same phenomenon can occur
when interactions gap the fermions as we remove the diagnostic fields. This
seems plausible, making the procedure described here for detecting massless
edge states relevant even when interactions are introduced, even though the
original motivation of looking at zeromodes of the free fermion operator is no
longer relevant. To understand this better, we look at the model of Ref.
Fidkowski:2009dba  in greater detail.

Consider $2N$ copies of the $1+1$ dimension Majorana fermion model discussed
in § 3. As shown in eq. (16) the Minkowski domain wall solutions in the free
theory with a step function mass take the form

$\displaystyle\psi_{i}(x,t)=\eta_{i}(t)e^{-m|x|}\begin{pmatrix}\phantom{-}1\\\
-1\end{pmatrix}\ ,$ (131)

and when quantized, the hermitian $\hat{\eta}_{i}$ operators obey the Clifford
algebra

$\displaystyle\\{\hat{\eta}_{i},\hat{\eta}_{j}\\}=2\delta_{ij}\ ,\qquad
i=1,\ldots,2N.$ (132)

From these we can construct the ladder operators

$\displaystyle\hat{c}_{a}=\frac{\hat{\eta}_{a}+i\hat{\eta}_{a+N}}{2}\
,\qquad\hat{c}^{\dagger}_{a}=\frac{\hat{\eta}_{a}-i\hat{\eta}_{a+N}}{2}\
,\qquad a=1,\ldots N\ ,$ (133)

which obey the usual fermion anticommutation relations

$\displaystyle\\{\hat{c}_{a},\hat{c}_{b}\\}=\\{\hat{c}^{\dagger}_{a},\hat{c}^{\dagger}_{b}\\}=0\
,\qquad\\{\hat{c}_{a},\hat{c}^{\dagger}_{b}\\}=\delta_{ab}\ .$ (134)

The $2^{N}$-fold degenerate edge states can then be constructed by acting with
$c_{a}^{\dagger}$ operators on a state $|0\rangle$, which is defined to be the
state annihilated by all of the $\hat{c}_{a}$ operators.

Since the $\hat{\eta}_{i}$ operators obey a Clifford algebra, they define an
$\mathfrak{so}$(2N) Lie algebra, under which the degenerate ground states
transform as the $2^{N}$ dimensional reducible spinor representation. When
considering interactions between the edge states, it is convenient to
represent the $\hat{\eta}_{i}$ operators as $2^{N}\times 2^{N}$ hermitian
Dirac gamma matrices which act on these states. Interactions between these
states can then be represented as a matrix consisting of sums of totally
antisymmetrized products of even numbers of gamma matrices, which we will call
$H_{\text{int}}$.

One constraint we will impose on the interactions is that they preserve time
reversal symmetry, since the gapless edge states in the free theory owe their
existence to that symmetry in the first place, as discussed in § 3.2. The
action of time reversal on the bulk states is $\psi\to\sigma_{1}\psi$ which
takes $\hat{\eta}_{i}\to-\hat{\eta}_{i}$. This sign is not interesting since
we will only be considering products of an even number of the $\hat{\eta}_{i}$
operators; however, in order for an interaction represented an antisymmetrized
product of $2k$ antisymmetrized gamma matrices to be hermitian it must be
proportional to $i^{k}$, which means that operators with odd $k$ flip sign
under the antiunitary time reversal transformation. So we restrict the
interaction to operators involving products of multiples of four fermion
fields.

One of the results of Ref. Fidkowski:2009dba  is that this time reversal
invariant $H_{\text{int}}$ can gap all of the edge states in this model if and
only if the number of flavors is a multiple of eight. Completely gapping the
edge states means that there is a unique, nondegenerate ground state – so to
prove this result we need to show that a sum of totally antisymmetrized
products of $4k$ $SO(2N)$ gamma matrices can only have a unique lowest
eigenvalue when $2N=0\mod 8$. This is easy to show and the argument is given
in Appendix B.

Consider the case with eight flavors of fermions and consider the interaction
defined in eq. (175) and eq. (176). On the domain wall,

$\displaystyle
H_{\text{int}}=\omega\left(\hat{c}_{1}\hat{c}_{2}\hat{c}_{3}\hat{c}_{4}+h.c+1\right)$
(135)

which for $\omega>0$ has the unique groundstate
$|\Omega\rangle=(|0000\rangle-|1111\rangle)/\sqrt{2}$ with energy $E=0$,
fourteen degenerate states with $E=\omega$, and an isolated state with energy
$E=2\omega$. Due to (Euclidian) time translation invariance, none of the
solutions to the Schrödinger equation are localized in time. If we now
subtract from the Hamiltonian (add to the Lagrangian) our diagnostic field
$\mu\epsilon(\tau)$ then for $2\omega<\mu_{0}$ all sixteen states are
localized, corresponding to the Fock space spanned by four Dirac or eight
Majorana fermions, and in this case we expect to find the index
$-\tfrac{1}{2}\int\partial_{\mu}{\cal J}_{\mu}=8$, connecting smoothly to our
free fermion result at $\omega=0$. However, for fixed nonzero interaction
strength $\omega$, as we remove the diagnostic field $\mu\to 0$ we are
eventually in the range $0<\mu<\omega$ where the only localized state is the
groundstate, and we expect the index calculation to give us zero. However, we
do not know what to expect for the index for the intermediate values
$\omega<\mu<2\omega$, for which there are fifteen localized states: the ground
state and the fourteen states one can create by acting on the vacuum with one,
two or three powers of the $\hat{c}_{i}$ operators. Nonetheless, this picture
suggests that in the presence of fully gapping interactions, the index as
defined by the integral of the divergence of the generalized Hall current can
correctly count massless edge states, although we have no argument that the
index can only take integer values when our diagnostic parameter $\mu$ is of
the same size as the interaction strength.

How do these interaction affect the actual calculation of the divergence of
the generalized Hall current in the doubled Euclidean version of the $1+1$
dimension Minkowski theory? We do not have a quantitative answer but believe
that the gapping of the single particle propagator in the fully interacting
theory will render the generalized Hall current divergenceless555For work on
the Greens function for theories with gapped edge states, see Refs.
you2014topological ; catterall2017novel ; you2018symmetric ; xu2021green .. In
the effective Euclidean $0+1$ dimension theory one can compute the 1-particle
propagator and find

$\displaystyle\langle\Omega|\eta_{i}e^{-\hat{H}_{\text{int}}\tau}\eta_{j}|\Omega\rangle=e^{-\omega\tau}\delta_{ij}\
,$ (136)

In the $1+1$ dimension bulk theory we expect the interactions act like the ’t
Hooft operator induced by instantons in Ref. t1976symmetry , saturating the
zeromodes and serving as topologically trivial infrared cutoff. In the absence
of interactions and diagnostic fields, the fermion propagator we use to
compute the generalized Hall current in the $N_{f}=8$ model defined as

$\displaystyle\frac{\int\prod_{i=1}^{8}d\Psi_{i}\,e^{-S}\,\Psi_{i}(y_{1})\bar{\Psi}_{j}(y_{2})}{\int\prod_{i=1}^{8}d\Psi_{i}\,e^{-S}}\
,\qquad S=\int d^{2}x\bar{\Psi}{\cal D}\Psi\ ,$ (137)

is divergent since the denominator vanishes due to the integration over
zeromodes of ${\cal D}$ which do not appear in the action.

Suppose we now add a fully gapping interaction, such as the one given in eq.
(176),

$\displaystyle{\cal
L}_{\text{int}}=\omega\left[\left(\psi_{1}+i\psi_{5}\right)^{T}C\gamma_{\chi}\left(\psi_{2}+i\psi_{6}\right)\right]\left[\left(\psi_{3}+i\psi_{7}\right)^{T}C\gamma_{\chi}\left(\psi_{4}+i\psi_{8}\right)\right]+h.c.\
.$ (138)

The natural extension to add in the doubled Euclidian theory is then

$\displaystyle{\cal
L}_{\text{int}}=\omega\left[\left(\bar{\Psi}_{1}+i\bar{\Psi}_{5}\right)\Gamma_{3}\left(\Psi_{2}+i\Psi_{6}\right)\right]\left[\left(\bar{\Psi}_{3}+i\bar{\Psi}_{7}\right)\Gamma_{3}\left(\Psi_{4}+i\Psi_{8}\right)\right]+h.c.\
,$ (139)

where $\Gamma_{3}=\sigma_{1}\times\gamma_{\chi}$ is the doubled version of
$\gamma_{\chi}$ given in eq. (19). When this term is added to the action in
eq. (137) the fermion propagator is no longer IR divergent since the zeromodes
of ${\cal D}$ now appear in the action. As a result, one should still find a
well defined index as the diagnostic fields are removed – but as the infrared
cutoff arising from the interaction presumably has trivial spatial topology,
there will presumably be a topological phase transition to a trivial phase
with vanishing divergence for the generalized Hall current, similar to that
seen in § 3.2. It would be interesting to develop a quantitative method to
perform the calculation in the presence of interactions, but is beyond the
scope of this paper.

##  7 Discussion

We have shown that the presence of the massless edge states in topological
matter manifests itself by the inflow of a current, which we call a
generalized Hall current, in a related system in Euclidian spacetime. The
divergence of this current indicates the existence of massless edge states
just as the Hall current inflow does for the Integer Quantum Hall Effect. But
this current appears in all topological classes in Minkowski spacetime,
including those that do not have conserved currents because of a lack of
continuous symmetries (such as for a topological superconductor), or whose
edges do not suffer from chiral anomalies because they lack chiral symmetry
(such as topological insulators in $1+1$ and $3+1$ dimensions). In this sense
one arrives at a unified picture for disparate manifestations of topological
matter.

Furthermore, while the original motivation was to study the index of the free
Euclidean Dirac operator, the generalized Hall currents can be computed for
interacting systems as well, and we gave qualitative arguments for why we
expect the utility of such currents to persist. Whether this idea can be put
on a firmer foundation is an open question. It is an attractive proposition to
able to compute analytically how interactions affect topological properties in
different systems in various dimensions. If a general theory for gapping
massless chiral edge states in $3+1$ dimensions can be derived, that might
shed light on what restrictions there are on the matter content of chiral
gauge theories regulated on a lattice; this is of obvious interest given that
the Standard Model is a chiral gauge theory.

##  8 Acknowledgements

We thank Lukasz Fidkowski and Michael Clancy for useful conversations. DBK is
supported in part by DOE Grant No. DE-FG02-00ER41132 and by the DOE QuantISED
program through the theory consortium “Intersections of QIS and Theoretical
Particle Physics” at Fermilab. SS acknowledges support from the U.S.
Department of Energy grant No. GR-024204-00001.

##  Appendix A Topology in momentum space for Dirac operators

We have seen repeatedly in this paper that the Feynman diagram for the
generalized Hall current can be expressed as a trace involving a unitary
matrix $\xi$ and its derivatives, contracted with an epsilon tensor, where
$\xi$ is the unitarized Euclidiean fermion operator in $d+1$ dimensions,
$\xi=\tilde{D}/(\det\tilde{D})^{2^{-k}}$ where $d+1=2k$ for even dimensions
and $d=2k$ for odd dimensions. $\xi$ in general takes the form

$\displaystyle\xi=a(p)+i\not{b}(p)\ ,\qquad a(p)^{2}+b_{\mu}(p)b_{\mu}(p)=1\
.$ (140)

For an ordinary Dirac fermion, we have

$\displaystyle a=\frac{m}{\sqrt{m^{2}+p^{2}}}\ ,\qquad
b_{\mu}=\frac{p_{\mu}}{\sqrt{m^{2}+p^{2}}}\ .$ (141)

We see that the $d+2$-component unit vector $\\{a,b_{\mu}\\}$ represents a map
from $(d+1)$-dimensional momentum space to the sphere $S^{d+1}$. Note,
however, that $\xi$ maps all of momentum space onto half of the sphere; as
$p\to\infty$. At $p=0$ we have $\xi=1$, while for infinite momentum, $\xi\to
i\hat{p}_{\mu}\gamma_{\mu}$ which can be thought of the equator of the sphere,
where the poles are represented by the matrices $\pm 1$. A map onto all of
$S^{d+1}$ is described by $U=\xi^{2}$, where for infinite momentum, $U\to-1$.
Evidently the topology for a Dirac fermion in $(d+1)$-dimensional momentum
space is described by the $O(d+1)$ nonlinear sigma model in $(d+1)$
dimensions.

A convenient way to describe this nonlinear sigma model is as an
$SO(d+2)/SO(d+1)$ sigma model, which is what we do here. By computing
$\sqrt{g}$, where $g_{\mu\nu}$ is the metric of the sigma model and $g$ is its
determinant, we can simply express the volume form in terms of the matrix
$\xi$, in a form easy to relate to the Feynman diagram calculations.

We define

$\displaystyle\xi$ $\displaystyle=$ $\displaystyle
e^{i\theta_{\mu}\gamma_{\mu}/4}=\cos\frac{\theta}{4}+i\hat{\not{\theta}}\sin\frac{\theta}{4}\
,$ (142) $\displaystyle U$ $\displaystyle=$
$\displaystyle\xi^{2}=e^{i\theta_{\mu}\gamma_{\mu}/2}=\cos\frac{\theta}{2}+i\hat{\not{\theta}}\sin\frac{\theta}{2}\
,$ (143) $\displaystyle A_{a}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\left(\xi^{\dagger}\partial_{a}\xi-\xi\partial_{a}\xi^{\dagger}\right)={\cal
A}_{a\mu}\gamma_{\mu}\ .$ (144)

For even $d+1$, it is easy to show that $U$ parametrizes an $SO(d+2)/SO(d+1)$
sigma model. We can write $U$ as

$\displaystyle U=e^{i\theta_{\mu}\sigma^{\prime}_{\mu,d+2}}\
,\qquad\mu=1,\ldots,d+1\ ,$ (145)

where

$\displaystyle\sigma_{\mu\nu}^{\prime}=\frac{i}{4}\left[\gamma_{\mu}^{\prime},\gamma_{\nu}^{\prime}\right]\
,\qquad\gamma_{\mu}^{\prime}=i\gamma_{d+2}\gamma_{\mu}\
,\qquad\gamma_{d+2}^{\prime}=\gamma_{d+2}\ ,$ (146)

since $\sigma^{\prime}_{\mu,d+2}=\gamma_{\mu}/2$. In this form we see that
$U(x)$ would describe the Goldstone bosons of the spontaneous symmetry
breaking pattern $SO(d+2)\to SO(d+1)$, where the $\sigma_{\mu\nu}^{\prime}$
generate $SO(d+2)$, and the subset $\sigma^{\prime}_{\mu,d+2}$ are the “broken
generators” which are not also generators of the $SO(d+1)$ subalgebra. A
similar construction can be made for odd $d+1$.

The metric for this sigma model is given by

$\displaystyle g_{\mu\nu}\propto{\rm
Tr\,}\partial_{\mu}U^{\dagger}\partial_{\nu}U\propto{\rm
Tr\,}A_{\mu}A_{\nu}={\cal A}_{\mu\alpha}{\cal A}_{\nu\beta}{\rm
Tr\,}\gamma_{\alpha}\gamma_{\beta}\propto\left({\cal A}{\cal
A}^{T}\right)_{\mu\nu}\ ,$ (147)

where the proportionalities are all constant. Thus we have

$\displaystyle\sqrt{g}=N\det{\cal A}\ .$ (148)

With the convention for both $SO(2k)$ and $SO(2k+1)$

$\displaystyle{\rm
Tr\,}\left[\gamma_{2k+1}\gamma_{\mu_{1}}\ldots\gamma_{\mu_{2k}}\right]=(2i)^{k}\epsilon_{\mu_{1}\ldots\mu_{2k}}\
,$ (149)

we can rewrite this as

$\displaystyle\sqrt{g}=N\epsilon_{\mu_{1}\ldots\mu_{d+1}}\times\begin{cases}{\rm
Tr\,}\gamma_{d+2}A_{\mu_{1}}\ldots A_{\mu_{d+1}}&\text{even }d+1\\\ {\rm
Tr\,}A_{\mu_{1}}\ldots A_{\mu_{d+1}}&\text{odd }d+1\end{cases}\ ,$ (150)

With the normalization condition

$\displaystyle\int d^{d+1}\theta\,\sqrt{g}=1.$ (151)

we have

$\displaystyle d+1=2:$ $\displaystyle\sqrt{g}$ $\displaystyle=$
$\displaystyle-\frac{i}{4\pi}\epsilon_{ij}{\rm
Tr\,}\gamma_{3}A_{i}A_{j}=\frac{i}{16\pi}\epsilon_{ij}{\rm
Tr\,}[\gamma_{\chi}(\partial_{i}U)(U^{\dagger}\partial_{j}U)$ (152)
$\displaystyle=$
$\displaystyle\frac{1}{8\pi}\left(\frac{\sin\theta/2}{\theta}\right)\ ,$
$\displaystyle d+1=3:$ $\displaystyle\sqrt{g}$ $\displaystyle=$
$\displaystyle\frac{i}{3\pi^{2}}\epsilon_{ijk}{\rm
Tr\,}A_{i}A_{j}A_{k}=\frac{1}{24\pi^{2}}\epsilon_{ijk}{\rm
Tr\,}(U^{\dagger}\partial_{i}U)(U^{\dagger}\partial_{j}U)(U^{\dagger}\partial_{k}U)$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\left(\frac{\sin\theta/2}{\theta}\right)^{2}\
,$ $\displaystyle d+1=4:$ $\displaystyle\sqrt{g}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}\epsilon_{ijk\ell}{\rm
Tr\,}\gamma_{5}A_{i}A_{j}A_{k}A_{\ell}=-\frac{1}{256\pi^{2}}\epsilon_{ijk\ell}{\rm
Tr\,}[\gamma_{5}(\partial_{i}U)(U^{\dagger}\partial_{j}U)(U^{\dagger}\partial_{k}U)(U^{\dagger}\partial_{\ell}U)$
$\displaystyle=$
$\displaystyle\frac{3}{16\pi^{2}}\left(\frac{\sin\theta/2}{\theta}\right)^{3}\
,$ $\displaystyle d+1=5:$ $\displaystyle\sqrt{g}$ $\displaystyle=$
$\displaystyle\frac{1}{15\pi^{3}}\epsilon_{ijk\ell m}{\rm
Tr\,}A_{i}A_{j}A_{k}A_{\ell}A_{m}$ $\displaystyle=$
$\displaystyle\frac{i}{480\pi^{3}}\epsilon_{ijk\ell m}{\rm
Tr\,}[(U^{\dagger}\partial_{i}U)(U^{\dagger}\partial_{j}U)(U^{\dagger}\partial_{k}U)(U^{\dagger}\partial_{\ell}U)(U^{\dagger}\partial_{m}U)$
$\displaystyle=$ $\displaystyle
i\frac{1}{2\pi^{3}}\left(\frac{\sin\theta/2}{\theta}\right)^{4}\ .$

The integral in eq. (151) has an interpretation other than as the normalized
volume integral: with the definitions in eq. (140) and eq. (144) it is the
winding number of a map $U(p)$ from momentum space to $SO(d+2)/SO(d+1)\cong
S^{d+1}$. Since $U(p)\to-1$ as $|{\mathbf{p}}|\to\infty$ in any direction,
momentum space is effectively compactified to $S^{d+1}$ and so the winding
number is an element of the homotopy group $\pi_{n}(S^{n})$ with $n=d+1$.

##  Appendix B Gapping the Majorana edge states with interactions in $1+1$
dimensions

Here we give a simple argument for a result of Fidkowski:2009dba , that one
can fully gap the edge states in the $1+1$ dimensional Majorana model when the
number of flavors is $2N=0\mod 8$, and not for other numbers of flavors. For a
pedagogical review, see https://doi.org/10.48550/arxiv.1506.05805

We have seen in § 6 that for $2N$ flavors, the most general time reversal
invariant interactions can be written as sums of totally antisymmetrized
products of $4k$ $SO(2N)$ gamma matrices acting on the $2^{N}$-dimensional
Hilbert space of degenerate ground states. Call this interaction matrix
$H_{\text{int}}$, and consider the effect on $H_{\text{int}}$ of two special
matrices: $\gamma_{2N+1}$, which anti-commutes with all the other gamma
matrices, and ${\widetilde{C}}$ which has the following properties666Note that
the conventional definition of the charge conjugation matrix $C$ satisfies
$C\gamma_{\mu}C=-\gamma_{\mu}^{T}$ whereas ${\widetilde{C}}$ defined here
satisfies that equation for $SO(4k+2)$ but
${\widetilde{C}}\gamma_{\mu}{\widetilde{C}}=+\gamma_{\mu}^{T}$ for $SO(4k)$;
both $C$ and ${\widetilde{C}}$ to conjugate the generators,
$\sigma_{\mu\nu}\to-\sigma_{\mu\nu}^{T}$, and the alternating sign better
serves our purpose here. For $SO(4k)$ the conventional $C$ is given by
multiplying ${\widetilde{C}}$ by the chiral matrix $\gamma_{4k+1}$. :

$\displaystyle{\widetilde{C}}$ $\displaystyle=$
$\displaystyle{\widetilde{C}}^{\dagger}={\widetilde{C}}^{-1}\ ,$ (153)
$\displaystyle{\widetilde{C}}\gamma_{\mu}{\widetilde{C}}$ $\displaystyle=$
$\displaystyle\begin{cases}-\gamma_{\mu}^{T}&2N=8k+2,8k+6\\\
+\gamma_{\mu}^{T}&2N=8k,8k+4\ .\end{cases}\ ,$ (156)
$\displaystyle{\widetilde{C}}^{T}$ $\displaystyle=$
$\displaystyle\begin{cases}+{\widetilde{C}}&2N=8k,\,8k+6\\\
-{\widetilde{C}}&2N=8k+2,\,8k+4\ .\end{cases}\ ,$ (159)
$\displaystyle{\widetilde{C}}\gamma_{2N+1}{\widetilde{C}}$ $\displaystyle=$
$\displaystyle\begin{cases}+\gamma_{2N+1}&2N=4k,\\\ -\gamma_{2N+1}&2N=4k+2\
.\end{cases}\ .$ (162)

An explicit representation of ${\widetilde{C}}$ can be easily found using the
following recursive definition for the gamma matrices of $SO(2N)$. For $SO(2)$
we take the Pauli matrices as our gamma matrices:

$\displaystyle SO(2):\qquad$ $\displaystyle\gamma^{(2)}_{i}=\sigma_{i}\
,\qquad i=1,\ldots,3\ ,$ (163)

and then for $SO(2N)$ for $N\geq 2$ we take

$\displaystyle\gamma_{i}^{(2N)}$ $\displaystyle=$
$\displaystyle\sigma_{1}\otimes\gamma^{(2N-2)}_{i}\ ,\qquad i=1,\ldots,2N-1\
,$ (164) $\displaystyle\gamma_{2N}^{(2N)}$ $\displaystyle=$
$\displaystyle\sigma_{2}\otimes 1\ ,$ (166)
$\displaystyle\gamma_{2N+1}^{(2N)}$ $\displaystyle=$
$\displaystyle\sigma_{3}\otimes 1\ .$ (168)

In this basis the matrix ${\widetilde{C}}$ is found to be a direct product of
matrices alternating between $\sigma_{2}$ and $\sigma_{3}$:

$\displaystyle{\widetilde{C}}^{(2)}=\sigma_{2}\ ,\
{\widetilde{C}}^{(4)}=\sigma_{3}\otimes\sigma_{2}\ ,\
{\widetilde{C}}^{(6)}=\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{2}\ ,\
{\widetilde{C}}^{(8)}=\sigma_{3}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{2}\
,$ (169)

and so on. It can be easily verified in this basis that ${\widetilde{C}}$
possesses the properties in eq. (162).

Now it is easy to prove that the eigenvalues of $H_{\text{int}}$ have to be at
least doubly degenerate for all $SO(2N)$ groups except $2N=8k$, where $k$ is
an integer. $H_{\text{int}}$ consists of sums of products of $4k$
$\gamma$-matrices totally antisymmetrized in their indices, with real
coefficients, it follows that

$\displaystyle H_{\text{int}}=H_{\text{int}}^{\dagger}\
,\qquad[H_{\text{int}},\gamma_{2N+1}]=0\
,\qquad{\widetilde{C}}H_{\text{int}}{\widetilde{C}}=H^{T}_{\text{int}}\ .$
(170)

Since $[H_{\text{int}},\gamma_{2N+1}]=0$ we can simultaneously diagonalize
$H_{\text{int}}$ and $\gamma_{2N+1}$ with eigenvectors $\psi_{n,\sigma}$ where

$\displaystyle H_{\text{int}}\psi_{n,\sigma}=\lambda_{n}\psi_{n,\sigma}\
,\qquad\gamma_{2N+1}\psi_{n,\sigma}=\sigma\psi_{n,\sigma}\
,\qquad\lambda_{n}\in{\mathbb{R}}\ ,\quad\sigma=\pm 1.$ (171)

Now consider their action on the vector
$\chi_{n,\sigma}\equiv{\widetilde{C}}\psi_{n,\sigma}^{*}$:

$\displaystyle H_{\text{int}}\chi_{n,\sigma}$ $\displaystyle=$
$\displaystyle{\widetilde{C}}({\widetilde{C}}H_{\text{int}}{\widetilde{C}})\psi_{n,\sigma}^{*}={\widetilde{C}}H_{\text{int}}^{T}\psi_{n}^{*}={\widetilde{C}}H_{\text{int}}^{*}\psi_{n}^{*}=\lambda_{n}{\widetilde{C}}\psi_{n}^{*}=\lambda_{n}\chi_{n,\sigma}\
,$ (172)

and

$\displaystyle\gamma_{2N+1}\chi_{n,\sigma}$ $\displaystyle=$
$\displaystyle{\widetilde{C}}({\widetilde{C}}\gamma_{2N+1}{\widetilde{C}})\psi_{n,\sigma}^{*}=\begin{cases}{\widetilde{C}}\gamma_{2N+1}\psi_{n,\sigma}^{*}=+\sigma\chi_{n,\sigma}&2N=4k\\\
-{\widetilde{C}}\gamma_{2N+1}\psi_{n,\sigma}^{*}=-\sigma\chi_{n,\sigma}&2N=4k+2\
.\end{cases}$ (173)

Thus $\chi_{n,\sigma}$ is an eigenstate of $H_{\text{int}}$ with eigenvalue
$\lambda_{n}$, and now we would like to know if it is proportional to
$\psi_{n,\sigma}$, in which case we have learned nothing, or orthogonal to
$\psi_{n,\sigma}$, in which case we have shown that the eigenvalue $\lambda$
is at least doubly degenerate.

First of all, we see that while
$\gamma_{2N+1}\psi_{n,\sigma}=\sigma\psi_{n,\sigma}$ we have
$\gamma_{2N+1}\chi_{n,\sigma}=-\sigma\chi_{n,\sigma}$ for
$2N=4k^{\prime}+2=8k+2,\,8k+6$, proving that $\psi_{n,\sigma}$ and
$\chi_{n,\sigma}$ are indeed orthogonal in these cases and the spectrum is
doubly degenerate. We can also directly compute their inner product and find

$\displaystyle\chi_{n,\sigma}^{\dagger}\psi_{n,\sigma}=\psi_{n,\sigma}^{T}{\widetilde{C}}\psi_{n,\sigma}$
(174)

which equals zero whenever ${\widetilde{C}}$ is antisymmetric, which occurs
for $2N=8k+2,\,8k+4$ – and so the spectrum is doubly degenerate in these cases
also.

Putting the two results together we see double degeneracy for
$2N=8k+2,\,8k+4,\,8k+6$, leaving only $2N=8k$ as a possible candidate for
$H_{\text{int}}$ to have unique eigenvalues.

Now we can ask: for $SO(8k)$ do we need to go beyond the $\gamma^{4}$ terms in
$H_{\text{int}}$ in order to gap all of the edge states? For this we will just
count parameters. $H_{\text{int}}$ is $2^{N}$ dimensional, so we should be
able to obtain $2^{N}$ different eigenvalues if we have at least $2^{N}$
parameters in $H_{\text{int}}$. (This is overkill, since we only need the
lowest eigenvalue to be unique). The number of independent antisymmetric
4-index tensors, whose indices can take $2N$ values is $2N!/[4!(2N-4)!]$ which
is greater than $2^{N}$ for $N\geq 3$. Therefore a purely 4-fermion
interaction can gap all the fermions for every $SO(8k)$.

A simple example in the eight flavor model of a time reversal invariant
interaction that gaps all of the edge states is

$\displaystyle
H_{\text{int}}=\omega\left(\hat{c}_{1}\hat{c}_{2}\hat{c}_{3}\hat{c}_{4}+h.c.+1\right)\
,$ (175)

where the $\hat{c}_{i}$ ladder operators were defined in eq. (133) and we
assume $\omega>0$. (Under time reversal,
$\hat{c}_{i}\leftrightarrow-\hat{c}_{i}^{\dagger}$). The eigenstates of
$H_{\text{int}}$ include a unique groundstate with eigenvalue $E=0$, fourteen
degenerate states with eigenvalue $E=\omega$, and a unique maximal state with
eigenvalue $2\omega$. The eigenstates corresponding to the minimum and maximum
energy are linear combinations of the empty and fully occupied states,
$(|0000\rangle\mp|1111\rangle)/\sqrt{2}$, where $\hat{c}_{i}|0000\rangle=0$,
and
$|1111\rangle=\hat{c}^{\dagger}_{4}\hat{c}^{\dagger}_{3}\hat{c}^{\dagger}_{2}\hat{c}^{\dagger}_{1}|0000\rangle$.

The interaction in eq. (175) can be realized in our $1+1$-dimension Lagrangian
by the term

$\displaystyle{\cal
L}_{\text{int}}=\omega\left[\left(\psi_{1}+i\psi_{5}\right)^{T}C\gamma_{\chi}\left(\psi_{2}+i\psi_{6}\right)\right]\left[\left(\psi_{3}+i\psi_{7}\right)^{T}C\gamma_{\chi}\left(\psi_{4}+i\psi_{8}\right)\right]+h.c.\
.$ (176)

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