A=d\omega+\mathopen{}\mathclose{{}\left[A,\omega}\right]$ (107)

we get

$\displaystyle Z\mathopen{}\mathclose{{}\left[B^{g^{-1}}}\right]$
$\displaystyle=\int\mathcal{D}\phi_{i}\;e^{iS[\phi_{i}]+i\int{\rm
Tr}\mathopen{}\mathclose{{}\left[B^{g^{-1}}\cdot J}\right]}$ (108)
$\displaystyle=\int\mathcal{D}\phi_{i}\;e^{iS[\phi_{i}]+i\int{\rm
Tr}\mathopen{}\mathclose{{}\left[B\cdot J^{g}+\omega dJ}\right]}.$

Now, let us make a change of variable $\phi\to\phi^{g}$. Allowing for a
possibly non-trivial (i.e. anomalous) Jacobian factor but assuming that the
action is invariant under this transformation, we get

$\displaystyle Z\mathopen{}\mathclose{{}\left[B^{g^{-1}}}\right]$
$\displaystyle=\int\mathcal{D}\phi_{i}^{g}\;J\mathopen{}\mathclose{{}\left[\frac{\partial\phi}{\partial\phi^{g}}}\right]\;e^{iS\mathopen{}\mathclose{{}\left[\phi_{i}^{g}}\right]+i\int{\rm
Tr}[B\cdot J^{g}+\omega dJ]}$ (109)
$\displaystyle=\int\mathcal{D}\phi_{i}\;e^{-i\int{\rm
Tr}[\omega\mathcal{A}(B)]}\;e^{iS[\phi_{i}]+i\int{\rm Tr}[B\cdot
J]}\;\mathopen{}\mathclose{{}\left(1+i\int{\rm Tr}[\omega dJ]}\right),$

where we have written the anomalous Jacobian factor as an anomaly phase
$e^{-i\int{\rm Tr}[\omega\mathcal{A}(B)]}$. Therefore, we see that the
condition that the partition function is invariant under a formal local
transformation $B\to B^{g}=g(B+d)g^{-1}$ is equivalent to the statement that
the global symmetry $G$ satisfies a complete set of (anomalous) Ward
identities: $\langle dJ\rangle=\mathcal{A}$. Furthermore, we can use this in a
slightly different way. Namely, we instead perform a homogeneous
transformation $B\to gBg^{-1}$. We see that under this, the partition function
will change by $Z[B]\to Z[B^{g}]=e^{-i\int\omega\mathcal{A}(B)}Z[B]$. In other
words, if the theory satisfies a non-anomalous Ward identity for $G$, we will
not see this anomaly phase, while the anomaly phase will show up whenever the
theory is actually anomalous under a $G$ transformation.

##  Appendix C Review of chiral anomalies

In this appendix, we review the standard treatment of fermion anomalies. After
reviewing the Wess-Zumino consistency condition that the fermion anomaly
should comply with, we introduce descent equations as a construction to solve
the consistency condition. We then discuss the BRST formulation, and explain
how chiral anomalies are defined as equivalence classes of BRST cohomology.

###  C.1 Wess-Zumino consistency condition

Let $W$ be the generating functional for connected diagrams of a theory in the
presence of external vector fields $A_{\mu}^{a}$ and other external (e.g.
scalar, pseudo-scalar) fields. Up to contributions associated with those other
external fields, the generators of a gauge transformation on $W$ are given by
eq. (102). The $X_{a}$’s fulfill the Lie algebra $\mathbf{g}$ of the gauge
group $G$:

$\mathopen{}\mathclose{{}\left[X_{a}(x),X_{b}(y)}\right]=f_{abc}X_{c}(x)\delta(x-y).$
(110)

The anomalous Ward identity is expressed as

$X_{a}(x)W\mathopen{}\mathclose{{}\left[A_{\mu},\dots}\right]=G_{a}(x).$ (111)

Here $G_{a}(x)$ is the anomaly associated with $X_{a}$. The Wess-Zumino
consistency condition follows by simply applying the Lie algebra to $W$:

$X_{a}(x)G_{b}(y)-X_{b}(y)G_{a}(x)=f_{abc}G_{c}(x)\delta(x-y).$ (112)

This seemingly very simple condition has played a very important role in
understanding deep aspects of anomalies. It made it possible to construct the
form of the non-Abelian anomaly from pure geometrical considerations, without
ever computing Feynman (triangle) diagrams. Along the way, it suggested a
possible deep connection between anomalies in $D$ spacetime dimensions and
higher-dimensional physics. It also allowed to successfully solve the
anomalous Ward identity of eq. (111), for known anomaly function $G(x)$, in
the presence of Nambu-Goldstone bosons in the spectrum. It clarified the role
of local counter terms in switching between consistent and covariant
anomalies, and in ensuring that anomalies associated to an unbroken subgroup
$H$ vanish.

###  C.2 Descent equations and anomaly polynomial

The manipulations of geometrical objects that lead to the construction of the
non-Abelian anomaly in $D=2n$ spacetime dimension are called _descent
equations_ , and the resulting anomaly in its simplest form is sometimes
dubbed as _canonical anomaly_. As we will see, chiral anomalies are defined up
to local counter terms (or equivalently up to choice of regulator). In order
to make the discussion self-contained, we give a brief review of this idea.

The goal is to construct an anomaly that satisfies the Wess-Zumino consistency
condition. To be concrete, we assume that the $2n$-dimensional spacetime
manifold is compact and closed (i.e. there is no boundary). We first consider
the following $(2n+2)$-dimensional form:

$\Omega_{2n+2}(A)={\rm Tr}\mathopen{}\mathclose{{}\left(F^{n+1}}\right),$
(113)

where $A$ is the gauge connection 1-form, and $F$ is its 2-form field
strength. It is easy to show using the Bianchi identity eq. (101) that
$\Omega_{2n+2}$ is a closed form:

$d{\rm Tr}F^{n+1}=(n+1){\rm
Tr}\mathopen{}\mathclose{{}\left(dFF^{n}}\right)=-(n+1){\rm
Tr}\mathopen{}\mathclose{{}\left([A,F]F^{n}}\right)=0.$ (114)

Also, clearly $\Omega_{2n+2}$ is gauge invariant. Since it is closed, at least
locally, we can write it as

$\Omega_{2n+2}(A)=d\omega_{2n+1}^{(0)}(A),$ (115)

where the superscript “0” indicates that it is zeroth order in the gauge
parameter. We note, however, that the fully general solution is

$\tilde{\omega}_{2n+1}^{(0)}(A)=\omega_{2n+1}^{(0)}(A)+dB_{2n}(A),$ (116)

where the additional $2n$-form $B_{2n}$ is a local counter term, which is
allowed simply because $d^{2}=0$. The gauge variation of
$\tilde{\omega}_{2n+1}^{(0)}$ is closed. This is because $\Omega_{2n+2}$ is
gauge invariant, and

$d\delta_{\alpha}\tilde{\omega}^{(0)}_{2n+1}=\delta_{\alpha}d\tilde{\omega}^{(0)}_{2n+1}=\delta_{\alpha}\Omega_{2n+2}=0.$
(117)

Therefore, at least locally, we can write

$\begin{gathered}\delta_{\alpha}\tilde{\omega}_{2n+1}^{(0)}=d\tilde{\omega}_{2n}^{(1)}\qquad\textnormal{and}\qquad\delta_{\alpha}\omega_{2n+1}^{(0)}=d\omega_{2n}^{(1)},\\\
\tilde{\omega}_{2n}^{(1)}(\alpha,A)=\omega_{2n}^{(1)}(\alpha,A)+\delta_{\alpha}B_{2n}(A)+d\hat{\omega}_{2n-1}^{(1)}(\alpha,A).\end{gathered}$
(118)

Here the superscript “1” means the corresponding object is
$\mathcal{O}\mathopen{}\mathclose{{}\left(\alpha^{1}}\right)$, and as before,
in order to be fully general, we added the exact form
$d\hat{\omega}_{2n-1}^{(1)}(\alpha,A)$. For spacetime manifolds without
boundary, however, this term will not play any role, and we will drop it in
the following.232323As we will discuss momentarily, the anomaly that solves
the Wess-Zumino consistency condition results from integrating
$\tilde{\omega}_{2n}^{(1)}(\alpha,A)$ over a manifold without boundary. Let us
now define two functionals. We consider the $(2n+1)$-dimensional disk
$D_{2n+1}$, with $\partial D_{2n+1}=S^{2n}$:

$\begin{gathered}\tilde{W}[A]\equiv\int_{D_{2n+1}}\tilde{\omega}_{2n+1}^{(0)}(A),\\\
\tilde{G}[\alpha,A]\equiv\int_{S^{2n}}\alpha^{a}(x)G_{a}(x)=\int_{S^{2n}}\tilde{\omega}_{2n}^{(1)}(\alpha,A).\end{gathered}$
(119)

The claim is that $\tilde{G}[\alpha,A]$ is the anomaly consistent with the
Wess-Zumino condition. To see this, first notice that

$\delta_{\alpha}\tilde{W}[A]=\tilde{G}[\alpha,A].$ (120)

In this sense, $\tilde{W}[A]$ is naively just like the effective action $W[A]$
we discussed above, but with the possible addition of local counter
terms.242424To be more precise, it is actually analogous to the Chern-Simons
action in one more spacetime dimension, whose anomaly inflows to the boundary
corresponding to the physical spacetime. Then, since
$\delta_{\alpha_{1}}\delta_{\alpha_{2}}-\delta_{\alpha_{2}}\delta_{\alpha_{1}}=\delta_{[\alpha_{1},\alpha_{2}]}$,

$\delta_{\alpha_{1}}\tilde{G}[\alpha_{2},A]-\delta_{\alpha_{2}}\tilde{G}[\alpha_{2},A]=\tilde{G}[[\alpha_{1},\alpha_{2}],A].$
(121)

This shows that $\tilde{G}[\alpha,A]$ is indeed a functional of gauge fields
which fulfills the Wess-Zumino consistency condition, and hence a candidate
for the non-Abelian anomaly. We further note that

$\begin{gathered}\tilde{G}[\alpha,A]=G[\alpha,A]+G_{\rm CT}[\alpha,A],\\\
G[\alpha,A]=\int_{S^{2n}}\omega_{2n}^{(1)}(\alpha,A)\qquad\textnormal{and}\qquad
G_{\rm CT}[\alpha,A]=\int_{S^{2n}}\delta_{\alpha}B_{2n}(A).\end{gathered}$
(122)

One can repeat the above proof with
$W[A]=\int_{D_{2n+1}}\omega_{2n+1}^{(0)}(A)$ to show that $G[\alpha,A]$ also
solves the Wess-Zumino consistency condition. This $G[\alpha,A]$ is called the
_canonical anomaly_ , and $G_{\rm CT}[\alpha,A]$ is the (shift in the) anomaly
due to the addition of the local counter term $W_{\rm
CT}[A]=\int_{D_{2n+1}}dB_{2n}(A)$. For obvious reasons, $\tilde{G}[\alpha,A]$
is then called the _shifted anomaly_.

###  C.3 BRST formalism

In this appendix, we adopt the BRST formalism for chiral anomalies. An
introduction to BRST formulation of chiral anomalies and descent equations can
be found in Weinberg:1996kr ; Harvey:2005it  or in the original paper
Manes:1985df .

##### Review of BRST invariance

Let us begin the discussion with a very brief summary of BRST invariance. The
path integral quantization of a gauge theory (with possibly matter fields
$\psi$) results in

$\mathcal{L}=\mathcal{L}_{\rm
matter}(\psi)-\frac{1}{4g^{2}}F_{a}^{\mu\nu}F_{a\mu\nu}-\partial_{\mu}\omega_{a}^{*}\partial^{\mu}\omega_{a}+f_{abc}(\partial_{\mu}\omega_{a}^{*})A^{\mu}_{c}\omega_{b}+h_{a}f_{a}+\frac{1}{2}\xi
h_{a}h_{a},$ (123)

where $\omega$ ($\omega^{*}$) is an (anti-)ghost field and $h_{a}$ an
auxiliary field that can be trivially integrated over in the path integral. A
typical choice of gauge fixing is for example
$f_{a}=\partial_{\mu}A^{\mu}_{a}$. This Lagrangian is not invariant under
gauge transformations. It is, however, invariant under BRST transformations,
with transformation parameter $\theta$ which anticommutes with ghost fields
and all fermionic matter fields:

$\begin{gathered}\delta_{\theta}\psi=-t_{a}\theta\omega_{a}\psi,\qquad\delta_{\theta}A_{a\mu}=\theta
D_{\mu}\omega_{a},\qquad\delta_{\theta}\omega_{a}=-\frac{1}{2}f^{abc}\omega_{b}\omega_{c},\\\
\delta_{\theta}\omega^{*}_{a}=-\theta
h_{a},\qquad\delta_{\theta}h_{a}=0.\end{gathered}$ (124)

Notice that for matter and gauge fields, this is just a gauge transformation
with transformation parameter $\theta\omega_{a}$. With proper assignment of
ghost number (1 for ghost fields themselves, 0 for regular matter and gauge
fields), one sees that a BRST transformation increases the ghost number by
one. This motivates the introduction of the BRST operator $s$, whose action
increases the ghost number by one, basically reproducing the above
transformation rules. In terms of $s$ we get252525Notice the notation used for
the anticommutator of the Grassmann fields, as explained in appendix A.

$\begin{gathered}sA=D(A)\omega,\qquad s\omega=-\frac{1}{2}[\omega,\omega],\\\
s^{2}=0,\qquad\\{s,d\\}=0.\end{gathered}$ (125)

One says that the above set of equations (called structure equations) defines
a graded commutative differential algebra. Here, $\omega$ plays the role of a
1-form along the “ghost” direction, and $s$ the “exterior derivative” along
the “ghost” direction.

##### WZ consistency condition revisited

In BRST language, the Wess-Zumino consistency condition is just the statement
that the chiral anomaly is BRST-closed. To see why, let’s define

$G[\omega,A]=\int_{S^{2n}}\omega^{a}(x)G^{a}[x,A],$ (126)

where we replaced the gauge parameter with the ghost field $\omega(x)$. Then
we get

$\displaystyle sG[\omega,A]$ $\displaystyle=\int
d^{2n}x\mathopen{}\mathclose{{}\left(-\frac{1}{2}\commutator{\omega(x)}{\omega(x)}^{a}}\right)G^{a}[x,A]-\int
d^{2n}x\omega^{a}(x)\mathopen{}\mathclose{{}\left(\omega\cdot
X}\right)G^{a}[x,A]$ (127) $\displaystyle=-\frac{1}{2}\int
d^{2n}xf^{abc}\omega^{b}(x)\omega^{c}(x)G^{a}[x,A]-\int
d^{2n}xd^{2n}y\omega^{a}(x)\omega^{b}(y)X^{b}(y)G^{a}[x,A]$
$\displaystyle=\frac{1}{2}\int
d^{2n}xd^{2n}y\omega^{a}(x)\omega^{b}(y)\Bigl{[}X^{a}(x)G^{b}[y,A]-X^{b}(y)G^{a}[x,A]-f^{abc}G^{c}[x,A]\delta(x-y)\bigr{]}.$

As expected, the Wess-Zumino consistency condition is indeed equivalent to the
statement $sG[\omega,A]=0$: consistent chiral anomalies are BRST-closed. One
of the immediate implications is that any addition of BRST-exact terms, i.e.
terms of the form $sB$, still result in a valid anomaly functional. Hence,
chiral anomalies are equivalence classes of BRST cohomology Manes:1985df .
This cohomology is defined by the operation $s$, and a BRST “form” takes as
its coefficients local functionals of $A$ and its derivative $dA$ (or
equivalently $A$ and $F$).

##### Descent equations revisited

We can now repeat the descent equations procedure in BRST language:

  * •

$\Omega_{2n+2}=d\omega_{2n+1}^{(0)}$ and $s\Omega_{2n+2}=0$.

Since $\Omega_{2n+2}$ has ghost number 0, the operator $s$ acts on it as a
usual gauge transformation with $\omega$ as transformation parameter.

  * •

$d\mathopen{}\mathclose{{}\left(s\omega_{2n+1}^{(0)}}\right)=-sd\omega_{2n+1}^{(0)}=-s\Omega_{2n+2}=0$
implies that $s\omega_{2n+1}^{(0)}=d\omega_{2n}^{(1)}$.

Now the superscript denotes the degree in ghost number.

  * •

$d\mathopen{}\mathclose{{}\left(s\omega_{2n}^{(1)}}\right)=-sd\omega_{2n}^{(1)}=-s^{2}\omega_{2n+1}^{(0)}=0$
implies that $s\omega_{2n}^{(1)}=d\omega_{2n-1}^{(2)}$.

In particular, this shows that
$s\int_{S^{2n}}\omega_{2n}^{(1)}=\int_{S^{2n}}d\omega_{2n-1}^{(2)}=0$ and, up
to a numerical coefficient, $G\sim\int_{S^{2n}}\omega_{2n}^{(1)}$ is a
consistent chiral anomaly.

The entire discussion can be repeated if we replace $\omega$ with
$\tilde{\omega}$, i.e. with the addition of local counter terms $B_{2n}$. In
fact, since the requirement for the consistent anomaly is $sG=0$, shifting
$\omega_{2n}^{(1)}\to\tilde{\omega}_{2n}^{(1)}=\omega_{2n}^{(1)}+sB_{2n}$ with
an arbitrary $2n$-form still satisfies the condition. At the level of
$\omega_{2n+1}^{(0)}$, this results in
$\omega_{2n+1}^{(0)}\to\tilde{\omega}_{2n+1}^{(0)}=\omega_{2n+1}^{(0)}+dB_{2n}$,
i.e. a shift in the Chern-Simons action.

##  Appendix D Cartan’s homotopy formula and shifted anomaly

###  D.1 Cartan’s homotopy formula

In the previous appendix, we talked about the graded algebra defined by $d$
and $s$. Roughly, there are spacetime forms with corresponding exterior
derivative $d$, and there are BRST forms with corresponding derivative $s$. In
order to get a closed algebra, we required a relation between $d$ and $s$,
which is given by $\\{d,s\\}=0$. For Cartan’s homotopy formula, we consider,
instead of $s$ (or maybe _in addition to_ $s$), auxiliary “directions” defined
by a family of continuous parameters $t=\\{t_{1},t_{2},\dots\\}$. We want to
construct the anti-derivation with respect to $t$, call it $d_{t}$, and
establish a closed algebra between $d$ and $d_{t}$. It turns out that in
addition to the _odd_ derivatives $d$ and $d_{t}$, we need an _even_ operator
$\ell_{t}$. Now, we have a family of connections smoothly parametrized by $x$
and $t$, $A_{t}(x)$. $A_{t}(x)$ is a $t$-form as well as an $x$-form. $d_{t}$
and $\ell_{t}$ are chosen such that the following graded algebra is satisfied:

$\\{d,d\\}=\\{d_{t},d_{t}\\}=\\{d,d_{t}\\}=0,\qquad[\ell_{t},d]=d_{t},\qquad[d_{t},\ell_{t}]=0.$
(128)

In a sense, the second equation, i.e. $[\ell_{t},d]=d_{t}$, is the only non-
trivial one: it means $\ell_{t}$ decreases the degree in $dx$ by one, while at
the same time increasing the degree in $dt$ by one. It provides a notion of
“$d^{-1}$”, which we need to determine the form of anomaly polynomials and
local counter terms. It is not hard to show that for a polynomial of
$\ell_{t}$, $f(\ell_{t})$, we get

$[f(\ell_{t}),d]=d_{t}f^{\prime}(\ell_{t})=f^{\prime}(\ell_{t})d_{t}.$ (129)

The next step is to determine the action of $\ell_{t}$ on the algebra
$\mathscr{Q}$ of polynomials generated by $\\{A_{t},F_{t}\equiv
dA_{t}+A_{t}^{2},d_{t}A_{t},d_{t}F_{t}\\}$.262626The reason we don’t include
$dF_{t}$ is that it is equivalent to $[A_{t},F_{t}]$ thanks to the Bianchi
identity. The unique action of $\ell_{t}$ on a polynomial $Q\in\mathscr{Q}$ is
determined by requiring that (i) its action is consistent with the above
graded algebra and (ii) the algebra $\mathscr{Q}$ is _stable_ under the action
of $d,d_{t}$ and $\ell_{t}$. The resulting action of $\ell_{t}$ is

$\ell_{t}A_{t}=0,\qquad\ell_{t}F_{t}=d_{t}A_{t}.$ (130)

We are now ready to derive the _differential form of the extended Cartan
homotopy formula_. It is obtained by applying eq. (129) to
$f(\ell_{t})=\ell_{t}^{p+1}/(p+1)!$. For a polynomial $Q\in\mathscr{Q}$, we
get

$d_{t}\frac{\ell_{t}^{p}}{p!}Q=\frac{\ell_{t}^{p+1}}{(p+1)!}dQ-d\frac{\ell_{t}^{p+1}}{(p+1)!}Q.$
(131)

A more useful formula for the study of anomalies is obtained by integrating
eq. (131) over the domain $T$ of $t$. In order to do this, we need to
establish a convention for the incomplete integration. We follow Manes:1985df
for the conventions: if $\alpha$ is a form of degree $(r,s)$ in $(dx,dt)$, we
have

$\begin{gathered}\int_{X}\alpha\equiv\int_{X}\alpha_{r,s}dx^{r}dt^{s}=\mathopen{}\mathclose{{}\left(\int_{X}\alpha_{r,s}dx^{r}}\right)dt^{s},\\\
\int_{T}\alpha\equiv\int_{T}\alpha_{r,s}dx^{r}dt^{s}=(-1)^{rs}\mathopen{}\mathclose{{}\left(\int_{T}\alpha_{r,s}dt^{s}}\right)dx^{r}.\end{gathered}$
(132)

It follows that

$\begin{gathered}d\int_{T}\alpha=(-1)^{s}\int_{T}d\alpha,\qquad
d_{t}\int_{X}\alpha=(-1)^{r}\int_{X}d_{t}\alpha,\\\
\int_{X}d\alpha=\int_{\partial
X}\alpha,\qquad\int_{T}d_{t}\alpha=\int_{\partial T}\alpha.\end{gathered}$
(133)

While it can be explicitly checked, the first two equations imply that
$\int_{X}$ $\mathopen{}\mathclose{{}\left(\int_{T}}\right)$ behaves like an
odd operation of degree $r$ ($s$). The last two show that Stokes’ theorem
works as usual. The _integral form of the extended Cartan homotopy formula_ is
obtained by integrating eq. (131):

$\int_{\partial
T}\frac{\ell_{t}^{p}}{p!}Q=\int_{T}\frac{\ell_{t}^{p+1}}{(p+1)!}dQ-(-1)^{p+q+1}d\int_{T}\frac{\ell_{t}^{p+1}}{(p+1)!}Q,$
(134)

where $q$ is the degree of $Q$ in $dt$. Notice that $\ell_{t}^{p+1}Q/(p+1)!$
has degree $(p+q+1)$ in $dt$, and therefore so does $\int_{T}$.

###  D.2 Derivation of the shifted anomaly and local counter terms

So far Cartan’s homotopy formula has been introduced as a pure mathematical
expression for a given algebra of $d,d_{t}$ and $\ell_{t}$. There are three
independent choices one can make when the formula is used: $p,q$, and the
choice of parametrization for $A_{t}$. Now remember that $\ell_{t}$ provides
an operation that behaves like a “$d^{-1}$”. From the descent equations, we
learn that $\Omega_{2n+2}=d\omega_{2n+1}^{(0)}$, and
$s\omega_{2n+1}^{(0)}=d\omega_{2n}^{(1)}$. While those tell us that objects
like $\omega_{2n+1}^{(0)}$ and $\omega_{2n}^{(1)}$ exist, we still need a way
to determine their explicit form. There is no _unique_ inverse of $d$ to
simply solve these equations. Different choices of “$d^{-1}$” will lead to
different $\omega_{2n+1}^{(0)}$ and $\omega_{2n}^{(1)}$, related to each other
by the ambiguity associated with local counter terms. $\ell_{t}$ provides one
such operation. Importantly, depending on the choice of $A_{t}$ (and in
principle also $p$, although we will exclusively take $p=0$), we can get
different answers.

We first mention a few possible options for $A_{t}$:

$\begin{gathered}A_{t}=tA,\;\;t\in[0,1],\qquad\qquad
A_{t}=tA_{1}+(1-t)A_{0},\;\;t\in[0,1],\\\
A_{t}=t_{1}A_{1}+t_{2}A_{2}+(1-t_{1}-t_{2})A_{0},\;\;t_{1},t_{2}\in[0,1].\end{gathered}$
(135)

The first connects $A$ to nothing between $t=0$ and $t=1$. It will turn out to
be a useful choice for a theory with simple symmetry group $G$ without any
symmetry breaking. The second connects two different configurations $A_{0}$
and $A_{1}$. We will find it useful when we need to distinguish a particular
part of the group (or algebra), notably in the case of $G/H$ or a product
group $G_{1}\times G_{2}$. The last is an example with more than one $t$, and
was used in Manes:1985df  to derive the “triangle formula” and other results.
In order to avoid confusion, in the following we will refer to the parameters
of the first and second parametrization as $t$ and $s$,
respectively.272727Notice that $s$ here is not the BRST operation. We will
never encounter both in one place. Let us now consider some examples:

  * (1)

$Q_{t}=\Omega_{2n+2,t}={\rm Tr}(F_{t}^{n+1}),\;A_{t}=tA,\;p=q=0$.

Cartan’s homotopy formula (in its integrated version) with these choices
results in

$\Omega_{2n+2,1}-\Omega_{2n+2,0}=\Omega_{2n+2}(A)=d\int_{T}\ell_{t}\Omega_{2n+2,t}\equiv
d\omega_{2n+1}^{(0)},$ (136)

where we used the subscript $t=0,1$ to denote that the corresponding
functional is in terms of $A_{t}$ with a specified $t$-value. An explicit
expression for $\omega_{2n+1}^{(0)}$ is then given by

$\omega_{2n+1}^{(0)}=\int_{T}\ell_{t}\Omega_{2n+2,t}=(n+1)\int_{0}^{1}dt\;{\rm
Tr}\mathopen{}\mathclose{{}\left[AF_{t}^{n}}\right].$ (137)

This expression was indeed obtained in Zumino:1983rz  with a different method.
According to the language used in Chu:1996fr , the chiral anomaly obtained
from this is the _canonical anomaly_.

  * (2)

$Q_{s}=\Omega_{2n+2,s},\;A_{s}=sA_{1}+(1-s)A_{0},\;p=q=0$.

This time the integrated form of Cartan’s homotopy formula gives

$\Omega_{2n+2,1}-\Omega_{2n+2,0}=d\int_{S}\ell_{s}\Omega_{2n+2,s}\equiv
d\tilde{\omega}_{2n+1}^{(0)}.$ (138)

Explicitly, $\tilde{\omega}_{2n+1}^{(0)}$ is

$\tilde{\omega}_{2n+1}^{(0)}(A_{0},A_{1})=(n+1)\int_{0}^{1}ds\;{\rm
Tr}\mathopen{}\mathclose{{}\left[(A_{1}-A_{0})F_{s}^{n}}\right].$ (139)

  * (3)

Relation between $\omega_{2n+1}^{(0)}$ and $\tilde{\omega}_{2n+1}^{(0)}$.

From (1), we know that each $\Omega_{2n+2,i},\;i=0,1$ appearing on the LHS of
eq. (138) can be expressed as $d\omega_{2n+1,i}^{(0)}$ by eq. (136). Roughly,
we get

$d\omega_{2n+1,1}^{(0)}-d\omega_{2n+1,0}^{(0)}=d\tilde{\omega}_{2n+1}^{(0)}\quad\Rightarrow\quad\tilde{\omega}_{2n+1}^{(0)}=\omega_{2n+1,1}^{(0)}-\omega_{2n+1,0}^{(0)}+dB_{2n},$
(140)

where the total derivative accounts for the presence of the local counter
term. How do we determine the form of $B_{2n}$? First of all, we define
$\omega_{2n+1,s}^{(0)}$ in the following way for $i=0,1$:

$\Omega_{2n+2,i}=d\omega_{2n+1,i}^{(0)},\qquad\omega_{2n+1,i}^{(0)}=\int_{T}\ell_{t}\Omega_{2n+2,it},\qquad
A_{it}=tA_{i}.$ (141)

Then we note that $A_{0}$ and $A_{1}$ can be homotopically connected by
extending $i$ to $s$, which results in the following parametrization:

$A_{st}=tA_{s}=t\mathopen{}\mathclose{{}\left(sA_{1}+(1-s)A_{0}}\right),\qquad
s\in[0,1].$ (142)

In particular, this means that we now have an object $\omega_{2n+1,s}^{(0)}$
given by

$\omega_{2n+1,s}^{(0)}=\int_{T}\ell_{t}\Omega_{2n+2,st},\;\;A_{st}=tA_{s}=t\mathopen{}\mathclose{{}\left(sA_{1}+(1-s)A_{0}}\right),$
(143)

which makes it possible to write down Cartan’s homotopy formula for
$\omega_{2n+1,s}^{(0)}$. It is given by

$\displaystyle\omega_{2n+1,1}^{(0)}-\omega_{2n+1,0}^{(0)}$
$\displaystyle=\int_{S}\ell_{s}d\omega_{2n+1,s}^{(0)}+d\int_{S}\ell_{s}\omega_{2n+1,s}^{(0)}$
(144)
$\displaystyle=\int_{S}\ell_{s}\Omega_{2n+2,s}+d\int_{S}\ell_{s}\omega_{2n+1,s}^{(0)}$
$\displaystyle=\tilde{\omega}_{2n+1}^{(0)}+d\int_{S}\ell_{s}\omega_{2n+1,s}^{(0)}.$

Comparing with eq. (140), this yields the result

$B_{2n}=-\int_{S}\ell_{s}\omega_{2n+1,s}^{(0)}.$ (145)

The expressions for the counter term and the related shifted Chern-Simons
action indeed agree with the ones given in Chu:1996fr .

  * (4)

Explicit expression for $B_{2n}$.

An explicit expression for $B_{2n}$ is obtained by evaluating the integral,
and was done in Chu:1996fr . So, instead of repeating the computation, we will
simply quote the result for $n=2$. $B_{4}$ is given by

$B_{4}(A_{h},A)=\frac{1}{2}{\rm
Tr}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(A_{h}A-AA_{h}}\right)\mathopen{}\mathclose{{}\left(F+F_{h}}\right)+AA_{h}^{3}-A_{h}A^{3}+\frac{1}{2}A_{h}AA_{h}A}\right],$
(146)

where $F_{h}=dA_{h}+A_{h}^{2}$.

  * (5)

Useful properties.

The shifted CS action $\tilde{\omega}_{2n+1}^{(0)}$ and the local counter term
$dB_{2n}$ satisfy the following properties, which are used frequently
throughout the paper and follow from eq. (139) and (144):

    * 1

$\tilde{\omega}_{2n+1}^{(0)}(A_{0}^{g},A_{1}^{g})=\tilde{\omega}_{2n+1}^{(0)}(A_{0},A_{1})$.

The shifted CS action is invariant under _simultaneous_ gauge transformations
of $A_{0}$ and $A_{1}$, $A_{i}\to
g\mathopen{}\mathclose{{}\left(A_{i}+d}\right)g^{-1}$. We emphasize that this
holds regardless of the relation (144) and, in particular, of whether
$\omega_{2n+1,0}^{(0)}$ vanishes (as in the $G/H$ case) or not (as with a
general $G_{1}\times G_{2}$). This is simply because, under simultaneous
transformations, $(A_{1}-A_{0})\to g(A_{1}-A_{0})g^{-1}$, i.e. the derivative
terms cancel and the whole term transforms as adjoint matter, so we can use
cyclicity of the trace in eq. (139).

    * 2

$\tilde{\omega}_{2n+1}^{(0)}(A_{0},A_{1}=A_{0})=0$.

    * 3

$dB_{2n}(A_{0},A_{1}=A_{0})=0$.

    * 4

$\tilde{\omega}_{2n+1}^{(0)}(A_{0}=0,A_{1})=\omega_{2n+1}^{(0)}(A_{1})$.

To see this, we note that with $A_{0}=0$, we get $A_{t}\to tA_{1}$, and eq.
(139) reduces to

$\tilde{\omega}_{2n+1}^{(0)}(A_{0}=0,A_{1})=(n+1)\int_{0}^{1}dt\;{\rm
Tr}\mathopen{}\mathclose{{}\left[A_{1}F_{t}^{n}}\right]=\omega_{2n+1}^{(0)}(A_{1}).$
(147)

    * 5

$dB_{2n}(A_{0}=0,A_{1})=0$.

    * 6

$\tilde{\omega}_{2n+1}^{(0)}(A_{0},A_{1}=0)=-\omega_{2n+1}{(0)}(A_{0})$.

    * 7

$dB_{2n}(A_{0},A_{1}=0)=0$.

The first property leads to many interesting and important consequences. We
discuss a few of them below.

  * (6)

$G/H$ symmetry group.

We consider a symmetric space. Let’s denote the algebra for the unbroken group
$H$ as $\mathbf{h}$, and that of the coset as $\mathbf{k}$. Referring to the
corresponding elements as $h\in\mathbf{h},\;k\in\mathbf{k}$, we want

$[h,h]\subseteq\mathbf{h}\;\;({\rm
subalgebra}),\quad[h,k]\subseteq\mathbf{k}\;\;({\rm
reductive}),\quad[k,k]\subseteq\mathbf{h}\;\;({\rm symmetric}).$ (148)

The gauge field $A$ and the (infinitesimal) group element
$\alpha\in\mathbf{g}$ can be decomposed into their $\mathbf{h}$ and
$\mathbf{k}$ parts:

$\begin{gathered}A=A_{h}+A_{k},\qquad\alpha=\beta+\gamma,\\\
A_{h},\beta\in\mathbf{h},\qquad A_{k},\gamma\in\mathbf{k}.\end{gathered}$
(149)

Under an infinitesimal gauge transformation,

$\delta_{\alpha}A=\underbrace{\mathopen{}\mathclose{{}\left(d\beta+[A_{h},\beta]+[A_{k},\gamma]}\right)}_{=\;\delta_{\alpha}A_{h}}+\underbrace{\mathopen{}\mathclose{{}\left(d\gamma+[A_{h},\gamma]+[A_{k},\beta]}\right)}_{=\;\delta_{\alpha}A_{k}}.$
(150)

For a general $\alpha\in\mathbf{g}$, $A$ and $A_{h}$ do not transform in the
same way. They do so, however, under $\beta\in\mathbf{h}$. Therefore, if we
set $A_{0}=A_{h}$ and $A_{1}=A$, the resulting shifted Chern-Simons action is
invariant under $H$. Or, equivalently, the modified effective action is
invariant under any $H$-transformation: for all $\beta\in\mathbf{h}$,
$G_{\beta}=\delta_{\beta}W=0$. This is indeed a crucial assumption in the
derivation of the original Wess-Zumino action. The current discussion allows
us to determine what local counter term needs to be added. If, in addition,
$H$ is an anomaly-free embedding (AFE), then
$\omega_{2n+1,0}^{(0)}=\omega_{2n+1}^{(0)}(A_{h})=0$ and

$\tilde{\omega}_{2n+1}^{(0)}(A_{h},A)=\omega_{2n+1}^{(0)}(A)+dB_{2n}(A_{h},A).$
(151)

  * (7)

Product group $G_{1}\times G_{2}$ and mixed anomaly.

Let’s comment on the case in which $G_{1}=U(1)$ and $G_{2}$ is non-Abelian
(e.g. $SU(N)$). In this example we have a mixed anomaly, and the role of the
local counter term is to attribute the anomaly to one of the following: the
$G_{1}$ current, the $G_{2}$ current, or a combination of the two. This is
exactly what the counter term $B_{2n}$ constructed above does, as it ensures
that the anomaly associated with a chosen subgroup vanishes. To see this, we
first formally embed $G_{1}\times G_{2}$ into a larger group (e.g. with
$G_{1}$ and $G_{2}$ as diagonal blocks), so that we can formally write gauge
fields and (infinitesimal) group elements as

$\begin{gathered}A=A_{1}+A_{2},\qquad
A_{1}\in\mathscr{G}_{1},\;A_{2}\in\mathscr{G}_{2},\\\
\alpha=\alpha_{1}+\alpha_{2},\qquad\alpha_{1}\in\mathscr{G}_{1},\;\alpha_{2}\in\mathscr{G}_{2}.\end{gathered}$
(152)

Then under gauge transformations of each factor group, we get

$\begin{gathered}({\rm Under}\;\alpha_{1}\in\mathscr{G}_{1})\quad A\to
A+d\alpha_{1},\quad A_{1}\to A_{1}+d\alpha_{1},\\\ ({\rm
Under}\;\alpha_{2}\in\mathscr{G}_{2})\quad A\to
A+d\alpha_{2}+[A,\alpha_{2}],\quad A_{2}\to
A_{2}+d\alpha_{2}+[A_{2},\alpha_{2}].\end{gathered}$ (153)

In the second line, we used the fact that
$\mathscr{G}=\mathscr{G}_{1}\oplus\mathscr{G}_{2}$, i.e.
$[A_{1},\alpha_{2}]=0$. Therefore, we see for example that if we take
$A_{0}\equiv A_{2}$ and $A_{1}\equiv A$ in the sense of the parametrization of
eq. 135, then those fields transform under $\alpha_{2}\in\mathscr{G}_{2}$ in
the same way, and $\tilde{\omega}_{2n+1}(A_{0},A_{1})$ will be invariant under
a $G_{2}$-transformation. In other words, the associated counter term moves
the entire anomaly to the $U(1)$ current. The opposite works just as well: if
we instead take $A_{0}\equiv A_{1}$ and $A_{1}\equiv A$, then this time all
the mixed anomaly is attributed to the $G_{2}$-current. Our discussion is very
general, and these statements hold for any choice of $G_{2}$ that has a mixed
anomaly with $G_{1}=U(1)$.

For the sake of completeness, we briefly comment on the algebra. While taking
$\mathscr{G}_{2}=\mathbf{h}$ and $\mathscr{G}_{1}=\mathbf{k}$ results in the
algebra of a symmetric space, this is not the case if we instead take
$\mathscr{G}_{1}=\mathbf{h}$ and $\mathscr{G}_{2}=\mathbf{k}$. This may have
potential implications for WZW terms.

##  Appendix E Wess-Zumino-Witten action

In this appendix, we first review the motivation for the Wess-Zumino-Witten
(WZW) action Wess:1971yu ; Witten:1983tw  following the original argument by
Wess and Zumino Wess:1971yu . We then present a construction for the WZW
action in terms of the shifted CS action $\tilde{\omega}^{(0)}_{2n+1}$
Chu:1996fr .

Suppose we have a theory with effective action (for connected diagrams)
$W[V_{\mu},\xi,\dots]$ which consists of (external) vector fields ($V_{\mu}$),
(external) (pseudo-)scalars $\xi$, and some other fields. We further assume
that the symmetry group $G$ is spontaneously broken down to $H$ and the theory
is anomalous under the gauge transformation:

$\delta_{\alpha}W[V_{h},V_{k},\xi]=\mathopen{}\mathclose{{}\left(X_{i}\beta_{i}+Y_{a}\gamma_{a}+Z_{A}\alpha_{A}+\dots}\right)W[V_{h},V_{k},\xi,\dots]=G_{\alpha}[V_{h},V_{k}],$
(154)

where we split the gauge fields into their $\mathbf{h}$ and $\mathbf{k}$
parts, $V=V_{h}+V_{k}$. We also do the same for the transformation parameter,
$\alpha=\beta+\gamma$, with $\beta\in\mathbf{h}$ and $\gamma\in\mathbf{k}$.
Accordingly, $X_{i}$ is the generator associated with $V_{h}$, $Y_{a}$ with
$V_{k}$, and $Z_{A}$ with the (pseudo-)scalars $\xi$. The associated anomaly
functional $G_{\alpha}$ is a functional of gauge fields only. In perturbation
theory, we know the effective action $W$ and can compute its variation, hence
the anomaly, by computing Feynman diagrams. The situation is much less clear
in strongly coupled theories, especially when the theory confines, and
generically $W$ is not known in the confined phase. If, on the other hand, one
knows the anomaly functional $G_{\alpha}$ (e.g. by computing it in the quark-
gluon phase and using RG invariance of the chiral anomaly), one may hope to
solve the above anomalous Ward identity to get the effective action $W$ in the
strongly coupled IR phase. Mathematically, the question Wess and Zumino
addressed in their work Wess:1971yu  is whether we can solve for $W$ once
$G_{\alpha}$ is known.

For an explicit construction of the WZW action, we first consider a spacetime
manifold $S^{2n}$. We assume that the gauge field $B$ on $S^{2n}$ can be
extended to $A$ on the disk $D_{2n+1}$, whose boundary is $S^{2n}$. The
boundary value of the gauge field on $D_{2n+1}$ is the gauge field we started
with on $S^{2n}$: $A|_{S^{2n}}=B$. Next, we consider $\Sigma=e^{-\xi}\in G/H$.
As we did for the gauge field, we need to extend this to $D_{2n+1}$ as well:
$\Sigma\to\Lambda$ in such a way that $\Lambda|_{S^{2n}}=\Sigma$. This is
possible if $\pi_{2n}(G/H)$ is trivial.

Now, the claim is that the WZW effective action can be written in terms of the
shifted CS action as

$W[\Sigma,B]=\int_{D_{2n+1}}\mathopen{}\mathclose{{}\left[\tilde{\omega}_{2n+1}^{(0)}(A_{h},A)-\tilde{\omega}_{2n+1}^{(0)}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(A^{\Lambda^{-1}}}\right)_{h},A^{\Lambda^{-1}}}\right)}\right].$
(155)

In order to confirm this, it suffices to check the following:

  * 1

Eq. (155) solves the anomalous Ward identity.

To see this, we first show that
$\tilde{\omega}_{2n+1}^{(0)}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(A^{\Lambda^{-1}}}\right)_{h},A^{\Lambda^{-1}}}\right)$
is _invariant_ under any $g\in G$ transformation. Indeed we have

$A^{\Lambda^{-1}}\to\mathopen{}\mathclose{{}\left(A^{g}}\right)^{(\Lambda^{g})^{-1}}=A^{h\circ\Lambda^{-1}\circ
g^{-1}\circ
g}=h(\Lambda,g)\mathopen{}\mathclose{{}\left(d+A^{\Lambda^{-1}}}\right)h^{-1}(\Lambda,g).$
(156)

In other words, under an arbitrary $g\in G$, the combination
$A^{\Lambda^{-1}}$ transforms under a particular $H$-element defined by
$g\Lambda=\Lambda^{g}h\mathopen{}\mathclose{{}\left(\Lambda,g}\right)$ as if
it were a single gauge field. This proves the statement since while
$\tilde{\omega}_{2n+1}^{(0)}(A_{h},A)$ is not invariant under an arbitrary $G$
transformation, it is invariant under any $H$ transformation. This also
implies that any non-trivial anomaly only comes from the first term in eq.
(155), which is to be expected since the usual shifted CS term solves the
anomalous Ward identity (with shifted anomaly).

  * 2

It vanishes as $\Sigma\to 1$.

This is easily seen by noting that $\Lambda\to 1$ as $\Sigma\to 1$. Hence,
there is no solution to the anomalous Ward identity which is a function of
gauge fields only.

  * 3

It is a function of the _boundary_ values, $\Sigma$ and $B$.

It is sufficient here to show that the integrand is a closed form:

$d\mathopen{}\mathclose{{}\left(\tilde{\omega}_{2n+1}^{(0)}(A_{h},A)-\tilde{\omega}_{2n+1}^{(0)}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(A^{\Lambda^{-1}}}\right)_{h},A^{\Lambda^{-1}}}\right)}\right)=\Omega_{2n+2}(A)-\Omega_{2n+2}\mathopen{}\mathclose{{}\left(A^{\Lambda^{-1}}}\right)=0,$
(157)

as $\Omega_{2n+2}$ is gauge invariant. Therefore, at least locally, the
integrand can be written as an exact $2n$-form, and the integral over the disk
depends only on the boundary values, $\Sigma$ and $B$.

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# Ab initio transport calculations: from normal to superconducting current

H. Ness  Department of Physics, King’s College London, Strand, London WC2R
2LS, UK  M. van Schilfgaarde  Department of Physics, King’s College London,
Strand, London WC2R 2LS, UK  National Renewable Energy Laboratory, Golden,
Colorado 80401, USA

###### Abstract

Applying the Bogoliubov-de Gennes equations with density-functional theory, it
is possible to formulate first-principles description of current-phase
relationships in superconducting/normal (magnetic)/superconducting trilayers.
Such structures are the basis for the superconducting analog of
Magnetoresistive random access memory devices (JMRAM). In a recent paper [Ness
_et al._ , 2022] we presented results from the first attempt to formulate such
a theory, applied to the Nb/Ni/Nb trilayers. In the present work we provide
computational details, explaining how to construct key ingredient (scattering
matrices $S_{N}$) in a framework of linear muffin-tin orbitals (LMTO).

##  I Introduction

In a recent paper [Ness  _et al._ , 2022], we have combined density functional
theory and the Bogoliubov-de Gennes equations to form a first-principles
approach to the study of transport in magnetic Josephson junctions (MJJ). This
method allowed us to predict and explain the properties of realistic MJJs such
as the period of oscillation and decay of the critical current oscillations
with the ferromagnet thickness. We applied our methodology to study realistic
material stacks of the Nb/Ni/Nb trilayer and established that suppression of
supercurrent is an intrinsic property of the junctions, even in absence of
disorder.

To determine the supercurrent in a superconductor–normal metal–superconductor
(S/N/S) junction from a density-functional approach (which is inherently
single-particle), one needs to “decompose” the entire scattering process into
different steps. First we use the Andreev approximation to account for
electron–hole scattering processes Beenakker (1992), the (spin-resolved)
Andreev reﬂection at each left S/N and right N/S interfaces is described by a
reflection matrix. Second, we assume that the main contribution to the
supercurrent comes from Andreev bound states localized in the junction (in the
short junction limit) in the energy window corresponding to the
superconducting gap. The energy spectrum of the Andreev bound states can be
obtained by solving an equation Beenakker (1992, 2004); van Heck  _et al._
(2014); de Jong and Beenakker (1995) which basically states the conservation
of incoming/outgoing particle fluxes following scattering in the normal state
region and Andreev reflections at the left S/N and right N/S interfaces. This
equation involves the Andreev reflection matrices (at the left S/N and right
N/S interfaces) as well as the scattering matrices $S_{N}$ for electron- and
$S_{N}^{*}$ hole-like waves in the central normal N region. These single-
particle (normal state) $S_{N}$ scattering matrices are needed the objects we
obtain from density-functional theory.

In the present paper, we show in detail how to obtain such scattering matrices
from the Questaal suite, which is an open access electronic structure code
based on the LMTO technique Que ; Pashov  _et al._ (2020).

##  II Transport in the normal state

The Questaal package Que ; Pashov  _et al._ (2020) calculates the single-
particle electronic structure and includes some many-body effect corrections
as well. It is based on the LMTO technique.

One of Questaal’s packages provides the ability to calculates the full non-
equilibrium (NE) transport properties of an infinite $L$-$C$-$R$ system
representing a central region $C$ cladded by two semi-infinite $L$ and $R$
leads Faleev  _et al._ (2005). The transport properties are obtained by using
the NE Green’s functions (GF) formalism Meir and Wingreen (1992). The
transport calculations are done with a LMTO basis-set in the so-called Atomic
Sphere Approximation. Owing to the finite range of the basis set, the
$L$-$C$-$R$ system can be described in terms of an infinite stack of
principal-layers (PLs) which interact only with their nearest neighboring PLs
(Fig. 1). The direction of the electronic current is perpendicular to the PLs,
and periodic boundary conditions are used within each PL. The discretization
of the surface Brillouin zone of the corresponding PL introduces a set of
transverse momentum $k_{\parallel}$. Before transport calculations are
performed, the density of the $L$-$C$-$R$ must be computed self-consistently
in a standard DFT framework. Apart from a constant shift, potentials in the
semi-infinite $L$\- and $R$\- layers are kept frozen at the potential of their
respective bulk systems. A dipole must form across the $C$ region so that the
$L$\- and $R$\- Fermi levels align. This is generated in the course of the
self-consistent cycle, and also determines the shift needed to align the $L$\-
and $R$\- Fermi levels. The first account of this method was presented in
Ref.[van Schilfgaarde and Lambrecht, 1998], and the formalism is described in
detail in Ref.[Faleev  _et al._ , 2005], including its implementation for the
non-equilibrium cases. Self-consistency can be obtained in the non-equilibrium
case, though it is not important here.

Figure 1: Schematic representation of the stacking of principal-layers PLs for
the calculation of the transport in two-terminal $L$-$C$-$R$ junctions.

The key quantity is the total transmission probability $T(E,V)$ from which the
non-linear current can be obtained with the conventional Landauer-like elastic
scattering framework. The linear conductance regime is simply described by
$T(E_{F},V=0)$ taken at the Fermi energy $E_{F}$ of the $L$-$C$-$R$ system at
equilibrium. The total transmission probability $T(E,V)$ is given by

$T(E,V)=\sum_{k_{\parallel},\sigma}w_{k_{\parallel}}T(E,V;k_{\parallel},\sigma)\
,$ (1)

where $w_{k_{\parallel}}$ is the weight associated with the transverse
momentum $k_{\parallel}$, and $\sigma$ is the spin of the electron.

The partial transmission probability $T(E,V;k_{\parallel},\sigma)$ are
obtained from NEGF Meir and Wingreen (1992) as follows:

$\begin{split}&T(E,V;k_{\parallel},\sigma)=\\\ &{\rm
Tr}_{C}\left[\Gamma_{LL}(E;k_{\parallel})\ G^{r}_{LR}(E;k_{\parallel})\
\Gamma_{RR}(E;k_{\parallel})\ G^{a}_{RL}(E;k_{\parallel})\right]\
,\end{split}$ (2)

where (for simplifying the notation, explicit dependence of spin $\sigma$ and
bias $V$ has been dropped) the trace is taken over the basis set of the
central $C$ region. The GF $G^{r/a}$ are the retarded/advanced NEGF of the $C$
region connected to the leads, i.e.
$G^{r/a}=[g^{r/a}-\Sigma^{r/a}_{LL}-\Sigma^{r/a}_{RR}]^{-1}$, where
$\Sigma^{r/a}_{LL}$ and $\Sigma^{r/a}_{RR}$ are the corresponding $L$ and $R$
lead self-energies respectively ($g^{r/a}$ is the GF of the disconnected $C$
region). The quantities $\Gamma_{LL/RR}$ are the imaginary part of the lead
self-energies, $\Gamma_{LL/RR}={\rm i}(\Sigma^{r}-\Sigma^{a})_{LL/RR}$.
Finally, $G_{LR}$ are the NEGF matrix elements connecting the most-left PL of
the $C$ region to the most-right PL of region $C$.

In order to calculate the supercurrent in the junctions, we actually need the
full scattering matrix $S_{N}$ of the central region. The $S_{N}$ matrix is
built from the transmission (reflection) coefficients between the $L$ and $R$
and not from the transmission (reflection) probability Eq. (2). This marks one
essential difference between describing superconducting and normal transport:
$S_{N}$ has additional information not needed for transmission in the normal
state.

Hence, we need to apply a transformation to Eq.(2) to be able to extract the
transmission coefficients. Instead of folding down the degrees of freedom of
the $L$ and $R$ leads into a closed form for the lead self-energies and
calculating the transmission probability from a trace over the degrees of
freedom of the central $C$ region, we have to unfold these degrees of freedom
and calculate the transmission probability from a trace over the these degrees
of freedom. The latter can form propagating waves in the $L$ and $R$ leads
which are linked by transmission and reflection coefficients as in the
original picture of Landauer-like scattering Fisher and Lee (1981).

Therefore we first need to determine the eigenmodes of propagation in the $L$
and $R$ leads, and then transform the lead self-energies into the eigenmode
basis of these propagating states.

###  II.1 Eigenmodes of propagation in the leads

####  II.1.1 Bulk GF

Deep inside the leads, one can calculate the eigenmodes of propagation by
solving a nearest-neighbor (in terms of PLs) tight-binding-like equation for
the bulk GF $g_{p,p^{\prime}}$

$-S_{0,-1}g_{-1,0}+(P-S)_{0,0}g_{0,0}-S_{0,1}g_{1,0}=1\ .$ (3)

In the LMTO language Pashov  _et al._ (2020), the quantity $(P-S)_{0,0}$ plays
the role of a local energy-dependent Hamiltonian in the PL (with index $p=0$),
where $P$ are the so-called potential functions and the structure constant
$S_{p,p^{\prime}}$ couples only adjacent PLs ($p-p^{\prime}=\pm 1$). The bulk
is translationally invariant in the direction perpendicular to the PLs; hence
$S_{0,-1}=S_{1,0}$ and $g_{-1,0}=g_{0,1}$.

One solves the equation

$-S_{1,0}g_{0,1}+(P-S)_{0,0}g_{0,0}-S_{0,1}g_{1,0}=1$ (4)

by expanding the wavefunction coefficients as a solution $\bm{\alpha}$ of a
quadratic equation Chen  _et al._ (1989). This quadratic equation can be
recast into a generalized eigenvalue problem
$\bm{A}\bm{x}=\lambda\bm{B}\bm{x}$ by introducing a new vector
$\bm{\beta}=\lambda\bm{\alpha}$ and working in a enlarged (doubled) vector
space Chen  _et al._ (1989); Fujimoto and Hirose (2003). The generalized
eigenvalue problem is written as:

$\left[\begin{array}[]{cc}-{S}_{1,0}&(P-S)_{0,0}\\\
{0}&{S}_{0,1}\end{array}\right]\left[\begin{array}[]{c}\bm{\alpha}\\\
\bm{\beta}\end{array}\right]=\lambda\left[\begin{array}[]{cc}{0}&{S}_{0,1}\\\
{S}_{0,1}&{0}\end{array}\right]\left[\begin{array}[]{c}\bm{\alpha}\\\
\bm{\beta}\end{array}\right]\ ,$ (5)

and can be solved from a set of two independent equations:

$-S_{1,0}Pr^{-1}P^{-1}+(P-S)_{0,0}-S_{0,1}PrP^{-1}=0$ (6)

and

$-S_{1,0}Qx^{-1}Q^{-1}+(P-S)_{0,0}-S_{0,1}QxQ^{-1}=0\ .$ (7)

Note that all matrices $M\equiv S,g,Q,P$ are dependent on variables $M\equiv
M(E;k_{\parallel},\sigma)$. The above equations need to be solved for each
energy $E$, each $k_{\parallel}$ and each spin $\sigma$.

The eigenvalue vectors $r$ and $x$ characterize the propagating (or decaying)
modes in the bulk. Their meaning becomes clear when written as Bloch-like
factors $e^{\pm ik_{z}a}$, where $a$ is the characteristic width of the PL and
$k_{z}$ is the (energy dependent) wave number normal to the interface. The
columns of the matrices $P$ and $Q$ are the corresponding eigenvectors. For
the propagating modes, the wave numbers $k_{z}(E;k_{\parallel},\sigma)$ are
real numbers; $k_{z}(E)$ contains a non-zero imaginary part for the decaying
modes. We choose the following convention: $|r_{i}|\leq 1$ (propagating and
decaying modes towards the right) and $|x_{i}|\geq 1$ (propagating and
decaying modes towards the left).

####  II.1.2 Reformulating the generalized eigenvalue problem

One can rewrite the central equations of the previous section in term of the
wavefunctions $\phi_{p}$ of the PL $p$ deep inside the leads:

$-S_{1,0}\phi_{p-1}+g_{b}^{-1}\phi_{p}+S_{0,1}\phi_{p+1}=0$ (8)

as the system is translation-invariant in the bulk of the leads. For notation
convenience, we use $g_{b}^{-1}=(P-S)_{0,0}$.

By introducing the ratio matrix ${R}_{p}^{\leftarrow}=\phi_{p-1}\phi_{p}^{-1}$
and by manipulation of the recursion relation

$\phi_{p}=g_{b}S_{1,0}\phi_{p-1}+g_{b}S_{0,1}\phi_{+1}\ ,$ (9)

one ends up with the following expression Fujimoto and Hirose (2003) for the
ratio matrix ${R}_{p+1}^{\leftarrow}$:

${R}_{p+1}^{\leftarrow}=\left({1}+g_{b}S_{1,0}[{1}-{R}_{p}^{\leftarrow}g_{b}S_{1,0}]^{-1}{R}_{p}^{\leftarrow}\right)g_{b}S_{0,1}\
.$ (10)

As the bulk of the lead is translation-invariant, we can use the generalized
Bloch condition $\phi_{p+1}=\lambda\phi_{p}$ between adjacent PLs (note that
this is reminiscent of the relation $\bm{\beta}=\lambda\bm{\alpha}$ used in
the previous section). The ratio matrix becomes
${R}_{p}^{\leftarrow}=\phi_{p-1}\phi_{p}^{-1}=\lambda^{-1}$, and Eq. (10) is
another formulation of the quadratic equation Chen  _et al._ (1989) discussed
previously.

However, we have found this expression can be used as an iterative scheme,
i.e.

$\lambda_{i+1}^{-1}=\left({1}+g_{b}S_{1,0}[\lambda_{i}-g_{b}S_{1,0}]^{-1}\right)g_{b}S_{0,1}$
(11)

to improve (at will) the accuracy (precision) of the eigenvalues
$\lambda\equiv(r,x)$. This is particularly crucial in the cases of degenerate
modes that might occur at particular $E$ and $k_{\parallel}$.

####  II.1.3 The case of singular matrices

We have also implemented the possibilities of dealing with singular matrices
that may occur in the generalized eigenvalue problem
$\bm{A}\bm{x}=\lambda\bm{B}\bm{x}$ which is not easily solvable when one, or
both, $\bm{A},\bm{B}$ are singular. An eigenvalue shift procedure can be used
to solve the linear generalized eigenvalue problem when both matrices are
singular and when the full eigensystem is required A. Jennings and Cole
(1978).

For that one adds the term $-\alpha\bm{B}\bm{x}$ on both sides of
$\bm{A}\bm{x}=\lambda\bm{B}\bm{x}$. If $\bar{\bm{A}}=\bm{A}-\alpha\bm{B}$ is
not singular, one can calculate $\bar{\bm{M}}=\bar{\bm{A}}^{-1}\bm{B}$ and
solve the conventional eigenvalue problem:

$\bar{\bm{M}}\bm{x}=\frac{1}{\lambda-\alpha}\bm{x}.$ (12)

A typical eigenvalue $\gamma$ of $\bar{\bm{M}}$ must be related to one of the
$\lambda$ values according to $\gamma=1/\mu$ where $\mu=\lambda-\alpha$, with
the corresponding eigenvector unchanged.

###  II.2 Surface Green’s functions

We can now build the surface GF of the leads which enters into the definition
of the lead self-energies $\Sigma^{r/a}_{RR}$. The $L$ and $R$ surface GFs are
obtained from only one part of the eigenmodes, i.e. from the propagating modes
and the modes that decay inside the bulk of the corresponding lead Chen  _et
al._ (1989).

The surface GF of the $R$ lead is obtained from

$g_{RR}=\left[g_{b}^{-1}-S_{0R}PrP^{-1}\right]^{-1}$ (13)

and the $L$ lead surface GF from

$g_{LL}=\left[g_{b}^{-1}-S_{0L}Qx^{-1}Q^{-1}\right]^{-1}\ .$ (14)

We have used an explicit notation for the leads’ structure constant, $S_{0R}$
and $S_{0L}$, as the two $L$ and $R$ leads need not be identical.

In comparison to the bulk case in Sec. II.1.1, we have $S_{0R}=S_{0,1}$ and
$S_{R0}=S_{1,0}$ for the $R$ region, and $S_{0L}=S_{1,0}$ and $S_{L0}=S_{0,1}$
for the $L$ region. Note the different ordering of the subscript of the bulk
structure constant for the $L$ and $R$ regions: the $R$ ($L$) surface GF is
built from “propagating” towards two different $R$ ($L$) directions.

By introducing Eq. (6) into Eq. (13), and Eq. (7) into Eq. (14), we find that
the matrices $P,P^{-1}$ ($Q,Q^{-1}$) are the transformations that diagonalize
$g_{RR}S_{R0}$ ($g_{LL}S_{L0}$) respectively. Indeed, we have

$g_{RR}=\left[S_{R0}Pr^{-1}P^{-1}\right]^{-1}\ ,$ (15)

or equivalently

$g_{RR}S_{R0}=PrP^{-1}\ ,$ (16)

and

$g_{LL}S_{L0}=Qx^{-1}Q^{-1}\ .$ (17)

The matrices $P,P^{-1}$ ($Q,Q^{-1}$) perform the change of basis set, from the
original basis to the basis of the propagation eigenmodes.

###  II.3 Transmission coefficients

We now proceed with the transformation of the quantities
$\Gamma_{\alpha\alpha}$ into the basis of the propagation eigenmodes. The
quantity $\Gamma_{\alpha\alpha}$ is related to the imaginary part of the lead
self-energy $\Sigma_{\alpha\alpha}$ ($\alpha{=}L,R$)

$\Gamma_{\alpha\alpha}=S_{0\alpha}\ {\rm
i}\left[g^{r}_{\alpha\alpha}-g^{a}_{\alpha\alpha}\right]S_{\alpha 0}\ .$ (18)

Using Eq. (17) and the relation
$S_{0L}g^{a}_{LL}=(g^{r}_{LL}S_{L0})^{\dagger}=(Q^{-1})^{\dagger}(x^{-1})^{\dagger}Q^{\dagger}$,
we find that

$\Gamma_{LL}={\rm
i}\left(S_{0L}Qx^{-1}Q^{-1}-(Q^{-1})^{\dagger}(x^{-1})^{\dagger}Q^{\dagger}S_{L0}\right)\
.$ (19)

Introducing the identity $1=QQ^{-1}$ ($1=(Q^{-1})^{\dagger}Q^{\dagger}$) to
the right (left) side of the second (first) term in Eq. (19), one ends up
with:

$\Gamma_{LL}=(Q^{-1})^{\dagger}v_{L}Q^{-1}$ (20)

where

$v_{L}={\rm
i}\left[Q^{\dagger}S_{0L}Qx^{-1}-\left(Q^{\dagger}S_{0L}Qx^{-1}\right)^{\dagger}\right]\
.$ (21)

Proceeding similarly for $\Gamma_{RR}$, we get

$\Gamma_{RR}=(P^{-1})^{\dagger}v_{R}P^{-1}$ (22)

where

$v_{R}={\rm
i}\left[P^{\dagger}S_{0R}Pr-\left(P^{\dagger}S_{0R}Pr\right)^{\dagger}\right]$
(23)

It is crucial to note that the matrices $v_{L}$ and $v_{R}$ correspond to the
expectation values of the current operator calculated in the basis set of the
eigenmodes Wimmer (2008).

For non-degenerate modes, the diagonal elements $v_{L,n}$ ($v_{R,m}$)
correspond to the group velocity $\partial E/\partial k$ of the propagating
mode $n$ ($m$) in the $L$ ($R$) region Fujimoto and Hirose (2003); Wimmer
(2008). For decaying modes the diagonal elements are simply zero. In the case
of degeneracy, the velocity matrices $v_{L}$ and $v_{R}$ are block diagonal.
Then we need to apply a further transformation to get diagonal matrices
$v_{L,R}\rightarrow v^{D}_{L,R}$ Wimmer (2008).

Once the velocity matrices $v_{L}$ and $v_{R}$ are diagonal, we can write
$v^{D}_{L,n}=\text{sgn}(v^{D}_{L,n})|v^{D}_{L,n}|^{1/2}|v^{D}_{L,n}|^{1/2}$
and
$v^{D}_{R,m}=\text{sgn}(v^{D}_{R,m})|v^{D}_{R,m}|^{1/2}|v^{D}_{R,m}|^{1/2}$,
and transform Eq. (2) as follows:

$\begin{split}&{\rm Tr}_{C}\left[\Gamma_{LL}\ G^{r}_{LR}\ \Gamma_{RR}\
G^{a}_{RL}\right]\\\ &={\rm
Tr}\left[(Q^{-1})^{\dagger}v^{D}_{L}Q^{-1}g^{r}_{LR}(P^{-1})^{\dagger}v^{D}_{R}P^{-1}g^{a}_{LR}\right]\\\
&={\rm Tr}_{L+R}\left[\ t_{LR}\ t_{LR}^{\dagger}\ \right]\ .\end{split}$ (24)

The transmission probability is now expressed in terms of transmission
coefficients $t_{n,m}$ linking the propagating modes $n$ of the $L$ lead to
the propagating modes $m$ of the $R$ lead. The transmission coefficients are
given by a generalized Fisher-Lee expression Fisher and Lee (1981):

$\begin{split}&t_{LR}\equiv\\\ &t_{n,m}(E;k_{\parallel},\sigma)={\rm
i}|v^{D}_{L,n}|^{1/2}\left[{Q^{-1}\ g^{r}_{LR}\
(P^{-1})^{\dagger}}\right]_{n,m}|v^{D}_{R,m}|^{1/2}\end{split}$ (25)

It is important to note that the original Meir and Wingreen expression Meir
and Wingreen (1992) involves a trace over the degrees of freedom in the $C$
region (first line in Eq. (24) ), while the Fisher and Lee expression Fisher
and Lee (1981) involves a trace over the propagating modes in the $L$ and $R$
leads (last line in Eq. (24) ).

###  II.4 Reflection coefficients

In analogy to the transmission probability, we can also define reflection
probabilities in the same lead:

$\begin{split}R_{L}(E)&={\rm Tr}_{C}\left[\Gamma_{LL}\ G^{r}_{LL}\
\Gamma_{LL}\ G^{a}_{LL}\right]\\\ R_{R}(E)&={\rm Tr}_{C}\left[\Gamma_{RR}\
G^{r}_{RR}\ \Gamma_{RR}\ G^{a}_{RR}\right]\end{split}$ (26)

Following the derivations given in the previous section we find (for the $L$
region):

$\begin{split}&{\rm Tr}_{C}\left[\Gamma_{LL}\ G^{r}_{LL}\ \Gamma_{LL}\
G^{a}_{LL}\right]\\\ &={\rm
Tr}\left[(Q^{-1})^{\dagger}v_{L}Q^{-1}g^{r}_{LL}(Q^{-1})^{\dagger}v_{L}Q^{-1}g^{a}_{LL}\right]\\\
&={\rm Tr}_{L}\left[r_{LL}(r_{LL})^{\dagger}\right]\end{split}$ (27)

The reflection coefficients $r_{LL}=r_{n,n^{\prime}}$ are now expressed in the
basis of the propagating modes $n$ and $n^{\prime}$ of the $L$ lead.

Similarly, one can find the reflection probability $R_{R}(E)={\rm
Tr}_{R}\left[r_{RR}(r_{RR})^{\dagger}\right]$ from the reflection coefficients
$r_{RR}=r_{m,m^{\prime}}$ expressed in the basis of the propagating modes $m$
and $m^{\prime}$ of the $R$ lead.

One should note that the reflection probability, defined in Eq. (26), contains
the contributions of both the $L$ ($R$) incoming wave(s) and the reflected
waves in the $L$ ($R$) region. In order to obtain the correct reflection
coefficients (and proper flux conservation), one needs to suppress the
contribution of the incoming wave in the $n$-th $L$ channel ($m$-th $R$
channel) in the $L$ ($R$) lead respectively.

Hence the expressions of the reflection coefficients are as follows:

$\begin{split}&r_{LL}=r_{n,n^{\prime}}(E;k_{\parallel},\sigma)=\\\ &{\rm
i}|v^{D}_{L,n}|^{1/2}\ \left[Q^{-1}\ g^{r}_{LL}\
(Q^{-1})^{\dagger}\right]_{n,n^{\prime}}|v^{D}_{L,n^{\prime}}|^{1/2}-\delta_{nn^{\prime}}\end{split}$
(28)

and

$\begin{split}&r_{RR}=r_{m,m^{\prime}}(E;k_{\parallel},\sigma)=\\\ &{\rm
i}|v^{D}_{R,m}|^{1/2}\ \left[P^{-1}\ g^{r}_{RR}\
(P^{-1})^{\dagger}\right]_{m,m^{\prime}}|v^{D}_{R,m^{\prime}}|^{1/2}-\delta_{mm^{\prime}}\end{split}$
(29)

##  III Full scattering matrix and supercurrent

The full scattering matrix $S_{N}$ is built from the reflection coefficients
$r_{LL}$ and $r_{LL}$, and from the transmission coefficients $t_{LR}$ and
$t_{RL}$ ($t_{RL}$ is the transpose of $t_{LR}$). All quantities are
explicitly dependent of $(E;k_{\parallel},\sigma)$.

For the calculations of the supercurrent, we construct the normal state
scattering matrix $S_{N}$ as follows:

$\displaystyle
S_{N}(E;k_{\parallel})=\left(\begin{array}[]{cc}\left[\begin{array}[]{cc}r_{LL}(\uparrow)&0\\\
0&r_{LL}(\downarrow)\end{array}\right]&\left[\begin{array}[]{cc}t_{LR}(\uparrow)&0\\\
0&t_{LR}(\downarrow)\end{array}\right]\\\ \\\
\left[\begin{array}[]{cc}t_{RL}(\uparrow)&0\\\
0&t_{RL}(\downarrow)\end{array}\right]&\left[\begin{array}[]{cc}r_{RR}(\uparrow)&0\\\
0&r_{RR}(\downarrow)\end{array}\right]\\\ \end{array}\right)$ (41) . (42)

The normal state scattering matrix $S_{N}(E_{F}+\varepsilon;k_{\parallel})$
characterizes electron (particle) transport for a positive energy
$\varepsilon$ above the Fermi level $E_{F}$. The transport of hole
(antiparticle) is given by the time-reserved symmetric
$S_{N}^{*}(E_{F}-\varepsilon;k_{\parallel})$ scattering matrix for (negative)
energy $-\varepsilon$ below $E_{F}$.

As mentioned above, the dc Josephson current in the junction is obtained from
the Andreev bound states formed in the junction. The spectrum of the Andreev
bound states can be calculated from a scattering matrix formalism Beenakker
(1992, 2004) In such an approach, the spatial separation of Andreev and normal
scattering is the key simplification which allows one to relate the Josephson
current directly to the normal-state scattering matrix of the junction.

We have successfully applied such an approach in a recent paper Ness  _et al._
(2022) to the study of the supercurrent decay and oscillation in magnetic
Josephson junctions made of Ni layers connected to two Nb leads.

##  IV Application

We now provide an illustrative example of our eigenmode approach applied to a
simple case. We consider that all PLs in the $L$-$C$-$R$ junction are
identical, i.e. we study the bulk (equilibrium) transport properties of a
“perfect” crystal. In this case, the transmission probability (at a given
energy $E$ and for a given $k_{\parallel}$ point) is simply given by the
number of “real” bands crossing that energy $E$ at that $k_{\parallel}$ point,
each “real” band corresponding to a propagating mode. We compare the results
obtained from the two Meir and Wingreen Meir and Wingreen (1992) and Fisher
and Lee Fisher and Lee (1981) expressions.

Figure 2 shows the transmission probability $T(E,V;k_{\parallel},\sigma)$
calculated at equilibrium $V{=}0$ and for spin $\uparrow$ of bulk Co. The
unique PL of bulk Co is made of 2 atoms of Co with 9 (_spd_) orbitals, i.e.
the size of the matrices corresponding to that PL is 18$\times$18 per spin.
The calculations performed from the transmission coefficients with only 5
propagating modes provide the very same results as the transmission
probability obtained from Eq.(2), as expected. Note that the maximum
transmission probability is 5, which is indeed the number of propagating modes
we found.

Figure 2: Transmission probability $T(E,V=0;k_{\parallel},\sigma=\uparrow)$
for an energy window around the equilibrium Fermi level $E_{F}=0$ and for
different $k_{\parallel}$ points (corresponding to different colors). The
solid lines are from the Meir and Wingreen expression. The open circles
correspond to the Fisher and Lee formula for which only 5 propagating modes
are taken into account.

We conclude the present paper by the following note: the size ($N\times N$) of
the velocity matrices $v_{\alpha}$ is given by the size of the matrices for
the structure constant $S_{0\alpha}$ and for $g_{b}^{-1}$ in the identical PLs
of the $\alpha=L,R$ leads (this is for each spin and for each value of
$k_{\parallel}$). We have found that the number of propagating modes (with non
zero value of the velocity) is much smaller than $N$. This reduces
considerably the size of the transmission (reflection) coefficient matrices
and hence improve the computational performances.

For bulk Co, we have seen that $N=18$ and only 5 propagating modes are
present. For other examples shown in Ness  _et al._ (2022), we get similar
trends. In the case of Nb(110)/Ni(111)/Nb(110) junctions, each PL of the
$\alpha=L,R$ leads contains 10 atoms of Nb with 9 (_spd_) orbitals, hence
$N=90$. However, there is only a maximum of 25 propagating modes in the
corresponding Nb leads. For Nb(110)/Ni(110)/Nb(110) junctions, there are 2
atoms of Nb in each PL, i.e. $N=18$, and only 7 propagating modes in the
leads. For Nb(110)/Fe(111)/Nb(110), there are 3 atoms of Nb in each PL, i.e.
$N=27$, and only 7 propagating modes in the leads Ness  _et al._ .

###### Acknowledgements.

HN and MvS acknowledge financial support from Microsoft Station Q via a
sponsor agreement between KCL and Microsoft Research. In the late stages of
this work MvS was supported by the U.S. Department of Energy, Ofﬁce of
Science, Basic Energy Sciences under Award # FWP ERW7906. The authors
acknowledge fruitful discussions and collaboration with Roman Lutchyn, Ivan
Sadovskyy and Andrey Antipov leading to Ref. [Ness  _et al._ , 2022].

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# A novel multimodal dynamic fusion network for disfluency detection in spoken
utterances

Disfluency, though originating from human spoken utterances, is primarily
studied as a uni-modal text-based Natural Language Processing (NLP) task.
Based on early-fusion and self-attention-based multimodal interaction between
text and acoustic modalities, in this paper, we propose a novel multimodal
architecture for disfluency detection from individual utterances. Our
architecture leverages a multimodal dynamic fusion network that adds minimal
parameters over an existing text encoder commonly used in prior art to
leverage the prosodic and acoustic cues hidden in speech. Through experiments,
we show that our proposed model achieves state-of-the-art results on the
widely used English Switchboard for disfluency detection and outperforms prior
unimodal and multimodal systems in literature by a significant margin. In
addition, we make a thorough qualitative analysis and show that, unlike text-
only systems, which suffer from spurious correlations in the data, our system
overcomes this problem through additional cues from speech signals. We make
all our codes publicly available on GitHub 111We will release code on paper
acceptance.

Index Terms—  computational paralinguistics, multimodality, disfluency
detection

##  1 Introduction

Unlike written text, in spoken conversations, humans often fail to pre-plan
what they are about to speak. This phenomenon, known as disfluency, is a para-
linguistic concept that is ubiquitous in human conversations. In the past
decade, with the rapid adoption of speech as an input to modern intelligent
Natural Language Understanding (NLU) systems, disfluency detection has become
increasingly popular as an emerging research topic as NLU systems trained on
fluent data can easily get misled due to the presence of disfluencies.

Fig. 1 shows an example of disfluency, whereby it can be divided into 3 main
parts, a reparandum, an optional interregnum, and a repair. Disfluency
detection as a Natural Language Processing (NLP) task generally focuses on
identifying and removing reparandum. Furthermore, disfluencies can primarily
be categorized into 5 types, namely, repetitions, restarts, repairs,
deletions, and substitutions (see Table 1 for examples). Repetition primarily
occurs when linguistic components repeat, usually in the form of partial
words, complete words, or short phrases. Substitution occurs when the same
linguistic components are replaced in order to clarify an underlying concept.
Deletions or false restart refers to abandoned linguistics components.

Fig. 1: A sentence from the English Switchboard corpus with disfluencies. RM=Reparandum, IM=Interregnum, RP=Repitition. The preceding RM is corrected by the following RP.  Table 1: Different types of disfluencies. Type | Example
---|---
Repair | [i do + i] ski yes
Repetition | but [i + i] grew up with cats
Restart | [you were +$\\{$uh$\\}$ ] he was waiting
| for what again
Deletion | [i that it just +] you know it ’s
| absolutely devastating
Substitution | the pen was kept [under + over]
| the table

In the past decade, neural architectures have shown promising results in the
task of disfluency detection, where the majority of prior work primarily
reports results on the SwitchBoard (SWBD) corpus [1]. The Switchboard corpus
consists of telephone conversations between speakers of American English and
is transcribed for fully spontaneous speech, which makes it suitable for this
task. However, a majority of these systems achieving state-of-the-art
performance, leverage the modality of text only to detect reparandums solving
a sequence tagging task in segmented transcripts from the SWBD corpus. Though
the modality of text has rich semantic and syntactic information and
contextualized token representations from transformer models aid systems in
recognizing out-of-linguistic-context words like disfluencies, we acknowledge
the fact that disfluency generally originates from spoken utterances and
acoustic cues like lexical, prosody, pitch, stutter, etc. which are largely
ignored can prove as important signals for disfluency detection. Though some
amount of prior work for this task leverage the modality of speech [2, 3, 4,
5], most of these systems suffer from 2 main problems: 1) They require hand-
engineering of certain acoustic and prosodic features and the performance
depends heavily on the quality of features mined. 2) They rely on a simple
concatenation of text and acoustic features and fail to pay attention to fine-
grained multimodal information, thus making them unable to capture inter-
modality interactions effectively.

Main Contribution: To address these gaps, we propose a novel multimodal
framework MDFN that leverages the modalities of both speech and text as input
without requiring the need for specific hand-engineered features. To achieve
this, we first adopt the use of pre-trained contextualized representation
models for both the text and speech modalities separately, where we make use
of BERT [6] architecture as the text encoder and wav2vec-2.0 [7] as the speech
encoder. Second, to better capture the implicit alignments between individual
text tokens and speech frames, we propose the use of a unique 2 branch
multimodal interaction module (MMI). MMI essentially couples the standard
Transformer layer with a cross-modal attention mechanism to produce a speech-
aware word representation and a word-aware speech representation for each
input word. Next, we concatenate the utterance representation obtained from
both branches to finally classify disfluent tokens in a span classification
setup. Our system achieves state-of-the-art results for disfluency on the SWBD
dataset. Additionally, we also make a thorough qualitative analysis to show
the effectiveness of speech signals in disfluency detection.

##  2 Related Work

Four major categories can be used to categorize disfluency detection models.
The first one uses noisy channel models [8, 9], which call for a transducer in
the channel model based on Tree Adjoining Grammar (TAG). The second group
makes use of phrase structure, which is frequently connected to transition-
based parsing but calls for annotated syntactic structure [10, 11]. The last
category uses end-to-end Encoder-Decoder models to automatically recognize
disfluent segments, while the third and most popular category frames the work
as a sequence tagging problem [12, 13]. In recent work in this area, the
authors of [14, 15] offer a self-supervised learning and data augmentation
technique to learning disfluencies that have been demonstrated to decrease the
gap with supervised training, reducing the dependence of disfluency detection
on human-annotated datasets.

Learning disfluency detection is often framed as a sequence tagging task [14,
15, 12, 16, 10, 17]. As evident from prior work in this domain, most
disfluency models leverage only text transcripts from spoken utterances, with
some exceptions being [2, 3, 4, 5] where the authors use acoustic and prosodic
features from speech together with token-level features from text to predict
disfluent words. Beyond reporting improvements in performance over just using
text, the authors also show how using prosodic features improves the detection
of complex, long, and fake disfluencies. However, these features are hand-
crafted. In recent times, Self-supervised learning (SSL) for speech has proven
to be very effective on a wide range of Spoken Language Processing (SLP) tasks
[18]. Inspired by these findings, we propose to use high-level speech
representations from SSL models for disfluency detection.

Fig. 2: Illustration of the MDFN architecture.

##  3 Methodology

###  3.1 Problem Formulation

Suppose we have a dataset $D$ with $N$ utterances {$u$$1$, $u$$2$, $u$$3$,
$\cdots$, $u$$N$} where each utterance $u_{i}$ has both speech cues $a_{i}$
and text cues $t_{i}$ available where $u_{i}$ $=$ ($a_{i}$,$t_{i}$). $t_{i}$
can be ASR transcripts or human-annotated transcripts. We denote the _i_ -th
sentence with _M_ tokens as _t i_ = {$w$$t$$|$ $t$ = 1, $\cdots$ $T$}. The
corresponding label set is defined by
{$d\textsubscript{1}$,$d\textsubscript{2}$,$d\textsubscript{3}$, $\cdots$
$d$$N$} where $d\textsubscript{i}$ is the label sequence for each sentence and
is denoted by $d\textsubscript{i}$ = {$y$$t$$|$ $t$ = 1, $\cdots$ $M$} where
$y$$t$ $\in\mathcal{Y}$ and $\mathcal{Y}$ $=$ $\\{I,O\\}$. $I$ here stands for
disfluent tokens and $O$ stands for fluent tokens.

###  3.2 Contextualized Representations

Text Representations: Following much of prior art [16, 17], we use BERT _BASE_
[19] from the transformers family as our contextualized text encoder to encode
the transcripts for human-annotated utterances and obtain rich contextualized
token representations. Thus for a total of $M$ tokens, the
$m\textsuperscript{th}$ contextualised embedding for each token in text
transcript $t_{i}$, of utterance $u_{i}$, is denoted by $e_{m}^{t_{i}}$ $\in$
$\mathbb{R}^{768}$.

Speech Representations: We obtain context-aware powerful speech
representations from wav2vec 2.0 [7]. Wav2vec 2.0 is pre-trained using SSL on
unlabeled raw audio. For more details on pre-training and architecture of
wav2vec 2.0, we refer our readers to [7]. Wav2vec 2.0 outputs $J$ hidden
states and we denote the $j\textsuperscript{th}$ hidden state or
contextualized embeddings from the raw audio input $a_{i}$ of utterance
$u_{i}$ as $e_{j}^{a_{i}}$ where $e_{j}^{a_{i}}$ $\in$ $\mathbb{R}^{768}$ .
$J$ depends on the length of the raw audio file and the CNN feature extraction
layer of wav2vec 2.0, which extracts frames from the raw audio with a stride
of 20ms and a hop size of 25ms.

###  3.3 Multimodal Interaction Module (MMI)

Our Multimodal Interaction Module (MMI) consists of 3 Cross-Modal Encoder
(CME) blocks annotated as $A$, $B$, and $C$ in Fig. 2. Each of these 3 CME
blocks is similar to a generic transformer layer [20], where each layer is
composed of an _h_ -head $\mathbf{CMA}$ module [21], residual connections, and
feed-forward layers.

Speech-Aware Word Representations: As shown in Fig.2, to learn token
representations that are aware or knowledgeable of the associated spoken
utterance, we feed wav2vec-2.0 embeddings $\mathbf{A}$ $\in$
$\mathbb{R}^{d\times J}$ as queries and token embeddings $\mathbf{T}$ $\in$
$\mathbb{R}^{d\times M}$ as keys and values into $\mathbf{CMA}$ module of
$\mathbf{CME}$ block $A$ as follows:

$\mathbf{CMA}(\mathbf{A},\mathbf{T})=\operatorname{softmax}\left(\frac{\left[\mathbf{W}_{\mathbf{q}_{\mathbf{i}}}\mathbf{A}\right]^{\top}\left[\mathbf{W}_{\mathbf{k}_{\mathbf{i}}}\mathbf{T}\right]}{\sqrt{d/m}}\right)\left[\mathbf{W}_{\mathbf{v}_{\mathbf{i}}}\mathbf{T}\right]^{\top}$
(1)

where {$\mathbf{W}_{\mathbf{q}_{\mathbf{i}}}$,
$\mathbf{W}_{\mathbf{k}_{\mathbf{i}}}$,
$\mathbf{W}_{\mathbf{v}_{\mathbf{i}}}$} $\in$ $\mathbb{R}^{d/m\times h}$
denote the query, key and value weight matrices respectively for the $i^{th}$
attention head. The final output representation of the $\mathbf{CME}$ block
$B$ is now $\mathbf{P}$ $=$ ($\mathbf{p_{0}}$, $\mathbf{p_{1}}$, $\cdots$,
$\mathbf{p_{m-1}}$). Post this step, to address the fact that each generated
representation $\mathbf{p}_{i}$ in the previous block corresponds to the
$i^{th}$ acoustic embedding and not the token embedding; we feed $\mathbf{P}$
to another $\mathbf{CME}$ block $B$, which treats the original token
embeddings $\mathbf{T}$ as queries and $\mathbf{P}$ as keys and values.
Finally, we now obtain the final Speech-Aware Word Representations as
$\mathbf{R}$ $=$ ($\mathbf{r_{0}}$,$\mathbf{r_{1}}$, $\cdots$,
$\mathbf{r_{j-1}}$).

Word-Aware Speech Representations: Next, to obtain speech representations that
are aware of their corresponding semantic token representations and align each
word to its closely related wav2vec-2.0 embeddings, we make use of another
$\mathbf{CME}$ block $D$ by treating $\mathbf{T}$ as queries and $\mathbf{A}$
as keys and values. The final representations obtained from the block can be
denoted as $\mathbf{Q}$ $=$ ($\mathbf{q_{0}}$, $\mathbf{q_{1}}$, $\cdots$,
$\mathbf{q_{j-1}}$).

Acoustic Gate: In practice, speech frames might encode redundant information
like random noise, and this makes it an important step to implement an
acoustic gate $E$ which can dynamically control the contribution of each
speech frame embedding. Following much work in literature, we implement an
acoustic gate $\mathbf{g}$ as follows:

$\mathbf{g}=\sigma\left(\mathbf{W}_{g}^{\top}[\mathbf{R};\mathbf{Q}]+\mathbf{B}_{g}\right)$
(2)

where $\sigma$ is the element-wise sigmoid function. Finally, based on the
gate output, the final word-aware speech representations are obtained by
$\mathbf{Q}$ = $\mathbf{g}.\mathbf{Q}$.

###  3.4 Span Classification

The final span representations from both branches of the MMI module are then
concatenated to obtain our final cross-modal MMI representations $\mathbf{M}$
$\in$ $\mathbb{R}^{2d}$ where $\mathbf{M}$ = [$\mathbf{Q}$ ; $\mathbf{R}$].
Inspired by [16], for our system, we choose to solve a span-level
classification task over token-level classification. Thus after we obtain
$\mathbf{M}$, we enumerate through all possible $k$ spans $J$ = {$j$1,
$\cdots$ , $j$i, $\cdots$ , $j$$k$} up to a maximum number of consecutive
tokens and then re-assign a label $y_{i}$ $\in$ $\\{I,O\\}$ for each span
$j_{i}$. We then formulate the vectorial representation of each span as the
concatenation of the representations of the starting token
$\mathbf{m}_{b_{i}}$ $\in\mathbb{R}^{2d}$, the ending token
$\mathbf{m}_{s_{i}}$ $\in\mathbb{R}^{2d}$, and a length embedding
$\mathbf{\ell}_{i}$ $\in\mathbb{R}^{len}$.The length embedding is implemented
as a look-up table similar to [16]. The final vector representation for each
span fed into the span prediction layer is now $\mathbf{j}_{i}$ $=$
[$\mathbf{m}_{b_{i}}$ ; $\mathbf{m}_{s_{i}}$ ; $\mathbf{\ell}_{i}$].
$\mathbf{j}_{i}$ is then passed through a linear transformation followed by a
$\operatorname{softmax}$ operation to find the final class among $\\{I,O\\}$
for each utterance. Post this step, we employ the heuristic decoding method
proposed by [16].

##  4 Experiments

###  4.1 Dataset

We evaluate our proposed system on the human-annotated disfluency annotations
[22] for the English Switchboard Dataset [1]. For fair evaluation, following
much of prior-art [23], we split the entire SWBD dataset into training
(sw23$[\star]$.dps), development (sw4[5-9]$[\star]$.dps), and test
(sw4[0-1]$[\star]$.dps) splits. Post this, following [13], we pre-process the
text and convert all all characters to lowercase and remove all punctuation
and partial words.

###  4.2 Experimental Setup

All our models are implemented using the PyTorch deep learning framework. We
use the BERTBASE model as our pre-trained contextualized token embedding model
and the robust wav2vec 2.0LARGE [24] fine-tuned on SWBD as our contextualized
acoustic embedding model. Pre-trained checkpoints and implementations for both
models are adopted from the Huggingface library. We train our MDFN model for
20 epochs with a batch size of 32 using adam optimizer with a learning rate of
$1\times 10^{-5}$. The dimension of our span length embedding
$\mathbb{R}^{len}$ is 100, and the morph embedding is 100.

###  4.3 Baselines and Compared Methods

For comparison to prior art, we resort to both unimodal ad multimodal sequence
tagging baselines evaluated under the _IO_ tagging scheme. For our text-only
unimodal baseline, we resort to a simple span classifier BERT system from the
transformers family also evaluated on the _IO_ tagging scheme.

##  5 Results

Table 2 shows the evaluation results on the SWBD test set compared to the
prior art on disfluency detection. Following much of prior art, we evaluate
the performance of our proposed MDFN model on _$F_{1}$_. As we clearly see,
the our proposed model outperforms the current state-of-the-art (SOTA) system
[16] in $F_{1}$ scores by 1.5%. With a slightly higher recall, our model is
more sensitive toward the positive class but nevertheless maintains high
precision and hence a high $F_{1}$ score. Additionally, in the next section,
we also make an effort to analyze the benefits of using multimodal interaction
information in our span classifier model.

Table 2: Evaluation results of our proposed model compared to the baselines and prior art on the Switchboard test set. The best scores are denoted in bold. Model | $\mathrm{P}$ | $\mathrm{R}$ | $\mathrm{F\textsubscript{1}}$
---|---|---|---
Semi-CRF [12] | $90.0$ | $81.2$ | $85.4$
Bi-LSTM [25] | $91.6$ | $80.3$ | $85.9$
Attention-based [26] | $91.6$ | $82.3$ | $86.7$
Transition-based [27] | $91.1$ | $84.1$ | $87.5$
Self-supervised [15] | $93.4$ | $87.3$ | $90.2$
Self-trained [10] | $87.5$ | $93.8$ | $90.6$
EGBC [28] | $95.7$ | $88.3$ | $91.8$
BERT fine-tune [28] | $94.7$ | $89.8$ | $92.2$
BERT-CRF-Aux [17] | $94.6$ | $91.2$ | $92.9$
ELECTRA-CRF-Aux [17] | $94.8$ | $91.6$ | $93.1$
Span Classification BERT-GCN[16] | $\mathbf{95.2}$ | $93.2$ | $94.2$
BERT (Baseline) | $95.1$ | $93$ | $94.1$
MDFN (ours) | $92.8$ | $\mathbf{98.7}$ | $\mathbf{95.7}$

##  6 Result Analysis

MDFN vs. Span Classification BERT-GCN: Highlighted words below represent the
predictions made by the state-of-the-art text-only _Span Classification BERT-
GCN_ (SCBG) model [16] from literature. In contrast, our MDFN model predicted
that the sentences are fluent, which is in line with the ground truth
annotations.

  1. 1.

i finally got impaneled on one case on my next to the last day

  2. 2.

and that is that money tends to stick where it lands first

As we see in both examples, the spans classified as disfluent by the SCBG are
not disfluencies according to ground truth and our model. We hypothesize that
although [16] captures and repairs the longer restarts, it wrongly classifies
spoken utterances which contain nuances like incorrect grammar and usage of
multiple conjunctions, which makes the structure of the sentence complex and
seemingly disfluent. For example “on” and “that” appears twice in immediate
succession in sentences 1 and 2, respectively. Thus we can conclude that text-
only [16] gets biased to such spurious correlations, which makes it identify
these words as repetitions. In such cases, the acoustic modality helps capture
the true nature of the sentence with the help of the confidence and tone of
the speaker in audio.

Highlighted words below represent the predictions made by our MDFN model. In
contrast, the SCBG model predicted the sentences as fluent, which is in line
with the ground truth annotations.

  1. 1.

end it ’s very easy

  2. 2.

any so how do you get most of your news

In the switchboard dataset, there are multiple instances of segmented
sentences which occur due to cut-offs or speaker overlap. Due to this, the
initial or end utterances can be wrongly classified as disfluent, as the MDFN
model doesn’t use linguistic rules to find the relation among words. These are
captured by parse trees used by the SCBG model, as seen in the above examples.

##  7 Conclusion

In this paper, we present a novel multimodal dynamic fusion network that
effectively utilizes both the modalities of speech and text for disfluency
detection. Our proposed system achieves state-of-the-art results in disfluency
detection on the very popular SWBD dataset. As part of future work, we would
like to propose better multimodal networks for disfluency detection that
better capture the implicit relations between the two modalities.
Additionally, we would also like to perform a human evaluation of our model
results for ASR-generated transcripts to evaluate the effectiveness of our
model over unimodal text approaches in such a setting.

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