\section{Introduction} \label{intro}

According to the Standard Model of particle physics and to some of its extensions, spontaneous symmetry breaking and phase transitions constitute a crucial aspect for the early evolution of the universe.

When the temperature starts decreasing because of the expansion of the universe, spontaneous symmetry breaking can be triggered so that the interactions among elementary particles undergo (dis)continuous jumps from one phase to another. New phases with a broken symmetry will form in many regions at the same time, and in each of them only one single vacuum state will be spontaneously chosen. Sufficiently separated spatial regions may not be in causal contact, so that it is quite natural to assume that the early universe is divided into many causally disconnected patches whose size is roughly given by the Hubble radius\footnote{The Hubble radius is given by $R_{\rm H}\sim H^{-1}=a(t)/\dot{a}(t)$ where $a(t)$ is the scale factor depending on the cosmic time $t$. More precisely speaking, the vacuum is chosen over the region with the correlation length of the fields at that time.}, and in each of which the vacuum is independently determined.
As the universe expands, it can eventually happen that patches with different vacua collide in such a way that boundaries begin to form between adjacent regions with a different vacuum state.
Since the field associated with the spontaneous breaking has to vary in a continuous way between different vacua, it must interpolate smoothly from one vacuum to another via the hill of the potential. This implies that finite-energy field configurations must form at the boundaries separating patches with different vacua, and must persist even after the phase transition is completed. These objects are called \textit{topological defects}~\cite{Vilenkin:2000jqa}, and their formation mechanism (in a cosmological context) is known as \textit{Kibble mechanism}~\cite{Kibble:1976sj,Kibble:1980mv}.

Topological defects can be of several type and different spatial dimensions, and their existence is in one-to-one correspondence with the topology of the vacuum manifold~\cite{Vilenkin:2000jqa}. Domain walls are two-dimensional objects that form when a discrete symmetry is broken, so that the associated vacuum manifold is disconnected. Strings are one-dimensional objects associated to a symmetry breaking whose corresponding vacuum manifold is not simply-connected, and their formation could be predicted both by some extensions of the Standard Model of particle physics and by some classes of Grand Unified Theory (GUT). Monopoles are zero-dimensional objects whose existence is ensured when the vacuum manifold is characterized by non-contractible two-spheres, and they constitute an inevitable prediction of GUT. Moreover, there exist other topological objects called textures that can form when larger groups are broken and whose vacuum-manifold topology is more complicated.

Since the existence of topological defects is intrinsically related to the particular topology of the vacuum manifold, they can naturally appear in several theories beyond the Standard Model that predict a spontaneous symmetry breaking at some high-energy scale.
For instance, the spontaneous breaking of $SU(5)$ symmetry in GUT leads to the formation of various topological defects.
Therefore, observations and phenomenology of topological defects are very important, and should be considered as unique test-benches to test and constrain theories of particle physics and of the early universe. This also means that for any alternative theory - e.g. that aims at giving a complete ultraviolet (UV) description of the fundamental interactions - it is worth studying the existence of topological defects, investigating how their properties differ with the respect to other models/theories, and putting them to the test with current and future experiments.

In this paper we discuss for the first time topological defects in the context of \textit{nonlocal field theories} in which the Lagrangians contain infinite-order differential operators. In particular, we will make a very detailed analysis of domain wall solutions.
The type of differential operator that we will consider do not lead to ghost degrees of freedom in the particle spectrum despite the presence of higher-order time derivatives in the Lagrangian.

The work is organized as follows:
\begin{enumerate}

\item[\textbf{Sec.~\ref{sec:nonlocal-review}:}] we introduce nonlocal field theories by discussing the underlying motivations and their main properties.

\item[\textbf{Sec.~\ref{sec-review}:}] we briefly review the domain wall solution in the context of standard (local) two-derivative theories by highlighting various features whose mathematical and physical meanings will be important for the subsequent sections.

\item[\textbf{Sec.~\ref{sec-nft-dw}:}] we analyze for the first time domain wall solutions in the context of ghost-free nonlocal field theories by focusing on the simplest choice for the infinite-order differential operator in the Lagrangian. Despite the high complexity of non-linear and infinite-order differential equations, we will be able to find an approximate analytic solution by relying on the fact that the topological structure of the vacuum manifold ensures the existence of an exact domain wall configuration.
Firstly, we analytically study the asymptotic behavior of the solution close to the two symmetric vacua.
Secondly, we find a linearized nonlocal solution by perturbing around the local domain wall configuration. We show that the linearized treatment agrees with the asymptotic analysis, and make remarks on the peculiar behavior close to the origin. We perform an order-of-magnitude estimation of width and energy per unit area of the domain wall. Furthermore, we derive a theoretical lower bound on the scale of nonlocality for the specific domain wall configuration under investigation.

\item[\textbf{Sec.~\ref{sec-other}:}] we briefly comment on other topological defects like string and monopole.

\item[\textbf{Sec.~\ref{sec-dis}:}] we summarize our results, and discuss both theoretical and phenomenological future tasks.

\item[\textbf{App.~\ref{sec-corr}:}] we develop a formalism to confirm the validity of the linearized solution close to the origin.

\item[\textbf{App.~\ref{app-em}:}] we find a compact expression for the energy-momentum tensor in a generic nonlocal (infinite-derivative) field theory, and apply it to the specific nonlocal scalar model analyzed in the main text.

\end{enumerate}

We adopt the mostly positive convention for the metric signature, $\eta=\diag(-,+,+,+),$ and work with the natural units system, $c=\hbar=1.$

\section{Nonlocal field theories}\label{sec:nonlocal-review}

The wording `nonlocal theories' is quite generic and, in principle, can refer to very different theories due to the fact that the nonlocal nature of fields can manifest in various ways. In this work with `nonlocality' we specifically mean that Lagrangians are made up of certain non-polynomial differential operators containing infinite-order derivatives.

A generic nonlocal Lagrangian contains both polynomial and non-polynomial differential operators, i.e. given a field $\phi(x)$ one can have
\begin{equation}
\mathcal{L}\equiv\mathcal{L}\left(\phi,\partial\phi,\partial^2\phi,\dots,\partial^n\phi,\frac{1}{\Box}\phi,{\rm ln}\left(- \Box/M_s^2\right)\phi,e^{\Box/M_s^2}\phi,\dots\right),\label{nonlocal-lagr}
\end{equation}
where $\Box=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$ is the flat d'Alambertian and $M_s$ is the energy scale at which nonlocal effects are expected to become important.
Non-analytic differential operators like $1/\Box$ and ${\rm log}(-\Box)$ are usually important at infrared (IR) scales, e.g. they can appear as contributions in the finite part of the quantum effective action in standard perturbative quantum field theories~\cite{Barvinsky:2014lja,Woodard:2018gfj}. Whereas analytic operators like $e^{\Box/M_s^2}$ are usually responsible for UV modifications and do not affect the IR physics. Such a transcendental differential operator typically appears in the context of string field theory~\cite{Witten:1985cc,Gross:1987kza,Eliezer:1989cr,Tseytlin:1995uq,Siegel:2003vt,Pius:2016jsl,Erler:2020beb} and p-adic string~\cite{Freund:1987kt,Brekke:1988dg,Freund:1987ck,Frampton:1988kr,Dragovich:2020uwa}.

We are interested in alternative theories that extend the Standard Model in the UV regime, therefore we will focus on analytic differential operators. In general, we can consider a scalar Lagrangian of the following type\footnote{To keep the formula simpler we do not write the scale of nonlocality in the argument of $F(-\Box)$ which, to be more precise, should read $F(-\Box/M_s^2).$}:
\begin{equation}
\mathcal{L}=-\frac{1}{2}\phi F(-\Box)\phi - V(\phi)\,,\label{analytic-Lag}
\end{equation}
where $V(\phi)$ is a potential term, and the the kinetic operator can be defined through its Taylor expansion
\begin{equation}
F(-\Box)=\sum\limits_{n=0}^\infty f_n(-\Box)^n\,,\label{nl-oper}
\end{equation}
where $f_n$ are constant coefficients. It should now be clear that the type of nonlocality under investigation manifests through the presence of infinte-order derivatives.

To recover the correct low-energy limit and avoid IR modifications, it is sufficient to require that the function $F(z),$ with $z\in \mathbb{C},$ does not contain any poles in the complex plane. Thus, we choose $F(-\Box)$ to be an \textit{entire function} of the d'Alembertian $\Box$.

By making use of the Weierstrass factorization theorem for entire functions we can write
\begin{equation}
F(-\Box)=e^{\gamma(-\Box)}\prod\limits_{i=1}^{N}(-\Box+m_i^2)^{r_i}\,,
\end{equation}
where $\gamma(-\Box)$ is another entire function, $m_i^2$ are the zeroes of the kinetic operator $F(-\Box),$ and $r_i$ is the multiplicity of the $i$-th zero. The integer $N\geq 0$ counts the number of zeroes and, in general, can be either finite or infinite.

To prevent the appearance of ghost degrees of freedom, it is sufficient to exclude the possibility to have extra zeroes besides the standard two-derivative one\footnote{It is worth mentioning that this is \textit{not} the unique possibility for ghost-free higher derivative theories. In fact, we can allow additional pairs of complex conjugate poles and still avoid ghost degrees of freedom and respect unitarity, in both local~\cite{Modesto:2015ozb,Modesto:2016ofr,Anselmi:2017yux,Anselmi:2017lia} and nonlocal theories~\cite{Buoninfante:2018lnh,Buoninfante:2020ctr}. Moreover, tree-level unitarity was shown to be satisfied also if one admits branch cuts in the bare propagator~\cite{Abel:2019ufz,Abel:2019zou}.}. We impose that the kinetic operator does not contain any additional zeroes, so that the effects induced by new physics are entirely captured by the differential operator $e^{\gamma(-\Box)}.$ Therefore, we consider the following Lagrangian:
\begin{equation}
\mathcal{L}=\frac{1}{2}\phi\, e^{\gamma(-\Box)}(\Box-m^2)\,\phi -V(\phi)\,,\label{exp-nonlocal}
\end{equation}
whose propagator reads
\begin{equation}
\Pi(p^2)=-i\frac{e^{-\gamma(p^2)}}{p^2+m^2}\,.\label{propag-nonlocal}
\end{equation}
From the last equation it is evident that no additional pole appears other than $p^2=-m^2,$ because $e^{-\gamma(p^2)}$ is an exponential of an entire function and as such does not have poles in the complex plane.

More generally, under the assumption that the transcendental function $e^{-\gamma(p^2)}$ is convergent in the limits $p^0\rightarrow\pm i\infty,$ it was shown that an S-matrix can be well defined for the Lagrangian~\eqref{exp-nonlocal}, and it can be proven to satisfy perturbative unitarity at any order in loop~\cite{Pius:2016jsl,Briscese:2018oyx,Chin:2018puw,Koshelev:2021orf}. Moreover, the presence of the exponential function can make loop-integrals convergent so that the scalar theory in Eq.~\eqref{exp-nonlocal} turns out to be finite in the high-energy regime~\cite{Krasnikov:1987yj,Moffat:1990jj,Tomboulis:2015gfa,Buoninfante:2018mre}. Very interestingly for such nonlocal theories, despite the presence of infinite-order time derivatives, a consistent initial value problem can be formulated in terms of a finite number of initial conditions~\cite{Barnaby:2007ve,Calcagni:2018lyd,Erbin:2021hkf}.

This type of transcendental operators with some entire function $\gamma(-\Box)$ have been intensely studied in the past years not only in the context of quantum field theories in flat space~\cite{Krasnikov:1987yj,Moffat:1990jj,Tomboulis:2015gfa,Buoninfante:2018mre,Boos:2018kir,Boos:2019fbu}, but also to formulate ghost-free infinite-derivative theories of gravity~\cite{Krasnikov:1987yj,Kuzmin:1989sp,Tomboulis:1997gg,Biswas:2005qr,Modesto:2011kw,Biswas:2011ar,Frolov:2015bia,Frolov:2015bta,Buoninfante:2018xiw,Buoninfante:2018stt,Buoninfante:2018xif,Koshelev:2016xqb,Koshelev:2017tvv,Koshelev:2020foq,Kolar:2020max}.

In this work we assume that fundamental interactions are intrinsically nonlocal, and that nonlocality becomes relevant in the UV regime. Thus, we consider nonlocal quantum field theories as possible candidates for UV-complete theories beyond the Standard Model. With this in mind, we will analyze topological defects in infinite-derivative field theories, and investigate the physical implications induced by nonlocality in comparison to standard (local) two-derivative theories.

In what follows, we work with the simplest ghost-free nonlocal model for which the entire function is given by
\begin{equation}
\gamma(-\Box)=-\frac{\Box}{M_s^2}\,.\label{entire-box^1}
\end{equation}

\section{Standard domain wall: a brief review} \label{sec-review}

Before discussing domain walls in the context of nonlocal quantum field theories, it is worth reminding some of their basic properties in standard two-derivative field theories, which will then be useful for the main part of this work.

In presence of a domain wall one has to deal with a static scalar field that only depends on one spatial coordinate, e.g. $x,$ and whose Lagrangian reads~\cite{Vilenkin:2000jqa,Saikawa:2017hiv}
\begin{align}
	{\cal L} = \frac{1}{2}\qty(\partial_x\phi)^2 -U(\phi)\,,\qquad U(\phi)=\frac{\lambda}{4}(\phi^2-v^2)^2\,,\label{local-Lag-wall}
\end{align}
which is $\mathbb{Z}_2$-symmetric as it is invariant under the transformation $\phi\rightarrow-\phi;$ $\lambda>0$ is a dimensionless coupling constant and $v>0$ is related to the symmetry-breaking energy scale. The quartic potential has two degenerate minima at $\phi=\pm v$ ($U(\pm v)=0$).

As mentioned in the Introduction, the discrete symmetry $\mathbb{Z}_2$ can be spontaneously  broken, for instance, in the early universe because of thermal effects. As a consequence, causally disconnected regions of the universe can be characterized by a different choice of the vacuum (i.e. $\phi=+v$ or $\phi=-v$), and when two regions with different vacua collide a continuous two-dimensional object -- called domain wall -- must form at the boundary of these two regions.

Let us now determine explicitly such a finite-energy configuration interpolating $\pm v.$
First of all, we impose the asymptotic boundary conditions
\begin{align}
	\phi(-\infty) = -v\,,
	\qquad
	\phi(\infty) = v\,.\label{boundary conditions}
\end{align}
The field configuration must be non-singular and of finite energy, therefore $\phi(x)$ must interpolate smoothly between the two vacua, this implies that there exists a point $x_0\in \mathbb{R}$ such that $\phi(x_0)=0.$ Without any loss of generality, we can choose the reference frame such that the centre of the wall is at the origin $x_0=0,$ i.e. $\phi(0)=0.$

The energy density can be computed as
\begin{align}
	\mathcal{E}(x) \equiv T_{0}^0(x)= \frac{1}{2}(\del_x\phi)^2 + \frac{\lambda}{4}(\phi^2-v^2)^2\,,\label{local-e-dens}
\end{align}
from which it follows  $\mathcal{E}(x)\geq U(0)=\lambda v^2/4,$ and this implies that  there exists a solution that does not dissipate at infinity. Hence, the topological structure of the vacuum manifold -- which is disconnected in the case of $\mathbb{Z}_2$ symmetry -- ensures the existence of a non-trivial field configuration of finite energy.

We can determine qualitatively the behavior of this field configuration by making an order-of-magnitude estimation of the width $R$ (along the $x$-direction), and of the energy per unit area $E$ of the wall.

In fact, we can define the width of the wall in three ways. The first one is to use the energy density in Eq.~\eqref{local-e-dens}. The lowest energy configuration interpolating the two vacua can be found by balancing the kinetic and potential term in the energy density $\mathcal{E}(x)$.

By approximating the gradient with the inverse of the width, $\partial_x\sim 1/R,$ and the field value with $\phi\sim v,$ Eq.~\eqref{local-e-dens} gives
\begin{align}
	\frac{1}{2} \frac{1}{R^2} v^2 \sim \frac{\lambda}{4} v^4\quad
		\Rightarrow\quad
	R \sim \sqrt{\frac{2}{\lambda}} \frac{1}{v}\,,\label{local-radius-estim}
\end{align}
from which it follows that the width of the wall is of the same order of the Compton wavelength $R\sim(\sqrt{\lambda}v)^{-1}\sim m^{-1}$.
Whereas, the energy per unit area can be estimated as
\begin{eqnarray}
	E&=& \int_\mathbb{R}{{\rm d}x}\qty[\frac{1}{2}(\del_x\phi)^2+\frac{\lambda}{4}(\phi^2-v^2)^2] \nonumber\\[2mm]
	 &\sim& (\text{width of the wall}) \times (\text{energy density}) \nonumber\\[2mm]
		& \sim& R \times \lambda v^4 \nonumber\\[2mm]
		&\sim&\sqrt{\lambda}v^3\,.\label{local-energ-estim}
\end{eqnarray}
The other two ways are to use the exact configuration (solution) of the domain wall.
The field equation
\begin{align}
	\partial_x^2 \phi(x) = \lambda \phi(\phi^2-v^2)\label{local-field-eq}
\end{align}
can be solved by quadrature, and an exact analytic solution can be found, and it satisfies all the qualitative properties discussed above.
The exact solution is sometime called `kink', and it reads~\cite{Vilenkin:2000jqa}
\begin{align}
	\phi(x) = v \tanh \left(\sqrt{\frac{\lambda}{2}}vx\right)\,.\label{local-dom-wal-sol}
\end{align}
Its asymptotic behavior is given by
\begin{align}
|x|\rightarrow \infty\quad \Rightarrow \quad \phi(x)\sim \pm v\qty(1-2e^{-\sqrt{2\lambda}vx})\,.
		\label{eq:asymp sol in local case}
\end{align}
Through the exact solution~\eqref{local-dom-wal-sol}, we can define the width of the wall in two ways. One way is to identify it with the typical length scale over which $\phi(x)$ changes in proximity of the origin, that is, the length scale $\ell$ defined as the inverse of the gradient at the origin, i.e. $\ell\sim v/(\partial_x\phi|_{x=0}),$ where the scale $v$ is introduced for dimensional reasons. From Eq.~\eqref{local-dom-wal-sol} we have
\begin{equation}
\del_x\phi(x)|_{x=0}=v^2\sqrt{\frac{\lambda}{2}},
\end{equation}
which yields
\begin{equation}
\ell\sim \sqrt{\frac{2}{\lambda}}\frac{1}{v} = R\,.\label{ell-linearized-local}
\end{equation}
The last way is to use the asymptotic behavior given in Eq.~\eqref{eq:asymp sol in local case}. The width of the wall, $\widetilde{R}$, can be defined as
\begin{align}
|x|\rightarrow \infty\quad \Rightarrow \quad \phi(x)\sim \pm v\qty(1-2e^{-\frac{2x}{\widetilde{R}}})\,,
		\label{eq:asymp radius}
\end{align}
which yields $\widetilde{R} \sim \sqrt{\frac{2}{\lambda}} \frac{1}{v} = R = \ell\,.$ In the local case, all of the three definitions give the same expressions and we need not discriminate them. But, as we will show, in the nonlocal case, all of the definitions would give different expressions in the sub-leading order, though two of them ($R$ and $\widetilde{R}$) have the similar feature. In fact, $R$ and/or $\widetilde{R}$ might be more appropriate as the definition of the width (or the radius) of a domain wall because $\ell$ is related to the behavior of the solution close to the origin and far from the vacuum.

We can obtain the energy per unit area as
\begin{eqnarray}
       E = \int_{\mathbb{R}} {\rm d}x\mathcal{E}(x) =\int_{\mathbb{R}} {\rm d}x\qty(\frac{{\rm d}\phi}{{\rm d}x})^2
		= \frac{4}{3} \sqrt{\frac{\lambda}{2}}v^3\,,
\end{eqnarray}
which is consistent with the estimation in Eq.~\eqref{local-energ-estim} up to an order-one numerical factor.

The discussion in this Section was performed for a local two-derivative theory in one spatial dimension. However, the essential concepts and methods, like the condition for the existence of a solution related to the non-trivial topology of the vacuum manifold, and the order-of-magnitude estimations, can be applied to nonlocal field theories and to higher dimensional cases (e.g. string and monopole).

\section{Domain wall in nonlocal field theories} \label{sec-nft-dw}

In this Section we analyze the domain wall solution for the nonlocal field theory~\eqref{exp-nonlocal} with the simplest choice of entire function given in~\eqref{entire-box^1}. Hence, we consider a nonlocal generalization of the Lagrangian~\eqref{eq:Lagrangian for DW} given by
\begin{align}
	{\cal L}
		= \frac{1}{2}\phi e^{-\partial_x^2/M_s^2}(\partial_x^2+\lambda v^2)\phi
				-\frac{\lambda}{4}\qty(\phi^4+v^4)\,,
	\label{eq:Lagrangian for DW}
\end{align}
whose field equation reads
\begin{align}
	e^{-\partial_x^2/M_s^2}(\partial_x^2+\lambda v^2)\phi = \lambda\phi^3\,.
		\label{eq:the equation of the motion for DW}
\end{align}
We can easily verify that in the limit $\partial_x^2/M_s^2\rightarrow 0$ we consistently recover the local two-derivative case, i.e. Eqs.~\eqref{local-Lag-wall} and~\eqref{local-field-eq}.

In this case the differential equation is non-linear and highly nonlocal, and finding a solution seems to be very difficult not only analytically but even numerically. However, despite the complexity of the scalar field equation, we can still find a domain wall configuration. First of all, the existence of such a solution is still guaranteed by the non-trivial topological structure of the vacuum manifold which is disconnected in the case of $\mathbb{Z}_2$ symmetry. Indeed, also in this case a finite-energy field configuration that smoothly interpolates between the two vacua $\phi=\pm v$ must exist. Furthermore, by relying on the fact that the presence of the exponential operator $e^{-\partial_x^2/M_s^2}$ should not change the number of degrees of freedom and of initial conditions~\cite{Barnaby:2007ve}, we can impose the same boundary conditions as done in the two-derivative case in Eq.~\eqref{boundary conditions}, i.e. $\phi(\pm \infty)=\pm v.$ We can set $x_0=0$ to be the centre of the wall such that $\phi(0)=0,$ and expect $\phi(x)$ to be a smooth odd function.

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.6]{schematic-illustration.jpg}
	\caption{Schematic illustration of the analysis made in this Section to study the domain wall configuration in nonlocal field theory, whose qualitative behavior is drawn consistently with the boundary conditions $\phi(\pm\infty)=\pm v$, and with the choice of the origin $\phi(0)=0.$ We analyze the behavior of the domain wall configuration in several regimes, and use different methods to find approximate analytic solutions. (i) We study the asymptotic behavior of the domain wall solution in the limit $|x|\rightarrow \infty.$ (ii) We find a linearized nonlocal solution by perturbing around the local domain wall configuration treated as a background, and analyse its behavior not only at infinity but also close to the origin. (iii) We make an order-of-magnitude estimation for the width and the energy per unit area of the wall, and verify the consistency with the analytic approximate solutions.}
	\label{fig:conceptual figure for analysis of nonlocal domain wall}
\end{figure}

Very interestingly, knowing that a domain wall solution must exist, and equipped with a set of boundary conditions, we can still find an approximate analytic solution by working perturbatively in some regime. We will proceed as follows. In Sec.~\ref{subsec-asym} we will determine the solution asymptotically close to the vacua $\pm v$ (i.e. for $|x|\rightarrow \infty$). In Sec.~\ref{subsec-pert} we will find a linearized nonlocal solution by perturbing around the known local `kink' configuration treated as a background. Finally, in Sec.~\ref{subsec-order} we will make an order-of-magnitude estimation for the width and the energy of the nonlocal domain wall, and check the consistency with the approximate analytic solutions. See Fig.~\ref{fig:conceptual figure for analysis of nonlocal domain wall} for a schematic illustration of our analysis.

\subsection{Asymptotic solution for $|x|\rightarrow \infty$} \label{subsec-asym}

As a first step we analyze the asymptotic behavior of the solution close to the two vacua $\phi=\pm v$, i.e. in the regime $|x|\rightarrow \infty.$ Let us first consider the perturbation around $\phi=+v,$ i.e. we write
\begin{equation}
    \phi(x)=v+\delta\phi(x)\,,\qquad \frac{|\delta\phi|}{v}\ll 1\,,
\end{equation}
so that the linearized field equation reads
\begin{align}
	e^{-\del_x^2/M_s^2} (\del_x^2+\lambda v^2) \delta\phi = 3\lambda v^2 \delta\phi\,.
\end{align}
Taking inspiration from the asymptotic behavior of the domain wall in the local case (see Eq.~\eqref{eq:asymp sol in local case}), as an ansatz we assume that $\phi(x)$ approaches the vacuum exponentially, i.e. we take
\begin{align}
	\delta\phi = \phi - v = A e^{-B x} \label{eq:def for asymp sol}
\end{align}
where $A,$ and $B>0$ are two constants.

Since the exponential is an eigenfunction of the kinetic operator, we can easily obtain an equation for $B,$
\begin{align}
	e^{-B^2/M_s^2} (B^2+\lambda v^2)  = 3\lambda v^2\,.\label{eq-for-B}
\end{align}
By using the principal branch $W_0(x)$ of the Lambert-W function (defined as the inverse function of $f(x)=xe^x$) we can solve Eq.~\eqref{eq-for-B} as follows
\begin{align}
	B^2 = - M_s^2\, W_0\qty(-\frac{3\lambda v^2}{M_s^2}e^{-\lambda v^2/M_s^2})-\lambda v^2\,.
		\label{eq:coeff B in asymp sol}
\end{align}
Before continuing let us make some remarks on the Lambert-W function. It is a multivalued function that has an infinite number of branches $W_n$ with $n\in \mathbb{Z}.$ The only real solutions are given by the branch $W_0(x)$ for $x\geq -1/e,$ and an additional real solution comes from the branch $W_{-1}(x)$ for $-1/e\leq x<0.$ In the equations above we have taken the so-called principal branch $W_0,$ and we will do the same in the rest of the paper. However, we will also comment on the branch $W_{-1}$ and the physical implications associated to it. Regarding higher order branches $n>0,$ they will generate non-physical complex values, and in some cases they do not even recover the local limit; therefore, we discard such solutions.

Note that, by means of the asymptotic analysis above the coefficient $A$ cannot be determined as it factors out from the field equation but, as we will explain in Sec.~\ref{subsec-pert}, we will be able to determine it up to order $\mathcal{O}(1/M_s^2)$.

\subsubsection{A theoretical constraint on the scale of nonlocality} \label{subsubsec-cond}

The use of the Lambert-W function to obtain the solution for $B$ in Eq.~\eqref{eq:coeff B in asymp sol} relied on the fact that  Eq.~\eqref{eq-for-B} could be inverted. As explained above this inversion is valid if and only if $c \in \{xe^x|x\in\mathbb{R}\}$, i.e. if $c\geq -1/e$.

Applying this condition to~\eqref{eq:coeff B in asymp sol}, we get
\begin{align}
	-\frac{3\lambda v^2}{M_s^2}e^{-\lambda v^2/M_s^2}\geq -\frac{1}{e}\,,\label{condition}
\end{align}
and inverting in terms of the (principal branch) Lambert-W function we obtain the following \textit{theoretical constraint}:
\begin{align}
M_s^2\geq  -\frac{\lambda v^2}{W_0\qty(-1/3e)}\,,	\label{eq:neccessary cond for asymp sol}
\end{align}
where we have used the fact that $W_0(x)$ is a monotonically increasing function. The inequality~\eqref{eq:neccessary cond for asymp sol} means that
the energy scale of nonlocality must be greater than the symmetry-breaking scale $\sqrt{\lambda} v.$ We can evaluate $W_0\qty(-1/3e)\simeq -0.14,$  so that the lower bound reads $M_s^2\gtrsim 7.14 \lambda v^2.$
One usually obtains constraints on the free parameters of a theory by using experimental data. In the present work, instead, we found a purely theoretical constraint. See Sec.~\ref{sec-dis} for further discussions on this feature.

Given the fact that $\lambda v^2/M_s^2< 0.14,$ we can expand~\eqref{eq:coeff B in asymp sol} and obtain
\begin{align}
	B^2
		= 2 \lambda v^2 \qty(1+\frac{3\lambda v^2}{M_s^2})
			+ {\cal O}\qty(\qty(\frac{\lambda v^2}{M_s^2})^2)\,,
	\label{eq:series expansion for coeff B}
\end{align}
or by taking the square root,
\begin{align}
	B
		= \sqrt{2 \lambda} v \qty(1+\frac{3}{2}\frac{\lambda v^2}{M_s^2})
			+ {\cal O}\qty(\qty(\frac{\lambda v^2}{M_s^2})^2)\,\,.
	\label{eq:series expansion for coeff B-sqrt}
\end{align}
From this expression, we can obtain the width of wall, $\widetilde{R}$, defined in Eq.~\eqref{eq:asymp radius} as
\begin{equation}
\widetilde{R} \sim \frac{2}{B}\sim \sqrt{\frac{2}{\lambda}}\frac{1}{v}\left(1-\frac32\frac{\lambda v^2}{M_s^2}\right)\,.\label{eq:tildeR}
\end{equation}
Also, from Eq.~\eqref{eq:series expansion for coeff B-sqrt} we can check that in the local limit $M_s\rightarrow \infty$ we recover the two-derivative case in Eq.~\eqref{eq:asymp sol in local case}:
\begin{align}
	\lim_{M_s\to\infty} B^2 = 2 \lambda v^2 = \qty(\sqrt{2\lambda}v)^2\equiv B_{\rm L}^2\,.
\end{align}
Furthermore, from~\eqref{eq:series expansion for coeff B} we can notice that $B\geq B_{\rm L}=\qty(\sqrt{2\lambda}v).$ This physically means that the nonlocal domain wall solution approaches the vacuum $\phi=+v$ \textit{faster} as compared to the local two-derivative case. This feature, which is manifest in Eq.~\eqref{eq:tildeR}, may also suggest that the width of the nonlocal domain wall is \textit{smaller} as compared to the local case; indeed this fact will also be observed with the expression of $R$ in the next Subsections.

So far we have only focused on the asymptotic solution for $x\rightarrow +\infty$ ($\phi(+\infty)=+v$) but the same analysis can be applied to the other asymptotic $x\rightarrow -\infty$ ($\phi(-\infty)=-v$), and the same results hold because of the $\mathbb{Z}_2$ symmetry.

\paragraph{Remark 1.} Before concluding this Subsection it is worth commenting on the validity of the asymptotic solution we determined. On one hand, we know that the existence of the domain wall is ensured by the topological structure of the vacuum manifold, and this should not depend on the value of $M_s.$ On the other hand, it appears that the asymptotic solution we found is only valid for some values of $M_s$ satisfying the inequality in Eq.~\eqref{eq:neccessary cond for asymp sol}, which seems to imply that the domain wall solution does not exist for other values of $M_s$. Is this a contradiction? The answer is no, all is consistent, and in fact the domain wall solution exists for any value of $M_s.$

First of all, we should note that to solve Eq.~\eqref{condition} in terms of $M_s$ we have used the principal branch $W_0(x)$ which is a monotonic increasing function, but an additional real solution can be found by using the branch $W_{-1}(x)$ which is, instead, a monotonically decreasing function. Thus, given the opposite monotonicity behavior of $W_{-1}$ as compared to $W_{0},$ if we solve Eq.~\eqref{condition} by means of $W_{-1}$ we get $M_s\leq -\lambda v^2/W_{-1}(-1/3e)\simeq 0.30\lambda v^2.$
Moreover, in the range of values $0.30 \lambda v^2 \lesssim M_s^2\lesssim 7.21 \lambda v^2$ the functional form in Eq.~\eqref{eq:def for asymp sol} does not represent a valid asymptotic behavior for the domain wall. In this case the domain wall configuration may be characterized by a completely different profile, but its existence is still guaranteed by the non-trivial topology.
Anyway, as already mentioned above, in this paper we only work with $W_0,$ therefore with values of $M_s$ satisfying the inequality~\eqref{eq:neccessary cond for asymp sol}. See also Sec.~\ref{sec-dis} for more discussion on this in relation to physical implications.

\subsection{Perturbation around the local solution} \label{subsec-pert}

Let us now implement an alternative method to determine the behavior of the nonlocal domain wall not only at infinity but also close to the origin.

We consider a linear perturbation around the standard two-derivative domain wall configuration $\phi_{\rm L}(x)=v\tanh(\sqrt{\lambda/2} vx)$. Let us define the deviation from the local solution as
\begin{align}
	\delta\phi (x) = \phi(x) - \phi_{\rm L}(x)\,, \qquad  \left|\frac{\delta\phi}{\phi_{\rm L}}\right|\ll 1\,, \label{pert-loc-nonloc}
\end{align}
in terms of which we can linearize the field equation~\eqref{eq:the equation of the motion for DW}:
\begin{eqnarray}
\left[e^{-\partial_x^2/M_s^2}(\partial_x^2+\lambda v^2)-3\lambda\phi_{\rm L}^2\right]\delta\phi=\lambda (1-e^{-\partial_x^2/M_s^2})\phi_{\rm L}^3\,.\label{linear-lnl}
\end{eqnarray}
Since the nonlocal scale appears squared in~\eqref{eq:the equation of the motion for DW}, we would expect that $\delta\phi\sim \mathcal{O}(1/M_s^2)$ such that the local limit is consistently recovered, i.e. $\delta\phi\to0$ when $M_s^2\to\infty$. We now write Eq.~\eqref{pert-loc-nonloc} up to order $\mathcal{O}(1/M_s^2)$ in order to extract the leading nonlocal correction to $\phi_{\rm L}.$ By expanding the nonlocal terms as follows
\begin{eqnarray}
	e^{-\del_x^2/M_s^2} \delta\phi&=&
	\delta\phi
	+ {\cal O}\qty(\frac{\del_x^2}{M_s^2}) \delta\phi\,,\\[2mm]
	e^{-\del_x^2/M_s^2} \phi_{\rm L}^3&=& \phi_{\rm L}^3-\frac{\partial_x^2}{M_s^2}\phi_{\rm L}^3
	+{\cal O}\qty(\qty(\frac{\del_x^2}{M_s^2})^2)\phi_{\rm L}^3\,,
\end{eqnarray}
we can write~\eqref{linear-lnl} up to order $\mathcal{O}(1/M_s^2):$
\begin{align}
	\qty[\partial_x^2+\lambda v^2-3\lambda\phi_{\rm L}^2(x)] \delta\phi
		= \frac{\lambda}{M_s^2}\del_x^2(\phi_{\rm L}(x)^3)\,.
		\label{lin-eq-nln-2}
\end{align}
This expansion is valid as long as the following inequality holds:
\begin{equation}
\frac{1}{|\delta\phi|}\left|\frac{\del_x^2}{M_s^2}\delta\phi\right| \ll 1\,.\label{ineq-2}
\end{equation}

We now introduce the dimensionless variable $s=\sqrt{\lambda/2}\,vx$ and the function $f(s)=\delta\phi(x)/v,$ so that we can recast Eq.~\eqref{lin-eq-nln-2} as
\begin{align}
	f^{\prime\prime}(s)+2(1-3\tanh^2 s)f(s) = \frac{\lambda v^2}{M_s^2}(\tanh^3 s)^{\prime\prime}\label{diff-eq-s}
\end{align}
where the prime $'$ denotes the derivative with respect to $s$. The above differential equation can be solved analytically, and its solution reads
\begin{eqnarray}
\!\!f(s)&=&
	\frac{C_1}{\cosh^2s}
	+ \frac{3C_2/2+27\lambda v^2/32M_s^2}{\cosh^2s} \log\frac{1+\tanh s}{1-\tanh s}
\nonumber\\[2mm]
	&&+\qty[
		\qty(2C_2+\frac{\lambda v^2}{8M_s^2})\cosh^2s
			+ \qty(3C_2-\frac{61\lambda v^2}{16M_s^2})
			+ \frac{2\lambda v^2}{M_s^2}(1+\tanh^2 s)
	] \tanh s\,,\,\,\,
\end{eqnarray}
where $C_1$ and $C_2$ are two integration constants to be determined.

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.48]{perturb-sol.pdf}
	\caption{
			%
			In this figure we show the linearized nonlocal domain wall solution $\phi(x)=\phi_{\rm L}(x)+\delta\phi(x)$ (solid blue line) in comparison with the local domain wall (orange dashed line). The nonlocal configuration approaches the asymptotic vacua at $x\rightarrow \pm \infty$ faster as compared to the local case.
			In the smaller plot we showed the behavior of the two solutions over a smaller interval in order to make more evident the differences between local and nonlocal cases. We can notice that when going from $x=0$ to $x\rightarrow \pm \infty$ the nonlocal curve slightly oscillates around the local one.

			We set $\lambda=2$, $v=1$ and $M_s^2=14.3,$ which are consistent with the theoretical constraint $M_s^2\geq -\lambda v^2/W_0(-1/3e)$ in Eq.~\eqref{eq:neccessary cond for asymp sol}.
			}
	\label{fig:illustration for the perturbative solution}
\end{figure}

The boundary conditions $\phi(\pm \infty)=\pm v$ in terms of the linearized deviation read $\delta\phi(\pm \infty)=0,$ or equivalently $f(\pm \infty)=0$. These are satisfied if and only if the algebraic relation $2C_2+\lambda v^2/8M_s^2=0$ holds true, which means that $C_2=-\lambda v^2/16M_s^2$.
Moreover, the constant $C_1$ must be zero because of the $\mathbb{Z}_2$-symmetry.
Thus, the solution for $\delta \phi$ is given by
\begin{align}
	\delta\phi
		= vf(s)
		= \frac{\lambda v^3}{M_s^2}\frac{1}{\cosh^2\sqrt{\frac{\lambda}{2}}vx}
			\qty[\frac{3}{4}\log\frac{1+\tanh\sqrt{\frac{\lambda}{2}}vx}{1-\tanh\sqrt{\frac{\lambda}{2}}vx}
					-2\tanh\sqrt{\frac{\lambda}{2}}vx]\,.
	\label{eq:exact sol for delta phi}
\end{align}
In Fig.~\ref{fig:illustration for the perturbative solution} we showed the behavior of the nonlocal domain wall solution $\phi=\phi_{\rm L}+\delta\phi$ in comparison with the local two-derivative one $\phi_{\rm L};$ we have set values for $v,$ $\lambda$ and $M_s$ consistently with the theoretical lower bound in Eq.~\eqref{eq:neccessary cond for asymp sol}. From the plot we can notice that the nonlocal solution approaches the vacua $\pm v$ faster as compared to the local case, which is in agreement with the asymptotic analysis in the previous Subsection. Indeed, we can expand the solution $\phi=\phi_{\rm L}+\delta\phi$ in the regime $|x|\rightarrow \infty$, and obtain
\begin{eqnarray}
\phi(x)&\simeq& \pm v\left[1-2\left(\frac{4v^2\lambda}{M^2_s}\right)e^{-\sqrt{2\lambda}vx}-2e^{-\sqrt{2\lambda}vx}\left(1-\frac{3}{2}\sqrt{2\lambda}\frac{\lambda v^3}{M_s^2}x\right)\right]\nonumber\\[2mm]
&\simeq& \pm v\left[1-2\left(1+\frac{4v^2\lambda}{M^2_s}\right)e^{-\sqrt{2\lambda}v(1+3\lambda v^2/2M_s^2)x}\right]\nonumber\\[2mm]
&\simeq& \pm v\left[1-2\left(1+\frac{4v^2\lambda}{M^2_s}\right)e^{-B x}\right]\,,
\end{eqnarray}
where to go from the first to the second line we have used the freedom to add negligible terms of order higher than $\mathcal{O}(1/M_s^2),$ i.e. $4v^2\lambda/M_s^2\simeq 4v^2\lambda/M_s^2(1-3\sqrt{2\lambda}\lambda v^3x/2M_s^2)$ and $1-3\sqrt{2\lambda}\lambda v^3/2M_s^2x\simeq e^{-(3\sqrt{2\lambda}\lambda v^3/2M_s^2)x}.$ Remarkably, the asymptotic behavior of the linearized solution perfectly matches the result obtained in Eq.~\eqref{eq:series expansion for coeff B-sqrt}, indeed the coefficient $B$ in the exponent turns out to be exactly the same in both approaches. Moreover, from the linearized solution we can also determine the coefficient $A$ up to order $\mathcal{O}(1/M_s^2)$, i.e. $A=-2v-8v^3\lambda/M_s^2,$ which could not be determined through the asymptotic analysis in Sec.~\ref{subsec-asym} (see Eq.~\eqref{eq:def for asymp sol}).

Furthermore, the behavior of the linearized solution close to the origin is quite peculiar as the nonlocal domain wall profile slightly oscillates around the local one. In other words, when going from $x=0$ to $x\rightarrow \infty$ the perturbation $\delta\phi$ is initially negative and then becomes positive; whereas the opposite happens when going from $x=0$ to $x\rightarrow-\infty.$
This property may suggest that the typical length scale $\ell$ over which $\phi(x)$ changes in proximity of the origin is larger as compared to the local case. As done for the local domain wall in Sec.~\ref{sec-review}, we can estimate such a length scale as the inverse of the gradient at the origin times the energy scale $v$, i.e.  $\ell\sim v/(\partial_x\phi|_{x=0}).$
By doing so, up to order $\mathcal{O}(1/M_s^2)$ we get
\begin{equation}
\del_x\phi(x)|_{x=0}=v^2\sqrt{\frac{\lambda}{2}}\left(1-\frac{\lambda v^2}{2M_s^2}\right)+\mathcal{O}\left(\frac{1}{M_s^4}\right)\,,\label{grad-zero}
\end{equation}
which yields
\begin{equation}
\ell\sim \sqrt{\frac{2}{\lambda}}\frac{1}{v}\left(1+\frac{\lambda v^2}{2M_s^2}\right)\,.\label{ell-linearized}
\end{equation}
Note that such a length scale does \textit{not} coincide with the width of the wall because it is related to the behavior of the solution close to the origin and far from the vacuum. In standard two-derivative theories the above computation would give a result for $\ell$ that coincides with the size of the wall, but this is just a coincidence. We will comment more on this in Sec.~\ref{subsec-order}.

\subsubsection{Validity of the linearized solution}

The above linearized solution was found perturbatively, and it is valid as long as the two inequalities in Eq.~\eqref{pert-loc-nonloc} and~\eqref{ineq-2} are satisfied. We now check when these conditions are verified.

By working with the variable $s=\sqrt{\lambda/2} vx$ and the field redefinition $\delta\phi=vf(s),$ the  inequality~\eqref{pert-loc-nonloc} reads:
\begin{equation}
  |H(s)|= \left|\frac{f(s)}{\tanh s}\right|\ll 1\,,
\end{equation}
where $H(s):=f(s)/\tanh s\,.$ By analyzing the behavior of $|H(s)|$ we can notice that it is always less than unity, thus supporting the validity of the linearized solution in Eq.~\eqref{eq:exact sol for delta phi}; see the left panel in Fig.~\ref{fig:graph for verifivation of perturbative condition}.

Let us now focus on the inequality~\eqref{ineq-2}. By introducing also in this case the variable $s=\sqrt{\lambda/2}vx$ we can write

\begin{eqnarray}
	\frac{\partial_x^2}{M_s^2}\delta\phi
	&=&\frac{\lambda v^2/2}{M_s^2} v f^{\prime\prime}(s) \nonumber\\[2mm]
	&=&\frac{\lambda v^3/2}{M_s^2} \frac{\lambda v^2}{M_s^2} \frac{{\rm d}^2}{{\rm d}s^2}
		\qty{
			\frac{1}{\cosh^2 s}
			\left(
				\frac{3}{4} \log\frac{1+\tanh s}{1-\tanh s} - 2\tanh s
			\right)
		 } \nonumber\\[2mm]
	&=&v \times
		\underbrace{\frac{1}{2}\qty(\frac{\lambda v^2}{M_s^2})^2\frac{{\rm d}^2}{{\rm d}s^2}
			\qty{
				\frac{1}{\cosh^2 s}
				\left(
					\frac{3}{4} \log\frac{1+\tanh s}{1-\tanh s} - 2\tanh s
				\right)
			}
		}_{=:\,g(s)}\,,
\end{eqnarray}
where $g(s):=\partial_x^2\delta\phi(x)/(M_s^2v)$.
In terms of the dimensionless functions $f(s)$ and $g(s)$ the inequality~\eqref{ineq-2} becomes
\begin{align}
	\left|\frac{\partial_x^2}{M_s^2}\delta\phi\right| \ll |\delta\phi|
		\quad\Leftrightarrow\quad
	|g(s)| \ll |f(s)|\,.
\end{align}
Therefore, we have to analyze the function
\begin{align}
	h(s):=\frac{g(s)}{f(s)}
				=\frac{\lambda v^2}{2M_s^2}
	 					\frac{\dps\frac{{\rm d}^2}{{\rm d}s^2}\qty[\frac{1}{\cosh^2 s}\qty(\frac{3}{4}\log\frac{1+\tanh s}{1-\tanh s}-2\tanh s)]}
			 						{\dps\frac{1}{\cosh^2 s}\qty(\frac{3}{4}\log\frac{1+\tanh s}{1-\tanh s}-2\tanh s)}\,,
\end{align}
and check for which values of $s$ its modulus $|h(s)|$ is less than unity.
In the right panel of Fig.~\ref{fig:graph for verifivation of perturbative condition} we have shown the behavior of $h(s);$ we have only plotted the region $s\geq 0$ as $h(s)$ is an even function in $s$.
We can notice that $h(s)$ diverges at the point $s\sim 1.03402$ where $f(s)$ vanishes; therefore, in the proximity of this point the linearized solution $\delta\phi(x)$ might not be valid.
However, for $|s|\rightarrow \infty$ and $s\rightarrow 0$ the inequality~\eqref{ineq-2} can be satisfied:

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.315]{validity-phi_L-delta-phi}\quad\includegraphics[scale=0.305]{validity-del-2-delta-phi}
	\caption{
			%
			(Left panel) behavior of $H(s)=f(s)/\tanh s$ as a function of $s=\sqrt{\lambda/2}vx.$ The modulus of the function is always less than unity, i.e. $|H(s)|< 1,$ supporting the validity of the linearized solution $\delta \phi (x) =v f(x)$ in Eq.~\eqref{eq:exact sol for delta phi}. $H(s)$ becomes smaller and smaller for larger values of $M_s.$
			(Right panel) behavior of the function $h(s)=g(s)/f(s).$
			As long as $|h(s)|\ll 1$ the linearized solution~\eqref{eq:exact sol for delta phi}  can be trusted as a good approximation of the true behavior of the nonlocal domain wall.
			We can notice that close to the asymptotics ($s\rightarrow \infty$) the function can be kept less than one, but there is a singularity at $s\sim 1.03402$ caused by the fact that $f(s)$ vanishes at this point.
            Moreover, the linearized approximation close to the origin and at infinity becomes better for larger values of $M_s.$
			In both panels we only showed the behavior for $s\geq 0$ because both functions $h(s)$ and $H(s)$ are even in $s.$
			We set $\lambda=2,$ $v=1$ and $M_s^2=14.3,$ which are consistent with the theoretical constraint $M_s^2\geq -\lambda v^2/W_0(-1/3e)$ in Eq.~\eqref{eq:neccessary cond for asymp sol}.
		}
	\label{fig:graph for verifivation of perturbative condition}
\end{figure}

\begin{align}
 	h(0)=\lim_{s\to0} h(s) = -7\frac{\lambda v^2}{M_s^2}\,,
 	\qquad
	h(\infty)=\lim_{s\to\infty} h(s) = 2\frac{\lambda v^2}{M_s^2}\,,\label{limits}
\end{align}
and by using the theoretical lower bound in Eq.~\eqref{eq:neccessary cond for asymp sol}, i.e. $\lambda v^2/M_s^2\leq-W_0^{-1}(-1/3e)\sim 0.14,$  it follows that both asymptotic limits are always less than unity, i.e. $|h(0)|< 0.98$ and $h(\infty)< 0.28,$ and the approximation becomes better for larger values of the scale of nonlocality $M_s$.

Let us now make two important remarks.

\paragraph{Remark 2.} In light of the remark at the end of Sec.~\ref{subsec-asym} we now understand that the linearized perturbative solution~\eqref{eq:exact sol for delta phi} would have not been valid if instead we had used $W_{-1}$ and the corresponding upper bound $M_s\leq -\lambda v^2/W_{-1}(-1/3e).$ In such a case a different domain wall solution with the same functional form of the asymptotic behavior is obviously guaranteed to exist, but we are not interested in it in the current work. Therefore, we emphasize again that we only work with a domain wall configuration consistent with the bound~\eqref{eq:neccessary cond for asymp sol}.

\paragraph{Remark 3.} We have noticed that the linearized approximation breaks down in the proximity of $s\sim 1.03402.$ It means that the boundary condition $\delta\phi(\pm\infty)=0$ imposed on the perturbation cannot be used to connect the behavior of the solution from $x=\pm\infty$ to $x=0$ because the boundary condition itself would break down. This might imply that apparently the linearized solution obtained in Eq.~\eqref{eq:exact sol for delta phi} can be trusted only close to the vacua, and that the above analysis is not enough to justify the behavior close to the origin.
However, by using a different and reliable perturbative expansion in the intermediate region around $s\sim 1.03402,$ and imposing junction conditions to glue different pieces of the solution defined in three different regions, we checked and confirmed that the behavior close to the origin found above is well justified. See App.~\ref{sec-corr} for more details.

\subsection{Estimation of width and energy} \label{subsec-order}

We now estimate the width and the energy per unit area of the nonlocal domain wall by performing an analogous analysis as the one made for the local two-derivative case in Sec.~\ref{sec-review}.

Let us approximate the field as $\phi\sim  v,$  and the gradient as $\del_x\sim R^{-1}$ where $R$ is the width of the wall.
From the expression of the energy-momentum  tensor~\eqref{final-stress-exp} (with $m^2=-\lambda v^2$), we can obtain the energy density of the wall:
\begin{align}
	\mathcal{E}(x)
		\equiv T_0^0(x)= -\frac{1}{2}\phi e^{-\partial_x^2/M_s^2}(\partial_x^2+\lambda v^2)\phi
			+\frac{\lambda}{4}(\phi^4+v^4)\,.
	\label{eq:energy density for DW}
\end{align}
The next step would be to impose the balance between the kinetic and the potential energy in~\eqref{eq:energy density for DW} for the lowest-energy configuration, and solve the resulting equation for the width $R.$ The presence of the infinite-derivative operator makes the procedure less straightforward as compared to the local case because we should first understand how to estimate $e^{-\del_x^2/M_s^2},$ i.e. whether to replace the exponent with $-1/M_s^2R^2$ or $+1/M_s^2R^2.$

The strategy to follow in order to avoid any ambiguities is to Taylor expand, recast the infinite-derivative pieces in terms of an infinite number of squared quantities, replace $\del_x\sim 1/R,$ $\phi\sim v,$ and then re-sum the series.

By Taylor expanding in powers of $\partial_x^2/M_s^2$ the infinite-derivative terms in Eq.~\eqref{eq:energy density for DW}, and neglecting total derivatives, we can write
\begin{align}
\!\phi e^{-\del_x^2/M_s^2}\del_x^2\phi&= -\del_x\phi e^{-\del_x^2/M_s^2}\del_x\phi\nonumber\\[2mm]
&= -\left[(\del_x\phi)^2-\frac{1}{M_s^2}\del_x\phi\del_x^2\del_x\phi+\frac{1}{2!M_s^4}\del_x\phi\del_x^4\del_x\phi-\cdots+\frac{(-1)^n}{n!M_s^{2n}}\del_x\phi\del_x^{2n}\del_x\phi+\cdots\right]  \nonumber\\[2mm]
&=-\left[(\del_x\phi)^2+\frac{1}{M_s^2}(\del_x^2\phi)^2+\frac{1}{2!M_s^4}(\del_x^3\phi)^2+\cdots+\frac{1}{n!M_s^{2n}}(\del_x^{n+1}\phi)^2  +\cdots\right]\nonumber\\[2mm]
&= -\sum\limits_{n=0}^\infty \frac{1}{n!}\left(\frac{1}{M_s^2}\right)^n \left(\del_x^{n+1}\phi\right)^2\,,
\end{align}
and
\begin{equation}
\lambda v^2\phi e^{-\del_x^2/M_s^2}\phi=\lambda v^2\sum\limits_{n=0}^\infty \frac{1}{n!}\left(\frac{1}{M_s^2}\right)^n \left(\del_x^{n}\phi\right)^2\,.
\end{equation}
Then, by using $\phi\sim v$ and $\del_x\sim 1/R,$ we get
\begin{align}
\phi e^{-\del_x^2/M_s^2}\del_x^2\phi\sim -\frac{v^2}{R^2}\sum\limits_{n=0}^\infty\frac{1}{n!}\left(\frac{1}{M_s^2R^2}\right)^n = -\frac{v^2}{R^2}e^{1/(M_sR)^2}\,,
\end{align}
and
\begin{equation}
\lambda v^2\phi e^{-\del_x^2/M_s^2}\phi\sim\lambda v^4 e^{1/(M_sR)^2}\,.
\end{equation}
Thus, we have shown that the correct sign in the exponent when making the estimation is the positive one\footnote{To further remove any possible ambiguity and/or confusion, it is worth mentioning that the same result would have been obtained if we would have started with a positive definite expression for the kinetic energy, for instance with the expression $\del_x\phi e^{-\del_x^2/M_s^2}\del_x\phi=(e^{-\del_x^2/2M_s^2}\del_x\phi)^2\geq 0,$ where we integrated by parts and neglected total derivatives. Also in this case one can show (up to total derivatives) that $(e^{-\del_x^2/2M_s^2}\del_x\phi)^2=\sum_{k,l=0}^\infty 1/(k!\,l!) (1/2M_s^2)^{k+l}\left(\del_x^{(k+l+1)}\phi\right)^2\sim (v^2/R^2)\left[\sum_{k=0}^\infty 1/k!(1/2M_s^2R^2)^n\right]^2= (v^2/R^2)e^{1/(M_sR)^2}$.}.

To make more manifest the consistency with the low-energy limit $M_s \to \infty$, it is convenient to separate the kinetic and the potential contributions in~\eqref{eq:energy density for DW} as follows:
\begin{align}
\mathcal{E}(x)=&
    \qty[
        -\frac{1}{2} \phi e^{-\del_x^2/M_s^2}\partial_x^2 \phi
        - \frac{1}{2} \lambda v^2 \phi \qty(e^{-\del_x^2/M_s^2}-1) \phi
    ]
    + \qty[
        \frac{\lambda}{4} \qty(\phi^2-v^2)^2
    ]\,,
\end{align}
so that the balance equation between  kinetic and  potential energies reads
\begin{align}
    \frac{1}{2} \frac{v^2}{R^2} e^{1/(M_s R)^2} - \frac{1}{2} \lambda v^4 \qty(e^{1/(M_s R)^2}-1)
        \sim
    \frac{\lambda}{4} v^4\,.
    \label{eq:full-balance-eq}
\end{align}
We are mainly interested in the leading nonlocal correction, thus we expand for $M_sR\gg 1$ up to the first relevant nonlocal contribution:
\begin{eqnarray}
 \frac{1}{2}\frac{v^2}{R^2}\left(1+\frac{1}{M_s^2R^2}\right)-\frac{1}{2}\frac{\lambda v^4}{M_s^2R^2}\sim \frac{1}{4}\lambda v^4\nonumber
	\label{eq:full-balance-eq-leading}
\end{eqnarray}
The solution up to order $\mathcal{O}(1/M_s^2)$ is given by
\begin{eqnarray}
\frac{1}{R^2}\sim\frac{\lambda v^2}{2}\left(1+\frac{\lambda v^2}{2M_s^2} \right)\,,
\end{eqnarray}
from which we obtain
\begin{eqnarray}
	R\sim\sqrt{\frac{2}{\lambda }}\frac{1}{v}\left(1-\frac{\lambda v^2}{4M_s^2} \right)\,.
\end{eqnarray}
Therefore, the width of the nonlocal domain wall, $R$, turns out to be \textit{thinner} as compared to the local two-derivative case. This is consistent with both the asymptotic analysis in Sec.~\ref{subsec-asym} and with the linearized solution in Sec.~\ref{subsec-pert}. In fact, in the previous Subsections we found that the nonlocal configuration approaches the vacua $\pm v$ \textit{faster} as compared to the local case, i.e. the coefficient $B$ in Eq.~\eqref{eq:series expansion for coeff B-sqrt} is larger than the corresponding local one. Then, as shown in Eq.~\eqref{eq:tildeR}, $\widetilde{R}$ becomes smaller in the nonlocal case, which is consistent with the behaviour of $R$. That is, the coefficient $B$ and the width of the wall $R$ should be inversely proportional to each other; this means that if $B$ increases then $R$ must decrease, and indeed this is what we showed.

We can also estimate the energy per unit area
\begin{eqnarray}
	E&=& \int_\mathbb{R}{{\rm d}x}\qty[-\frac{1}{2}\phi e^{-\partial_x^2/M_s^2}(\partial_x^2+\lambda v^2)\phi
			+\frac{\lambda}{4}(\phi^4+v^4)] \nonumber\\[2mm]
	 &\sim& (\text{width of the wall}) \times (\text{energy density}) \nonumber\\[2mm]
		& \sim& R \times \lambda v^4 \nonumber\\[2mm]
		&\sim& \sqrt{\frac{\lambda}{2}}v^3\left(1-\frac{\lambda v^2}{4M_s^2}\right) \,,\label{energ-estim}
\end{eqnarray}
which is also \textit{decreased} as compared to the local case.

It is worth to emphasize that the expansion for small $\lambda v^2/M_s^2$ is well justified for the domain wall solution satisfying the bound in Eq.~\eqref{eq:neccessary cond for asymp sol} which was obtained by using the principal branch $W_0$ of the Lambert-W function.

\paragraph{Remark 4.} In Sec.~\ref{subsec-pert} we have estimated an additional scale $\ell$ in addition to the width ($\ell> R, \widetilde{R}$). In standard local theories all of the three scales are the same because there is only one physical scale, $\ell_{\rm L}=R_{\rm L}=\widetilde{R}_{\rm L} \sim (\sqrt{\lambda}v)^{-1}.$  In fact, in general $\ell$ and $R$ ($\widetilde{R}$) represent two different physical scales, and this becomes manifest in the nonlocal theory under investigation. The length scale $R$ ($\widetilde{R}$) is the one that contains the information about the size of the wall because it is proportional to $1/B,$ and it is related to how fast the field configuration approaches the vacuum. Whereas, the scale $\ell$ is related to how fast the field changes in the proximity of the origin, indeed it is inversely proportional to the gradient at $x=0$ (see Eqs.~\eqref{grad-zero} and~\eqref{ell-linearized}), and we have $\ell>\ell_{\rm L}.$ The difference between $\ell$ and $R$ ($\widetilde{R}$) is caused by the oscillatory behavior of the nonlocal solution around the local one.

\section{Comments on other topological defects} \label{sec-other}

So far we have only focused on the domain wall configuration. However, it would be very interesting if one could repeat the same analysis also in the case of other topological defects, like \textit{string} and \textit{monopole} which can appear in nonlocal models characterized by continuous-symmetry breaking. A full study of these topological defects in nonlocal field theories goes beyond the scope of this paper, however we can make some important comments.

First of all, the existence of such finite-energy configurations is always guaranteed by the non-trivial topological structure of the vacuum manifold.\footnote{Of course, they can exist only dynamically in the local case because of Derrick's theorem~\cite{Vilenkin:2000jqa}. But, in a nonlocal case, even Derrick's theorem might be circumvented. This issue will be left for a future work.}
Knowing that a solution must exist, then we could ask how some of their properties would be affected by nonlocality. Actually, a similar order-of-magnitude estimation as the one carried out in Sec.~\ref{subsec-order} can be applied to these other global topological defects. In particular, by imposing the balance between kinetic and potential energy one would obtain that the radius of both string and monopole are \textit{smaller} as compared to the corresponding ones in the local case.

We leave a more detailed investigation of higher dimensional topological defects,  including stabilizing gauge fields, for future tasks.

\section{Discussion \& conclusions}\label{sec-dis}

\paragraph{Summary.} In this paper, we studied for the first time topological defects in the context of the nonlocal field theories. In particular, we mainly focused on the domain wall configuration associated to the $\mathbb{Z}_2$-symmetry breaking in the simplest nonlocal scalar field theory with nonlocal differential operator $e^{-\Box/M_s^2}$. Despite the complexity of non-linear infinite-order differential equations, we managed to find an approximate analytic solution. Indeed, we were able to understand how nonlocality affects the behavior of the domain wall both asymptotically close to the vacua and around the origin.

Let us briefly highlight our main results:
\begin{itemize}

    \item We showed that the nonlocal domain wall approaches the asymptotic vacua $\pm v$ faster as compared to the local two-derivative case. We confirmed this feature in two ways: (i) studying the behavior of the solution towards infinity ($|x|\rightarrow \infty$); (ii) analyzing a linearized nonlocal solution found through perturbations around the local domain wall configuration.

    \item Such a faster asymptotic behavior also means that the width of the wall, $R$ ($\widetilde{R}$), is smaller than the corresponding local one. This physically means that the boundary separating two adjacent casually disconnected spatial regions with two different vacua (i.e. $+v$ and $-v$) becomes thinner as compared to the local case. We confirmed this property by making an order-of-magnitude estimation involving the balance equation between kinetic and potential energy. As a consequence, also the energy per unit area can be shown to be smaller.

    \item We noticed that the nonlocal domain wall has a very peculiar behavior around the origin, i.e. in the proximity of $x\sim 0$. We found that the linearized nonlocal solution, $\phi=\phi_{\rm L}+\delta \phi,$ oscillates around the local domain wall when going from $x=0$ to $|x|\rightarrow \infty.$ In other words, the perturbation $\delta\phi$ changes sign: when going from $x=0$ to $x=+\infty$ it is first negative and then positive, and vice-versa when going from $x=0$ to $x=-\infty$ . We confirmed the validity of the solution close to the origin in App.~\ref{sec-corr}.

    \item The specific nonlocal domain wall solution analyzed in this paper can exist only if the nonlocal scale $M_s$ satisfies the lower bound $M_s\gtrsim \sqrt{\lambda} v,$ namely if the energy scale of nonlocality is larger than the symmetry-breaking scale.

\end{itemize}

\paragraph{Discussion \& Outlook.}
Here we have only dealt with nonlocal field theories in flat spacetime without assuming any specific physical scenario. However, it might be interesting to understand how to embed our analysis in a cosmological context where we could expect also gravity to be nonlocal; see Refs.~\cite{Biswas:2005qr,Koshelev:2016xqb,Koshelev:2017tvv,Koshelev:2020foq} and references therein.

In particular, in Refs.~\cite{Koshelev:2017tvv,Koshelev:2020foq} inflationary cosmology in nonlocal (infinite-derivative) gravity was investigated, and the following experimental bound on the scale of nonlocality was obtained: $M_s\gtrsim H,$ where $H$ is the Hubble constant during inflation, i.e. $H\sim 10^{14}$GeV.
Our theoretical lower bound is consistent with the experimental constraint derived in~\cite{Koshelev:2017tvv,Koshelev:2020foq} for the gravity sector. Indeed, some symmetry breaking is expected to happen after inflation, i.e. at energies $v\lesssim H,$ which is consistent with the theoretical lower bound $M_s\gtrsim \sqrt{\lambda} v$ in Eq.~\eqref{eq:neccessary cond for asymp sol}.

Hence, in a cosmological context one would expect the following hierarchy of scales\footnote{We are implicitly assuming that there exists only one scale of nonlocality $M_s$ for both gravity and matter sectors.}:
\begin{equation}
M_s\gtrsim H\gtrsim v\,.\label{set-ineq}
\end{equation}
Very interestingly, this cosmological scenario can be used to rule out some topological-defect solutions in nonlocal field theory. For instance, in light of the discussions a the end of Sec.~\ref{subsec-asym} and Sec.~\ref{subsec-pert}, there must exist at least another domain wall configuration that is valid for $M_s^2\lesssim \lambda v^2.$ In such a case the set of inequalities~\eqref{set-ineq} would be replaced by $v\gtrsim M_s\gtrsim H,$ which implies that the symmetry breaking would happen before inflation. Thus, if we are interested in domain wall formation after inflation, then we can surely discard any configuration valid in the regime $M_s^2\lesssim \lambda v^2.$

It would be very interesting if one would consider other topological defects, like strings and monopoles, which can appear in nonlocal models characterized by continuous-symmetry breaking. In fact, global string might play important roles like axion emissions in an expanding universe. In the local case, the topological defects that are formed from global-symmetry breaking should be unstable because of Derrick's theorem~\cite{Vilenkin:2000jqa} which excludes the existence of stationary stable configurations in dimensions greater than one. Then, they can exist only dynamically e.g. in an expanding universe. However, Derrick's theorem might not apply to a nonlocal case thanks to the nonlocality. It would be also interesting to investigate whether such stationary stable configurations could exist in a nonlocal case or not.

As another potential future direction to follow we can consider another class of models characterized by gauge symmetries as well as global ones. In fact, among the possible physical applications that can be studied in relation to topological defects, we have gravitational waves, e.g. the ones emitted by cosmic strings. We would expect that the presence of nonlocality would change the dynamics in such a way to modify non-trivially the gravitational wave-form.
This type of investigations will provide powerful test-benches to test nonlocal field theories, and to further constrain the structure of the nonlocal differential operators in the Lagrangian and the value of the nonlocal scale. In this work we only focused on the simplest model with $F(-\Box)=e^{-\Box/M_s^2},$ but one could work with more generic operators. Actually, the class of viable differential operators is huge (e.g. see~\cite{Buoninfante:2020ctr}), and it would be interesting to reduce it by means of new phenomenological studies.

As yet another future work we also wish to consider other type of field theoretical objects. An important phenomenon is the \textit{false vacuum decay}~\cite{Coleman:1977py,Callan:1977pt,Coleman:1980aw} according to which the false vacuum -- which corresponds to a local minimum of the potential -- has a non-zero probability to decay through quantum tunneling into the true vacuum -- which corresponds to a global minimum. This tunneling process consists in interpolating between the false and the true vacuum through an instanton (a bounce solution).
It may be very interesting to generalize the standard analysis done for a local two-derivative theory to the context of nonlocal field theories, and to understand how nonlocality would affect this phenomenon, e.g. how the tunneling probability would change. Another interesting direction is to consider non-topological solitons like Q-balls and oscillons/I-balls. The existence of these objects is also related to the presence of a bounce solution. But, one should notice that, different from topological defects, the presence of these bounce solutions is not guaranteed in nonlocal theories. In fact, it is very difficult to guarantee the presence of such a bounce solution in a nonlocal theory, different from the local cases. Therefore, even if one would obtain (possible) approximate solutions somehow, one cannot make any argument based on such approximate solutions without the proof of the existence of exact bounce solutions. This is the reason why we dealt only with topological defects in this paper, and left a study of non-topological field theoretical objects for future work.

Finally, we should emphasize that non-linear and infinite-order differential equations are not only difficult to solve analytically but even numerically. Indeed, up to our knowledge no numerical technique to find domain wall solutions is currently known. Some techniques to solve nonlinear equations involving infinite-order derivatives have been developed in the last decades~\cite{Moeller:2002vx,Arefeva:2003mur,Volovich2003,Joukovskaya:2008cr,Calcagni:2008nm,Frasca:2020ojd}, but none of them seem to be useful for the type of field equations considered in this paper. Therefore, as a future task it will be extremely interesting to develop new numerical and analytic methods to find topological-defect solutions. This will also be important to investigate the stability of topological defects in nonlocal field theory, something that we have not done in this paper.

\subsection*{Acknowledgements}

The authors are grateful to Sravan Kumar for useful discussions.
Y.~M. acknowledges the financial support from Advanced Research Center for Quantum Physics and Nanoscience, Tokyo Institute of Technology.
M.~Y. acknowledges financial support from JSPS Grant-in-Aid
for Scientific Research No. JP18K18764, JP21H01080, JP21H00069.


\section{Introduction}
\label{Intro}
In the field of network science and big data, it is necessary to expand the scope beyond classical time signal analysis and processing to accommodate the signals defined on the graph \cite{ramakrishna2021grid, chen2021graph, zhou2022novel, 4, 13}. The irregular structure of the underlying graph is different from the regular structure of classical signal processing, which brings great challenges to the analysis and processing of graph signal. Graph signal processing (GSP) offers effective tools to process such network data \cite{liao2022joint, gama2020graphs,morency2021graphon}. For instance, graph-supported signals can model vehicle trajectories over road networks \cite{deri2016new}. Research in GSP has only recently begun, but it is growing rapidly. The main contributions include wavelet and Fourier transforms \cite{2011Wavelets, Moura2013IEEE, 13, 25, 27}, sampling and reconstruction of graph signals \cite{kim2022quantization, yang2021efficient, chen2015discrete, romero2016kernel, yang2022reconstruction}, uncertainty principles \cite{tsitsvero2016signals, pasdeloup2019uncertainty}, filtering of graph signals \cite{xiao2021distributed, ozturk2021optimal}, etc.

Different transforms of graph signal are still the core of GSP. Among them, the graph Fourier transform (GFT) acts as a cornerstone \cite{4,13}. In the literature, there are two frameworks for frequency analysis and processing of graph signals: (i) based on Laplacian matrix \cite{4}, (ii) based on adjacency matrix. The classical Laplacian based approach is limited to graph signals located on undirected graphs \cite{13}. In this method, frequency ordering is based on quadratic form and small eigenvalues correspond to low frequencies and vice versa. On the other hand, the adjacency based approach builds on the shift operator of the graph. This approach constructs the Fourier basis by using generalized eigenvectors of the adjacency matrix. Recently, some unique methods have been proposed to extend the GFT to directed graphs \cite{singh2016graph,sardellitti2017graph, sevi2018harmonic, shafipour2018directed, shafipour2019windowed, shafipour2018digraph, barrufet2021orthogonal,furutani2019graph, marques2020signal, seifert2021digraph, chen2022graph}. \cite{singh2016graph} uses the in-degree matrix and weight matrix to define the directed Laplacian. \cite{sardellitti2017graph} proposes an alternative approach that builds the graph Fourier basis as the set of orthonormal vectors that minimize a continuous extension of the graph cut size, known as the Lov{\'a}sz extension. \cite{sevi2018harmonic} introduces a novel harmonic analysis for functions defined on the vertices of a strongly connected directed graph. \cite{shafipour2018directed} studies the problem of constructing a GFT for directed graphs, which decomposes graph signals into different modes of variation with respect to the underlying network. \cite{shafipour2019windowed} proposes a methodology to carry out vertex-frequency analyses of graph signals on directed graph based on \cite{shafipour2018directed}. \cite{barrufet2021orthogonal} also defines new transform for directed graph, which using the Schur decomposition and leads to a series of embedded invariant subspaces for which orthogonal basis are available. \cite{furutani2019graph} extends graph signal processing to directed graphs based on the Hermitian Laplacian. \cite{marques2020signal} provides an overview of the current landscape of signal processing on directed graphs. Although these methods can construct Fourier bases with desirable properties, Fourier bases cannot completely retain information about the structure of the underlying graph.

In order to extract the local information of the graph signal, a new research direction has been proposed in graph signal processing (GSP), that is, fractional order \cite{25,26,27,28,ozturk2021optimal,yan2021windowed, ge2022optimal, kartal2022joint, yan2022multi}. The fractional order has gained considerable attention in the last 20 years in classical signal processing, and the application of fractional order to graphs has also aroused the interest of researchers \cite{ozturk2021optimal,yan2021windowed, ge2022optimal, kartal2022joint}. The graph fractional domain is a combination of the graph spectrum domain and fractional transform domain. The graph fractional Fourier transform (GFRFT) related to graph adjacency matrix is proposed in \cite{25}. Furthermore, a new spectral graph Fractional Fourier Transform (SGFRFT) related to graph Laplacian matrix is proposed in \cite{27}. GFRFT and SGFRFT show advantages in revealing the local characteristics of the graph signal. However, for directed graph, both of these transforms have their drawbacks. The Laplacian matrix for SGFRFT is constructed by an undirected graph, so SGFRFT does not apply to directed graphs. Although the GFRFT can use for directed graph, it has some potential problems. First, the basis comes from the Jordan decomposition is not orthonormal, and the Parseval’s identity does not hold and inner products are not preserved in the vertex domain and graph fractional domains. Second, numerical calculations of Jordan decomposition also often produce numerical instability even for medium-sized matrices \cite{golub1976ill}.

The presence of directionality plays a crucial role when it comes to modeling social networks, technological networks,
biological and biomedical networks, as well as information networks \cite{newman2003structure, han2012extended, chui2018representation}.
For directed graphs, the existing studies in graph fractional domain are all based on adjacency matrix \cite{25,26}. In the continuous setting, fractional Fourier transform seeks the orthogonal bases which are eigenfunctions of the fractional Laplacian operator. The background naturally leads us to consider the eigenvectors of the fractional graph Laplacian operator in the discrete setting. Thus, we believe the Laplacian-based construction is more natural. In this paper, we aim to generalize SGFRFT to directed graphs. In the new transform, the fractional Laplacian operator is a simple extension of the Hermitian Laplacian discussed in \cite{zhang2021magnet} to directed graphs. Moreover, the new definition of SGFRFT links the existing Laplacian based approach to directed graph. The paper is organized as follows. We first review the GFT and SGFRFT as our foundation in Section \ref{Preliminary}. Section \ref{main} introduces the procedure about how to design spectral graph fractional Fourier transform for signal on directed graph. Then, an ideal filter and frequency selective filter are presented in Section \ref{filter}. For the last part, we present experiments on real directed graph, and an application of signal denoising using the filtering in the previous section.

\section{Preliminaries}
\label{Preliminary}

\subsection{Spectral graph theory}
An undirected weighted graph $\mathcal{G}=\{\mathcal{V}, \mathcal{E}, \mathcal{W}\}$ consists of a finite set of vertices $\mathcal{V}=\{v_0, \cdots, v_{N-1}\}$, where $N=|\mathcal{V}|$ is the number of vertices, a set of edges $\mathcal{E}=\{(i,j)|i,j\in\mathcal{V},j\sim i\}\subseteq\mathcal{V}\times\mathcal{V}$, and a weighted adjacency matrix $\mathcal{W}$. If the values of $\mathcal{W}$ are all in ${0, 1}$ then $\mathcal{W}$ is called an adjacency matrix \cite{chui2018representation}.  $\mathcal{W}=[{\mathcal{W}_{ij}}]\in\mathbb{R}^{N\times N}$ is defined as $\mathcal{W}_{ij}=w_{ij}$ if $(i,j)\in\mathcal{E}$ and $\mathcal{W}_{ij}=0$ otherwise. The non-normalized graph Laplacian is a symmetric difference operator $\mathcal{L}=D-\mathcal{W}$ \cite{13}, where $D:=diag(d_1, ..., d_N)$ is a diagonal degree matrix of $\mathcal{G}$, and $d_i:=\sum^{N}_{j=1}w_{ij}$. Let $\{\chi_0, \chi_1, \cdots, \chi_{N-1}\}$ be the set of orthonormal eigenvectors. Suppose that the corresponding Laplacian eigenvalues are sorted as $0=\lambda_0<\lambda_1\leq\lambda_2\leq\cdots\leq\lambda_{N-1}:=\lambda_{max}$.
Therefore
\begin{equation}
	\mathcal{L}=\mathbf{\chi}\mathbf{\Lambda} \mathbf{\chi}^H,
\end{equation}
where
\begin{equation}\label{tezheng}
	\mathbf{\chi} =
		[\chi_0, \chi_1, \cdots, \chi_{N-1}],
\end{equation}
and the diagonal matrix is $\mathbf{\Lambda}=diag([\lambda_0, \lambda_1, \cdots, \lambda_{N-1}])$. The superscript $H$ represents the conjugate transpose of matrix.

The graph signal $f$ is defined as binding a scalar value to each vertex through the function $f:\mathcal{V}\rightarrow\mathbb{R}$.
Using the definition of (inverse) graph Fourier transform (GFT) as in \cite{13}, the GFT of $f$ is
\begin{equation}
	\widehat{f}(\ell)=\langle f,\chi_{\ell}\rangle=\sum^{N}_{n=1}f(n)\chi^{*}_{\ell}(n),  \ell=0,1,\cdots,N-1,
\end{equation}
where $*$ is complex conjugate.
The inverse GFT is given by
\begin{equation}\label{IGFT}
	\begin{split}
		f(n)=\sum^{N-1}_{\ell=0}\widehat{f}(\ell)\chi_{\ell}(n), n=0,1,\cdots,N-1.
	\end{split}
\end{equation}

\subsection{Spectral graph Fractional Fourier Transform}
The graph fractional Laplacian operator $\mathcal{L}_{\alpha}$ is defined by $\mathcal{L}_{\alpha}=\mathbf{\kappa}R\mathbf{\kappa}^{H}$, where $\alpha$ is the fractional order, $0<\alpha\leq1$,
$\ell=0,1,\cdots,N-1$ \cite{27}.
Note that
\begin{equation}\label{fractionallapalcian}
	\mathbf{\kappa} = \begin{bmatrix}
		\kappa_0,& \kappa_1,& \cdots,& \kappa_{N-1}
	\end{bmatrix}={\chi}^\alpha,
\end{equation}
and
\begin{equation}
	R=\text{diag}({\begin{bmatrix}
			r_0,& r_1,& \cdots,& r_{N-1}
	\end{bmatrix}})
	=\mathbf{\Lambda}^\alpha,
\end{equation}
so that
\begin{equation}
	r_\ell=\lambda_\ell^\alpha.
\end{equation}
In the follow-up part of this paper, computing the $\alpha$ power of a matrix always uses matrix power function.

The spectral graph Fractional Fourier Transform (SGFRFT) of any signal $f$ building on the graph $\mathcal{G}$ is defined by \cite{27}:
\begin{equation}
	\begin{split}
		\widehat{f}_{\alpha}(\ell)=\langle f,\kappa_{\ell}\rangle=\sum^{N}_{n=1}f(n)\kappa^{*}_{\ell}(n),  \ell=0,1,\cdots,N-1,
	\end{split}
\end{equation}
when $\alpha=1$, the SGFRFT degenerates into standard GFT.

The inverse SGFRFT is given by
\begin{equation}\label{IGFRFT}
	\begin{split}
		f(n)=\sum^{N-1}_{\ell=0}\widehat{f}_{\alpha}(\ell)\kappa_{\ell}(n), n=0,1,\cdots,N-1.
	\end{split}
\end{equation}

\begin{remark}
	In this paper, to distinguish directed graphs from undirected graphs,  we use $\mathcal{G}$ represents undirected graphs whereas $G$ represents directed graphs.
\end{remark}

\section{Spectral graph fractional Fourier transform for directed graph (DGFRFT)}
\label{main}

The orthogonality of eigenvectors gives the algebraic ideal properties of the SGFRFT, which makes the graph signal processing of undirected graphs develop well. However, since the eigenvectors of the fractional Laplacian operator of digraphs are usually not orthonormal, it is difficult to extend graph signal processing to digraphs simply. Our goal is to find a new way of defining fractional Laplacian matrix to keep the orthogonality of the eigenvectors and avoid the calculation of Jordan decomposition, i.e., Hermitian fractional Laplacian matrix. The Hermitian Laplacian is a complex matrix obtained
from an extension of the graph Laplacian \cite{zhang2021magnet, furutani2019graph, fanuel2018magnetic, f2020characterization}. It preserves the edge directionality and Hermitian property. Here we consider a directed graph $G=(V,E,W)$. $V$ is the set of $N$ vertices and $E$ is the set of directed edges. $W$ is the weight matrix of the graph, its element is defined as $w_{ij}$ which represents the weight of the directed edge from vertex $i$ to $j$. For directed graphs, the integers $d_i^{in}$ and $d_i^{out}$ specify the number of arrowheads directed toward and away from vertex $i$ respectively, \cite{hakimi1965degrees}. The in-degree of vertex $i$ is calculated as $d_i^{in}=\sum^N_{j=1}w_{ij}$, whereas, the out-degree is $d_i^{out}=\sum^N_{j=1}w_{ji}$. We define a new weight matrix as $W_s=[w_{ij,s}]$, where $w_{ij,s}=\frac{1}{2}(w_{ij}+w_{ji})$. $W_s=[w_{ij,s}]$ and $E_s$ (ignoring the directionality of $E$) uniquely
determine the corresponding undirected graph $G_s = (V, E_s, W_s)$ for directed $G$. The diagonal degree matrix $D_s$ of $G_s$ is $D(i,i):=\sum^{N}_{j=1}w_{ij,s}$. Then a Hermitian Laplacian of $G$ is defined as:
\begin{definition}(Hermitian graph Laplacian matrix)
\begin{equation}
	L=D_s-\Gamma_q\odot W_s,
\end{equation}
where $\odot$ is the hadamard product \cite{horn1990hadamard}, and $\Gamma_q$ is a Hermitian matrix which encodes the edge directionality of $G$.
\end{definition}
$\Gamma_q$ can take many forms, here's a simple example. Define $\Gamma_q=[\gamma(i,j)]$ that satisfies $\gamma(i,j)=\gamma(i,j)^H$.
$\gamma$ is a map from $V\times V$ to a unitary group of degree 1 and is written by \cite{f2020characterization}:
\begin{equation}
	\gamma_q{i,j}= e^{2\pi iq(w_{ij}-w_{ji})},
\end{equation}
when $w_{ij}=1$, $\gamma_q{i,j}= e^{2\pi iq}$ represents the edge from vertex $i$ to $j$ and $0\leq q<1$ is a rotation parameter.

As $D_s$ and $W_s$ are real symmetric matrices, $L$ is Hermitian matrix. Let $v_k$ and $u_k$ be respectively the k-th eigenvalue and eigenvector of
the Hermitian Laplacian $L$. The eigendecomposition of $L$ can be written as:
\begin{equation}
	L=UVU^H,
\end{equation}
where $U=[u_0, u_1, \cdots, u_{N-1}]$. $V=diag([v_0, v_1, \cdots, v_{N-1}])$ is a real diagonal matrix, and Hermitian Laplacian eigenvalues are sorted as $0\leq v_0<v_1\leq v_2\leq\cdots\leq v_{N-1}:=v_{max}$.

The graph Hermitian fractional Laplacian matrix for directed graph is given by $L_{\alpha,d}=PQP^H$, in which
\begin{equation}\label{fractionallapalcian}
	P = \begin{bmatrix}
		p_0,& p_1,& \cdots,& p_{N-1}
	\end{bmatrix}={U}^\alpha,
\end{equation}
and
\begin{equation}
	Q=\text{diag}({\begin{bmatrix}
			\xi_0,& \xi_1,& \cdots,& \xi_{N-1}
	\end{bmatrix}})
	=V^\alpha,
\end{equation}
that is
\begin{equation}
	\xi_\ell=v_\ell^\alpha, \quad l=0,1,\cdots,N-1.
\end{equation}

We select two different $\alpha$ as examples to show the spectral properties of Hermitian fractional Laplacian matrix $L_{\alpha,d}$ in Fig. \ref{eigenvaluefigure}.  When the rotation parameter $q=0$, the Hermitian fractional Laplacian matrix degrades to normal fractional Laplacian matrix as each element of $\Gamma_q$ equals to $1$. Therefore, when the values of $q$ are relatively small, the spectrum of the Hermitian fractional Laplacian is similar to the spectrum of the fractional Laplacian, otherwise, when $q$ is large, there will be oscillation.

\begin{figure}
	\centering
	\subfigure[Eigenvalues of Hermitian fractional Laplacian when $\alpha=0.8$]{
		\label{alpha=0.8}
		\includegraphics[width=0.45\textwidth]{Spec_for_alpha0.8.pdf}}
	\subfigure[Eigenvalues of Hermitian fractional Laplacian when $\alpha=0.6$]{
		\label{alpha=0.6}
		\includegraphics[width=0.45\textwidth]{Spec_for_alpha0.6.pdf}}
	\caption{Eigenvalues of Hermitian fractional Laplacian with respect to different fractional orders.}
	\label{eigenvaluefigure}
\end{figure}

The total variation for graph signals is defined as an absolute sum of the
discrete difference of a graph signal \cite{ono2015total}. It was first introduced in as an extension of the original total variation (TV) \cite{chan2001digital}. We define the total variation of a graph signal $f$ as
\begin{equation}
\operatorname{TV}(\boldsymbol{f}):=\sum_{(i, j) \in \mathcal{E}}|f(i)-f(j)|^2.
\end{equation}
to measure the smoothness of eigenvectors of $L_{\alpha,d}$. The total variation has an intuitive interpretation: it compares how the signal varies with time or space and calculates a cumulative magnitude of the signal. The smaller the difference between the original signal $f(i)$ and $f(j)$, the lower the signal's variation. Fig. \ref{TV} shows total variations of eigenvectors of $L_{\alpha,d}$
on a random directed graph with 50 nodes when the fractional parameter $\alpha=0.8$ and $0.6$. To construct a random directed graph, we fix $n$ nodes and for each pair of nodes we generate a direct edge with probability $p$ ($n=50, p=0.1$).

\begin{figure}[t]
	\centering
	\subfigure[Total variations of eigenvectors of Hermitian fractional Laplacian when $\alpha=0.8$]{
		\label{TValpha=0.8}
		\includegraphics[width=0.45\textwidth]{Total_Var_for_alpha0.8.pdf}}
	\subfigure[Total variations of eigenvectors of Hermitian fractional Laplacian when $\alpha=0.6$]{
		\label{TValpha=0.6}
		\includegraphics[width=0.45\textwidth]{Total_Var_for_alpha0.6.pdf}}
	\caption{Total variations of eigenvectors of Hermitian fractional Laplacian with respect to different fractional orders.}
	\label{TV}
\end{figure}

Then we prove that $L_{\alpha,d}$ is a positive semi-definite Hermitian matrix.
\begin{proposition}\label{Hermitian}
	For any fractional order $\alpha$, the $L_{\alpha,d}$ is a Hermitian matrix:
	\begin{equation}
		(L_{\alpha,d})^{H}=L_{\alpha,d}.
	\end{equation}
\end{proposition}
\begin{proof}
	Since $V$ is a real diagonal matrix, $(V^\alpha)^H=V^\alpha$.
	\begin{equation}
	   (L_{\alpha,d})^{H}=(PQP^H)^H=(U^\alpha V^\alpha(U^\alpha)^H)^H=U^\alpha V^\alpha(U^\alpha)^H=PQP^H=L_{\alpha,d}.
	\end{equation}
\end{proof}

\begin{proposition}
	For any fractional order $\alpha$, the $L_{\alpha,d}$ is a positive semi-definite matrix.
\end{proposition}
\begin{proof}
	From Proposition \ref{Hermitian}, we know $V^\alpha$ is real diagonal matrix. Let $J=V^{\alpha/2}(U^\alpha)^H$, so $L_{\alpha,d}=J^HJ$. For any signal $f\in\mathbb{C}^N$, note that
	\begin{equation}
		f^HL_{\alpha,d}f=f^HU^\alpha V^\alpha(U^\alpha)^Hf=f^HJ^HJf=(Jf)^HJf\geq0
	\end{equation}
\end{proof}

Clearly, in this case, this Hermitian fractional Laplacian matrix has a set of orthonormal eigenvectors. This orthogonality allows the basic concepts of graph signal processing of undirected graphs to be extended directly to those of directed graphs.

\begin{definition}(Directed graph fractional Fourier transform)
The spectral graph Fractional Fourier Transform for directed graph (DGFRFT) of any signal $f$ building on the graph $G$ is defined as:
\begin{equation}
	\begin{split}
		\widehat{f}_{\alpha,d}(\ell)=\langle f,p_{\ell}\rangle=\sum^{N}_{n=1}f(n)p^{*}_{\ell}(n), \ell=0,1,\cdots,N-1.
	\end{split}
\end{equation}
By the matrix form, the DGFRFT is
\begin{equation}
		\widehat{f}_{\alpha,d}=P^Hf,
\end{equation}
when $\alpha=1$, the DGFRFT degenerates into GFT for directed graph.

The inverse DGFRFT is given by
\begin{equation}\label{IGFRFT}
	\begin{split}
		f(n)=\sum^{N-1}_{\ell=0}\widehat{f}_{\alpha,d}(\ell)p_{\ell}(n), n=0,1,\cdots,N-1.
	\end{split}
\end{equation}
\end{definition}

Now we have the definition of DGFRFT, it has some useful properties.
\begin{property} Unitarity: $P^{-1}=(U^\alpha)^{-1}=U^{-\alpha}=P^H$.
\end{property}
\begin{property} Index additivity: $U^{-\alpha}\circ U^{-\beta}=U^{-\beta}\circ U^{-\alpha}=U^{-(\alpha+\beta)} $.
\end{property}
\begin{property} Reduction to SGFRFT when $w_{ij}=w_{ji}$.
\end{property}
\begin{property} Parseval relation holds, for any signal $f$ and $g$ defined on the directed graph $G$ we have:
\begin{equation}
	\begin{split}
		\langle f,g\rangle=\langle \widehat{f}_{\alpha,d},\widehat{g}_{\alpha,d}\rangle.
	\end{split}
\end{equation}
If $f=g$, then
\begin{equation}\label{parseval}
	\begin{split}
		&\sum^{N}_{n=1}|f(n)|^{2}=\|f\|_{2}^{2}=\langle f,f\rangle\\
		&=\langle \widehat{f}_{\alpha,d},\widehat{f}_{\alpha,d}\rangle=\|\widehat{f}_{\alpha,d}\|_{2}^{2}=\sum^{N-1}_{\ell=0}|\widehat{f}_{\alpha,d}(\ell)|^{2}.
	\end{split}
\end{equation}
\end{property}

DGFRFT is a new transform focused on directed graph in graph fractional domain. Compared with GFRFT, it has several significant advantages.
First, the columns of U are linearly independent eigenvectors, and at the same time they are orthogonal. This results in DGFRFT that preserves the inner product when passed from the vertex domain to the graph fractional domain. In addition, DGFRFT preserves edge directionality. Finally, computing the GFRFT needs to do Jordan decomposition, when the size of the graph exceeds the median value, the calculation of Jordan decomposition will lead to serious and difficult numerical instability. As our new Laplacian matrix for directed graph is Hermitian matrix, the Jordan decomposition can be avoided in the calculation of DGFRFT.

\section{Directed graph filtering}
\label{filter}
\subsection{Spectral graph filtering}
In classical signal processing, filtering can be defined by convolution. The convolution of signal a with b is the result of signal a filtered by b. Therefore, to define the directed graph filtering in graph fractional domain, first we need to define the convolution operator. Convolution in the time domain is equivalent to multiplication in the Fourier domain. For directed graph, the graph fractional convolution operator is defined as the following form consistent with the classical convolution by using the directed graph fractional Laplacian eigenvector.
\begin{definition}(Convolution operator)
For any graph signal $f$ and $g$ which underlying graph structure is directed, their graph fractional convolution $*$ is
	\begin{equation}\label{convolution}
		(f*g)(n)=\sum_{l=0}^{N-1}	\widehat{f}_{\alpha,d}(\ell)	\widehat{g}_{\alpha,d}(\ell)p_{\ell}(n).
	\end{equation}
\end{definition}

A graph filter is a system which takes a graph signal as input and produces another graph signal as output \cite{chen2014signal}. Given a input directed graph signal $f_in$, the filtering is defined by the convolution of $f_in$ and a filter $h$. Thus the spectral directed graph filtering in vertex domain is:
\begin{equation}
		f^{out}(n)=(f*h)(n)=\sum^{N-1}_{\ell=0}\widehat{f}_{\alpha,d}^{in}(\ell)\widehat{h}_{\alpha,d}(\ell)p_{\ell}(n), n=0,1,\cdots,N-1,
\end{equation}
and in graph fractional domain:
\begin{equation}
		\widehat{f}_{\alpha,d}^{out}(\ell)=\widehat{f}_{\alpha,d}^{in}(\ell)\widehat{h}_{\alpha,d}(\ell), \ell=0,1,\cdots,N-1.
\end{equation}

\subsection{Frequency selective filtering}
The above spectral graph filtering is an ideal filter. It retains all frequencies within a given range and completely removes those that are out of range. Because the graph has a finite number of points, it is impossible to design such a filter. A frequency selective filter is a system that isolates the specific frequency components and excludes all others. It's easier to achieve in application. The ideal response of the frequency selective filter is \cite{iacobucci2005frequency}
\begin{equation}
	h_d(l)=\left\{\begin{array}{l}
		1,l\in\text{passband}\\
		0,l\in\text{stopband}.
	\end{array}\right.
\end{equation}
Lowpass, bandpass and highpass filters are three common frequency selective filters.

Using $h_d(l)$, the graph signal $f$ is filtered in graph fractional domain by:
\begin{equation}
		\widehat{f}_{\alpha,d}^{out}=J_d\widehat{f}_{\alpha,d},
\end{equation}
where $J_d=diag([h_d(\xi_1), h_d(\xi_2), \cdots, h_d(\xi_N)])$, and $P^{-1}f=\widehat{f}_{\alpha,d}$ is the DGFRFT of $f$.

Or, equivalently, vertex domain filtering can be obtained by inverse DGFRFT:
\begin{equation}
	f^{out}=P\widehat{f}_{\alpha,d}^{out}=PJ_d\widehat{f}_{\alpha,d}=PJ_dP^{-1}f=Hf.
\end{equation}
where $H=PJ_dP^{-1}$ represents the transfer matrix.

\section{Application}
\begin{figure}[t]
	\centering
	\includegraphics[width=0.7\textwidth]{direct_original.pdf}
	\caption{Average temperature data measured by 50 meteorological state stations in the US.}
	\label{USTEM}
\end{figure}

\begin{figure}[t]
	\centering
	\subfigure[original signal.]{
		\label{direct_original}
		\includegraphics[width=0.4\textwidth]{direct_original.pdf}}
	\subfigure[noisy signal.]{
		\label{direct_noisy}
		\includegraphics[width=0.4\textwidth]{direct_noisy.pdf}}
	\subfigure[denoised signal using Hermitian GFT \cite{furutani2019graph}.]{
		\label{direct_denoise_HL}
		\includegraphics[width=0.4\textwidth]{direct_denoise_HL.pdf}}
	\subfigure[denoised signal using DGFRFT.]{
		\label{direct_denoise_FrHL}
		\includegraphics[width=0.4\textwidth]{direct_denoise_FrHL.pdf}}
	\caption{Original, noisy and denoised temperature signals on directed US graph.}
	\label{directed}
\end{figure}

\begin{figure}[t]
	\centering
	\subfigure[original signal.]{
		\label{undirect_original}
		\includegraphics[width=0.3\textwidth]{undirect_original.pdf}}
	\subfigure[noisy signal.]{
		\label{undirect_noisy}
		\includegraphics[width=0.3\textwidth]{undirect_noisy.pdf}}
	\subfigure[denoised signal.]{
		\label{undirect_denoise_FrHL}
		\includegraphics[width=0.3\textwidth]{undirect_denoise.pdf}}
	\caption{Original, noisy and denoised temperature signals on undirected US graph.}
	\label{undirected}
\end{figure}

Here we assess the performance of DGFRFT via simulations on two graphs. In Simulation 1, we use the US temperature data to test that  directionality is an important key in application. For a graph that starts out as a directed graph, we apply DGFRFT to denoise. Then ignore the directionality of the graph, do the same denoising step, and observe the results. In graph fractional domain, DGFRFT and GFRFT \cite{25} are suitable for directed graph. The former is based on Laplacian matrix, and the latter uses adjacency matrix. In Simulation 2, for a real dataset, we use DGFRFT and GFRFT to perform the same filtering and denoising task, and then compare the results. The experimental results show that our DGFRFT introduces smaller errors and has higher robustness.

\subsection{Simulation 1: \textbf{Directed} VS \textbf{Undirected}}
In this section, real temperature data measured by 50 state meteorological stations in the United States (US) are used to make experiment. We consider signal denoising on a directed graph. First, based on a map of the US, we define a digraph using geographic locations and latitude. In this graph, the vertexes represent the 50 states of the US. The directed edges between states are based on latitudinal assignments from states with low latitude to states with high latitude. Only when two states share a border are they connected by an edge. The average annual temperature in each state is viewed as a graph signal. States at lower latitudes have higher average temperatures. Therefore, there is a correlation between latitude and temperature, and it is reasonable to use latitude to define a digraph. Fig. \ref{USTEM} shows the US temperature signal on a directed graph. The data is available in \sloppy\url{https://www.currentresults.com/Weather/US/average-annual-state-temperatures.php}.

We generate a noisy signal $g= f + n$, where $f$ is the original signals representing average temprature on the graph and $n$ is the noise vector whose coordinates are independently sampled from the Gaussian distribution with zero mean and standard deviation $\sigma = 10$. We apply GFT \cite{4}, Hemitian GFT \cite{furutani2019graph}, DGFRFT to recover the original signal from the noisy signal. For each method, we use a low-pass filter kernel
\begin{equation*}
	\hat{h_d}(\lambda) = \frac{1}{1+c\lambda},
\end{equation*}
where $\lambda$ is the eigenvalue corresponding to graph Laplacian, Hermitian graph Laplacian, graph Hermitian fractional Laplacian respectively. The denoised signal can be calculated as
\begin{equation*}
	\tilde{f} = U\hat{H}U^*g,
\end{equation*}
where $\hat{H} = \text{diag}\{\hat{h_d}(\lambda_0),\ldots,\hat{h_d}(\lambda_{N-1})\}$ and $U$ is the GFT, Hermitian GFT, DGFRFT transform matrix. In this experiment, we set $c=0.02$, $q=0.5$ and $\alpha=0.9$. Note that we can only apply GFT \cite{4} to undirected graphs. To verify the importance of the directionality, we use the above denoising scheme using the same US graph in Fig. \ref{USTEM}, where we omit the direction of edges. Fig. \ref{directed} and \ref{undirected} give an intuitive example of the original, noisy, and denoised temperature signals on directed and undirected US graphs respectively. In the directed case, the performance difference of Hermitian GFT and DGFRFT is observed in Fig. \ref{direct_denoise_HL} and \ref{direct_denoise_FrHL}. In the undirected case, the denoising result after GFT is shown in Fig. \ref{undirect_denoise_FrHL}.

In addition, we calculate the root mean square error (RMSE) between original signal and the denoised signal obtained using GFT \cite{4}, Hermitian GFT \cite{furutani2019graph} and DGFRFT approaches. The results are described in Table \ref{tab:RE}. As the results demonstrate, these three transforms to graph signal denoising perform well, leading to small average errors. The RMSE obtained by our method is about 6.3828, while this quantity changes to 6.5685 and 6.5657 by using classical GFT based on undirected graph \cite{4} and another kind of GFT based on directed graph \cite{furutani2019graph} for denoising respectively. In conclusion, the proposed approach DGFRFT on directed graph outperforms other two ways, which highlights its practical usefulness in data denoising.

\begin{table*}[htbp]
	\centering
	\caption{Average RMSE between denoised signal and original signal.}
	\begin{tabular}{|c|c|c|}
		\hline
		Graph type & Transform method & RMSE\\
		\hline
		undirected & GFT \cite{4} & 6.5685\\
		directed & Hermitian GFT \cite{furutani2019graph} & 6.5657\\
		directed & DGFRFT & 6.3828\\
		\hline
	\end{tabular}
	\label{tab:RE}
\end{table*}

\subsection{Simulation 2: \textbf{DGFRFT} VS \textbf{GFRFT}}
Next, compared to GFRFT \cite{25}, which is also applicable to digraph, we study a real brain graph to demonstrate the superiority of DGFRFT in denoising tasks. The datasets represent the macaque large-scale visual and sensorimotor area corticocortical connectivity \cite{rubinov2010complex}. It has 47 vertexes and 505 edges (121 edges are directed). The vertexes represent cortical areas and edges represent large corticocortical tracts or functional associations. The data is available in \sloppy\url{https://sites.google.com/site/bctnet/}.

\begin{figure}[t]
	\centering
	\includegraphics[width=0.7\textwidth]{curves.pdf}
	\caption{A sample realization of the original, noisy, and recovered signals of DGFRFT and GFRFT}
	\label{compared}
\end{figure}

\begin{figure}[t]
	\centering
	\subfigure[Relative recovery error of DGFRFT]{
		\label{laperror}
		\includegraphics[width=0.45\textwidth]{box_lap.pdf}}
	\subfigure[Relative recovery error of GFRFT]{
		\label{adjerror}
		\includegraphics[width=0.45\textwidth]{box_adj.pdf}}
	\caption{Relative recovery error

		of DGFRFT and GFRFT when $\alpha=0.9$ and $q=0.2$.}
	\label{error}
\end{figure}

Let $U$ be the orthonormal DGFRFT or GFRFT basis. We construct a synthetic graph signal by $x_f=e^{-f}$. We add Gaussian noise to the original signal to obtain noisy signal $g=x_f+n$. We use a series of filter kernels constructed by $H := \text{diag}(h)$, where $h_i = 1[i \le \ell]$ and $\ell$ is a tuning parameter to control the number of diagonal elements. The parameter $\ell$ can be viewed as spectral window size and when $\ell=47$ the filter kernel is just the identity matrix. The recovered signal is given by
\begin{equation*}
	\tilde{x}_f = UHU^*g.
\end{equation*}
Fig. \ref{compared} shows an example of original, noisy, and recovered signals of DGFRFT and GFRFT with window size $5$. Moreover, we compute the relative recovery error with respect to the true error. The recovery error is defined as $e_f = \|\tilde{x}_f - x_f\|/\|x_f\|$. The true error is defined as $e = \|n\|/\|x\|$, and the relative recovery error is defined as $e_f/e$. Fig. \ref{error} are boxplots depicting $e_f/e$ versus $\ell$ averaged over 1000 Monte-Carlo simulations, and demonstrates the effectiveness of adopting filters along with the proposed two methods. We can see DGFRFT is much more stable compared with GFRFT. The Jordan decomposition required in GFRFT is numerically unstable, which may be responsible for the large reconstruction errors. Therefore, compared with GFRFT, DGFRFT has certain advantages when dealing with graph signals whose underlying structure is directed graph.

\section{Conclusions}
Signals defined on directed graphs have important practical significance. This paper proposes a methodology to carry out graph signals processing on directed graph in spectral graph fractional domain. First, we introduce a method to construct a new fractional Laplacian matrix for directed graph and prove that it is a positive semi-definite Hermitian matrix. Equipped with this Hermitian  fractional Laplacian matrix, a new transform named DGFRFT is presented. Then, to highlight the utility of DGFRFT, we propose two basic filtering method and use it to signal denoising on real data. Finally, the effectiveness of the DGFRFT construction is illustrated through tests on directed real world graphs.