# Weak deflection angle by Kalb-Ramond Traversable Wormhole in Plasma and Dark
Matter Mediums

Wajiha Javed<EMAIL_ADDRESS>Department of Mathematics, Division of
Science and Technology, University of Education, Lahore-54590, Pakistan  Hafsa
Irshad<EMAIL_ADDRESS>Department of Mathematics, Division of
Science and Technology, University of Education, Lahore-54590, Pakistan
Reggie C. Pantig<EMAIL_ADDRESS>Physics Department, De La Salle
University, 2401 Taft Avenue, Manila, 1004 Philippines  Physics Department,
Mapúa University, 658 Muralla St., Intramuros, Manila 1002, Philippines  Ali
Övgün<EMAIL_ADDRESS>Physics Department, Eastern Mediterranean
University, Famagusta, 99628 North Cyprus via Mersin 10, Turkey.

###### Abstract

This paper is devoted to compute the weak deflection angle for Kalb-Ramond
traversable wormhole solution in plasma and dark matter mediums by using the
method of Gibbons and Werner. To acquire our results, we evaluate Gaussian
optical curvature by utilizing the Gauss-Bonnet theorem in the weak field
limits. We also investigate the graphical influence of deflection angle
$\tilde{\alpha}$ with respect to the impact parameter $\sigma$ and minimal
radius $r_{0}$ in plasma medium. Moreover, we derive the deflection angle by
using different method known as Keeton and Petters method. We also examine
that if we remove the effects of plasma and dark matter, the results become
identical to that of non-plasma case.

General Relativity; Gravitational Lensing; Gauss-Bonnet Theorem; Plasma
Medium; Dark Matter; Kalb-Ramond Traversable Wormhole; Keeton and Petters
Method

###### pacs:

95.30.Sf, 98.62.Sb, 97.60.Lf

##  I Introduction

A visual illustration of a black hole (BH) shows that it dissipates energy
through radiation, then compresses and sooner or later dissolves Javed:2017saf
. Albert Einstein first estimated the presence of a BH by his theory of
General Relativity Einstein . A BH is considered as an area of space having
too strong gravitational pull that the fastest moving objects even light
cannot get out of it. Four types of BHs that exist are intermediate BHs,
stellar BHs, miniature BHs and supermassive BHs. Black holes are certainly
very simple as they have two primary parts, event horizon and the singularity.
Event horizon is the boundary that indicates the limit of a BH. On event
horizon, the escape velocity becomes equal to the velocity of light. While
singularity is a point in space where the existing mass have infinite density.
According to GR, spacetime singularities rise various issues, both
scientifically and physically Javed:2020fli .

Just like BHs, wormholes (WHs) appear as effective solutions to the Einstein
field equations. The simplest solution to Einstein field equations is
Schwarzschild solution. The idea of a WH was first given by Flamm in $1916$,
soon after the invention of Schwarzschild’s BH solution. In general, WH is the
link between two separate regions of space that are far away from each other
via a tunnel 1 . In $1935$, Einstein and Rosen Einstein  additionally
investigated the theory of inter-universe connections. These spacetime
connections were known to be “Einstein-Rosen Bridges”. However, the idea of
“WH” was formulated by Wheeler in $1957$ Wheeler:1955zz ; Fuller:1962zza .
After that, he illustrated that WH would be unsteady and non-traversable for
even a photon.

Later on, the term traversable WH was developed by Morris and Thorne in 1988
Morris:1988cz . They established flat traversable WHs with exotic matter that
do not satisfy the null energy conditions Hochberg:1998ii . Exotic matter
creates problems for making stable WHs. Morris, Thorne and Yurtsever also
showed that traversable WH can be made stable and flat by applying the Casimir
effect. Another type of traversable WH, a thin-shell WH was introduced by
Matt-Visser in 1989 Visser:1989kh  in which a path through the WH may be made
in such a way that the traversing path does not cross the region of exotic
matter. Also, the metric of Ellis WH was considered firstly in Ellis:1973yv
and analysis of Ellis WH returned to standard work of Morris and thorne where
they introduced the traversable WHs. The deflection of light initially
suggested by Chetouani and Clement in Ellis WH Chetouani . Nakajima and Asada
also studied Gravitational lensing by Ellis WH Nakajima:2012pu .

It is a known fact that weak and strong gravitational lensing (GL) is a very
productive area to find not only dark and heavy objects but also BHs and WHs.
To identify a WH, a possible way is the implementation of GL. The GL by WHs
was reviewed on a large scale in the literature of theoretical physics as well
as astrophysics Kuhfittig:2013hva -31 .

Gravitational lensing suggested by Soldner for the first time in the
background of Newtonian theory. The basic theory of GL is formed by Liebes,
Refsdal and many other scientists 32 -Pantig:2022gih . When light released by
distant galaxies, passed through the heavy objects in the universe, the
gravitational attraction by those heavy objects can deviate the light from its
pathway. This process is known as GL. Gravitational lensing is very applicable
method to understand dark matter, galaxies and universe. Three types of GL
listed in the literature are: (i) strong GL (ii) weak GL (iii) micro-GL. The
strong GL indicates the approximate magnification, location and time delay of
the images with the aid of BHs. Strong GL is also helpful to see various
objects such as boson stars, fermion stars and monopoles Barriola . The weak
GL is used to discover the mass of astronomical objects without demanding
their changing nature. Weak lensing also differentiates dark energy from
modified gravity and investigates how universe is expanding rapidly.
Gravitational lensing has been calculated for many spacetimes by applying
various methods 19 -Gibbons:2008rj . In the past few years, we examine many
researches that relate GL with the Gauss-Bonnet theorem (GBT).

Using GBT, Gibbons and Werner Gibbons:2008rj  revealed that it is possible to
compute the deflection angle in weak field limits using Gauss-bonnet theorem.
After that, Werner expanded this technique to Kerr BHs Werner:2012rc  by using
Nazim’s method for Rander-Finsler metric. Then, Ono et al. enlarged the study
of weak GL in stationary axisymmetric spacetimes using finite distance method
42 -Takizawa:2020egm . The theory of GL comprises of three physical processes
named as (i) geometric optics (ii) thin lens approximation (iii) perturbation
theory of GL Keeton:2005jd ; Keeton:2006sa ; Sereno .

For a distant observer from a source, the bending angle of light can be
determined by using GBT in the weak field limits Gibbons:2008rj . By
considering an oriented surface, let us describe a domain $D_{R}$ surrounded
by the beam of light with a circular boundary $C_{R}$ having Euler
characteristic element $\mathcal{X}$ and metric $g$ at the focus area where
light rays coincide with source and the viewer. So, when the GBT is applied
within the optical metric, it provides us the bending angle of light stated as
Gibbons:2008rj :

$\int\int_{\mathcal{D}_{R}}\mathcal{K}dS+\oint_{\partial\mathcal{D}_{R}}kdt+\sum_{n}\theta_{n}=2\pi\mathcal{X}(\mathcal{D}_{R}),\\\
$

where, $D_{R}$ is the region that comprises of the source of light waves,
observer and the focal point of the lens, $\mathcal{K}$ shows the Gaussian
optical curvature, $dS$ is surface element, $k$ is known as geodesic curvature
and $\partial\mathcal{D}_{R}$ shows that this portion is surrounded by the
outermost light rays. The asymptotic bending angle of light can be computed as
Gibbons:2008rj :

$\tilde{\alpha}=-\int\int_{D_{\infty}}\mathcal{K}dS,\\\ $

where $\tilde{\alpha}$ represents the bending angle and $D_{\infty}$ indicates
the infinite domain bounded by the rays of photon, apart from the lens.
Moreover, By utilizing GBT, bending angle for static and axisymmetric rotating
Teo WH was investigated by Jusufi and Övgün JusufiOvgun . Recently, Övgün has
studied deflection angle by Damour-Solodukhin WHs Ovgun:2018fnk .

An approximate form of GL by the help of spherically symmetric lenses upto
post-post-Newtonian (PPN) is newly originated by Keeton and Petters
Keeton:2005jd ; Keeton:2006sa . Sereno and de Luca Sereno  expanded the PPN
approximation to Kerr BHs. The Keeton and Petters method allows the
calculations of those observable quantities that can be basically independent
of coordinates and therefore physically applicable.

Dark matter comprises of those particles that do not absorb, reflect, or emit
light, so that they cannot be detected by observing electromagnetic
radiations. Dark matter is the substance that cannot visualize directly. Dark
matter produces upto $27\%$ of overall mass-energy of the universe and the
remaining part consists of dark energy. Dark matter can only be noticed by
their gravitational interaction. Although, they have weak non-gravitational
interaction and are of non-relativistic nature Feng . Some of the dark matters
exist are: weakly relating massive particles (WIMPs), axions, sterile
neutrinos, super WIMPs etc. Dark matter particles are formed by non-baryonic
particles. The neutrino was the only familiar non-baryonic particle and it is
taken as the first dark matter candidate. Refractive index used in dark matter
maintains the propagation speed. According to the $20^{th}$ century report, we
have seen a lot of times that mostly matter consists of protons and neutrons.
But that matter which we see is not suitable form of matter. In this universe,
there are some other types of matter whose masses are five times greater than
the regular matter. Such unknown matter is named as “dark matter”. We cannot
detect its existence in the laboratory till now. For the proper understanding
of dark matter, we have to understand and then use various branches of
astronomy and physics such as particle physics is important in describing the
relation between dark matter and standard matter. Cosmology, General
Relativity and astrophysics are applicable for the study of dark matter on a
wide range in literature Latimer:2013rja ; Latimer:2017lwm . For the
inspection of bending angle through dark matter, we take the refractive index
as Latimer:2013rja ; Latimer:2017lwm :

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

where, $\omega$ is the frequency of photon. It is noticed here that
$\beta=\frac{\rho_{0}}{4m^{2}\omega^{2}}$ represents the mass density of the
scattered dark matter particles, $A_{0}=-2e^{2}\epsilon^{2}$ and $A_{2}j\geq
0$. The higher order terms like $O(\omega^{2})$ and onward are linked with
polarizability of the dark matter candidate. The terms $O(\omega^{2})$ stand
for the neutral dark matter candidate while $O(\omega^{-2})$ denote the
charged dark matter candidates.

The main goal of this paper is to examine the deflection of light for Kalb-
Ramond traversable WH solution by using various methods.

This paper is organized as; In section 2, we analyze the Kalb-Ramond
traversable WH solution in detail. In section 3, we compute the deflection
angle for Kalb-Ramond traversable WH solution in plasma medium. Section 4, is
related to the graphical inspection of deflection angle of Kalb-Ramond
traversable WH solution in the framework of plasma medium. In section 5, we
investigate the bending angle of Kalb-Ramond traversable WH solution by using
Keeton and Petters method. In section 6, we enlarge our observations and
derive the value of deflection angle in dark matter medium. In section 7, we
conclude all the results.

##  II Kalb-Ramond Traversable Wormhole Solution

This section is devoted to describe some properties of a traversable WH
solution. The metric is spherically symmetric and independent of time for the
sake of simplification. Wormhole is like a tunnel connecting two regions of
space, have no horizon because the presence of horizon does not allow two-
sided travelling. The time taken by the traveller to pass through WH should be
limited and fairly short. The traveller experiencing the gravitational force
inside the WH should be slightly small.
Local Lorentz violation effects also result in modified gravitational
dynamics. Although the Lorentz symmetry may be violated close to the Planck
scale. When one or more of the tensor fields gain nonzero vacuum expectation
values (VEV), Lorentz symmetry breaks spontaneously. The Kalb-Ramond (KR)
field, an antisymmetric tensor field $B\mu\nu$ that appears in string
theories, is another cause for spontaneous Lorentz symmetry breaking (LSB). A
non-vanishing VEV breaks the gauge and the Lorentz symmetry by allowing a
self-interaction potential. The tensor antisymmetric VEV can be split into two
vectors, called pseudo-electric and pseudo-magnetic vectors, in the same way
that the electromagnetic field strength can. A background KR field recently
altered a static and spherically symmetric wormhole. The parameter-dependent
power-law correction to the Schwarszchild solution resulted from a non-minimal
interaction between the KR VEV and the Ricci tensor. The effects of this LSB
solution on the WH temperature and the shadows were also investigated Lessa .
The static spherically symmetric spacetime of a Morris-Thorne traversable WH
is expressed by the metric 54 :

$ds^{2}=-e^{2{\phi(r)}}dt^{2}+\frac{dr^{2}}{1-\frac{\Omega(r)}{r}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2},$ (1)

where the value of $\Omega(r)$ is defined as: Lessa:2020imi

$\Omega(r)=r(\frac{r}{r_{0}})^{\frac{2}{1-2\lambda}}.$

Here, $\lambda$ is the Lorentz violating parameter $\lambda=|b|^{2}\xi_{2}$ ,
$b$ and $\xi_{2}$ are constants Lessa:2020imi , $\phi(r)$ is redshift function
and $\Omega(r)$ is the shape function of WH. Both of these functions are
adjustable. Redshift function is finite everywhere. The shape function
$\Omega(r)$ calculates the shape of the WH or describes the WH physically. The
radial variable $r$ has a minimum value of $r_{0}$ and a maximum value of
infinity, or $r_{0}\leq r<\infty$. The $r_{0}$ is a positive constant as it
decreases from positive infinity to a least value and after that starts moving
towards positive infinity.

After putting the value of shape function, the spacetime metric becomes:

$ds^{2}=-dt^{2}+\frac{dr^{2}}{1-(\frac{r}{r_{0}})^{\frac{2}{1-2\lambda}}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (2)

Now, for Kalb-Ramond traversable WH solution, we can write our metric in
general form as:

$ds^{2}=-g_{tt}dt^{2}+g_{rr}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2},$ (3)

where, $g_{tt}=1$ and
$g_{rr}=\frac{1}{1-(\frac{r}{r_{0}})^{\frac{2}{1-2\lambda}}}$. For the
calculation of deflection angle by Kalb-Ramond traversable WH solution, the
optical path can be obtained by the null geodesic condition $ds^{2}=0$. Also,
we assume that the photon is moveable in equatorial plane
($\theta=\frac{\pi}{2}$). In order to calculate the deflection angle, one can
obtain the optical path metric in explicit form given as follows:
Lessa:2020imi

$dt^{2}=\frac{dr^{2}}{1-(\frac{r}{r_{0}})^{\frac{2}{1-2\lambda}}}+r^{2}d\phi^{2}.$
(4)

The above metric will be used to calculate the deflection angle of light by
Kalb-Ramond travesable WH solution which is described.

##  III Deflection Angle in Plasma Medium

In this section, we evaluate impact of plasma medium on the weak GL for Kalb-
Ramond traversable WH solution (4). The value of refractive index $n(r)$ for
the given WH solution is defined as 55 :

$n(r)=\sqrt{1-\frac{\omega_{e}^{2}(r)}{\omega_{\infty}^{2}(r)}f(r)}.$ (5)

In refractive index $n(r)$, $\omega_{e}$ indicates the plasma frequency of
electron while $\omega_{\infty}$ denotes the frequency of photon that is
noticed by the observer at infinity. So, optical metric can be written as:

$dt^{2}=g^{opt}_{lm}dx^{l}dx^{m}=n^{2}(r)\left[{\frac{dr^{2}}{f(r)}+r^{2}d\phi^{2}}\right],$
(6)

where, $f(r)$ is defined as

$f(r)=\frac{1}{1-(\frac{r}{r_{0}})^{\frac{2}{1-2\lambda}}}.$ (7)

Now, the relative Guassian optical curvature can be determined by the
expression present in 57 ; 58 :

$\mathcal{K}=\frac{R}{2},$ (8)

where $R$ is Ricci scalar. We calculate the Guassian optical curvature for
Kalb Ramond traversable WH solution as:

$\mathcal{K}\simeq-\frac{r_{0}}{2r^{3}}+\frac{5}{4}\frac{r_{0}^{2}}{r^{4}}\frac{w_{e}^{2}}{w_{\infty}^{2}}-\frac{r_{0}}{r^{3}}\frac{w_{e}^{2}}{w_{\infty}^{2}}+\mathcal{O}({r_{0}^{3}}).$
(9)

The Guassian curvature depends on minimal radius $r_{0}$ and radial parameter
$r$. For the sake of simplicity, we consider the optical curvature upto order
$2$ as well as to match our results given in non-plasma medium case.

The infinitesimal surface element for Kalb-Ramond traversable WH solution can
be computed as:

$dS=\sqrt{g}drd\phi=\left(r-\frac{rw_{e}^{2}}{w_{\infty}^{2}}\right)drd\phi+\mathcal{O}({r_{0}^{3}}).$

To find the deflection angle in plasma medium, we utilize the GBT. As the beam
of light approaches from the infinity upto a large distance and remember that
we are present in weak field limitations. Here, beam of light becomes almost
straight. Hence, we utilize the straight line approximation
$r=\frac{\sigma}{\sin\phi}$, where $\sigma$ expresses the impact parameter,
GBT is stated as Gibbons:2008rj :

$\tilde{\alpha}=-\int_{0}^{\pi}\int_{\frac{\sigma}{\sin\phi}}^{\infty}\mathcal{K}dS.$
(10)

The resulting expression can be expressed for $\lambda=\frac{3}{2}$. The
Lorentz violating parameter must be restricted to the range $\lambda>1/2$ to
achieve an asymptotically flat spacetime. If we take $\lambda\simeq 0$, the
geometry of metric will not be asymptotically flat. For this value of
$\lambda$, deflection angle $\tilde{\alpha}$ for Kalb-Ramond traversable WH
can be obtained as:

$\tilde{\alpha}\simeq\frac{r_{0}}{\sigma}+\pi\left(\frac{r_{0}}{4\sigma}\right)^{2}+\frac{w_{e}^{2}}{w_{\infty}^{2}}\frac{r_{0}}{\sigma}+\frac{r_{0}^{2}\pi}{8\sigma^{2}}\frac{w_{e}^{2}}{w_{\infty}^{2}}+\mathcal{O}({r_{0}^{3}}).$
(11)

The above result shows that the light rays are moving in plasma medium. If we
remove plasma effects, this angle will convert into the non-plasma medium. The
deflection angle $\tilde{\alpha}$ depends on minimal radius $r_{0}$ and the
impact parameter $\sigma$, where bending angle is directly proportional to
$r_{0}$ and inversely proportional to $\sigma$.

##  IV Graphical Behavior of Deflection Angle

This section is based on the explanation of the graphical analysis of
deflection angle for Kalb-Ramond traversable WH solution in plasma medium. For
this purpose, we analyze the deflection angle with respect to the impact
parameter $\sigma$ and the minimal radius $r_{0}$.

###  IV.1 $\tilde{\alpha}$ versus $\sigma$

Figure 1: $\tilde{\alpha}$ versus $\sigma$ .

Figure 1 shows the behavior of $\tilde{\alpha}$ with respect to the impact
parameter $\sigma$ for plasma impact
$\frac{\omega_{e}}{\omega_{\infty}}=10^{-1}$. For $r_{0}\leq 3$, we observe
that, in first graph as $\sigma$ increases, $\tilde{\alpha}$ decreases
exponentially and shows a convergent behavior; which converges to zero. While
on the other hand, we evaluate that with the increase of $r_{0}$, the
deflection angle also increases. Similarly for $r_{0}\geq 3$ in second plot,
the angle shows a decreasing behavior when value of impact parameter
increases.

Figure 2: $\tilde{\alpha}$ versus $\sigma$.

Figure 2 shows that for $r_{0}\leq 3$, $r_{0}>3$ by assuming plasma impact
$\frac{\omega_{e}}{\omega_{\infty}}=10^{-2}$, one can analyze the similar
behavior like that of $\frac{\omega_{e}}{\omega_{\infty}}=10^{-1}$. We also
investigate that for small values of $r_{0}$, as $\sigma$ increases,
$\tilde{\alpha}$ exponentially decreases. Moreover, it is to be observed that
as the value of $\sigma$ increases, $\tilde{\alpha}$ decreases. It shows that
deflection angle has inverse relation with $\sigma$ and direct relation with
$r_{0}$.

###  IV.2  $\tilde{\alpha}$ versus $r_{0}$

Figure 3: $\tilde{\alpha}$ versus $r_{0}$.

Figure 3 shows the behavior of deflection angle $\tilde{\alpha}$ of light with
respect to $r_{0}$ for the impact parameter $0<\sigma<50$ and plasma impact
$\frac{\omega_{e}}{\omega_{\infty}}=10^{-1}$. We examine that for
$0<\sigma<50$, the left graph indicates the linear behavior as $r_{0}$
increases the bending angle $\tilde{\alpha}$ also increases and as $\sigma$
increases, deflection angle decreases. For larger $\sigma$ angle approaches
towards zero. The similar case is with the right plot for $\sigma>50$ that
angle increases with increasing the values of $r_{0}$ and vice versa.

Figure 4: $\tilde{\alpha}$ versus $r_{0}$.

Figure 4 shows that for $\sigma>50$ and $0<\sigma\leq 50$ having plasma
impact’s value $\frac{\omega_{e}}{\omega_{\infty}}=10^{-2}$, we can analyze
similar behavior as for case $\frac{\omega_{e}}{\omega_{\infty}}=10^{-1}$. We
evaluate that as values of $r_{0}$ increases, deflection angle also increases
and as $\sigma$ increases, bending angle decreases which shows a divergent
behavior of graphs. Furthermore, we investigate that bending angle
$\tilde{\alpha}$ has direct relation with impact parameter $r_{0}$ and inverse
relation with $\sigma$.

##  V Deflection Angle using Keeton and Petters Method

Keeton and Petters established a completely beneficial framework for computing
corrections in a standard asymptotically flat metric theory of gravity
Keeton:2005jd ; Keeton:2006sa . The central focus is to illustrate a way to
manage lensing in computing gravity theories using PPN corrections upto third
order.

The non-linear spherically symmetric and asymptotically Minkowski spacetime is
defined by:

$ds^{2}=-A\left(r\right)dt^{2}+B\left(r\right)dr^{2}+C\left(r\right)d\Omega^{2}.$
(12)

Due to the spherical symmetry, geodesics of Eq.(12) lie in equatorial plane
while $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ represents the
standard unit metric. The above equation becomes:

$ds^{2}=-A\left(r\right)dt^{2}+B\left(r\right)dr^{2}+C\left(r\right)d\theta^{2}.$
(13)

For $A\left(r\right)\rightarrow 1$, $B\left(r\right)\rightarrow 1$ and
$C\left(r\right)\rightarrow r^{2}$, the spacetime metric is flat in the
absence of lens and we suppose that $A$, $B$ and $C$ are all +ve in the
outside region of the lens from where the required light rays pass. The Kalb-
Ramond traversable WH solution can be presented as:

$ds^{2}=-dt^{2}+\frac{dr^{2}}{1-(\frac{r}{r_{0}})^{\frac{2}{1-2\lambda}}}+r^{2}d\theta^{2},$
(14)

where the metric functions are:

$A\left(r\right)=1,~{}~{}~{}~{}~{}~{}$ (15)

and

$~{}~{}~{}~{}~{}~{}~{}\\\
B\left(r\right)=\frac{1}{1-(\frac{r}{r_{0}})^{\frac{2}{1-2\lambda}}}.$ (16)

To find the PPN coefficients, we compare the coefficients of the extended form
of metric function with the coefficients of standard form of general metric in
a PPN series to third order. The general form of PPN series for Eq.(5.1) given
as Keeton:2005jd :

$A(r)=1+2a_{1}\left(\frac{\phi}{c^{2}}\right)+2a_{2}\left(\frac{\phi}{c^{2}}\right)^{2}+2a_{3}\left(\frac{\phi}{c^{2}}\right)^{3}...$
(17)
$B(r)=1-2b_{1}\left(\frac{\phi}{c^{2}}\right)+4b_{2}\left(\frac{\phi}{c^{2}}\right)^{2}-8b_{3}\left(\frac{\phi}{c^{2}}\right)^{3}...$
(18)

Here, $\phi$ denotes the three dimensional Newtonian prospective. In our
paper, we consider,

$\left(\frac{\phi}{c^{2}}\right)=\frac{r_{0}}{r}.$ (19)

We get the values of the coefficients of PPN metric as

$a_{1}=0,~{}~{}a_{2}=0,~{}~{}a_{3}=0,~{}~{}b_{1}=\frac{1}{2},~{}~{}b_{2}=\frac{1}{4},~{}~{}b_{3}=\frac{1}{8}.$

After finding the PPN coefficients, we then determine the coefficients in the
extended form of the bending angle. The extended form of light deflection
angle can be written as follows

$\alpha\left(b\right)=A_{1}\left(\frac{m}{b}\right)+A_{2}\left(\frac{m}{b}\right)^{2}+A_{3}\left(\frac{m}{b}\right)^{3}+O\left(\frac{m}{b}\right)^{4}.$
(20)

We take $r_{0}=m$ and $\sigma=b$. To compute the coefficients of bending
angle, we utilize the following equation

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle
2\left(a_{1}+b_{1}\right),$ $\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\left(2a_{1}^{2}\
-a_{2}+a_{1}b_{1}+b_{2}-\frac{b_{1}^{2}}{4}\right)\pi,$ $\displaystyle A_{3}$
$\displaystyle=$
$\displaystyle\frac{2}{3}[35a_{1}^{3}+15a_{1}^{2}b_{1}-3a_{1}\left(10a_{2}+b_{1}^{2}-4b_{2}\right)$
(21) $\displaystyle+$ $\displaystyle
6a_{3}+b_{1}^{3}-6a_{2}b_{1}-4b_{1}b_{2}+8b_{3}].$

Using the Eq.(21), we can determine the values of coefficient presented as
follows

$A_{1}=1,~{}~{}~{}~{}~{}A_{2}=\frac{3\pi}{16}~{}~{}~{}~{},A_{3}=\frac{5}{12}.$

After substituting the values of coefficients in Eq.(20), we get the resulting
bending angle of Kalb-Ramond traversable WH as:

$\tilde{\alpha}\left(\sigma\right)=\left(\frac{r_{0}}{\sigma}\right)+\frac{3\pi}{16}\left(\frac{r_{0}}{\sigma}\right)^{2}+\frac{5}{12}\left(\frac{r_{0}}{\sigma}\right)^{3}+O\left(\frac{r_{0}}{\sigma}\right)^{4}.$
(22)

This expression shows the deflection angle by using Keeton and Petters method
which is dependent upon $r_{0}$ and $\sigma$. The bending angle is directly
proportional to $r_{0}$ and inversely proportional to $\sigma$.

##  VI Deflection Angle of Photon in Dark Matter Medium

Here, in this section, we examine how dark matter influences the weak
deflection angle. For this purpose, we take the refractive index for dark
matter medium Latimer:2013rja ; Latimer:2017lwm ; Ovgun:2020yuv :

$n(\omega)=1+{\beta}A_{0}+A_{2}{\omega}^{2}.$ (23)

The 2-dimensional optical geometry for Kalb-Ramond traversable WH solution is:

$dt^{2}=n^{2}\left(\frac{dr^{2}}{1-(\frac{r}{r_{0}})^{\frac{2}{1-2\lambda}}}+r^{2}d\phi^{2}\right),$
(24)

with the condition:

$\frac{dt}{d\phi}|_{C_{R}}=n\left(r^{2}\right)^{\frac{1}{2}}.$ (25)

Applying this condition to a non-singular domain $C_{R}$ outside of the light
ray will yield the deflection angle. Hence, with the help of GBT we can
calculate the weak deflection angle for Kalb-Ramond traversable WH solution in
dark matter medium.

$\lim_{R\rightarrow\infty}\int_{0}^{\pi+\alpha}\left[\mathcal{K}_{g}\frac{dt}{d\phi}\right]|_{C_{R}}d\phi=\pi-\lim_{R\rightarrow\infty}\int\int_{D_{R}}\mathcal{K}dS.$
(26)

We compute the Guassian optical curvature as follows:

$\mathcal{K}=\frac{r_{0}}{2r_{3}\left(1+{\beta}A_{0}+A_{2}{\omega}^{2}\right)}.$
(27)

After this, we determine

$\lim_{R\rightarrow\infty}\mathcal{K}_{g}\frac{dt}{d\phi}|_{C_{R}}=1.$ (28)

Now, when we apply limit $R\rightarrow\infty$, then we can evaluate the
deflection angle for Kalb-Ramond traversable WH solution by using GBT as below

$\tilde{\alpha}=-\int_{0}^{\pi}\int_{\frac{\sigma}{\sin\phi}}^{\infty}\mathcal{K}dS.$
(29)

We get the weak deflection angle of Kalb-Ramond traversable WH solution in
dark matter medium after putting the value of $\mathcal{K}$ and $dS$

$\tilde{\alpha}\simeq\frac{r_{0}}{\sigma\left(1+\beta
A_{0}+A_{2}\omega^{2}\right)^{6}}+\frac{r_{0}^{2}\beta^{6}A_{0}^{12}\pi}{16\sigma^{2}\left(1+\beta
A_{0}+A_{2}\omega^{2}\right)^{6}}|+\mathcal{O}(r_{0}^{3}).$ (30)

This result shows that when the effect of dark matter vanishes, the bending
angle reduces to the deflection angle in non-plasma medium. The effect of dark
matter shows large deflection than in general case. The deflection angle
$\tilde{\alpha}$ depends upon $\sigma$ and $r_{0}$. We can see that dark
matter gives a large deflection angle than in general case.

##  VII Conclusion

This paper is concerned to calculate the bending angle $\tilde{\alpha}$ for
Kalb-Ramond traversable WH solution in plasma and dark matter mediums. We have
evaluated the metric and explained its mathematical interpretation. For the
calculation of bending angle, we have obtained the Guassian optical curvature.
After that, we applied GBT and derived the value of deflection angle for Kalb-
Ramond traversable WH solution in plasma medium Eq.(11) for
$\lambda=\frac{3}{2}$.

If we neglect or approach the value of the plasma effect
$\frac{\omega_{e}}{\omega_{\infty}}$ to zero, then effect of the plasma
vanished and the resulting expression of bending angle for plasma medium
reduced to the non-plasma case.

Graphical behavior of deflection angle in plasma medium has also been
calculated.

$\tilde{\alpha}$ with respect to impact parameter $\sigma$:

For $r_{0}\leq 3$ and $r_{0}>3$ having effect of plasma
$\frac{\omega_{e}}{\omega_{\infty}}=10^{-1}$, the deflection angle
$\tilde{\alpha}$ gradually approaches towards zero as $\sigma$ increases, in
both graphs. We have also observed that $\tilde{\alpha}$ exponentially
decreases and indicate a convergent behavior for large values of $\sigma$. We
have examined that for $r_{0}\leq 3$ and $r_{0}>3$ having plasma effect
$\frac{\omega_{e}}{\omega_{\infty}}=10^{-2}$ as $r_{0}$ approaches to
infinity, $\tilde{\alpha}\rightarrow\infty$ at smaller $\sigma$. We have also
investigated that $\tilde{\alpha}$ has inverse relation with $\sigma$ and
direct relation with $r_{0}$ .

$\tilde{\alpha}$ with respect to minimal radius $r_{0}$:

We have analyzed that for $0<\sigma\leq 50$ and $\sigma>50$ supposing
$\frac{\omega_{e}}{\omega_{\infty}}=10^{-1}$, the deflection angle
$\tilde{\alpha}$ increases as value of $r_{0}$ increases. A linear behavior of
both the graphs has examined. We have also investigated that as $\sigma$
increases, angle decreases and vice versa. We have reviewed that for
$0<\sigma\leq 50$ and $\sigma>50$ at
$\frac{\omega_{e}}{\omega_{\infty}}=10^{-2}$, deflection angle
$\tilde{\alpha}$ exhibited similar behavior as that for
$\frac{\omega_{e}}{\omega_{\infty}}=10^{-1}$. We have also examined that
$\tilde{\alpha}$ increases for larger $r_{0}$ and $\tilde{\alpha}$ decreases
as $\sigma$ increases. We have investigated a divergent behavior in both
cases. We have evaluated that $\tilde{\alpha}$ has direct relationship with
$r_{0}$ and inverse relation with $\sigma$.

Furthermore, we have also derived the deflection angle by using Keeton and
Petters method which is the approximate form of GL and used spherically
symmetric spacetime metric. For this purpose, we have found coefficients of
PPN metric by comparing the expanded metric function with standard PPN metric.
Later on, we have determined the coefficients of the bending angle and again
by comparing them with general form of Schwarzschild metric to get final
results given in Eq.(22).

Lastly, we have calculated the value of deflection angle $\tilde{\alpha}$ for
Kalb-Ramond traversable WH solution in the dark matter medium. Dark matter
contained such particles which cannot absorb, emit or reflect light rays. For
this purpose, we have calculated Gaussian optical curvature Eq.(27) and
determined the deflection angle (30).

In plasma, Keeton and Petters method and Dark matter mediums case, the results
reduced to the non plasma case. The resulting equations depend upon the
minimal radius $r_{0}$ and impact parameter $\sigma$ showing that deflection
angle is directly proportional to $r_{0}$ and inversely to $\sigma$. The plots
between angle and minimal radius or impact parameter demonstrates that when
$r_{0}$ increases, bending angle also move towards infinity. Also we observe a
convergent and divergent behavior w.r.t $\sigma$ and $r_{0}$ respectively.
When we remove the effect of plasma and dark matter, the resultant expression
becomes similar to that of Bumblebee traversable wormhole solution. In case of
Keeton and Petters method only the first term becomes similar to the general
case.

###### Acknowledgements.

A. Ö. and R. P. would like to acknowledge networking support by the COST
Action CA18108 - Quantum gravity phenomenology in the multi-messenger approach
(QG-MM).

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# Supplemental Material for “Design of supercurrent diode by vortex phase
texture”

Yuri Fukaya  CNR-SPIN, I-84084 Fisciano (SA), Italy, c/o Universitá di
Salerno, I-84084 Fisciano (SA), Italy  Maria Teresa Mercaldo  Dipartimento di
Fisica “E. R. Caianiello”, Università di Salerno, IT-84084 Fisciano (SA),
Italy  Daniel Margineda  NEST Istituto Nanoscienze-CNR and Scuola Normale
Superiore, I-56127, Pisa, Italy  Alessandro Crippa  NEST Istituto Nanoscienze-
CNR and Scuola Normale Superiore, I-56127, Pisa, Italy  Elia Strambini  NEST
Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127, Pisa, Italy
Francesco Giazotto  NEST Istituto Nanoscienze-CNR and Scuola Normale
Superiore, I-56127, Pisa, Italy  Carmine Ortix  Dipartimento di Fisica “E. R.
Caianiello”, Università di Salerno, IT-84084 Fisciano (SA), Italy  Mario Cuoco
CNR-SPIN, I-84084 Fisciano (SA), Italy, c/o Universitá di Salerno, I-84084
Fisciano (SA), Italy

###### Abstract

In the Supplemental Material we provide details about the model and the
methodology. We present how the supercurrent rectification gets modified for
mirror and time-reversal symmetric vortex configurations. We discuss and
evaluate the general conditions to maximize the amplitude of the supercurrent
rectification for a current phase relation that includes only first and second
harmonics. We present the dependence of the first harmonic of the supercurrent
by considering the lowest order Cooper pairs tunneling process.

##  I Model Hamiltonian and methodology

We present the model and the methodology employed to calculate the Josephson
current of the superconducting weak-link in the presence of vortices with
different winding number. The Josephson junction is schematically shown in
Fig. 1 (a) with the system size given by
$L_{y}\times(L^{\mathrm{L}}_{x}+L^{\mathrm{N}}_{x}+L^{\mathrm{R}}_{x})$ and
the superconducting order parameter
$\hat{\Delta}_{\mathrm{L,R}}e^{i\varphi_{\mathrm{L,R}}}$. Hereafter, $L$, $R$,
and $N$ stand for the indices of left and right-side superconductors, and the
normal metal layer, respectively. The Hamiltonian is written as

$\displaystyle\hat{\mathcal{H}}=\sum_{\langle
x,y;x^{\prime},y^{\prime}\rangle}\hat{C}^{\dagger}(x,y)\hat{H}(x,y;x^{\prime},y^{\prime};\varphi)\hat{C}(x^{\prime},y^{\prime}),$
(1)

with $\varphi=\varphi_{\mathrm{L}}-\varphi_{\mathrm{R}}$ being the phase
difference between the superconducting order parameters in the two sides of
the junction and $\bm{r}=(x,y)$ indicating the coordinate in the real space.
Here, $\langle x,y;x^{\prime},y^{\prime}\rangle$ indicates the summation
within the nearest-neighbor hopping and
$\hat{C}^{\dagger}(x,y)=[c^{\dagger}_{x,y,\uparrow},c^{\dagger}_{x,y,\downarrow},c_{x,y,\uparrow},c_{x,y,\downarrow}]$
denotes the creation operator. Then
$\hat{H}(x,y;x^{\prime},y^{\prime};\varphi)$ is given by

$\displaystyle\hat{H}(x,y;x^{\prime},y^{\prime};\varphi)$
$\displaystyle=\hat{H}^{\mathrm{L}}(x,y;x^{\prime},y^{\prime};\varphi)+\hat{H}^{\mathrm{R}}(x,y;x^{\prime},y^{\prime})+\hat{H}^{\mathrm{N}}(x,y;x^{\prime},y^{\prime})+\hat{H}^{\mathrm{L}}_{J}(x,y;x^{\prime},y)+\hat{H}^{\mathrm{R}}_{J}(x,y;x^{\prime},y),$
(2)

with the Hamiltonian in the Nambu space $\hat{H}^{\mathrm{L,N,R}}$ that is
constructed by the on site energy term $\hat{u}$, the energy gap function with
spin-singlet $s$-wave state
$\hat{\Delta}_{\mathrm{L,R}}e^{\varphi_{\mathrm{L,R}}}$, the nearest neighbor
hopping terms $\hat{t}_{x,y}$, and the tunneling part
$\hat{H}^{\mathrm{L,R}}_{J}$. The local term in the normal state is

$\displaystyle\hat{u}(x,y)=\begin{pmatrix}-\varepsilon&0\\\
0&-\varepsilon\end{pmatrix},$ (3)

and the nearest-neighbor hopping,

$\displaystyle\hat{t}_{x}(x,y;x+1,y)$
$\displaystyle=\hat{t}^{\dagger}_{x}(x,y;x-1,y)=\begin{pmatrix}-t&0\\\
0&-t\end{pmatrix},$ (4) $\displaystyle\hat{t}_{y}(x,y;x,y+1)$
$\displaystyle=\hat{t}^{\dagger}_{x}(x,y;x,y-1)=\begin{pmatrix}-t&0\\\
0&-t\end{pmatrix}.$ (5)

The superconducting order parameter $\hat{\Delta}_{\mathrm{L,R}}$ has spin-
singlet/$s$-wave symmetry in both superconductors:

$\displaystyle\hat{\Delta}_{\mathrm{X}}(x,y)$
$\displaystyle=\begin{pmatrix}0&\Delta_{\mathrm{X}}\\\
-\Delta_{\mathrm{X}}&0\end{pmatrix},$ (6)

with X=L,R. In the present study, the pairing amplitude are on site and
included in the local term of Hamiltonian for the left and right-side
superconductors. The tunneling Hamiltonian is expressed as

$\displaystyle\hat{t}_{J}(x,y;x+1,y)=\hat{t}^{\dagger}_{J}(x,y;x-1,y)=t_{\mathrm{int}}\begin{pmatrix}-t&0\\\
0&-t\end{pmatrix},$ (7)

with $t_{int}$ the charge transfer amplitude at the interface setting out the
transparency of the junction.

Figure 1: (a) Sketch of the Josephson junction in the $xy$-plane. Blue,
yellow, and red circles denote the atomic sites in the real space
$\bm{r}=(x,y)$. The system size in Josephson junctions is set by
$L_{y}\times(L^{\mathrm{L}}_{x}+L^{\mathrm{N}}_{x}+L^{\mathrm{R}}_{x})$. The
orange circle is an example of a phase vortex in superconductors. The blue and
red sites indicate the superconducting domain, while the yellow region stands
for the normal metal layers. (b) Phase of the vortex
$\varphi^{\mathrm{X}}_{\mathrm{v}}(\bm{r})$ and corresponding superconducting
order parameter $\Theta^{\mathrm{X}}(\bm{r})$ in the real space with X=L,R. We
also show the corresponding notations for the phase vortex for each side.
$\xi_{0}$ indicates the size of the vortex core wheareas the amplitude
superconducting order parameter vanishes.

The phase of the superconducting order parameter associated to the vortex
configuration is represented in Fig. 1 (b). At the position $\bm{r}=(x,y)$,
the phase vortex in the left-side superconductor is given by

$\displaystyle\varphi^{\mathrm{L}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})=V^{\mathrm{L}}_{0}\arg[(x-x^{\mathrm{L}}_{0})+i(y-y^{\mathrm{L}}_{0})],$
(8)

with the position of the core
$\bm{r}^{\mathrm{L}}_{0}=(x^{\mathrm{L}}_{0},y^{\mathrm{L}}_{0})$. In a
similar way, one can introduce the phase vortex at the core
$\bm{r}^{\mathrm{R}}_{0}=(x^{\mathrm{R}}_{0},y^{\mathrm{R}}_{0})$ in the
right-side superconductor as

$\displaystyle\varphi^{\mathrm{R}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{R}}_{0})$
$\displaystyle=\pi-\bar{\varphi}^{\mathrm{R}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{R}}_{0}),$
(9)
$\displaystyle\bar{\varphi}^{\mathrm{R}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{R}}_{0})$
$\displaystyle=V^{\mathrm{R}}_{0}\arg[(x-x^{\mathrm{R}}_{0})+i(y-y^{\mathrm{R}}_{0})],$
(10)

due to the mirror symmetry in the $yz$-plane. Here, $V^{\mathrm{X}}_{0}$
denotes the vortex winding.

The pair potential is described as

$\displaystyle\Delta_{\mathrm{X}}=\begin{cases}\Theta(\bm{r},\bm{r}^{\mathrm{X}}_{0})e^{i\varphi^{\mathrm{X}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{X}}_{0})}&\text{with
phase vortex}\\\ |\Delta_{0}|&\text{without phase vortex}\end{cases},$ (11)

with the energy gap amplitude $|\Delta_{0}|$. In the former case, the energy
gap amplitude $\Theta(\bm{r})$ is modified by the position:

$\displaystyle\Theta^{\mathrm{X}}(\bm{r},\bm{r}^{\mathrm{X}}_{0})$
$\displaystyle=|\Delta_{0}|\tilde{\Theta}^{\mathrm{X}}(\bm{r},\bm{r}^{\mathrm{X}}_{0}),$
(12)
$\displaystyle\tilde{\Theta}^{\mathrm{X}}(\bm{r},\bm{r}^{\mathrm{X}}_{0})$
$\displaystyle=\tanh\left[\frac{|\bm{r}-\bm{r}^{\mathrm{X}}_{0}|}{\xi^{\mathrm{X}}_{0}}\right],$
(13)

with the size of the vortex $\xi^{\mathrm{X}}_{0}$. We set the parameters as
$\varepsilon=-0.25t$, $|\Delta_{0}|=0.02t$, and $t_{\mathrm{int}}=0.90$.

We introduce the formulations to calculate the Josephson current for the
superconducting weak-link. The free energy in a Josephson junction at zero
temperature is evaluated as

$\displaystyle F(\varphi)=\frac{1}{L_{y}L_{x}}\sum_{E<0}E(\varphi),$ (14)

with the total length along the $x$-direction
$L_{x}=L^{\mathrm{L}}_{x}+L^{\mathrm{N}}_{x}+L^{\mathrm{R}}_{x}$ and along the
$y$-direction $L_{y}$. Here, $E(\varphi)$ stands for the eigenvalue of
$\hat{H}(x,y;x^{\prime},y^{\prime};\varphi)$. Then, the current phase relation
of the Josephson current in this system is evaluated by a variation of the
free energy with respect to the phase bias across the junction:

$\displaystyle I(\varphi)=\frac{\partial F(\varphi)}{\partial\varphi}.$ (15)

In this study, we adopt the effective coordinate as shown in Fig. 2 (a). The
reduced coordinates $\tilde{x}$ and $\tilde{y}$ are defined as

$\displaystyle\tilde{x}$
$\displaystyle=\begin{cases}\frac{x^{\mathrm{L}}}{L^{\mathrm{L}}_{x}}+\tilde{x}^{\mathrm{N}}_{L}&\tilde{x}<0\\\
\frac{x^{\mathrm{R}}}{L^{\mathrm{R}}_{x}}-\tilde{x}^{\mathrm{N}}_{R}&\tilde{x}>0\end{cases},$
(16) $\displaystyle\tilde{y}$ $\displaystyle=\frac{y}{L_{y}},$ (17)

with

$\displaystyle\tilde{x}^{\mathrm{N}}_{L}$
$\displaystyle=\frac{L^{\mathrm{N}}_{x}/2}{L^{\mathrm{L}}_{x}+L^{\mathrm{N}}_{x}/2},$
(18) $\displaystyle\tilde{x}^{\mathrm{N}}_{R}$
$\displaystyle=\frac{L^{\mathrm{N}}_{x}/2}{L^{\mathrm{R}}_{x}+L^{\mathrm{N}}_{x}/2},$
(19)

and the effective size of the vortex:

$\displaystyle\tilde{\xi}^{\mathrm{L}}_{0}(x)$
$\displaystyle=\frac{\xi^{\mathrm{L}}_{0}}{L^{\mathrm{L}}_{x}},\hskip
5.69054pt\tilde{\xi}^{\mathrm{L}}_{0}(y)=\frac{\xi^{\mathrm{L}}_{0}}{L_{y}},$
$\displaystyle\tilde{\xi}^{\mathrm{R}}_{0}(x)$
$\displaystyle=\frac{\xi^{\mathrm{R}}_{0}}{L^{\mathrm{R}}_{x}},\hskip
5.69054pt\tilde{\xi}^{\mathrm{R}}_{0}(y)=\frac{\xi^{\mathrm{R}}_{0}}{L_{y}}.$

We also define the effective coordinate for the position of the vortex core as

$\displaystyle\tilde{x}^{\mathrm{L}}_{0}$
$\displaystyle=\frac{x^{\mathrm{L}}_{0}}{L^{\mathrm{L}}_{x}}+\tilde{x}^{\mathrm{N}}_{L},$
(20) $\displaystyle\tilde{x}^{\mathrm{R}}_{0}$
$\displaystyle=\frac{x^{\mathrm{R}}_{0}}{L^{\mathrm{R}}_{x}}-\tilde{x}^{\mathrm{N}}_{R},$
(21) $\displaystyle\tilde{y}^{\mathrm{L,R}}_{0}$
$\displaystyle=\frac{y_{0}}{L_{y}}.$ (22)

In this study, we choose the size as $L^{\mathrm{N}}_{x}=10$,
$L^{\mathrm{R}}_{x}=L^{\mathrm{L}}_{x}=aL_{y}$, $L_{y}=30$ with the aspect
ratio $a$, and $\xi^{\mathrm{L}}_{0}=\xi^{\mathrm{R}}_{0}=10$.

##  II Supercurrent rectification: role of mirror and time-reversal symmetries

Figure 2: (a) Schematic illustration of the superconducting weak link in the
representation of reduced coordinates. $\tilde{x}^{\mathrm{N}}_{0}$ denotes
the effective width of the normal state layer in a junction. We explicitly use
the effective positions
$(\tilde{x}^{\mathrm{L}}_{0},\tilde{y}^{\mathrm{L}}_{0})$ and
$(\tilde{x}^{\mathrm{R}}_{0},\tilde{y}^{\mathrm{R}}_{0})$ in the left and
right-side superconductors, respectively. Black circles correspond to the
position of a vortex core in (b)-(i). (b)-(e) Free energy as a function of the
phase differences in the case of one vortex in (b)(c)(d) the left and (e) the
right-side superconductors at (b)
$(\tilde{x}^{\mathrm{L}}_{0},\tilde{y}^{\mathrm{L}}_{0})=(-0.52,-0.017)$,
(c)(d)
$(\tilde{x}^{\mathrm{L}}_{0},\tilde{y}^{\mathrm{L}}_{0})=(-0.52,0.017)$, and
(e) $(\tilde{x}^{\mathrm{R}}_{0},\tilde{y}^{\mathrm{R}}_{0})=(0.52,0.017)$. We
choose the winding number as (b)(c)(e) $V^{\mathrm{L}}_{0}=+1$ and (d)
$V^{\mathrm{L}}_{0}=-1$. The minimal value of the free energy shifts from
$\varphi=\varphi_{\mathrm{L}}-\varphi_{\mathrm{R}}=0$ in the presence of the
core away from the $x$-axis. The free energy is antisymmetric for the mirror
operations in the $yz$ and $zx$-planes and the reversion of the vortex
chirality. It indicates that the free energy is protected by the mirror
symmetry in Josephson junctions. (f)-(i) Current phase relations evaluated by
the corresponding free energy (b)-(e). The current phase relation is not
perfectly odd in the phase differences $\varphi$. The nonzero rectification is
generated by a phase vortex in one superconductor. The rectification reverses
by the mirror operation in Josephson junctions and the opposite vortex
chirality. We choose the aspect ratio as $a=3/2$ and the normalized
supercurrent as $I_{\mathrm{0}}=0.012|\Delta_{0}|$.

In this section, we show the relation for the value of the rectification for
mirror symmetric vortex configurations. We consider the Josephson junction in
the $xy$-plane as described in Fig. 2 (a) and investigate configurations for
which the vortex states are related by mirror symmetries through the $yz$ and
$zx$\- mirror planes, i.e. they are linked by $x\rightarrow-x$ and
$y\rightarrow-y$ transformations, or time-reversal symmetry by
$\hat{\Delta}\rightarrow\bar{\Delta}^{*}$, respectively. Then, one can show
that, in general, the following relations for the rectification amplitude
$\eta$ holds when considering vortex (arbitrary winding amplitude $V_{0}>0$)
or antivortex ($V_{0}<0$) configurations with a given position of the core
$(\tilde{x}_{0},\tilde{y}_{0})$ and mirror related core positions:

$\displaystyle\eta(\tilde{x}_{0},\tilde{y}_{0},V_{0})=-\eta(-\tilde{x}_{0},\tilde{y}_{0},V_{0}),$
(23)
$\displaystyle\eta(\tilde{x}_{0},\tilde{y}_{0},V_{0})=-\eta(\tilde{x}_{0},-\tilde{y}_{0},V_{0}),$
(24)
$\displaystyle\eta(\tilde{x}_{0},\tilde{y}_{0},V_{0})=-\eta(\tilde{x}_{0},\tilde{y}_{0},-V_{0})\,.$
(25)

This result can be also confirmed by explicit numerical evaluation of the
rectification amplitude for the superconducting weak-link. For instance, we
choose the following representative vortex core positions:
$\bm{r}^{\mathrm{L}}_{0}$ as
$(\tilde{x}^{\mathrm{L}}_{0},\tilde{y}^{\mathrm{L}}_{0})=(-0.52,-0.017)$ in
Fig. 2 (b),
$(\tilde{x}^{\mathrm{L}}_{0},\tilde{y}^{\mathrm{L}}_{0})=(-0.52,0.017)$ in
Fig. 2 (c)(d), and
$(\tilde{x}^{\mathrm{R}}_{0},\tilde{y}^{\mathrm{R}}_{0})=(0.52,0.017)$ in Fig.
2 (e). We also select the winding number as $V^{\mathrm{L}}_{0}=+1$ in Fig. 2
(b)(c)(e) and $V^{\mathrm{L}}_{0}=-1$ in Fig. 2 (d). We check the free energy
$F(\varphi)$ [Fig. 2 (b)-(e)] and the corresponding current phase relations
[Fig. 2 (f)-(i)]. Equations (23), (24), and (25) are obtained by comparing
Figs. 2 (c)(g) with Figs. 2 (e)(i), Figs. 2 (b)(f) and Figs. 2 (d)(h),
respectively. Based on the results in Fig. 2, regarding a combination of
mirror and time-reversal symmetries, one can also obtain that a change from
vortex to antivortex and of the vortex core position does not alter the sign
and amplitude of the rectification:

$\displaystyle\eta(\tilde{x}_{0},\tilde{y}_{0},V_{0})=\eta(-\tilde{x}_{0},\tilde{y}_{0},-V_{0}).$
(26)

##  III Condition to maximize the amplitude of the supercurrent rectification

Figure 3: Rectification $\eta$ of $I(\varphi)=I_{1}\sin\varphi+I_{2}\sin
2\varphi+J_{1}\cos\varphi$ (a) for $(I_{1},J_{1})$ space and (b) as a function
of $I_{2}$. We fix the parameters as (a) $I_{2}=I_{0}$ and (b)
$I_{1}=J_{1}=I_{0}$.

In this section, we calculate the optimal conditions to maximize the
supercurrent rectification starting from a generic current phase relation that
includes the first harmonics (even and odd parity) and the second harmonic
(odd parity). Let us then assume the following current phase relation:

$\displaystyle I(\varphi)=I_{1}\sin\varphi+I_{2}\sin
2\varphi+J_{1}\cos\varphi.$

In order to find the optimal regimes for maximal rectification, one can
evaluate $\eta$ as a function of the amplitudes $I_{1}$, $I_{2}$, and $J_{1}$.
First, we calculate $\eta$ for $(I_{1},J_{1})$ at $I_{2}=I_{0}$ in Fig. 3 (a).
The maximum value of $|\eta|$ appears around $|I_{1}|=|J_{1}|\sim I_{0}$.
Based on this point, we also calculate $\eta$ as a function of $I_{2}$ at
$I_{1}=J_{1}=I_{0}$ [Fig. 3 (b)]. The amplitude of $\eta$ is enhanced around
$I_{2}=\pm 0.5I_{0}$. Thus, both $|I_{1}|$ and $|J_{1}|$ should be comparable
to maximize the rectification in Josephson current.

##  IV Evaluation of first-harmonic component of supercurrent

The first-harmonic components of the Josephson current at a position $\bm{r}$
can be estimated by considering the direct Cooper pair tunnelling process as

$\displaystyle j(\varphi;\bm{r},\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle\propto\mathrm{Im}[\Delta_{\mathrm{L}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})\Delta^{*}_{\mathrm{R}}],$
(27)

where $\bm{r}^{\mathrm{L}}_{0}$ is the position of vortex core in the left-
side superconductor. In the presence of a vortex in the left-side
superconductor, the superconducting order parameter is written as

$\displaystyle\Delta_{\mathrm{L}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=|\Delta_{0}|\tilde{\Theta}(\bm{r},\bm{r}^{\mathrm{L}}_{0})e^{i\varphi^{\mathrm{L}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})}e^{i\varphi_{\mathrm{L}}},$
(28) $\displaystyle\tilde{\Theta}(\bm{r},\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=\tanh\left[\frac{|\bm{r}-\bm{r}^{\mathrm{L}}_{0}|}{\xi^{\mathrm{L}}_{0}}\right],$
(29)
$\displaystyle\varphi^{\mathrm{L}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=V^{\mathrm{L}}_{0}\arg[(x-x_{0})+i(y-y_{0})],$ (30)

with the number of the phase winding $V^{\mathrm{L}}_{0}$, the size of the
phase vortex $\xi^{\mathrm{L}}_{0}$, the phase of the superconducting order
parameter $\varphi_{\mathrm{L}}$, and the energy gap amplitude $\Delta_{0}$.

Figure 4: (a) $\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$ and (b)
$\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$ for $\bm{r}^{\mathrm{L}}_{0}$ space at
$V^{\mathrm{L}}_{0}=+3$ and $a=3/2$. $\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$
and $\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$ for $V^{\mathrm{L}}_{0}=+1,+2$ are
shown in Fig. 3 of the main part. (b)(e)(g)
$\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$ and (d)(f)(h)
$\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$ for $\bm{r}^{\mathrm{L}}_{0}$ space at
(c)(d) $V^{\mathrm{L}}_{0}=+1$, (e)(f) $+2$, and (g)(h) $+3$. In (c)-(h), we
set the aspect ratio as $a=2/3$. Figure 5: (a)(c)
$\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$ and (b)(d)
$\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$ for $\bm{r}^{\mathrm{L}}_{0}$ space at
$V^{\mathrm{L}}_{0}=+3$ and $a=1$. $\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$ and
$\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$ for $V^{\mathrm{L}}_{0}=+3$ are shown
in Fig. 3 of the main part.

For the right-side we have
$\Delta_{\mathrm{R}}=|\Delta_{0}|e^{i\varphi_{\mathrm{R}}}$ with the
superconducting phase bias expressed by $\varphi_{\mathrm{R}}$. Then, by
expanding the expression for $j(\phi)$ we obtain

$\displaystyle j(\varphi;\bm{r},\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle\propto\mathrm{Im}\left[|\Delta_{0}|^{2}\tanh\left[\frac{|\bm{r}-\bm{r}^{\mathrm{L}}_{0}|}{\xi^{\mathrm{L}}_{0}}\right]e^{i(\varphi_{\mathrm{L}}-\varphi_{\mathrm{R}})}e^{i\varphi^{\mathrm{L}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})}\right]$
$\displaystyle=|\Delta_{0}|\sin\varphi\tanh\left[\frac{|\bm{r}-\bm{r}^{\mathrm{L}}_{0}|}{\xi^{\mathrm{L}}_{0}}\right]\cos\varphi^{\mathrm{L}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})d\bm{r}+|\Delta_{0}|\cos\varphi\tanh\left[\frac{|\bm{r}-\bm{r}^{\mathrm{L}}_{0}|}{\xi^{\mathrm{L}}_{0}}\right]\sin\varphi^{\mathrm{L}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})d\bm{r},$
(31)

with $\varphi=\varphi_{\mathrm{L}}-\varphi_{\mathrm{R}}$. The resulting first-
harmonic Josephson current can be described by the spatial average of
$j(\varphi;\bm{r},\bm{r}^{\mathrm{L}}_{0})$ over the area of the
superconducting leads:

$\displaystyle\tilde{I}(\varphi;\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=\int_{S_{\mathrm{L}}}j(\bm{r},\bm{r}^{\mathrm{L}}_{0})d\bm{r}$
$\displaystyle\propto\sin\varphi\int_{S_{\mathrm{L}}}\tanh\left[\frac{|\bm{r}-\bm{r}^{\mathrm{L}}_{0}|}{\xi^{\mathrm{L}}_{0}}\right]\cos\varphi^{\mathrm{L}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})d\bm{r}+\cos\varphi\int_{S_{\mathrm{L}}}\tanh\left[\frac{|\bm{r}-\bm{r}^{\mathrm{L}}_{0}|}{\xi^{\mathrm{L}}_{0}}\right]\sin\varphi^{\mathrm{L}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})d\bm{r},$
(32)

with $S_{\mathrm{L}}$ being the area of the left-side superconductor. Based on
this analysis, we get that

$\displaystyle\tilde{I}(\varphi;\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle\propto
F_{\mathrm{c}}(\bm{r}^{\mathrm{L}}_{0})\sin\varphi+F_{\mathrm{s}}(\bm{r}^{\mathrm{L}}_{0})\cos\varphi,$
(33)

with

$\displaystyle F_{\mathrm{c}}(\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=\int_{S_{\mathrm{L}}}f_{\mathrm{c}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})d\bm{r},$
(34) $\displaystyle F_{\mathrm{s}}(\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=\int_{S_{\mathrm{L}}}f_{\mathrm{s}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})d\bm{r},$
(35)

where $S$ is the area of the superconducting lead.
$f_{\mathrm{c}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})$ and
$f_{\mathrm{s}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})$ are given by

$\displaystyle f_{\mathrm{c}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=\tanh\left[\frac{|\bm{r}-\bm{r}^{\mathrm{L}}_{0}|}{\xi^{\mathrm{L}}_{0}}\right]\cos\varphi_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0}),$
(36) $\displaystyle f_{\mathrm{s}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=\tanh\left[\frac{|\bm{r}-\bm{r}^{\mathrm{L}}_{0}|}{\xi^{\mathrm{L}}_{0}}\right]\sin\varphi_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0}).$
(37)

The contribution due to the
$\tilde{\Theta}(\bm{r},\bm{r}^{\mathrm{L}}_{0})=\tanh[{|\bm{r}-\bm{r}^{\mathrm{L}}_{0}|}/{\xi^{\mathrm{L}}_{0}}]$
term does not affect the main results for
$F_{\mathrm{c}}(\bm{r}^{\mathrm{L}}_{0})$ and
$F_{\mathrm{s}}(\bm{r}^{\mathrm{L}}_{0})$. Thus, we adopt
$\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})=\int_{S_{\mathrm{L}}}\cos\varphi_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})$
and
$\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})=\int_{S_{\mathrm{L}}}\sin\varphi_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})$
to proceed with the analysis of the first-harmonic content of the
supercurrent.

Firstly, we would like to point out that the analytical solutions of
$\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$ and
$\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$ can be explicitly derived for
$\int_{S_{\mathrm{L}}}\rightarrow\int^{L_{y}/2}_{-L_{y}/2}\int^{0}_{-L^{\mathrm{L}}_{x}}$
and they are given by:

$\displaystyle\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=\frac{\bar{R}_{1}-\bar{R}_{2}}{2}\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)+\frac{\bar{R}_{3}-\bar{R}_{4}}{2}\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)$
$\displaystyle-\frac{(-x^{\mathrm{L}}_{0})^{2}}{4}\left[\ln\frac{\bar{R}_{4}+\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}{\bar{R}_{4}-\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}-\ln\frac{\bar{R}_{1}+\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}{\bar{R}_{1}-+\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}\right]$
$\displaystyle-\frac{1}{4}(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{x})^{2}\left[\ln\frac{\bar{R}_{2}+\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}{\bar{R}_{2}-\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}-\ln\frac{\bar{R}_{3}+\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}{\bar{R}_{3}-\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}\right],$
(38) $\displaystyle\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=-\frac{\bar{R}_{4}-\bar{R}_{1}}{2}(-x^{\mathrm{L}}_{0})-\frac{\bar{R}_{2}-\bar{R}_{3}}{2}(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})$
$\displaystyle-\frac{1}{4}\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}\left[\ln\frac{\bar{R}_{1}-(-x^{\mathrm{L}}_{0})}{\bar{R}_{1}+(-x^{\mathrm{L}}_{0})}-\ln\frac{\bar{R}_{2}-(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})}{\bar{R}_{2}+(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})}\right]$
$\displaystyle-\frac{1}{4}\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}\left[\ln\frac{\bar{R}_{3}-(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})}{\bar{R}_{3}+(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})}-\ln\frac{\bar{R}_{4}-(-x^{\mathrm{L}}_{0})}{\bar{R}_{4}+(-x^{\mathrm{L}}_{0})}\right],$
(39)

for $V^{\mathrm{L}}_{0}=+1$,

$\displaystyle\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=-(-x^{\mathrm{L}}_{0})^{2}[\bar{\theta}_{4a}-\bar{\theta}_{1a}]+\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}[\bar{\theta}_{1a}-\bar{\theta}_{2a}]$
$\displaystyle-(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})^{2}[\bar{\theta}_{2a}-\bar{\theta}_{3a}-2\pi]+\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}[\bar{\theta}_{3a}-\bar{\theta}_{4a}],$
(40) $\displaystyle\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=-(-x^{\mathrm{L}}_{0})^{2}\ln\frac{\bar{R}_{4}}{\bar{R}_{1}}-(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})^{2}\ln\frac{\bar{R}_{2}}{\bar{R}_{3}}$
$\displaystyle+\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}\ln\frac{\bar{R}_{1}}{\bar{R}_{2}}+\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}\ln\frac{\bar{R}_{3}}{\bar{R}_{4}},$
(42)

for $V^{\mathrm{L}}_{0}=+2$, and

$\displaystyle\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=-\frac{3}{4}(-x^{\mathrm{L}}_{0})^{2}\ln\frac{\bar{R}_{1}+\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}{\bar{R}_{1}-\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}+\frac{3}{4}(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})^{2}\ln\frac{\bar{R}_{2}+\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}{\bar{R}_{2}-\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}$
$\displaystyle-\frac{3}{4}(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})^{2}\ln\frac{\bar{R}_{3}+\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}{\bar{R}_{3}-\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}+\frac{3}{4}(-x^{\mathrm{L}}_{0})^{2}\ln\frac{\bar{R}_{4}+\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)}{\bar{R}_{4}-\left(-\frac{L_{y}}{2}+y^{\mathrm{L}}_{0}\right)}$
$\displaystyle+\frac{5}{2}\frac{1}{\bar{R}_{1}}\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{3}+\frac{5}{2}\frac{1}{\bar{R}_{1}}(-x^{\mathrm{L}}_{0})^{2}\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)$
$\displaystyle-\frac{5}{2}\frac{1}{\bar{R}_{2}}\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{3}-\frac{5}{2}\frac{1}{\bar{R}_{2}}(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})^{2}\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)$
$\displaystyle+\frac{5}{2}\frac{1}{\bar{R}_{3}}\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{3}+\frac{5}{2}\frac{1}{\bar{R}_{3}}(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})^{2}\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)$
$\displaystyle-\frac{5}{2}\frac{1}{\bar{R}_{4}}\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{3}-\frac{5}{2}\frac{1}{\bar{R}_{4}}(-x^{\mathrm{L}}_{0})^{2}\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right),$
(43) $\displaystyle\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=-\frac{3}{4}\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}\ln\frac{\bar{R}_{1}-(-x^{\mathrm{L}}_{0})}{\bar{R}_{1}+(-x^{\mathrm{L}}_{0})}+\frac{3}{4}\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}\ln\frac{\bar{R}_{2}-(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})}{\bar{R}_{2}+(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})}$
$\displaystyle-\frac{3}{4}\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}\ln\frac{\bar{R}_{3}-(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})}{\bar{R}_{3}+(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})}+\frac{3}{4}\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}\ln\frac{\bar{R}_{4}-(-x^{\mathrm{L}}_{0})}{\bar{R}_{4}+(-x^{\mathrm{L}}_{0})}$
$\displaystyle-\frac{5}{2}\frac{1}{\bar{R}_{1}}(-x^{\mathrm{L}}_{0})^{3}-\frac{5}{2}\frac{1}{\bar{R}_{1}}(-x^{\mathrm{L}}_{0})\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}$
$\displaystyle+\frac{5}{2}\frac{1}{\bar{R}_{2}}(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})^{3}+\frac{5}{2}\frac{1}{\bar{R}_{2}}(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}$
$\displaystyle-\frac{5}{2}\frac{1}{\bar{R}_{3}}(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})^{3}-\frac{5}{2}\frac{1}{\bar{R}_{3}}(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}$
$\displaystyle+\frac{5}{2}\frac{1}{\bar{R}_{4}}(-x^{\mathrm{L}}_{0})^{3}+\frac{5}{2}\frac{1}{\bar{R}_{4}}(-x^{\mathrm{L}}_{0})\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2},$
(44)

for $V^{\mathrm{L}}_{0}=+3$, with

$\displaystyle\bar{R}_{1}$
$\displaystyle=\sqrt{(-x^{\mathrm{L}}_{0})^{2}+\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}},$
(45) $\displaystyle\bar{R}_{2}$
$\displaystyle=\sqrt{(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})^{2}+\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}},$
(46) $\displaystyle\bar{R}_{3}$
$\displaystyle=\sqrt{(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})^{2}+\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}},$
(47) $\displaystyle\bar{R}_{4}$
$\displaystyle=\sqrt{(-x^{\mathrm{L}}_{0})^{2}+\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)^{2}},$
(48)

and

$\displaystyle\bar{\theta}_{1a}$
$\displaystyle=\arg\left[-x^{\mathrm{L}}_{0}+i\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)\right],$
(49) $\displaystyle\bar{\theta}_{2a}$
$\displaystyle=\arg\left[(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})+i\left(\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)\right],$
(50) $\displaystyle\bar{\theta}_{3a}$
$\displaystyle=\arg\left[(-L^{\mathrm{L}}_{x}-x^{\mathrm{L}}_{0})+i\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)\right],$
(51) $\displaystyle\bar{\theta}_{4a}$
$\displaystyle=\arg\left[-x^{\mathrm{L}}_{0}+i\left(-\frac{L_{y}}{2}-y^{\mathrm{L}}_{0}\right)\right].$
(52)

These equations indicate that the vanishing conditions for the first harmonics
are given by: $\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})=0$ at
$x^{\mathrm{L}}_{0}=-0.5L^{\mathrm{L}}_{x}$ for $V^{\mathrm{L}}_{0}=+1,+3$ and
$\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})=0$ for $V^{\mathrm{L}}_{0}=+2$ by
$\bar{R}_{2}=\bar{R}_{1}$ and $\bar{R}_{4}=\bar{R}_{3}$, and
$\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})=0$ for $V^{\mathrm{L}}_{0}=+1,+2,+3$ at
$y^{\mathrm{L}}_{0}=0$ by $\bar{R}_{4}=\bar{R}_{1}$ and
$\bar{R}_{3}=\bar{R}_{2}$. Since these nodal lines are protected by the two-
fold rotational symmetry $C_{2}$ of the square/rectangle shape, they are
independent of the aspect ratio $a$. Only at $a=1$, that is, the square shape,
since
$\bar{\theta}_{4a}-\bar{\theta}_{1a}=\bar{\theta}_{1a}-\bar{\theta}_{2a}$ and
$\bar{\theta}_{2a}-\bar{\theta}_{3a}-2\pi=\bar{\theta}_{3a}-\bar{\theta}_{4a}$
of $\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$ for $V^{\mathrm{L}}_{0}=+2$, the
nodal structures for the first harmonics amplitudes appear along the diagonal
directions protected by the four-fold rotational symmetry $C_{4}$ shown in
Fig. 5 (c).

Here, we present another configuration for
$\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$ and
$\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$ shown in Figs. 4 and 5. At the aspect
ratio $a=3/2$, we show $\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$ and
$\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$ for $V^{\mathrm{L}}_{0}=+3$, which are
similar to those for $V^{\mathrm{L}}_{0}=+1$, in Fig. 4 (a)(b). This is the
reason why we discuss the case for $V^{\mathrm{L}}_{0}=+3$ at $a=1$ in the
main text. To confirm the role of the aspect ratio $a$ in this study, we also
calculate $\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$ and
$\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$ at $a=2/3$ for each
$V^{\mathrm{L}}_{0}=+1,+2,+3$ shown in Fig. 4 (c)-(h). As well as the results
at $a=3/2$, nodal lines of $\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$ and
$\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$ on the high-symmetric position are
fixed even at $a=2/3$, and vice versa. At $a=1$, we obtain the similar
distributions of $\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$ for
$V^{\mathrm{L}}_{0}=+1$ and $\bar{J}_{1}(\bm{r}^{\mathrm{L}}_{0})$ for
$V^{\mathrm{L}}_{0}=+1,+2$ in Fig. 5 (a)(b)(d), except for
$\bar{I}_{1}(\bm{r}^{\mathrm{L}}_{0})$ of $V^{\mathrm{L}}_{0}=+2$ [Fig. 5
(c)]. Owing to the lack of the four-fold rotational symmetry at $a\neq 1$
[Fig.4 (e)], the nodal lines along the diagonal direction at $a=1$ do not
cross at the center
$(\tilde{x}^{\mathrm{L}}_{0},\tilde{y}^{\mathrm{L}}_{0})=(-0.5,0)$ by $a\neq
1$. Thus, the aspect ratio $a$ also affects the determination of the
rectification in the Josephson current. We note that this discussion can be
applied to other cases such as a phase vortex in the right-side superconductor
and a few phase vortices in a superconductor.

For completeness, we also note that the second harmonic Josephson current can
be evaluated by considering the second order Cooper pair tunneling processes:

$\displaystyle\tilde{I}_{2}(\varphi;\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle\propto\int_{S}\mathrm{Im}[\Delta_{\mathrm{L}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})\Delta^{*}_{\mathrm{R}}\Delta_{\mathrm{L}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})\Delta^{*}_{\mathrm{R}}]d\bm{r}$
(53) $\displaystyle=|\Delta_{0}|^{4}\sin
2\varphi\int_{S}\tanh^{2}\left[\frac{|\bm{r}-\bm{r}^{\mathrm{L}}_{0}|}{\xi^{\mathrm{L}}_{0}}\right]\cos
2\varphi^{\mathrm{L}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})d\bm{r}+|\Delta_{0}|^{4}\cos
2\varphi\int_{S}\tanh^{2}\left[\frac{|\bm{r}-\bm{r}^{\mathrm{L}}_{0}|}{\xi^{\mathrm{L}}_{0}}\right]\sin
2\varphi^{\mathrm{L}}_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})d\bm{r}.$

Then, we can also find

$\displaystyle I_{2}(\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=|\Delta_{0}|^{4}\int_{S}\tanh^{2}\left[\frac{|\bm{r}-\bm{r}^{\mathrm{L}}_{0}|}{\xi^{\mathrm{L}}_{0}}\right]\cos
2\varphi_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})d\bm{r},$ (54)
$\displaystyle J_{\mathrm{2}}(\bm{r}^{\mathrm{L}}_{0})$
$\displaystyle=|\Delta_{0}|^{4}\int_{S}\tanh^{2}\left[\frac{|\bm{r}-\bm{r}^{\mathrm{L}}_{0}|}{\xi^{\mathrm{L}}_{0}}\right]\sin
2\varphi_{\mathrm{v}}(\bm{r},\bm{r}^{\mathrm{L}}_{0})d\bm{r},$ (55)

for $I_{2}(\bm{r}^{\mathrm{L}}_{0})$ and $J_{2}(\bm{r}^{\mathrm{L}}_{0})$.
Since both $I_{2}$ and $J_{2}$ are the $|\Delta_{0}|^{4}$ order, they are
generally smaller in amplitude than $I_{1}$ and $J_{1}$, as they are
proportional to $|\Delta_{0}|^{2}$. Then, close to the nodal lines of the
first harmonics, the second harmonic components are comparable in amplitude.