# Anomalous Josephson Hall effect in doped topological insulators with the
nematic superconductivity

R.S. Akzyanov  Dukhov Research Institute of Automatics, Moscow, 127055 Russia
Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, 141700
Russia  Institute for Theoretical and Applied Electrodynamics, Russian Academy
of Sciences, Moscow, 125412 Russia  A.L. Rakhmanov  Dukhov Research Institute
of Automatics, Moscow, 127055 Russia  Institute for Theoretical and Applied
Electrodynamics, Russian Academy of Sciences, Moscow, 125412 Russia

###### Abstract

We study the physics of the Josephson effect in nematic superconductors with
$E_{u}$ odd parity in the Ginzburg-Landau approach. Two-component vector
superconducting order parameter makes this effect rather unusual. We get that
the Meissner kernel has off-diagonal components. We derive current-phase
relations for different configurations of the junction, crystallographic axes
of the sample, and nematicity direction. We show that an anomalous Josephson
Hall effect can be observed in such a system without any magnetization. That
is, for definite orientations of the junction and crystal axes, a component of
the Josephson current along the junction is induced by the order parameter
phase difference across the contact. We also calculate the magnetic field
dependence of the maximum current through the junction. We find that the
period of the Fraunhofer oscillations of the maximum Josephson current depends
on the geometry of the junction, direction of the magnetic field, and
nematicity vector.

##  I Introduction

The Josephson effect is a general feature of superconducting phenomena [1, 2,
3]. It manifests itself as a generation of the supercurrent between two
superconducting pieces that are separated by a thin barrier. The value of this
current is controlled by the external gating that induces phase difference
between superconductors. This effect is not only of importance for many
superconducting devices but also provides important information about the
physical nature of the superconducting order. The discovery of topological
superconductivity opens a new prospect in the Josephson physics [4]. In
particular, the study of the Josephson currents in superconductor-
ferromagnetic (SF) heterostructures gives rise to the observation of a set of
new physical phenomena and the development of a new family of superconducting
devices [5]. An important property of the SF systems is the possibility to
govern the Josephson current by fine-tuning of magnetization of the F layer.

Recently, a significant interest arises to a “transverse” Josephson currents
that flows along the junction [6, 7, 8, 9, 10, 11, 12, 13, 14]. This
phenomenon is called Josephson Hall effect, anomalous Josephson Hall effect,
or tunnel Josephson Hall effect by an analogy with the anomalous Hall effect
in the normal state [15]. The Josephson Hall effect arises due to non-trivial
interplay between the magnetization and spin-orbit interaction [7, 9, 10, 11,
12, 13, 14] or unconventional superconductivity [6, 8]. Such a transverse
Josephson current can be comparable with the usual “longitudinal” Josephson
current [14].

Experiments with doped topological insulators such as CuxBi2Se3 [16, 17, 18,
19, 20, 21, 22], SrxBi2Se3 [23, 24, 25, 26, 27, 28, 29], NbxBi2Se3 [30, 31,
32, 33] reveal in them a superconductivity below critical temperature
$T_{c}\sim 3$K. The breaking of the rotational symmetry in these
superconductors has been observed experimentally [20, 32, 27, 25, 21] as well
as the spin triplet character of the Cooper’s pairing [22]. Such properties
are best described if we assume that the superconducting order parameter is a
two-component vector with $E_{u}$ representation, which is usually called
nematic superconducting state [34, 35, 36, 37].

The nematic superconductivity generated a great interest due to its unusual
properties such as existence of Majorana Kramer’s pairs [38, 39], vestigial
nematic order [40], surface Andreev bound states [41], unconventional
collective modes [42], spontaneous strain [43], unusual magnetic response
[44], and anisotropic quasiparticle interference [45, 46]. A major of these
effects are related to the vector nature of the order parameter. Orientation
of the order parameter vector is often referred to as a nematicity direction
and it controls the anisotropy axis of the system. Naturally, we expect an
interesting Josephson physics in such superconductors, which depends
significantly on the mutual orientations of the nematicity axis and the
junction plane.

In this work, we perform a study of the Josephson physics in the nematic
topological superconductors of the type ABi2Se3 based on the phenomenological
Ginzburg-Landau (GL) approach [35]. In Section II, we derive the GL equations
for such superconductors. We get that the Meissner kernel that shows the
response of the supercurrent to the vector potential has off-diagonal
components. In Section III, we consider a superconductor-insulator-
superconductor (SIS) Josephson junction and derive current-phase relations for
different orientations of the junction plane and crystallographic axes. We
show that the anomalous Josephson Hall effect (that is, the Josephson current
along the junction) can be observed in the system for a certain orientation of
the contact plane to the crystal axes. In Section IV, we analyze the
electromagnetic properties of the junction and derive the dependence of the
maximum Josephson current through the junction on the magnetic field. In
Section V, we discuss the obtained results.

##  II Ginzburg-Landau equations

We start with the GL free energy $F$ for the vector superconducting order
parameter $\vec{\eta}=(\eta_{1},\eta_{2})$

$F=F_{0}+F_{D}+F_{H},$ (1)

where $F_{0}$ is a “homogeneous” contribution, $F_{D}$ is the gradient term,
and $F_{H}$ is a contribution due to the electromagnetic field. Following
seminal paper by Fu, we write down $F_{0}$ in the form [35]

$\displaystyle F_{0}$ $\displaystyle=$ $\displaystyle
a(|\eta_{1}|^{2}+|\eta_{2}|^{2})+B_{1}(|\eta_{1}|^{2}+|\eta_{2}|^{2})^{2}$
$\displaystyle+$ $\displaystyle
B_{2}|\eta_{1}^{*}\eta_{2}-\eta_{1}\eta_{2}^{*}|^{2},$

where $a\propto T/T_{c}-1<0$ and $B_{1,2}$ are the GL coefficients, $T_{c}$ is
the critical temperature. The gradient term for the doped topological
insulator of the type ABi2Se3 can be chosen as [35, 47]

$\displaystyle F_{D}$ $\displaystyle=$ $\displaystyle
J_{1}(D_{i}\eta_{\alpha})^{*}D_{i}\eta_{\alpha}+J_{3}(D_{z}\eta_{\alpha})^{*}D_{z}\eta_{\alpha}+J_{4}[(D_{x}\eta_{1})^{*}D_{x}\eta_{1}-(D_{y}\eta_{1})^{*}D_{y}\eta_{1})+(D_{y}\eta_{2})^{*}D_{y}\eta_{2}$
(3) $\displaystyle-$
$\displaystyle(D_{x}\eta_{2})^{*}D_{x}\eta_{2}+(D_{x}\eta_{1})^{*}D_{y}\eta_{2}+(D_{x}\eta_{2})^{*}D_{y}\eta_{1}+(D_{y}\eta_{1})^{*}D_{x}\eta_{2}+(D_{y}\eta_{2})^{*}D_{x}\eta_{1}].$

Here $J_{n}$ are corresponding GL coefficients,
$D_{j}=-i\hbar\nabla_{j}-2eA_{j}/c$, summation over repeating indices $i=x,y$
and $\alpha=1,2$ is assumed, and $\mathbf{A}=(A_{x},A_{y},A_{Z})$ is the
vector-potential. The magnetic part of the free energy reads

$F_{H}=\frac{(\textrm{curl}{\bf A})^{2}}{8\pi}-\frac{\textrm{curl}{\bf
A}\cdot{\bf H_{0}}}{4\pi},$ (4)

where $\mathbf{H_{0}}$ is the applied field.

###  II.1 First GL equations

The first GL equations can be obtained from the variation $\delta
F/\delta\eta^{*}_{i}=0$. This was done in many papers (see, e.g., Ref. [36]).
We present here the result for the sake of completeness

$\displaystyle\left\\{a\sigma_{0}+2B_{1}\eta^{2}\sigma_{0}+2B_{2}\textrm{Im}\,(\eta^{*}_{1}\eta_{2})\sigma_{y}+(J_{1}D_{x}^{2}+J_{1}D_{y}^{2}+J_{3}D_{z}^{2})\sigma_{0}+J_{4}(D_{x}^{2}-D_{y}^{2})\sigma_{z}+J_{4}[D_{x},D_{y}]\sigma_{x})\right\\}\vec{\eta}=0.$
(5)

Here $\sigma_{j}$ ($j=0,x,y,z$) are the Pauli matrices that act in
$\vec{\eta}=(\eta_{1},\eta_{2})$ space,
$\eta^{2}=|\eta_{1}|^{2}+|\eta_{2}|^{2}$, and
$[D_{x},D_{y}]=D_{x}D_{y}-D_{y}D_{x}$. Boundary conditions for the first GL
equations at the superconductor-vacuum interface read

$\displaystyle J_{1}n_{i}D_{i}\eta_{1}+J_{3}n_{z}D_{z}\eta_{1}+$ (6)
$\displaystyle
J_{4}\left[n_{x}D_{x}\eta_{1}-n_{y}D_{y}\eta_{1}+n_{x}D_{y}\eta_{2}+n_{y}D_{x}\eta_{2}\right]=0,$
$\displaystyle J_{1}n_{i}D_{i}\eta_{2}+J_{3}n_{z}D_{z}\eta_{2}+$ (7)
$\displaystyle
J_{4}\left[n_{y}D_{y}\eta_{2}-n_{x}D_{x}\eta_{2}+n_{x}D_{y}\eta_{1}+n_{y}D_{x}\eta_{1}\right]=0,$

where $n_{i}=n_{x,y}$ and $n_{z}$ are corresponding components of the external
normal to the sample surface.

###  II.2 Second GL equations

To derive the second GL equations, we perform variation of the GL functional
with respect to the components of the vector potential. Here we present the
results in the form convenient for calculation of the Josephson current. We
start with variation of $F_{H}$ and introduce, as usual, the current
components

$\displaystyle\delta_{\mathbf{A}}F_{H}=\frac{1}{c}\delta{\mathbf{A}}\mathbf{j}_{s},\qquad{\mathbf{j}}_{s}=\frac{c}{4\pi}\textrm{curl}\,\textrm{curl}{\mathbf{A}}.$
(8)

In so doing, we obtain the second GL equations from the condition $\delta_{\bf
A}F=0$ in the form

$\displaystyle j_{sz}$ $\displaystyle=$ $\displaystyle-2e\hbar
J_{3}\left[F_{11}^{z}+F_{22}^{z}+\frac{4\pi}{\Phi_{0}}A_{z}(|\eta_{1}|^{2}+|\eta_{2}|^{2})\right],$
(9) $\displaystyle j_{sx}$ $\displaystyle=$
$\displaystyle-2e\hbar\left\\{\\!(J_{1}\\!+\\!J_{4})\\!\left(F_{11}^{x}\\!+\\!\frac{4\pi}{\Phi_{0}}A_{x}\eta_{1}^{*}\eta_{1}\right)\\!+\\!(J_{1}\\!-\\!J_{4})\\!\left(F_{22}^{x}\\!+\\!\frac{4\pi}{\Phi_{0}}A_{x}\eta_{2}^{*}\eta_{2}\right)\\!+\\!J_{4}\left[F_{12}^{y}\\!+\\!F_{21}^{y}\\!+\\!\frac{4\pi}{\Phi_{0}}A_{y}(\eta_{1}\eta_{2}^{*}\\!+\\!\eta_{1}^{*}\eta_{2})\right]\\!\right\\},$
$\displaystyle j_{sy}$ $\displaystyle=$
$\displaystyle-2e\hbar\left\\{\\!(J_{1}\\!-\\!J_{4})\\!\left(F_{11}^{y}\\!+\\!\frac{4\pi}{\Phi_{0}}A_{y}\eta_{1}^{*}\eta_{1}\right)\\!+\\!(J_{1}\\!+\\!J_{4})\\!\left(F_{22}^{y}\\!+\\!\frac{4\pi}{\Phi_{0}}A_{y}\eta_{2}^{*}\eta_{2}\right)\\!+\\!J_{4}\left[F_{12}^{x}\\!+\\!F_{21}^{x}\\!+\\!\frac{4\pi}{\Phi_{0}}A_{x}(\eta_{1}\eta_{2}^{*}\\!+\\!\eta_{1}^{*}\eta_{2})\right]\\!\right\\}.$

Here
$F_{\alpha\beta}^{j}=i\eta_{\alpha}^{*}\nabla_{j}\eta_{\beta}-i\eta_{\beta}\nabla_{j}\eta_{\alpha}^{*}$
and $\Phi_{0}=\pi\hbar c/e$ is the magnetic flux quantum. We introduce phases
of the vector order parameter as $\eta_{1}=|\eta_{1}|e^{i\varphi_{1}}$,
$\eta_{2}=|\eta_{2}|e^{i\varphi_{2}}$, $\varphi=(\varphi_{1}+\varphi_{2})/2$,
and $\delta=(\varphi_{1}-\varphi_{2})/2$. We also express the order parameter
components through a nematicity angle $\alpha$ as
$|\eta_{1}|=\eta\cos{\alpha}$ and $|\eta_{2}|=\eta\sin{\alpha}$,
$\alpha\in[0,\pi/2]$. In these notations we have

$\displaystyle F_{11}^{j}$ $\displaystyle=$
$\displaystyle-2\eta^{2}\cos^{2}{\alpha}\nabla_{j}\varphi_{1},\quad
F_{22}^{j}=-2\eta^{2}\sin^{2}{\alpha}\nabla_{j}\varphi_{2},\quad$
$\displaystyle F_{12}^{j}$ $\displaystyle+$ $\displaystyle
F_{21}^{j}=-2\eta^{2}\left(\sin{2\alpha}\cos 2\delta\nabla_{j}\varphi-\sin
2\delta\nabla_{j}\alpha\right).$

As a result, Eqs. (9) reads

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!j_{sz}$ $\displaystyle=$
$\displaystyle 4e\hbar
J_{3}\eta^{2}\left(\nabla_{z}\varphi+\cos{2\alpha}\nabla_{z}\delta-\frac{2\pi}{\Phi_{0}}A_{z}\right),$
(10) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!j_{sx}\\!$ $\displaystyle=$
$\displaystyle\\!4e\hbar\eta^{2}\\!\left\\{\\!(J_{1}\\!+\\!J_{4}\cos{2\alpha})\\!\left(\\!\nabla_{x}\varphi\\!-\\!\frac{2\pi}{\Phi_{0}}A_{x}\\!\right)\\!+\\!(J_{1}\cos{2\alpha}\\!+\\!J_{4})\nabla_{x}\delta\\!+\\!J_{4}\\!\left[\sin{2\alpha}\cos{2\delta}\\!\left(\\!\nabla_{y}\varphi\\!-\\!\frac{2\pi}{\Phi_{0}}A_{y}\\!\right)\\!-\\!\sin{2\delta}\nabla_{y}\alpha\\!\right]\\!\right\\},$
(11) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!j_{sy}\\!$ $\displaystyle=$
$\displaystyle\\!4e\hbar\eta^{2}\\!\left\\{\\!(J_{1}\\!-\\!J_{4}\cos\\!{2\alpha})\\!\left(\\!\nabla_{y}\varphi\\!-\\!\frac{2\pi}{\Phi_{0}}A_{y}\\!\right)\\!+\\!(J_{1}\cos{2\alpha}\\!-\\!J_{4})\nabla_{y}\delta\\!+\\!J_{4}\\!\left[\sin{2\alpha}\cos{2\delta}\\!\left(\\!\nabla_{x}\varphi\\!-\\!\frac{2\pi}{\Phi_{0}}A_{x}\\!\right)\\!-\\!\sin{2\delta}\nabla_{x}\alpha\\!\right]\\!\right\\}.$
(12)

These rather cumbersome equations can be rewritten in a compact form

$\displaystyle
j_{si}=4e\hbar\eta^{2}\left[\nu_{i}\left(\nabla_{i}\varphi-\frac{2\pi}{\Phi_{0}}A_{i}\right)\\!+\\!\gamma_{i}\nabla_{i}\delta+J_{4}\sin
2\alpha\cos
2\delta\left(\nabla_{\bar{i}}\varphi-\frac{2\pi}{\Phi_{0}}A_{\bar{i}}\right)-J_{4}\sin{2\delta\nabla_{\bar{i}}}\alpha\right].$
(13)

Here $i=(x,y,z)$, $\bar{i}=(y,x,0)$, which mean that two last terms are absent
in the formula for $j_{sz}$, $(\nu_{x},\nu_{y},\nu_{z})=(J_{1}+J_{4}\cos
2\alpha,J_{1}-J_{4}\cos 2\alpha,J_{3})$, and
$(\gamma_{x},\gamma_{y},\gamma_{z})=(J_{1}\cos 2\alpha+J_{4},J_{1}\cos
2\alpha-J_{4},J_{3}\cos 2\alpha)$.

Making a gradient transformation ${\bf A}\rightarrow{\bf A}+\Phi_{0}\nabla
f/2\pi$ we restore the same Eq. 13 after substitution
$\varphi\rightarrow\varphi+f$. This means that $\varphi_{i}$ transform as
$\varphi_{i}\rightarrow\varphi_{i}+f/2$. Note also, that the phase difference
$\delta$ in the considered topological superconductors is a function of its
internal properties and applied strain or magnetic field [35, 36, 43]. For
example, if applied strain and/or magnetic field are absent we have a pure
nematic phase with $\delta=0$ when $B_{2}>0$ and a pure chiral phase with
$\delta=\pi/2$ when $B_{2}<0$. The applied strain and/or magnetic field can
transform the pure phase to some intermediate state with $0<\delta<\pi/2$
[43].

From the obtained formulas it follows, in particular, that the supercurrent
along the $z$ direction is generated by the $z$ component of the vector
potential and/or the phase gradient along the $z$ axis. However, in contrast
to “usual” superconductors, the supercurrents along $x$ and $y$ directions are
generated by both $A_{x}$ and $A_{y}$ components of the vector potential and
$x$\- and $y$-components of the phase gradient, as well.

If we neglect variations of $\eta_{1,2}$ in the bulk of the superconductor we
obtain the London equation for the nematic superconductor. In such a limit, it
is convenient to rewrite Eq. (13) in the matrix form

$\mathbf{j}_{s}=-\frac{c}{4\pi}\hat{K}\mathbf{A},\quad\hat{K}=\frac{32\pi
e^{2}\eta^{2}}{c^{2}}\begin{pmatrix}\nu_{x}&\bar{J}_{4}&0\\\
\bar{J}_{4}&\nu_{y}&0\\\ 0&0&\nu_{z}\end{pmatrix},$ (14)

where $\bar{J}_{4}=J_{4}\sin 2\alpha\cos{2\delta}$. As we see, Meissner kernel
$K_{\alpha\beta}$ has off-diagonal components $K_{xy}$ which is unusual. The
$K_{xy}$ terms in the Meissner kernel are inherent for superconductors
anisotropic in the $xy$-plane. The considered materials are
crystallographically isotropic in this plane. The anisotropy arises in the
superconducting state due to the vector nature of the order parameter. Note,
that the existence of the off-diagonal components of the Meissner kernel has
been missed in previous microscopic calculations [48, 49].

Substituting Eq. (14) in the corresponding Maxwell equation, we obtain the
London equation for the nematic superconductor

$\mathbf{H}+\textrm{curl}\left(\hat{K}^{-1}\textrm{curl}\right)\,\mathbf{H}=0.$
(15)

Since $\hat{K}^{-1}$ has off-diagonal components, we have a mixing between
$H_{x}$ and $H_{y}$ components of the magnetic field.

##  III Current-phase relations

In this section, we obtain relations between the Josephson current through a
SIS junction and the phases of the order parameter components when the
external electromagnetic field is absent. The junction consists of two
superconductors with the nematic pairing. Superconducting correlations are
induced by the proximity effect inside the junction $-d/2<z<d/2$. We assume
that these correlations are weak [50] and we can keep only quadratic in
$\eta_{i}\eta_{j}$ terms neglecting terms of the fourth order that leads to
the linearized GL equations that we obtain from (1). Following a standard
approach, we assume that the GL coefficient in the junction $a=a_{n}$ is
positive.

###  III.1 Junction transverse to z direction

Figure 1: Junction transverse to $z$ direction.

Consider a junction transverse to $z$ direction, which we will call further
$z$-junction Fig. 1. Inside the junction $-d/2<z<d/2$ the linearized GL
equation (5) reads as

$\nabla_{z}^{2}\vec{\eta}=\frac{a_{N}}{\hbar^{2}J_{3}}\vec{\eta}.$ (16)

We will use subscript $L$ for the parameters at the left ($z<0$) of the
junction and $R$ for the parameters at the right ($z>0$). We set the boundary
conditions $\eta_{j}(+d/2)=\eta_{jR}$ ans $\eta_{j}(-d/2)=\eta_{jL}$ and
obtain

$\displaystyle\\!\\!\\!\eta_{j}\\!=\\!\frac{(\eta_{jR}\\!+\\!\eta_{jL})\cosh\kappa_{z}z}{2\cosh(\kappa_{z}d/2)}\\!+\\!\frac{(\eta_{jR}\\!-\\!\eta_{jL})\sinh\kappa_{z}z}{2\sinh(\kappa_{z}d/2)},$
(17)

where $\kappa_{z}=\sqrt{a_{N}/\hbar^{2}J_{3}}$. We substitute this formula
into Eq. (13) for the supercurrent along $z$ axis and derive

$\displaystyle\\!\\!\\!\\!\\!\\!j_{z}\\!=\\!j_{cz}\left[\cos{\alpha_{R}}\cos{\alpha_{L}}\sin{\theta_{1}\\!}+\\!\sin{\alpha_{R}}\sin{\alpha_{L}}\sin{\theta_{2}}\right],$
(18)

where $\theta_{j}=\varphi_{jR}-\varphi_{jL}$ and

$j_{cz}=\frac{4e\hbar J_{3}\kappa_{z}\eta_{R}\eta_{L}}{\sinh\kappa_{z}d}.$
(19)

We see that the Josephson current through the junction is a sum of
contributions from two components of the order parameter. This relation
differs from a general current-phase relation for usual two-band
superconductors [51] since the phases of the order parameter components in the
nematic superconductors are not independent [35]. It is reasonable to express
the Josephson current in terms of the sum and difference of the phases of the
order parameter components:

$\displaystyle j_{z}=j_{cz}\times$ (20)
$\displaystyle\left[\cos(\alpha_{R}\\!-\\!\alpha_{L})\sin{\theta}\cos{\phi}\\!+\\!\cos(\alpha_{R}\\!+\\!\alpha_{L})\cos{\theta}\sin{\phi}\right],$

where $\theta=\varphi_{R}-\varphi_{L}$ and $\phi=\delta_{R}-\delta_{L}$. We
can tune the value of the Josephson current varying the nematicity direction.
For example, in the case of the Josephson junction between two purely nematic
superconductors $\delta_{L}=\delta_{R}=0$ expression for the Josephson current
takes a simple form $j_{z}=j_{cz}\cos(\alpha_{R}\\!-\\!\alpha_{L})\sin\theta$.
We see that if the vector order parameters at different sides of the contact
are orthogonal, $\alpha_{R}-\alpha_{L}=\pi/2$ there is no current through the
junction, $j_{z}=0$. In other terms, $j_{z}=0$ if
$\vec{\eta}_{R}=\eta_{R}(1,0)$ and $\vec{\eta}_{L}=\eta_{L}(0,1)$ or vice
versa $\vec{\eta}_{R}=\eta_{R}(0,1)$ and $\vec{\eta}_{L}=\eta_{L}(1,0)$.

As usual for the Josephson effect, we assume that the density of
superconducting current in the bulk is much larger than the Josephson critical
current density [1, 2, 3]. That is, the London penetration depth $\lambda$ is
much larger than a Josephson penetration depth $\lambda_{J}$. To estimate
$\lambda_{J}$ we need equations for the junction in the magnetic field
obtained below in Section IV. Here we only note that the condition
$\lambda\ll\lambda_{J}$ requires $\kappa d\gg 1$, that is, the contact is far
from a superconducting transition, and the value $\eta(z=0)$ is much smaller
than the equilibrium order parameter in the bulk, $\eta(|z|>d/2)$.

###  III.2 Junction transverse to x direction

Figure 2: Junction transverse to $x$ axis. Nematicity angles $\alpha_{L}$ and
$\alpha_{R}$ show direction of the vector of the order parameter,
$\delta_{L(R)}=0$.

Here we consider a junction transverse to the $x$ axis, $x$-junction (see Fig.
2). The case of $y$-junction is a similar. The linearized first GL equation
(5) in the junction $-d/2<x<d/2$ is

$\nabla_{x}^{2}\vec{\eta}=\frac{(J_{1}{\color[rgb]{0,0,1}-}J_{4}\sigma_{z})\,a_{N}}{\hbar^{2}(J_{1}^{2}-J_{4}^{2})}\vec{\eta}.$
(21)

We set boundary conditions $\eta_{j}(+d/2)=\eta_{jR}$ and
$\eta_{j}(-d/2)=\eta_{jL}$ and derive

$\\!\\!\\!\eta_{j}\\!=\\!\frac{(\eta_{jR}\\!+\\!\eta_{jL})\cosh\kappa_{j}x}{2\cosh(\kappa_{j}d/2)}\\!+\\!\frac{(\eta_{jR}\\!-\\!\eta_{jL})\sinh\kappa_{j}x}{2\sinh(\kappa_{j}d/2)},$
(22)

where $\kappa_{1}=\sqrt{a_{N}/\hbar(J_{1}+J_{4})}$ and
$\kappa_{2}=\sqrt{a_{N}/\hbar(J_{1}-J_{4})}$. We substitute Eq. (22) in the
formula for the components of the supercurrent, Eqs. (13). We observe that in
the considered geometry the Josephson current through the junction has two
components, usual current through the junction $j_{x}$ and transverse or Hall
current along it, $j_{y}$. For the current through the contact we obtain

$\displaystyle j_{x}$ $\displaystyle=$ $\displaystyle
j_{c1}\cos{\alpha_{R}}\\!\cos{\alpha_{L}}\\!\sin{\theta_{1}}\\!\\!+\\!j_{c2}\sin{\alpha_{R}}\sin{\alpha_{L}}\\!\sin{\theta_{2}},$
(23) $\displaystyle j_{c1}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{4e\hbar(J_{1}\\!+\\!J_{4})\kappa_{1}\eta_{L}\eta_{R}}{\sinh\kappa_{1}d},\,\,\,j_{c2}\\!=\\!\frac{4e\hbar(J_{1}\\!-\\!J_{4})\kappa_{2}\eta_{L}\eta_{R}}{\sinh\kappa_{2}d}.$

As in the case of the $z$ junction, $j_{x}$ consists of two terms, which
corresponds to two components of the order parameter, but in contrast to the
previous case, the critical values of the current for $\eta_{1}$ and
$\eta_{2}$ are different. Similar to Eq. (20), we can rewrite the latter
formula in terms of $\theta$ and $\phi$

$\displaystyle
j_{x}=\left(j_{c1}\cos{\alpha_{R}}\\!\cos{\alpha_{L}}\\!+\\!j_{c2}\sin{\alpha_{R}}\\!\sin{\alpha_{L}}\right)\sin{\theta}\cos{\phi}$
$\displaystyle+\left(j_{c1}\cos{\alpha_{R}}\\!\cos{\alpha_{L}}\\!-\\!j_{c2}\sin{\alpha_{R}}\\!\sin{\alpha_{L}}\right)\cos{\theta}\sin{\phi}.$
(24)

As in the case of the $z$-junction, we can tune the current value and even
“turn off” the contact (that is, to put $j_{x}=0$) by a proper choice of the
nematicity direction. As an example, consider a junction formed by two purely
nematic superconductors, $\delta_{R}=\delta_{L}=\phi=0$. In this case,
$j_{x}=0$ if the nematicity direction in right and left sides of the junction
are perpendicular, just similar to the case of the $z$-junction.

According to Eq. (13), in addition to $j_{x}$, we have a Josephson Hall
current $j_{y}$ flowing along the junction. Presence of such a transverse
current is a direct consequence of the non-diagonal Meissner kernel and is an
analog of the anomalous Josephson Hall effect [6, 7, 8, 9, 10, 11, 12, 13,
14]. Using Eqs. (9) and (22) we obtain

$\displaystyle j_{y}=4e\hbar
J_{4}\textrm{Im}\left(\eta_{1}^{*}\nabla_{x}\eta_{2}+\eta_{2}^{*}\nabla_{x}\eta_{1}\right).$
(25)

The current $j_{y}$ depends on $x$. In the case of two nematic
superconductors, the function $j_{y}(x)$ can be presented as
$j_{y}(x)=j_{y,s}(x)\sin(\alpha_{R}+\alpha_{L})+j_{y,as}(x)\sin(\alpha_{R}-\alpha_{L})$
where $j_{y,s}(x)=j_{y,s}(-x)$ is a symmetric component of the current and
$j_{y,as}(x)=-j_{y,as}(-x)$ is antisymmetric. As we see, tuning of
$\alpha_{R}$ and $\alpha_{L}$ can give us purely symmetric or antisymmetric
current. In general, the transverse current $j_{y}$ can be of the same order
or even larger (for $\delta>0$) than $j_{x}$. Moreover, $j_{y}$ can be non-
zero when the current through the junction $j_{x}$ is zero. The coordinate
dependence of the symmetric and antisymmetric parts of the current $j_{y}(x)$
is illustrated in Fig. 3. The dependence of these components of the current on
the junction thickness $d$ is shown in Fig. 4. It is interesting to note that
$j_{y,as}(d)$ first grows with $d$, attains a maximum, and finally decreases,
while $j_{y,s}(d)$ decreases monotone from maximum at small $d$ to zero, when
$\kappa_{1}d\gg 1$, as it is common for Josephson currents [1].

Figure 3: Dependence of the symmetric $j_{y,s}$ (red line) and antisymmetric
$j_{y,as}$ (green dashed line) components of the transverse current
$j_{y}(x)/j_{y0}$ on the coordinate $x$ for $J_{4}/J_{1}=3/4$, $k_{1}d=5$.
Here $j_{y0}=4e\hbar J_{4}$ Figure 4: Dependence of the symmetric $j_{y,s}$
(red line) and antisymmetric $j_{y,as}$ (green dashed line) components of the
transverse current $j_{y}(x)/j_{y0}$ on the $k_{1}d$ for $J_{4}/J_{1}=3/4$ at
the left side of the contact $x=-d/2$. Here $j_{y0}=4e\hbar J_{4}$

Consider the case of the junction between two nematic superconductors,
$\delta_{R}=\delta_{L}=0$. To compare $j_{y}$ and $j_{x}$, we introduce an
average value

$\bar{j}_{y}=\frac{1}{d}\int_{-d/2}^{d/2}j_{y}(x)dx$ (26)

After a straightforward calculations we derive

$\bar{j}_{y}=j_{cy}\sin{\theta},$ (27)

where

$\displaystyle
j_{cy}=\sin(\alpha_{R}+\alpha_{L})\frac{8e\hbar\eta^{2}\sqrt{J_{1}^{2}-J_{4}^{2}}}{d\sinh{(\kappa_{1}d)}\sinh{(\kappa_{2}d)}}$
(28) $\displaystyle\times(\cosh{\kappa_{2}d}-\cosh{\kappa_{1}d})$

In the case of two nematic superconductors $\delta_{R}=\delta_{L}=0$, $j_{y}$
and $j_{x}$ are typically of the same order for $\kappa_{i}d\sim 1$ and
$j_{cy}/j_{cx}\sim\tan{\alpha}/\kappa_{1}d$ for $\kappa_{i}d\gg 1$. The value
of the Hall current $\bar{j}_{y}$ is zero in the follows cases: (i) if
$J_{4}=0$ (that is, $\kappa_{1}=\kappa_{2}$) and the anisotropy is absent in
$(x,y)$ plane; (ii) if $\alpha_{R}+\alpha_{L}=0$ or
$\alpha_{R}+\alpha_{L}=\pi$.

##  IV Junction in the magnetic field

In this section, we consider the junction in the magnetic field $\mathbf{H}$.
As usual for Josephson physics, we assume that $\mathbf{H}$ lies in the
contact plane. We consider here only the case of two purely nematics
superconductors, $\delta_{L}=\delta_{R}=0$. Non-zero $\delta$ results in the
existence of an additional magnetization in the superconductor, see Ref. [43],
which, in general, should be included in the Maxwell equations. Also, finite
$\delta$ leads to the presence of the antisymmetric component in the current
along the junction, see the text after Eq. (25). These effects deserve a
separate detailed study in future work.

###  IV.1 z-junction

First, we consider the junction, which lies in the plane $(x,y)$ transverse to
z-axis. Since the problem has rotation symmetry in the $(x,y)$ plane we can
choose the $y$ axis along the direction of the applied magnetic field,
${\mathbf{H}}_{0}=(0,H_{0},0)$. The magnetic field in the junction, $z=0$, is
${\mathbf{H}}(x,y)=[0,H(x,y),0]$ and

$\nabla_{x}H(x,y)=\frac{4\pi}{c}j_{z},$ (29)

where $j_{z}$ is the current through the junction.

First, we consider the case without Josephson currents. The magnetic field in
the bulk of the superconductor, $\mathbf{H}(z)=[H_{x}(z),H_{y}(z),0]$ we can
find using London equation (15). In this limit we have

$\displaystyle\nabla^{2}_{z}H_{x}$ $\displaystyle=$ $\displaystyle\frac{32\pi
e^{2}\eta^{2}}{c^{2}}\left(\nu_{y}H_{x}-\bar{J}_{4}H_{y}\right),$
$\displaystyle\nabla^{2}_{z}H_{y}$ $\displaystyle=$ $\displaystyle\frac{32\pi
e^{2}\eta^{2}}{c^{2}}\left(\nu_{x}H_{y}-\bar{J}_{4}H_{x}\right),$

and the dependence of $\mathbf{H}$ on $x$ is weak. The solution of these
system decays when $z\rightarrow\pm\infty$ and $\mathbf{H}(z=0)=(0,H(x,y),0)$.
In the considered case $\delta=0$ (purely nematic superconductor) we obtain
inside the superconductor $|z|>d/2$

$\displaystyle H_{y}$ $\displaystyle=$ $\displaystyle
H(x,y)\\!\left(\sin^{2}\\!{\alpha_{L(R)}}\,e^{\frac{d/2-|z|}{\lambda_{1}}}+\cos^{2}\\!{\alpha_{L(R)}}\,e^{\frac{d/2-|z|}{\lambda_{1}}}\right),$
$\displaystyle H_{x}$ $\displaystyle=$
$\displaystyle\frac{H(x,y)\sin{2\alpha_{L(R)}}\\!\left(e^{\frac{d/2-|z|}{\lambda_{2}}}\\!-\\!e^{\frac{d/2-|z|}{\lambda_{1}}}\right)\textrm{sign}(z)}{2},$
$\displaystyle\lambda_{1,2}$ $\displaystyle=$
$\displaystyle\frac{c}{4\sqrt{2\pi(J_{1}\pm J_{4})}\,e\eta}.$ (30)

Thus, we obtain an estimate for the London penetration depths $\lambda_{1,2}$
and observe that the magnetic field in the $y$-direction in the junction
generates the $x$-component of the field in the bulk. The latter property is a
characteristic of anisotropic superconductors [52]. In the considered geometry
the anisotropy in the $(x,y)$ plane arises due to the vector nature of the
order parameter.

The relation between the phase difference in the junction and magnetic field
can be derived by different methods giving the same result [53, 54, 1, 3]. We
outline briefly one of them.

From the Maxwell equation we have

$\displaystyle j_{x}=-\frac{c}{4\pi}\nabla_{z}H_{y},\quad
j_{y}=\frac{c}{4\pi}\nabla_{z}H_{x}.$ (31)

We integrate the components of the magnetic field across the junction and
using Eqs. (IV.1) obtain

$\int_{-\infty}^{+\infty}H_{y}(z)dz=Hd_{1},\quad\int_{-\infty}^{+\infty}H_{x}(z)dz=Hd_{2},$
(32)

where

$\displaystyle d_{1}$ $\displaystyle=$ $\displaystyle
d\\!+\\!(\sin^{2}{\alpha_{R}}\\!+\\!\sin^{2}{\alpha_{L}})\lambda_{1}\\!+\\!(\cos^{2}{\alpha_{R}}\\!+\\!\cos^{2}{\alpha_{L}})\lambda_{2},$
$\displaystyle d_{2}$ $\displaystyle=$ $\displaystyle
d/2+(\sin{2\alpha_{R}}-\sin{2\alpha_{L}})(\lambda_{2}-\lambda_{1})/2.$ (33)

The first of these equations is a formal definition of the effective “magnetic
thickness” of the junction $d_{1}$. We assume here for definiteness that
$\sin{2\alpha_{R}}-\sin{2\alpha_{L}}>0$ since the choice of the right and left
sides of the junction is arbitrary.

We differentiate Eqs. (31) with respect to z having in mind that
$\nabla_{z}\varphi\approx\theta\delta(z)$ if $\lambda_{J}\gg\lambda_{1,2}$
[here $\delta(z)$ is delta-function]. Then, we integrate the result with
respect to $z$ from $z>-z_{1}$ to $z<z_{1}$, where $z_{1}\gg d_{1}/2$. Since
$H_{x,y}(z)$ tends to zero with all its derivatives when $z\rightarrow\infty$,
we come to relations

$\nabla_{i}\theta=\frac{2\pi}{\Phi_{0}}\int_{-\infty}^{+\infty}\nabla_{z}A_{i}dz,$
(34)

where $i=x,y$. Taking into account that $H_{x}=-\nabla_{z}A_{y}$,
$H_{y}=\nabla_{z}A_{x}$, after integration we derive

$\displaystyle\nabla_{x}\theta=\frac{2\pi
d_{1}}{\Phi_{0}}H(x,y),\quad\nabla_{y}\theta=\frac{2\pi
d_{2}}{\Phi_{0}}H(x,y).$ (35)

Finally, using Eqs. (20) and (29) we obtain the equation for $\theta$ in the
form

$\displaystyle\lambda_{J}^{2}\theta^{\prime\prime}_{xx}=\sin{\theta},$ (36)
$\displaystyle\theta^{\prime}_{y}=\frac{d_{2}}{d_{1}}\theta^{\prime}_{x}$ (37)

where we introduce the Josephson length in the usual form [1]

$\lambda_{J}=\sqrt{\frac{c\Phi_{0}}{8\pi^{2}d_{1}j_{cz}\cos{(\alpha_{R}-\alpha_{L})}}}.$
(38)

Note, that for $\alpha_{R}=\alpha_{L}$ we have $d_{2}=0$ and
$\theta^{\prime}_{y}=0$.

Equation (36) is often called Ferrell-Prange equation [53]. The first of Eqs
(35) relays $\theta$ and the magnetic field and also gives boundary condition
for $\theta$ at the edges of the junction where $H=H_{0}$. The values
$\alpha_{R,L}$ are constants, which depend on the superconducting state in the
bulk. We can rewrite the applicability condition of our approach
$\lambda_{J}\gg\lambda_{1,2}$ in the explicit form using Eqs. (19), (IV.1),
and (38)

$\frac{(J_{1}\pm J_{4})\sinh{\kappa_{z}d}}{J_{3}\kappa_{z}d_{1}}\gg 1.$ (39)

Now we derive a dependence of the maximum persistent current through the
junction $I_{max}$ on the applied magnetic field $H_{0}$ or on the value of
the captured in the junction magnetic flux $\Phi$ [1]. As usual, we consider a
short contact in $x$ and $y$ directions $L_{x,y}\ll 2\lambda_{J}$ placed in a
sufficiently large magnetic field, $H_{0}\gg\Phi_{0}/2\pi\lambda_{J}d_{1}$,
where $L_{i}$ is the junction length in $i$-th direction. Within these limits,
the magnetic field in the junction is homogeneous and equals the applied
field. We integrate the first of equations (35) taking into account that
$H(x,y)=H_{0}$ and obtain

$\theta=\frac{2\pi\Phi}{\Phi_{0}}\left(\frac{x}{L_{x}}+\frac{d_{2}}{d_{1}}\frac{y}{L_{x}}\right)+C,$
(40)

where $\Phi=L_{x}d_{1}H_{0}$ is the magnetic flux captured in the junction and
$C$ is a constant. We substitute the latter expression in Eq. (20) for the
current and integrate it along the junction from $x=-L_{x}/2$ to $x=L_{x}/2$
and $y=-L_{y}/2$ to $y=L_{y}/2$. We obtain the value of the current through
the contact $I$ in the form

$\frac{I}{I_{cz}}=\cos(\alpha_{L}-\alpha_{R})\sin{C}\frac{\sin{(\pi\Phi/\Phi_{0})}}{\pi\Phi/\Phi_{0}}\frac{\sin{(\kappa\pi\Phi/\Phi_{0})}}{\kappa\pi\Phi/\Phi_{0}},$

where $I_{cz}=j_{cz}dL_{x}$, $\kappa=d_{2}L_{y}/d_{1}L_{x}$. Thus, the maximum
of $I$ is

$\\!\\!\\!\frac{I_{max}}{I_{cz}}\\!=\\!\left|\cos(\alpha_{L}\\!-\\!\alpha_{R})\frac{\sin{(\pi\Phi/\Phi_{0})}}{\pi\Phi/\Phi_{0}}\frac{\sin{(\kappa\pi\Phi/\Phi_{0})}}{\kappa\pi\Phi/\Phi_{0}}\right|,$
(41)

The dependence $I_{max}(\Phi)$ is shown in Fig. 5. It has a typical
Fraunhofer-like pattern, but details of this pattern differ from that for a
usual $s$-wave superconductor (that corresponds to $d_{2}=0$ for the
considered problem). Mathematically, this difference occurs due to dependence
of the phase $\theta$ on two coordinates, $\theta=\theta(x,y)$. One of the
prominent feature of the observed picture is that the maximal critical current
decays as $I_{max}\propto 1/H^{2}$ instead of $I_{max}\propto 1/H$ for a the
$s$-wave superconductors. Period of the current $I_{max}(\Phi)$ oscillations
depends on the parameters of the system, in particular, on the aspect ratio of
the junction $L_{y}/L_{x}$. The latter feature is illustrated in Fig. 6. The
variation of the oscillationa period and $1/H^{2}$ decay of the maximum
current occurs due to additional factor $\sin(\kappa f)/\kappa f$ in Eq. 41
where $f=\pi\Phi/\Phi_{0}$. If $L_{y}\ll L_{x}$, we get $\kappa\ll 1$ and
$I_{max}\propto|\sin f/f|$, which corresponds to the case of the s-wave
superconductor. If $L_{y}\gg L_{x}$ and $\kappa\gg 1$, then, the period of the
oscillations becomes significantly smaller than $\Phi_{0}$ when the applied
magnetic field is low. Note, that we can control behavior of $I_{max}(\Phi)$
rotating the applied field in the junction plane if values $L_{x}$ and $L_{y}$
are significantly different.

Figure 5: Dependence of the normalized maximum critical current on the
magnetic flux for different nematicity angles $\alpha_{R}$ and $\alpha_{L}$.
We set $J_{4}/J_{1}=3/4$, $L_{y}=L_{x}$ and $d\ll\lambda_{1}$. Figure 6:
Dependence of the normalized maximum critical current on the magnetic flux for
different values of the aspect ratio of the contact $L_{y}/L_{x}$. We set
$J_{4}/J_{1}=3/4$, $\alpha_{R}=\pi/4$, $\alpha_{L}=0$ and $d\ll\lambda_{1}$.

###  IV.2 x-junction

Here we consider a junction transverse to the $x$-axis and the magnetic field
lying in the $(y,z)$ plane, $\mathbf{H}(x=0)=(0,H_{y},H_{z})$, where $H_{y,z}$
depend slowly on $y$ and $z$. Using London equation (15) and following a
similar procedure as for the $z$-junction, we derive for the magnetic field in
the bulk of the superconductor

$\displaystyle H_{y}(x)$ $\displaystyle=$ $\displaystyle
H_{y}e^{(d/2-|x|)/\lambda_{y}},\,\,H_{z}(x)=H_{z}e^{(d/2-|x|)/\lambda_{z}},$
$\displaystyle\lambda_{y}$ $\displaystyle=$
$\displaystyle\frac{c}{4e\eta\sqrt{2\pi J_{3}}},$ (42)
$\displaystyle\lambda_{z,L(R)}$ $\displaystyle=$
$\displaystyle\frac{c}{4e\eta}\sqrt{\frac{J_{1}+J_{4}\cos{2\alpha_{L(R)}}}{\\!2\pi\\!\left[J_{1}^{2}\\!-\\!J_{4}^{2}(1\\!-\\!\sin^{2}{2\alpha_{L(R)}}\sin^{2}{2\delta_{L(R)}})\right]}},$

where $\lambda_{y,z}$ are corresponding London penetration depths
($\lambda_{y,L}=\lambda_{y,R}=\lambda_{y}$). In the considered case the
components of the field decay independently. Similar to the $z$-junction, we
can estimate the effective thickness of the $x$-junction as
$d_{i}=d+\lambda_{i,L}+\lambda_{i,R}$ ($i=y,z$), $d_{y}$ corresponds to the
applied field along $y$ axis and $d_{z}$ to the field along $z$ axis.

In the case of the $x$-junction, the Josephson current has the component along
the contact $j_{y}(x)$, Eq. (25). According Eqs. (14), this current induces a
$z$-component of the magnetic field $H_{z}\propto j_{y}^{\prime}(x)$. In
general, this field is not negligibly small since $j_{y}^{\prime}(x)\propto
j_{y}/d$. To avoid this difficulty we consider here only the purely nematic
superconductors ($\delta_{R.L}=0$) and the short junction in a sufficiently
large magnetic field, when the magnetic field in the junction is equal to the
applied magnetic field. Similar to the case of the $z$-junction we can write
the conditions of applicability of such an approach as [see Eqs. (III.2) and
(38)] $H_{i}\gg\Phi_{0}/2\pi\lambda_{J,i}d_{i}$ and
$\lambda_{i}\gg\lambda_{J,i}$, where

$\lambda_{J,i}=\sqrt{\frac{c\Phi_{0}}{8\pi^{2}d_{i}j_{c1}F(\alpha_{R},\alpha_{L})}},$
$F(\alpha_{R},\alpha_{L})=\cos{\alpha_{L}}\cos{\alpha_{R}}+\frac{j_{c2}}{j_{c1}}\sin{\alpha_{L}}\sin{\alpha_{R}}.$
(43)

Now we can derive the dependence of the maximum Josephson current on the
magnetic field in the case of the $x$-junction. We present here the results
for two different orientations of the applied magnetic field, along $y$ and
$z$ axes. Following the same procedure as in the previous subsection, we
readily obtain

$\nabla_{i}\theta=\frac{2\pi}{\Phi_{0}}\int_{-\infty}^{+\infty}\nabla_{x}A_{i}dx,$
(44)

where $i=y,z$, $H_{y}=-\nabla_{x}A_{z}$ and $H_{z}=\nabla_{x}A_{y}$. After
integration over $x$, we derive

$\displaystyle\nabla_{y}\theta=\frac{2\pi
d_{y}}{\Phi_{0}}H_{z},\quad\mathbf{H}_{0}=(0,0,H_{z}),$
$\displaystyle\nabla_{z}\theta=-\frac{2\pi
d_{z}}{\Phi_{0}}H_{y},\quad\mathbf{H}_{0}=(0,H_{y},0).$ (45)

These equations relates the phase difference on the junction and the applied
magnetic field. Similar to the case of the $z$-junction, we can derive the
dependence of the maximum current on the captured flux in the form

$\displaystyle\frac{I_{x\,max}(\Phi_{i})}{I_{c1x}}=F(\alpha_{R},\alpha_{L})\,\frac{\sin{(\pi\Phi_{i}/\Phi_{0})}}{\pi\Phi_{i}/\Phi_{0}},$
(46)

where $i=y,z$ and $I_{c1x}=j_{c1}L$.

##  V Discussion

One of the key result that we present in this work is that the superconductors
with the nematic superconductivity in $E_{u}$ representation have off-diagonal
components $K_{xy}$ in the Meissner kernel, see Eq. (14). As a result, we
observe the anomalous Josephson Hall effect in the $x$-junction, that is, the
supercurrent flowing along the contact, see Eq. (25) and Fig. 3. We believe
that the non-diagonal Meissner kernel can give rise to other peculiarities in
electromagnetic properties of the Josephson junctions.

The current in the junction is controlled by the vector order parameter at
each side of the contact $\vec{\eta}=(\eta_{1R(L)},\eta_{1R(L)})$. It is
convenient to parametrize the order parameter as
$\vec{\eta}=e^{i\varphi}\eta(\cos\alpha e^{i\delta/2},\sin\alpha
e^{-i\delta/2})$, where nematicity direction $\alpha$ and phase difference
between the order parameters components $\delta$ determine the vector
properties of the order parameter. When $\delta=0$, the nematicity direction
$\alpha$ indicates direction of the order parameter vector in the coordinate
space $(x,y)$. The orientation of the order parameter vectors to the contact
significantly affects the current in the junction and we can control the
Josephson current by tuning the nematicity in the bulk of the superconductors.
The simplest strategy to control nematicity direction is the rotation of the
superconductor in $(x,y)$ plane. Experiments show that large samples of the
doped topological insulators are typically multidomain with different
orientations of the nematicity vector in each domain. Weak links between
different domains are natural objects to study the influence of $\alpha_{R,L}$
on the Josephson current. The control of the nematicity angle $\alpha$ by the
external strain was demonstrated experimentally, see, e.g., Refs. 55, 26.
Thus, we can hope that the variation of the nematicity direction by the
applied strain is a feasible task, which opens new possibilities for
superconducting devices.

Another parameter that controls the current is the phase difference between
order parameters components $\delta$. In the case of non-zero $\delta$ an
additional cosine term in the current-phase relations arises, see Eqs. (20)
and (III.2). For the pure nematic superconductor, this term vanishes,
$\delta=0$, while for the pure chiral superconductors $\delta=\pi/2$ and the
cosine is of importance. Theoretical analysis shows that the chiral
superconducting phase can exist in the systems with an open Fermi surface [56,
44], in thin films [57] or under magnetic doping of the nematic superconductor
[58, 59]. It is argued that in some experiments such a chiral phase with a
finite magnetization has been observed [28, 30]. However, it is debated in
other work [33]. As we show in our previous work, the non-zero $\delta$ can be
caused in the nematic phase by the external magnetization [43]. The value of
$\delta$ induced by the magnetization is small [44]. However, measurements of
the currents in the Josephson junctions can be performed with high accuracy.
We believe that the observation of the cosine term in the current that is
caused by the magnetization is experimentally possible.

We observe that in the $x$-junction the Josephson Hall current along the
junction, $j_{y}$, can be generated as well as the current across the
junction, $j_{x}$. In some cases $j_{y}$ can be larger than $j_{x}$. Consider
a junction made of two pure nematic superconductors with the order parameters
$\vec{\eta_{L}}=\eta(1,0)$ and $\vec{\eta_{R}}=\eta e^{i\theta}(0,1)$ that
orthogonal to each other and one of them is parallel to the contact plane. In
this case longitudinal current vanishes $j_{x}=0$ [see Eq. (III.2)] but
transverse current is non-zero $j_{y}=j_{cy}\sin\theta$ [see Eq. (27)].

An analysis of Eq. (25) shows that under definite conditions the Josephson
Hall current does not necessary decay exponentially with the increase of the
junction thickness $d$ if $\delta_{L}$ and/or $\delta_{R}$ is non-zero. In the
leading approximation with respect to $\kappa_{i}d\gg 1$, we obtain that
$d\bar{j}_{y}\propto
J_{4}\,\textrm{Im}(\eta_{1L}^{*}\eta_{2L}-\eta_{1R}^{*}\eta_{2R})$, which is
non-zero for $d\rightarrow+\infty$. For example, such a situation takes place
when $\delta\neq 0$ only in one side of the junction, which can be due to its
magnetization or when one side of the junction is the chiral superconductor
$\vec{\eta_{L}}\propto(1,i)$ and another side is nematic. It is worth
mentioning that the Hall current flows even we consider a single boundary of
the chiral superconductor, e.g. $\vec{\eta_{R}}=(0,0)$. Moreover, if $\delta$
in the superconductor is non-zero, then a finite current flows along the edge,
$d\bar{j}_{y}\propto J_{4}\eta_{L}^{2}\sin 2\alpha_{L}\sin\delta_{L}$. Note,
that the similar type of superconducting anomalous Hall effect has been
predicted for a chiral p-wave superconductor [60].

In Refs. [6, 7, 8, 9, 10, 11, 12, 13, 14] the anomalous Josephson Hall effect
have been considered in the materials with a finite magnetization. This
magnetization brings asymmetry to the Andreev bound states spectra that
results in a finite Josephson Hall current. In our case, this supercurrent
exist without magnetization. Thus, we establish that the anomalous Josephson
Hall effect can be realized without any magnetization while time-reversal
symmetry breaking occurs due to the phase difference across the contact.

One of the distinct features of London equations is the lack of gauge
invariance for the current that leads to the lack of the charge conservation
[61, 62]. Usually, it is not a problem when the Meissner kernel is diagonal
since we can always choose the proper gauge for a vector potential where the
charge is locally conserved. In our case, we get formally that the charge is
not conserved in case of the the $x$-junction if $\theta=\theta(y)$, which is
the case for $z$ direction of the magnetic field. This drawback cannot be
cured in a GL formalism. Instead, we should perform microscopic calculations
of the current that include vertex corrections for a current operator that
arises due to mean-field self-energy and Coulomb interaction. After that a
global gauge invariance is restored and reliable results are obtained for any
gauge of the vector potenial [61, 62]. In the case of the anomalous
superconducting Hall effect restoring the gauge invariance leads to the large
additional quasiparticle contribution to the edge current [63]. It means that
the anomalous Hall supercurrent is significantly renormalized by the vertex
corrections. Thus, we expect a strong renormalization by the vertex
corrections of the Josephson Hall current $j_{y}$ in the $x$-contact as well.
However, microscopic calculations that include vertex corrections are beyond
the scope of this paper.

In conclusion, we study electromagnetic properties of Josephson junction
betwee two nematic superconductors in the Ginzburg-Landau approach. We derive
the London equations for the nematic superconductor with odd $E_{u}$ pairing
using the second GL equation. We observe that the Meissner kernel has off-
diagonal components. Using this result, we obtain current-phase relations for
different orientations of the junction plane, crystallographic axes of the
sample, and the nematicity vector. We show that the anomalous Josephson Hall
effect can be observed in such in the absenct of magnetization. We calculate
the magnetic field dependence of the maximum current through the junction. We
show that the period of the maximum current oscillations with the captured in
the junction magnetic flux depends on the geometry of the junction, direction
of the magnetic field, and the nematicity vector.

## References

  * Barone and Paterno [1982] A. Barone and G. Paterno,  _Physics and applications of the Josephson effect_ , Vol. 1 (Wiley Online Library, 1982).
  * Tinkham [2004] M. Tinkham,  _Introduction to Superconductivity_, 2nd ed. (Dover Publications, 2004).
  * Likharev [1979] K. K. Likharev, Superconducting weak links, Rev. Mod. Phys. 51, 101 (1979).
  * Tafuri [2019] F. Tafuri,  _Fundamentals and frontiers of the Josephson effect_ , Vol. 286 (Springer Nature, 2019).
  * Mel’nikov  _et al._ [2021] A. S. Mel’nikov, S. V. Mironov, A. V. Samokhvalov, and A. I. Buzdin, Superconducting spintronics: state of the art and perspectives, Physics–Uspekhi , 64 (2021).
  * Wang  _et al._ [2011] J. Wang, L. Hao, Y. H. Yang, and K. S. Chan, Josephson hall current in a noncentrosymmetric superconductor/ferromagnet/superconductor junction, Journal of Applied Physics 110, 113717 (2011), https://doi.org/10.1063/1.3666055 .
  * Vedyayev  _et al._ [2013] A. Vedyayev, N. Ryzhanova, N. Strelkov, and B. Dieny, Spontaneous anomalous and spin hall effects due to spin-orbit scattering of evanescent wave functions in magnetic tunnel junctions, Phys. Rev. Lett. 110, 247204 (2013).
  * Yokoyama [2015] T. Yokoyama, Anomalous josephson hall effect in magnet/triplet superconductor junctions, Phys. Rev. B 92, 174513 (2015).
  * Matos-Abiague and Fabian [2015] A. Matos-Abiague and J. Fabian, Tunneling anomalous and spin hall effects, Phys. Rev. Lett. 115, 056602 (2015).
  * Dang  _et al._ [2015] T. H. Dang, H. Jaffrès, T. L. Hoai Nguyen, and H.-J. Drouhin, Giant forward-scattering asymmetry and anomalous tunnel hall effect at spin-orbit-split and exchange-split interfaces, Phys. Rev. B 92, 060403 (2015).
  * Mironov and Buzdin [2017] S. Mironov and A. Buzdin, Spontaneous currents in superconducting systems with strong spin-orbit coupling, Phys. Rev. Lett. 118, 077001 (2017).
  * Mal’shukov [2019] A. G. Mal’shukov, Ac anomalous hall effect in topological insulator josephson junctions, Phys. Rev. B 100, 035301 (2019).
  * Costa and Fabian [2020] A. Costa and J. Fabian, Anomalous josephson hall effect charge and transverse spin currents in superconductor/ferromagnetic-insulator/superconductor junctions, Phys. Rev. B 101, 104508 (2020).
  * Maistrenko  _et al._ [2021] O. Maistrenko, B. Scharf, D. Manske, and E. M. Hankiewicz, Planar josephson hall effect in topological josephson junctions, Phys. Rev. B 103, 054508 (2021).
  * Nagaosa  _et al._ [2010] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous hall effect, Rev. Mod. Phys. 82, 1539 (2010).
  * Hor  _et al._ [2010] Y. S. Hor, A. J. Williams, J. G. Checkelsky, P. Roushan, J. Seo, Q. Xu, H. W. Zandbergen, A. Yazdani, N. P. Ong, and R. J. Cava, Superconductivity in ${\mathrm{cu}}_{x}{\mathrm{bi}}_{2}{\mathrm{se}}_{3}$ and its implications for pairing in the undoped topological insulator, Phys. Rev. Lett. 104, 057001 (2010).
  * Sasaki  _et al._ [2011] S. Sasaki, M. Kriener, K. Segawa, K. Yada, Y. Tanaka, M. Sato, and Y. Ando, Topological superconductivity in ${\mathrm{cu}}_{x}{\mathrm{bi}}_{2}{\mathrm{se}}_{3}$, Phys. Rev. Lett. 107, 217001 (2011).
  * Kirzhner  _et al._ [2012] T. Kirzhner, E. Lahoud, K. B. Chaska, Z. Salman, and A. Kanigel, Point-contact spectroscopy of cu0.2bi2se3 single crystals, Phys. Rev. B 86, 064517 (2012).
  * Kawai  _et al._ [2020] T. Kawai, C. G. Wang, Y. Kandori, Y. Honoki, K. Matano, T. Kambe, and G. qing Zheng, Direction and symmetry transition of the vector order parameter in topological superconductors CuxBi2se3, Nature Communications 11, 10.1038/s41467-019-14126-w (2020).
  * Yonezawa  _et al._ [2016] S. Yonezawa, K. Tajiri, S. Nakata, Y. Nagai, Z. Wang, K. Segawa, Y. Ando, and Y. Maeno, Thermodynamic evidence for nematic superconductivity in CuxBi2se3, Nature Physics 13, 123 (2016).
  * Tao  _et al._ [2018] R. Tao, Y.-J. Yan, X. Liu, Z.-W. Wang, Y. Ando, Q.-H. Wang, T. Zhang, and D.-L. Feng, Direct visualization of the nematic superconductivity in ${\mathrm{cu}}_{x}{\mathrm{bi}}_{2}{\mathrm{se}}_{3}$, Phys. Rev. X 8, 041024 (2018).
  * Matano  _et al._ [2016] K. Matano, M. Kriener, K. Segawa, Y. Ando, and G. qing Zheng, Spin-rotation symmetry breaking in the superconducting state of CuxBi2se3, Nature Physics 12, 852 (2016).
  * Shruti  _et al._ [2015] Shruti, V. K. Maurya, P. Neha, P. Srivastava, and S. Patnaik, Superconductivity by sr intercalation in the layered topological insulator ${\mathrm{bi}}_{2}{\mathrm{se}}_{3}$, Phys. Rev. B 92, 020506(R) (2015).
  * Liu  _et al._ [2015] Z. Liu, X. Yao, J. Shao, M. Zuo, L. Pi, S. Tan, C. Zhang, and Y. Zhang, Superconductivity with topological surface state in SrxBi2se3, Journal of the American Chemical Society 137, 10512 (2015).
  * Kuntsevich  _et al._ [2018] A. Y. Kuntsevich, M. A. Bryzgalov, V. A. Prudkoglyad, V. P. Martovitskii, Y. G. Selivanov, and E. G. Chizhevskii, Structural distortion behind the nematic superconductivity in sr x bi2se3, New Journal of Physics 20, 103022 (2018).
  * Kuntsevich  _et al._ [2019] A. Y. Kuntsevich, M. A. Bryzgalov, R. S. Akzyanov, V. P. Martovitskii, A. L. Rakhmanov, and Y. G. Selivanov, Strain-driven nematicity of odd-parity superconductivity in ${\mathrm{sr}}_{x}{\mathrm{bi}}_{2}{\mathrm{se}}_{3}$, Phys. Rev. B 100, 224509 (2019).
  * Pan  _et al._ [2016] Y. Pan, A. M. Nikitin, G. K. Araizi, Y. K. Huang, Y. Matsushita, T. Naka, and A. de Visser, Rotational symmetry breaking in the topological superconductor srxbi2se3 probed by upper-critical field experiments, Scientific Reports 6, 28632 (2016).
  * Neha  _et al._ [2019] P. Neha, P. K. Biswas, T. Das, and S. Patnaik, Time-reversal symmetry breaking in topological superconductor ${\mathrm{sr}}_{0.1}{\mathrm{bi}}_{2}{\mathrm{se}}_{3}$, Phys. Rev. Materials 3, 074201 (2019).
  * Bannikov  _et al._ [2021] M. I. Bannikov, R. S. Akzyanov, N. K. Zhurbina, S. I. Khaldeev, Y. G. Selivanov, V. V. Zavyalov, A. L. Rakhmanov, and A. Y. Kuntsevich, Breaking of ginzburg-landau description in the temperature dependence of the anisotropy in a nematic superconductor, Phys. Rev. B 104, L220502 (2021).
  * Qiu  _et al._ [2015] Y. Qiu, K. N. Sanders, J. Dai, J. E. Medvedeva, W. Wu, P. Ghaemi, T. Vojta, and Y. S. Hor, Time reversal symmetry breaking superconductivity in topological materials (2015), arXiv:1512.03519 [cond-mat.supr-con] .
  * Kurter  _et al._ [2018] C. Kurter, A. D. K. Finck, E. D. Huemiller, J. Medvedeva, A. Weis, J. M. Atkinson, Y. Qiu, L. Shen, S. H. Lee, T. Vojta, P. Ghaemi, Y. S. Hor, and D. J. V. Harlingen, Conductance spectroscopy of exfoliated thin flakes of NbxBi2se3, Nano Letters 19, 38 (2018).
  * Asaba  _et al._ [2017] T. Asaba, B. J. Lawson, C. Tinsman, L. Chen, P. Corbae, G. Li, Y. Qiu, Y. S. Hor, L. Fu, and L. Li, Rotational symmetry breaking in a trigonal superconductor nb-doped ${\mathrm{bi}}_{2}{\mathrm{se}}_{3}$, Phys. Rev. X 7, 011009 (2017).
  * Das  _et al._ [2020] D. Das, K. Kobayashi, M. P. Smylie, C. Mielke, T. Takahashi, K. Willa, J.-X. Yin, U. Welp, M. Z. Hasan, A. Amato, H. Luetkens, and Z. Guguchia, Time-reversal invariant and fully gapped unconventional superconducting state in the bulk of the topological compound ${\mathrm{nb}}_{0.25}{\mathrm{bi}}_{2}{\mathrm{se}}_{3}$, Phys. Rev. B 102, 134514 (2020).
  * Fu and Berg [2010] L. Fu and E. Berg, Odd-parity topological superconductors: Theory and application to ${\mathrm{cu}}_{x}{\mathrm{bi}}_{2}{\mathrm{se}}_{3}$, Phys. Rev. Lett. 105, 097001 (2010).
  * Fu [2014] L. Fu, Odd-parity topological superconductor with nematic order: Application to ${\mathrm{cu}}_{x}{\mathrm{bi}}_{2}{\mathrm{se}}_{3}$, Phys. Rev. B 90, 100509(R) (2014).
  * Venderbos  _et al._ [2016a] J. W. F. Venderbos, V. Kozii, and L. Fu, Identification of nematic superconductivity from the upper critical field, Phys. Rev. B 94, 094522 (2016a).
  * Yonezawa [2018] S. Yonezawa, Nematic superconductivity in doped bi2se3 topological superconductors, Condensed Matter 4, 2 (2018).
  * Wu and Martin [2017] F. Wu and I. Martin, Majorana kramers pair in a nematic vortex, Phys. Rev. B 95, 224503 (2017).
  * Akzyanov and Rakhmanov [2021] R. S. Akzyanov and A. L. Rakhmanov, Strain-induced spin vortex and majorana kramers pairs in doped topological insulators with nematic superconductivity, Phys. Rev. B 104, 094511 (2021).
  * Hecker and Schmalian [2017] M. Hecker and J. Schmalian, Vestigial nematic order and superconductivity in the doped topological insulator cuxbi2se3, npj Quantum Mater. 3, 26 (2017).
  * Hao and Ting [2017] L. Hao and C. S. Ting, Nematic superconductivity in ${\mathrm{cu}}_{x}{\mathrm{bi}}_{2}{\mathrm{se}}_{3}$: Surface andreev bound states, Phys. Rev. B 96, 144512 (2017).
  * Uematsu  _et al._ [2019] H. Uematsu, T. Mizushima, A. Tsuruta, S. Fujimoto, and J. A. Sauls, Chiral higgs mode in nematic superconductors, Phys. Rev. Lett. 123, 237001 (2019).
  * Akzyanov  _et al._ [2020] R. S. Akzyanov, A. V. Kapranov, and A. L. Rakhmanov, Spontaneous strain and magnetization in doped topological insulators with nematic and chiral superconductivity, Phys. Rev. B 102, 100505(R) (2020).
  * Khokhlov and Akzyanov [2021a] D. A. Khokhlov and R. S. Akzyanov, Pauli paramagnetism of triplet cooper pairs in a nematic superconductor, Phys. Rev. B 104, 214514 (2021a).
  * Chen  _et al._ [2018] M. Chen, X. Chen, H. Yang, Z. Du, and H.-H. Wen, Superconductivity with twofold symmetry in Bi2Te3/FeTe0.55Se0.45 heterostructures, Science Advances 4, eaat1084 (2018).
  * Khokhlov and Akzyanov [2021b] D. A. Khokhlov and R. S. Akzyanov, Quasiparticle interference in doped topological insulators with nematic superconductivity, Physica E: Low-dimensional Systems and Nanostructures 133, 114800 (2021b).
  * Zyuzin  _et al._ [2017] A. A. Zyuzin, J. Garaud, and E. Babaev, Nematic skyrmions in odd-parity superconductors, Phys. Rev. Lett. 119, 167001 (2017).
  * Schmidt  _et al._ [2020] J. Schmidt, F. Parhizgar, and A. M. Black-Schaffer, Odd-frequency superconductivity and meissner effect in the doped topological insulator ${\mathrm{bi}}_{2}{\mathrm{se}}_{3}$, Phys. Rev. B 101, 180512(R) (2020).
  * Akzyanov [2021] R. S. Akzyanov, Lifshitz transition in dirty doped topological insulator with nematic superconductivity, Phys. Rev. B 104, 224502 (2021).
  * Buzdin [2008] A. Buzdin, Direct coupling between magnetism and superconducting current in the josephson ${\varphi}_{0}$ junction, Phys. Rev. Lett. 101, 107005 (2008).
  * Sasaki  _et al._ [2020] A. Sasaki, S. Ikegaya, T. Habe, A. A. Golubov, and Y. Asano, Josephson effect in two-band superconductors, Physical Review B 101, 184501 (2020).
  * Kogan [1981] V. G. Kogan, London approach to anisotropic type-ii superconductors, Phys. Rev. B 24, 1572 (1981).
  * Ferrell and Prange [1963] R. A. Ferrell and R. E. Prange, Self-field limiting of josephson tunneling of superconducting electron pairs, Phys. Rev. Lett. 10, 479 (1963).
  * Kulik and Yanson [1972] I. O. Kulik and I. K. Yanson,  _Josephson Effect in Superconducting Tunneling Structures_ (John Wiley and Sons, Inc., New York, 1972).
  * Kostylev  _et al._ [2020] I. Kostylev, S. Yonezawa, Z. Wang, Y. Ando, and Y. Maeno, Uniaxial-strain control of nematic superconductivity in srxbi2se3, Nature Communications 11, 4152 (2020).
  * Venderbos  _et al._ [2016b] J. W. F. Venderbos, V. Kozii, and L. Fu, Odd-parity superconductors with two-component order parameters: Nematic and chiral, full gap, and majorana node, Phys. Rev. B 94, 180504(R) (2016b).
  * Chirolli [2018] L. Chirolli, Chiral superconductivity in thin films of doped ${\mathrm{bi}}_{2}{\mathrm{se}}_{3}$, Phys. Rev. B 98, 014505 (2018).
  * Chirolli  _et al._ [2017] L. Chirolli, F. de Juan, and F. Guinea, Time-reversal and rotation symmetry breaking superconductivity in dirac materials, Phys. Rev. B 95, 201110(R) (2017).
  * Yuan  _et al._ [2017] N. F. Q. Yuan, W.-Y. He, and K. T. Law, Superconductivity-induced ferromagnetism and weyl superconductivity in nb-doped ${\mathbf{bi}}_{2}{\mathbf{se}}_{3}$, Phys. Rev. B 95, 201109(R) (2017).
  * Furusaki  _et al._ [2001] A. Furusaki, M. Matsumoto, and M. Sigrist, Spontaneous hall effect in a chiral p-wave superconductor, Phys. Rev. B 64, 054514 (2001).
  * Nambu [1960] Y. Nambu, Quasi-particles and gauge invariance in the theory of superconductivity, Phys. Rev. 117, 648 (1960).
  * Schrieffer [1964] J. R. Schrieffer,  _Theory of Superconductivity_ (W.A. Benjamin, San Francisco, 1964).
  * Stone and Roy [2004] M. Stone and R. Roy, Edge modes, edge currents, and gauge invariance in ${p}_{x}{+ip}_{y}$ superfluids and superconductors, Phys. Rev. B 69, 184511 (2004).