# Anomalous flux periodicity in proximitised quantum spin Hall constrictions

Lucia Vigliotti1, Alessio Calzona3, Björn Trauzettel2,3, Maura Sassetti1,4,
Niccolò Traverso Ziani1,4 1Dipartimento di Fisica, Università degli Studi di
Genova, Via Dodecaneso 33, 16136, Genova, Italy  2Institute for Theoretical
Physics and Astrophysics, University of Würzburg, D-97074 Würzburg, Germany
3Würzburg-Dresden Cluster of Excellence ct.qmat, Germany  4CNR SPIN, Via
Dodecaneso 33, 16136, Genova, Italy<EMAIL_ADDRESS>
###### Abstract

We theoretically analyse a long constriction between the helical edge states
of a two-dimensional topological insulator. The constriction is laterally
tunnel-coupled to two superconductors and a magnetic field is applied
perpendicularly to the plane of the two-dimensional topological insulator. The
Josephson current is calculated analytically up to second order in the tunnel
coupling both in the absence and in the presence of a bias (DC and AC
Josephson currents). We show that in both cases the current acquires an
anomalous $4\pi$-periodicity with respect to the magnetic flux that is absent
if the two edges are not tunnel-coupled to each other. The result, that
provides at the same time a characterisation of the device and a possible
experimental signature of the coupling between the edges, is stable against
temperature. The processes responsible for the anomalous $4\pi$-periodicity
are the ones where, within the constriction, one of the two electrons forming
a Cooper pair tunnels between the two edges.

  *

##  1 Introduction

The quest for ballistic electronic channels in the absence of magnetic fields
pushed intense scientific efforts into the development of topologically
protected edge states beyond the quantum Hall effect [1, 2]. Based on the
theoretical proposals by Kane and Mele [3, 4] and by Bernevig, Hughes, and
Zhang [5], in 2007 such edge states were detected in HgTe-CdTe
heterostructures [6, 7]. The state of electronic matter that was created is
called quantum spin Hall effect (QSHE). In the following years many other
platforms added up to HgTe-CdTe heterostructures, ranging from InAs-GaSb
heterostructures [8, 9], to bismuthene [10], 1T’-WTe2 [11], and jacutingaite
[12].

The QSHE is characterised by the fact that at low energy the electrons
circulate around the edges, as in the quantum Hall effect. However,
interestingly, they have well defined helicity, meaning that electrons with
opposite spin projection move in opposite directions [3, 4, 13, 14]. This fact
opens unprecedented possibilities in spintronics [15, 16, 17, 18]. Additional
technological applications of the QSHE are in topologically protected quantum
computation [19, 20, 21, 22], through the engineering of Majorana fermions
[23, 24, 25, 26, 27, 28, 29, 30, 31] and parafermions [32, 33, 34, 35], and in
superconducting spintronics [36, 37, 38, 39, 40]. At a more fundamental level,
electron quantum optics experiments [41, 42, 43] and fractional soliton
physics [44, 45, 46, 47, 48] could be inspected.

To fully exploit the potential of the QSHE, nanostructuring must be performed
in order to manipulate the edge states. In this direction, a substantial
effort has been devoted to the proximisation with superconductors [49, 50, 51,
52] and to the creation of constrictions between the edges [53, 54, 55].
Indeed, the combination of the two could allow - in complex setups - for the
creation of Majorana fermions, parafermions, Floquet bound states, and equal
spin pairing [56, 57, 58, 59, 60, 61]. Thermal and thermoelectric properties
have also been addressed [62, 63]. Most remarkably, the engineering of long
constrictions (on the scale of the inverse Fermi momentum) makes ferromagnetic
barriers, that were never implanted on QSHE systems, unnecessary. However,
surprisingly, the main building block of the aforementioned proposals, that is
the long topological constriction with superconducting contacts, is still
largely unexplored from the theoretical point of view and has never been
experimentally realised. Moreover, the only result reported in literature
dealing with a long topological constriction, in the absence of
superconductivity, is about an interaction induced reduction of the
conductance [53], that could disappear in the presence of superconductors, due
to screening. It is hence even difficult to pinpoint a clear signature of the
formation of the constriction in the presence of superconductors.

In this work, we calculate the current-voltage relation for the long
constriction with superconducting contacts and, at the same time, we provide
an accessible scheme for demonstrating the formation of topological
constrictions in the presence of superconductors. More specifically, we
characterise theoretically the Josephson current in presence of a magnetic
field perpendicular to the plane defined by the edges. What we find is that,
due to the constriction, the current acquires a $4\pi$-periodicity in the flux
related to the magnetic field. Such a periodicity is visible both in the zero
bias and in the finite bias regime. We also show that the signature is robust
- or even increased - at nonzero temperature. It is here worth noticing that
the anomalous periodicity we find is not strictly related to crossings in the
Andreev bound states and Majorana physics [19]. Anomalous periodicities have
been reported in [64], but based on different physics in a Josephson junction
with a nanowire as non-superconducting element and not a quantum spin Hall
constriction.

The rest of the article is structured as follows: in Sec. 2 we present our
model; in Sec. 3 we outline the formalism we use for the calculation of the
transport properties. Sec. 4 is devoted to the presentation of the results,
and finally in Sec. 5 we draw our conclusions and discuss the experimental
parameters. Several technical details are presented in the Appendices.

##  2 Model

The figure below (Fig. 1) shows the setup we are going to inspect, consisting
of a Josephson junction made of a two-dimensional topological insulator (2DTI)
sample of length $L$ and width $W$ tunnel-coupled to two superconductors
(SCs), right and left. The superconducting part is a proximitised region of
the topological structure. Halfway in the 2DTI region, a constriction between
the helical edge states, with length $\ell$ ($\ell<L$, $\ell>{k_{F}}^{-1}$,
with $k_{F}$ the Fermi momentum) and width $w\ll W$, is present. A magnetic
field $B$ applied perpendicularly to the plane of the 2DTI and a bias $V$ are
also included.

Figure 1: Schematic of the setup: a sample of 2DTI of length $L$ and width $W$
is laterally tunnel-coupled to two superconductors (right, $r$ and left, $l$);
a magnetic field $B$ applied perpendicularly to the plane of the 2DTI and a
bias $V$ are also included. Halfway in the 2DTI region, there is a
constriction of length $\ell$ and width $w$ between the helical edges.

In the following we are going to analyse one by one all the mentioned
elements.

###  2.1 The constriction

The basic ingredient is represented by the constriction between the helical
edges of the 2DTI. They extend for $-L/2<x<L/2$ and are $W$ apart from each
other at the interfaces with the SCs, while in the narrow constriction of
length $\ell<L$, $\ell>k_{F}^{-1}$, their separation reduces to $w\ll W$ (see
Fig. 2). As $L\gg k_{F}^{-1}$, the system under inspection acquires
translational invariance and the momentum $k$ becomes a good quantum number.
The associated Hamiltonian $\hat{H}_{E}^{0}$ is given by

$\hat{H}_{E}^{0}=\sum_{k}\bm{\hat{c}}^{\dagger}_{k}\mathcal{H}_{E}^{0}\bm{\hat{c}}_{k}.$
(1)

Here, we have

$\bm{\hat{c}}_{k}=\left(\hat{c}_{k,11},\hat{c}_{k,-11},\hat{c}_{k,-1-1},\hat{c}_{k,1-1}\right)^{T},$
(2)

with $\hat{c}_{k,\rho\tau}$ the Fermi operator that annihilates an electron
with momentum $k$ propagating in the $\rho$-direction channel of the $\tau$
edge. We set $\rho=1(-1)$ for the right (left) direction of motion and
$\tau=1(-1)$ for the upper (lower) edge. Due to the spin-momentum locking,
these two indices completely define the edge states, since the spin
polarisation is determined by their helical nature, as shown in Fig. 2.
Moreover, we define

$\mathcal{H}_{E}^{0}=\mathcal{H}_{kin.}+\mathcal{H}_{f.s.},$ (3)

with

$\mathcal{H}_{kin.}=\hbar
v_{F}k\tau_{3}\otimes\rho_{3}-\mu\tau_{0}\otimes\rho_{0}$ (4)

representing the kinetic energy with $v_{F}$ the Fermi velocity and $\mu$ the
chemical potential. In the equation above, $\rho_{i}$ and $\tau_{i}$,
$i=0,1,2,3$ represent the identity and the three Pauli matrices acting on
right/left mover and upper/lower edge space, respectively. Furthermore,

$\mathcal{H}_{f.s.}=f\tau_{1}\otimes\rho_{1}$ (5)

describes a forward scattering tunnelling term across the edges parametrised
by $f$. More specifically, it represents the processes where one electron
changes the edge while preserving the direction of motion, and hence flips its
spin. Note that the Hamiltonian is time-reversal invariant. A remark is here
in order: according to the spatial separation of the helical edges, we assume
the tunnelling to take place only in the constriction, where $w\ll W$.
However, in the long constriction case we inspect, we only consider tunnelling
events that conserve momentum $k$, meaning that we neglect finite size effects
related to $\ell$.

A backward scattering term of the type
$\mathcal{H}_{b.s.}=b\tau_{1}\otimes\rho_{0}$ is not considered here, as well
as any gapping of the edges due to intra-edge mechanisms; indeed,
backscattering (and small gaps opened by intra-edge mechanisms) do not
significantly affect the system away from the Dirac point. Since the latter is
located, in the thick heterostructure case, deep in the valence band [53], we
can ignore these effects. They could be included perturbatively but we do not
expect them to affect our results.
Supported by the coherent transport over long distances in state of the art
samples, we assume the absence of impurities [18]. For a schematic of the
system and of the couplings we consider, see Fig. 2.

Figure 2: Schematic of the constriction of length $\ell$ and width $w$. The
direction-conserving couplings between the upper and lower edge of the 2DTI,
with amplitude $f$, are also shown.

Diagonalising Eq. (1), we obtain

$\hat{H}_{E}^{0}=\sum_{i=1}^{4}\sum_{k}E_{A_{i}}(k)\hat{A}_{k,i}^{\dagger}\hat{A}_{k,i},$
(6)

with

$\displaystyle\hat{A}_{k,1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(-\hat{c}_{k,-11}+\hat{c}_{k,-1-1}\right)$
$\displaystyle\hat{A}_{k,2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\hat{c}_{k,-11}+\hat{c}_{k,-1-1}\right)$
$\displaystyle\hat{A}_{k,3}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(-\hat{c}_{k,11}+\hat{c}_{k,1-1}\right)$
$\displaystyle\hat{A}_{k,4}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\hat{c}_{k,11}+\hat{c}_{k,1-1}\right),$
(7)

and

$\displaystyle E_{A_{1}}(k)$ $\displaystyle=$ $\displaystyle-f-\hbar
v_{F}k-\mu$ $\displaystyle E_{A_{2}}(k)$ $\displaystyle=$ $\displaystyle
f-\hbar v_{F}k-\mu$ $\displaystyle E_{A_{3}}(k)$ $\displaystyle=$
$\displaystyle-f+\hbar v_{F}k-\mu$ $\displaystyle E_{A_{4}}(k)$
$\displaystyle=$ $\displaystyle f+\hbar v_{F}k-\mu.$ (8)

These four eigenstates have well-defined chirality (left for $\hat{A}_{k,1}$,
$\hat{A}_{k,2}$ and right for $\hat{A}_{k,3}$, $\hat{A}_{k,4}$). We highlight
the $1/\sqrt{2}$ weight in each of the combination $\hat{A}_{k,i}$, which
means that the new eigenstates are an equal superposition of an upper-edge
state and a lower-edge state. The role of $f$ is hence to split, in energy,
the dispersion. Notice the analogy with the Rashba coupling in quantum wires
[65, 66, 67, 68, 69].

In the following, when discussing the tunnelling processes, we will also
introduce a magnetic flux $\phi$ through the plane of the 2DTI.

###  2.2 Superconducting leads and effective Hamiltonian

The edges just described are proximitised, for $x<-L/2$ and $x>L/2$, with
standard BCS superconductors. The left SC, extending for $x<-L/2$, is indexed
by $j=-1$ while the right one, extending for $x>L/2$, by $j=1$. The
superconducting pairing is denoted by $\Delta$ and the bare superconducting
pairing phases - that we keep distinguished in principle - by
$\varphi_{j}^{0}$. Moreover, we assume the chemical potential to be the same
in the two SCs. Each SC is described by a Hamiltonian $\hat{H}^{j}_{S}$ and
the coupling between the SCs and the constriction is modeled by means of a
time-reversal invariant Hamiltonian [70]
$\hat{H}_{T}=\sum_{j}\hat{H}_{T}^{j}$. The explicit form of $\hat{H}^{j}_{S}$
and $\hat{H}^{j}_{T}$ can be found in Appendix A.

To obtain an effective Hamiltonian of the proximitised system, we integrate
out the SCs. This calculation is performed in direct space and in the absence
of the magnetic field. Eventually, we reinsert the magnetic field keeping in
mind the gauge invariance of the phase [70].
The extra term appearing in the edge Hamiltonian can be approximately written
as [70]

$\delta\hat{H}_{E}\approx\sum_{\zeta_{1},\zeta_{2},j}\left[\Gamma_{\zeta_{1},\zeta_{2},j}\hat{\psi}_{\zeta_{1}}(x_{j}^{-})\hat{\psi}_{\zeta_{2}}(x_{j}^{+})+h.c.\right],$
(9)

with $\hat{\psi}_{\rho\tau}(x)=1/\sqrt{L}\sum_{k}\hat{c}_{k,\rho\tau}e^{ikx}$.
In Eq. (9), $\zeta_{1},\zeta_{2}$ are collective indices standing for
$\rho_{1}\tau_{1},\rho_{2}\tau_{2}$, and
$x_{j}^{\pm}=jL/2\pm\delta_{\zeta_{1},\zeta_{2}}\xi/2$, where $\xi=\hbar
v_{F}/\Delta$ is the coherence length in the edges, which represents the short
distance cutoff of our system, $\xi\ll L$. The approximation is done for the
regime $E\ll\Delta$. In this regime, transport between the SCs and the edges
is essentially carried out by Cooper pairs (CPs) and there is no contribution
of single quasi-particles.
Eq. (9) accommodates every possible process of injection of the two electrons
(see Fig. 3): either in a spin-singlet or spin-triplet state (the triplet
processes being proportional to a factor
$\tilde{f}_{T}=f_{T}/\sqrt{1+f_{T}^{2}}$111$f_{T}$ is the ratio of spin-
flipping tunnelling processes over spin-conserving ones. It is reasonable to
include such a parameter in the model since it allows to take into account the
Rashba coupling in the material, which makes spin flips possible. Typically,
$f_{T}\ll 1$.); either into the same edges (“direct Andreev reflection”, DAR)
or into different ones (“crossed Andreev reflection”, CAR). CAR is possible as
we assume $\xi_{S}>W$, where $\xi_{S}=\hbar v_{F,S}/\Delta$ (with $v_{F,S}$
the Fermi velocity in the SCs) is the coherence length of the SCs, and its
suppression with respect to DAR will be denoted by $f_{C}$222Here,
$f_{C}\sim\mathsf{f}(k_{F,S}W)e^{-W/\xi_{S}}$, with $\mathsf{f}$ an
oscillatory and decaying function depending on the spatial dimension of the
SCs and $k_{F,S}$ the Fermi momentum in the SCs. This quantity naturally
emerges while integrating out the SCs, see [70] for the details.. Note that
the splitting $x_{j}^{\pm}$ makes tunnelling of spin-triplet CPs into or out
of the same edge possible.

Figure 3: Possible processes of injection of two electrons: in a spin-singlet
or spin-triplet state (left or right panels) and either into the same edges or
into different ones (upper or lower panels).

The summation in Eq. (9) has been antisymmetrised for each
$\zeta_{1}\neq\zeta_{2}$. Following this requirement, the summed terms are
reduced to ten:

$\displaystyle\Gamma_{11,11,j}$ $\displaystyle\Gamma_{11,-11,j}$
$\displaystyle\Gamma_{11,-1-1,j}$ $\displaystyle\Gamma_{11,1-1,j}$
$\displaystyle\Gamma_{-11,-11,j}$ $\displaystyle\Gamma_{-11,-1-1,j}$
$\displaystyle\Gamma_{-11,1-1,j}$ $\displaystyle\Gamma_{-1-1,-1-1,j}$
$\displaystyle\Gamma_{-1-1,1-1,j}$ $\displaystyle\Gamma_{1-1,1-1,j}.$

In momentum space, Eq. (9) becomes

$\delta\hat{H}_{E}=\sum_{\zeta_{1},\zeta_{2},j}\left[\Gamma_{\zeta_{1},\zeta_{2},j}\frac{1}{L}\sum_{k_{1}}\sum_{k_{2}}\hat{c}_{k_{1},\zeta_{1}}e^{ik_{1}x_{j}^{-}}\hat{c}_{k_{2},\zeta_{2}}e^{ik_{2}x_{j}^{+}}+h.c.\right]=\delta\hat{H}^{r}_{E}+\delta\hat{H}^{l}_{E}.$
(10)

Each of the $\Gamma_{\zeta_{1},\zeta_{2},j}$ coefficients contains all the
details specifying the tunnelling process, i.e. species of the electrons
forming the CP, spin-flipping, direct/crossed Andreev reflection. They are all
proportional to the tunnelling rate $\Gamma=\pi\mathfrak{T}^{2}N_{S}$, with
$N_{S}$ the normal density of states per spin at the Fermi-level in the
superconductors and $\mathfrak{T}$ the tunnelling coefficient related to the
opacity of the barrier. We report the full expression for the
$\Gamma_{\zeta_{1},\zeta_{2},j}$ coefficients, but not the whole calculation
to obtain them, since they are a result already known in literature [70]333The
minus sign in Eq. (11), not present in [70], is due to a different choice of
the order for our edges basis, naturally leading to the definition of
$\Gamma_{-1-1,1-1,j}\equiv\Gamma_{-1-1,1-1,j}-\Gamma_{1-1,-1-1,j}$ instead of
$\Gamma_{1-1,-1-1,j}\equiv\Gamma_{1-1,-1-1,j}-\Gamma_{-1-1,1-1,j}$ as in the
original paper.:

$\Gamma_{\zeta_{1},\zeta_{2},j}=(-1)^{\delta_{\zeta_{1},-1-1}\delta_{\zeta_{2},1-1}}\Gamma\left(\tilde{f}_{T}\right)^{\delta_{\rho_{1}*\tau_{1},\rho_{2}*\tau_{2}}}(f_{C})^{\delta_{\tau_{1},-\tau_{2}}}e^{i[\frac{j}{2}k_{F}L(\rho_{1}+\rho_{2})-\varphi^{0}_{j}]}.$
(11)

As expected, the factor $f_{C}$ is present only if $\tau_{1}\neq\tau_{2}$
(CAR) and $\tilde{f}_{T}$ is present only if the two spins of the electrons of
the CP are the same (triplet injection).

It is necessary to modify the rates in order to encode the effects of the
magnetic flux $\phi$ piercing the junction perpendicularly and of an applied
bias $V$ across the two SCs. The former introduces an Aharonov-Bohm (AB) phase
in the processes involving a direct Andreev reflection, while the latter
brings a time dependence into the superconducting phase difference. We neglect
the Zeeman coupling since it only provides a small energy splitting of the
energy bands while not significantly contributing to our results. The orbital
effect is encoded by the minimal coupling
$-i\hbar\nabla\rightarrow-i\hbar\nabla+e\mathbf{A}$, with $e>0$ the absolute
value of the electron charge. We use the Landau gauge to set
$\mathbf{A}=B(-y,0,0)$, so that the states on the upper and lower edge acquire
opposite contributions $\mp\frac{ev_{F}\phi}{2L}$ to the energy. In the long
junction limit $L\gg(\hbar v_{F})/\mu$, the orbital contribution can hence
also be neglected.

The gauge-invariant phase difference experienced by a CP going from the left
to right has the generic form
$\varphi_{r}-\varphi_{l}=(\varphi^{0}_{r}-\varphi^{0}_{l})+\gamma^{AB}+\omega_{J}t$,
where $\omega_{J}=2eV/\hbar$ is the Josephson frequency and $\gamma^{AB}$
takes into account the AB phase picked by the two electrons of the CP.

It is well known [70, 71] that a single electron travelling (from left to
right) on the upper/lower edge all along the junction acquires an AB phase
$\gamma^{AB}_{e}=\pm\pi\phi/2\phi_{0}$, with $\phi_{0}=h/2e$ the
superconducting flux quantum. Therefore, in the absence of inter-edge
tunnelling, $\gamma^{AB}=\pm\pi\phi/\phi_{0}$ for a CP that enters the 2DTI
via a DAR process and exits with another DAR process on the upper/lower edge.
By contrast, if a CP is injected and extracted via two CAR processes, its AB
phase is $\gamma^{AB}=0$, since the AB phases picked by each electron cancels
with each other.

The intriguing peculiarity of our system, however, is that we allow for an
arbitrary number of inter-edge tunnellings in the constriction, and this
allows for novel processes that transfer a CP between the SCs. To characterise
these possibilities, it is important to distinguish between even and odd
numbers of tunneling events.
With even number of tunnellings, a CP can now enter the junction with a DAR
process on the upper/lower edge and leave it with a DAR process on the
opposite (i.e. lower/upper) one. In this case, the AB phase is zero, since the
phase picked by the two electron on the upper edge is cancelled by the one
picked on the lower one.
With an odd number of tunnellings, by contrast, a CP can enter the junction
with a DAR and exit with a CAR (or viceversa). In this scenario, the
unconventional AB phase picked by the CP is given by
$\gamma^{AB}=\pm\pi\phi/2\phi_{0}$. Let us clarify this point with an example,
considering a CP that enters the junction from the left with a DAR in the
upper edge and exits it to the right with a CAR process. In this case, the two
electrons pick the same phase during their transit on the upper edge, from the
left SC to the constriction. However, after the constriction, they travel on
opposite edges and the phases that they pick between the constriction and the
right SC cancel with each other. As a result, $\gamma^{AB}=\pi\phi/2\phi_{0}$.
The opposite sign is obtained when the DAR process happens on the lower edge.

In deriving the expressions of the AB phases above, we neglected the flux
enclosed in the constriction. This approximation, which is fully justified by
the assumption $\ell w\ll WL$, allows us not to care about the exact number of
tunnelling events and their exact location within the constriction. If the
flux enclosed in the constriction were included in our calculation, it would
just add a weak Fraunhofer-like decay to the flux-dependence, but our results
would be qualitatively unchanged.

Interestingly, we can include in our previous description all the AB phases
described before, just by adding flux-dependent factors to the tunnelling
amplitudes in Eq. (11):

$e^{-i\varphi^{0}_{j}}\rightarrow
e^{-i\left[\varphi^{0}_{j}+j\frac{1}{2}\left(\omega_{J}t+\frac{\pi\phi(\tau_{1}+\tau_{2})}{2\phi_{0}}\right)\right]}.$
(12)

Notice that the gauge-invariant phase difference mentioned above is among the
SCs, and hence involves one tunnelling with $j=-1$
($\Gamma^{*}_{\zeta_{1},\zeta_{2},-1}$, from the left SC to the edges) and one
with $j=1$ ($\Gamma_{\zeta_{3},\zeta_{4},1}$, from the edges to the right SC).
Eq. (12) returns the correct phase differences we discussed: for two CAR
processes ($\tau_{1}=-\tau_{2}$), there is no flux dependence; for two DAR
processes on the same edge $\tau$, we do get a flux dependent phase
$\tau\pi\phi/\phi_{0}$; for two DAR on opposite edges we get zero; for a CAR
and a DAR we get the anomalous term, with the sign depending on the $\tau$ of
the DAR.

Finally, we make the unitary transformation inverse to the one of Eq. (7). Let
$a_{\zeta_{i},i}$, with $\zeta_{i}=11,\,-11,\,-1-1,\,1-1$ and
$i=1,\,2,\,3,\,4$, be the elements of such unitary matrix. Then, for each of
the four operators $\hat{c}_{k,\zeta_{i}}$, we have

$\hat{c}_{k,\zeta_{i}}=a_{\zeta_{i},1}\hat{A}_{k,1}+a_{\zeta_{i},2}\hat{A}_{k,2}+a_{\zeta_{i},3}\hat{A}_{k,3}+a_{\zeta_{i},4}\hat{A}_{k,4}.$
(13)

Substituting Eq. (13) in Eq. (10), we obtain

$\displaystyle\delta\hat{H}_{E}$
$\displaystyle=\sum_{i_{1},i_{2}}\sum_{k_{1},k_{2}}\sum_{\zeta_{1},\zeta_{2},j}\frac{1}{L}\left[\Gamma_{\zeta_{1},\zeta_{2},j}a_{\zeta_{1},i_{1}}a_{\zeta_{2},i_{2}}e^{ik_{1}x_{j}^{-}}e^{ik_{2}x_{j}^{+}}\hat{A}_{k_{1},i_{1}}\hat{A}_{k_{2},i_{2}}+h.c.\right]$
$\displaystyle\equiv\sum_{i_{1},i_{2},j}\sum_{k_{1},k_{2}}\frac{1}{L}\left[\Gamma_{i_{1},i_{2},j}(k_{1},k_{2})\hat{A}_{k_{1},i_{1}}\hat{A}_{k_{2},i_{2}}+h.c.\right],$
(14)

where we have defined

$\Gamma_{i_{1},i_{2},j}(k_{1},k_{2})\equiv\sum_{\zeta_{1},\zeta_{2}}\Gamma_{\zeta_{1},\zeta_{2},j}a_{\zeta_{1},i_{1}}a_{\zeta_{2},i_{2}}e^{ik_{1}x_{j}^{-}}e^{ik_{2}x_{j}^{+}}.$
(15)

For now, the summation over $i_{1},i_{2}$ runs over 16 terms. We can reduce
them up to 10 by antisymmetrising the coefficients when $i_{1}\neq i_{2}$,
introducing the new coefficients $\alpha_{i_{1},i_{2}}$444E.g., for
$i_{1},i_{2}=1,2$:
$\displaystyle\sum_{k_{1},k_{2}}\Gamma_{1,2,j}(k_{1},k_{2})\hat{A}_{k_{1},1}\hat{A}_{k_{2},2}+\sum_{k_{1},k_{2}}\Gamma_{2,1,j}(k_{1},k_{2})\hat{A}_{k_{1},2}\hat{A}_{k_{2},1}=$
$\displaystyle=$
$\displaystyle\sum_{k_{1},k_{2}}\Gamma_{1,2,j}(k_{1},k_{2})\hat{A}_{k_{1},1}\hat{A}_{k_{2},2}+\sum_{k_{1},k_{2}}\Gamma_{2,1,j}(k_{2},k_{1})\hat{A}_{k_{2},2}\hat{A}_{k_{1},1}=$
$\displaystyle=$
$\displaystyle\sum_{k_{1},k_{2}}\Gamma_{1,2,j}(k_{1},k_{2})\hat{A}_{k_{1},1}\hat{A}_{k_{2},2}-\sum_{k_{1},k_{2}}\Gamma_{2,1,j}(k_{2},k_{1})\hat{A}_{k_{1},1}\hat{A}_{k_{2},2}=$
$\displaystyle=$
$\displaystyle\sum_{k_{1},k_{2}}\left(\Gamma_{1,2,j}(k_{1},k_{2})-\Gamma_{2,1,j}(k_{2},k_{1})\right)\hat{A}_{k_{1},1}\hat{A}_{k_{2},2}\equiv$
$\displaystyle\equiv$
$\displaystyle\sum_{k_{1},k_{2}}\alpha_{1,2,j}(k_{1},k_{2})\hat{A}_{k_{1},1}\hat{A}_{k_{2},2}.$
. We obtain

$\delta\hat{H}_{E}=\sum_{j}\delta\hat{H}_{E}^{j}=\sum_{j}\sum_{i_{1},i_{2}}\sum_{k_{1},k_{2}}\frac{1}{L}\left[\alpha_{i_{1},i_{2},j}(k_{1},k_{2})\hat{A}_{k_{1},i_{1}}\hat{A}_{k_{2},i_{2}}+h.c.\right],$
(16)

where now
$\alpha_{i_{1},i_{2}}=\alpha_{1,1}\quad\alpha_{1,2}\quad\alpha_{1,3}\quad\alpha_{1,4}\quad\alpha_{2,2}\quad\alpha_{2,3}\quad\alpha_{2,4}\quad\alpha_{3,3}\quad\alpha_{3,4}\quad\alpha_{4,4}$.

We hence have the effective Hamiltonian $\hat{H}$ of the edges, which reads

$\hat{H}=\hat{H}_{E}^{0}+\delta\hat{H}_{E}.$ (17)

In the next section, we briefly set up the formalism we use for the evaluation
of the Josephson current.

##  3 Formalism for the transport properties

Let $\hat{N}=\sum_{i}\sum_{k}\hat{A}^{\dagger}_{ki}\hat{A}_{ki}$ be the total
number operator relative to the electrons. The net change in the number of
electrons, different from zero due to the coupling to the SCs, is given by
$\dot{\hat{N}}=\dot{\hat{N}}^{r}+\dot{\hat{N}}^{l}$, with

$\displaystyle\dot{\hat{N}}^{r}$
$\displaystyle=\frac{i}{\hbar}[\hat{H}_{E}^{0}+\delta\hat{H}_{E}^{r},\hat{N}]=\frac{i}{\hbar}[\delta\hat{H}_{E}^{r},\hat{N}]$
$\displaystyle\dot{\hat{N}}^{l}$
$\displaystyle=\frac{i}{\hbar}[\hat{H}_{E}^{0}+\delta\hat{H}_{E}^{l},\hat{N}]=\frac{i}{\hbar}[\delta\hat{H}_{E}^{l},\hat{N}]$
(18)

in the Heisenberg picture.

The operator $\hat{I}^{j}$, relative to the current flowing in the edges reads
as555We remind that the perturbation $\delta\hat{H}_{E}^{j}$ \- and therefore
also $\hat{I}^{j}$ \- acquires a time dependence in the
$\Gamma_{\zeta_{1},\zeta_{2},j}$ coefficients as the bias $V$ is non-zero (see
after Eq. (10) and Appendix A). $\hat{I}^{j}(t)=e\dot{\hat{N}}^{j}$. According
to this convention, $\hat{I}^{r}(t)$ and $\hat{I}^{l}(t)$ are the currents
injected from the superconducting leads in the edges, as shown in Fig. 4 by
the blue arrows. The total current is

$\hat{I}^{tot}(t)=\hat{I}^{r}(t)-\hat{I}^{l}(t),$ (19)

flowing in the direction shown in the figure (yellow arrow).

Figure 4: Direction of the right, left and total current flow according to our
conventions.

Since the procedure is independent of the choice of the lead, in the following
we sketch the calculation of the generic $j^{th}$ term.

Carrying out the anticommutation leads to

$\hat{I}^{j}=\frac{2ie}{\hbar
L}\sum_{i_{1},i_{2}}\sum_{k_{1},k_{2}}\frac{1}{L}\left[\alpha_{i_{1},i_{2},j}(k_{1},k_{2})\hat{A}_{k_{1},i_{1}}\hat{A}_{k_{2},i_{2}}-h.c.\right].$
(20)

Assuming the coupling between the lead $j$ and the edges to be weak (namely
$\Gamma N_{E}\ll 1$, with $N_{E}$ the density of states at the Fermi-level in
the edge system), $\delta\hat{H}_{E}$ can be regarded as a small perturbation.
We want to compute the expectation value

${I}^{j}(t)\equiv\left\langle\hat{U}(-\infty,t)\hat{I}^{j}(t)\hat{U}(t,-\infty)\right\rangle,$
(21)

taken with respect to the unperturbed edge state system in the far past. Here,
$\hat{U}(-\infty,t)=\hat{U}^{\dagger}(t,+\infty)=\hat{T}_{+}e^{-\frac{i}{\hbar}\int_{-\infty}^{t}d\tau\delta\hat{H}_{E}(\tau)}$
is the time-evolution operator in the interaction picture representation, with
$\hat{T}_{+}$ the time-ordering operator.

According to the linear response theory [72], that is to say up to linear
order, in $\delta\hat{H}_{E}$ (second order in $\Gamma$ in this case), we get

${I}^{j}(t)\approx\frac{i}{\hbar}\int_{-\infty}^{t}d\tau\left\langle\left[\delta\hat{H}_{E}(\tau),\hat{I}^{j}(t)\right]\right\rangle_{0}.$
(22)

The subscript “0” serves as a reminder of the fact that we are considering an
equilibrium average calculated with respect to the unperturbed system
(described by $\hat{H}_{E}^{0}$, see Eq. (6)) in the past. In the calculation
of $I^{j}(t)$, the relevant perturbation is the one induced by the $-j^{th}$
superconductor, namely $\delta\hat{H}^{-j}_{E}$. This is easily understood by
looking at the expressions of the $\Gamma_{\zeta_{1},\zeta_{2},j}$
coefficients: since they keep trace of the phase of the superconductor to whom
they are related, the SCs phase difference-dependent supercurrent originates
from the $-j,\,j$ terms.

Moving forward with the calculations (see Appendix B for the details), we get

$\displaystyle{I}^{j}(t)=$
$\displaystyle\frac{8e}{\hbar^{2}}\text{Im}\bigg{\\{}\int_{-\infty}^{\infty}dt^{\prime}\theta(t^{\prime})\sum_{i_{1},i_{2}}\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}}\frac{1}{L^{2}}\alpha_{i_{1},i_{2},j}(k_{1}^{\prime},k_{2}^{\prime},t)\alpha^{*}_{i_{1},i_{2},-j}(k_{1},k_{2},t-t^{\prime})$
$\displaystyle\left\langle\left[\hat{A}_{k_{1}^{\prime},i_{1}}(t)\hat{A}_{k_{2}^{\prime},i_{2}}(t),\hat{A}^{\dagger}_{k_{2},i_{2}}(t-t^{\prime})\hat{A}^{\dagger}_{k_{1},i_{1}}(t-t^{\prime})\right]\right\rangle_{0}\bigg{\\}},$
(23)

where we have recovered the explicit time-dependence of the coefficients
$\alpha$ and the correlation functions from now on will be time-ordered.

The unperturbed edge system being time-translation invariant, Eq. (23) is
equivalent to

$\displaystyle{I}^{j}(t)=$
$\displaystyle\frac{8e}{\hbar^{2}}\text{Im}\bigg{\\{}\int_{-\infty}^{\infty}dt^{\prime}\theta(t^{\prime})\sum_{i_{1},i_{2}}\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}}\frac{1}{L^{2}}\alpha_{i_{1},i_{2},j}(k_{1}^{\prime},k_{2}^{\prime},t)\alpha^{*}_{i_{1},i_{2},-j}(k_{1},k_{2},t-t^{\prime})$
$\displaystyle\left\langle\left[\hat{A}_{k_{1}^{\prime},i_{1}}(t^{\prime})\hat{A}_{k_{2}^{\prime},i_{2}}(t^{\prime}),\hat{A}^{\dagger}_{k_{2},i_{2}}(0)\hat{A}^{\dagger}_{k_{1},i_{1}}(0)\right]\right\rangle_{0}\bigg{\\}},$
(24)

where the time-evolved operators in the expectation value are simply given by

$\hat{A}_{k_{1}^{\prime},i_{1}}(t^{\prime})=\hat{A}_{k_{1}^{\prime},i_{1}}e^{-iE_{A_{i_{1}}}(k_{1}^{\prime})t^{\prime}/\hbar},\,\hat{A}_{k_{2}^{\prime},i_{2}}(t^{\prime})=\hat{A}_{k_{2}^{\prime},i_{2}}e^{-iE_{A_{i_{2}}}(k_{2}^{\prime})t^{\prime}/\hbar}.$
(25)

Making use of Wick’s theorem and recalling that

$\left\langle\hat{A}_{k_{1},i_{1}}(t^{\prime})\hat{A}^{\dagger}_{k_{2},i_{2}}(0)\right\rangle_{0}=\delta_{i_{1},i_{2}}\delta_{k_{1},k_{2}}\frac{1}{1+e^{\beta
E_{A_{i_{1}}}(k)}},$ (26)

with $\beta=1/k_{B}T$, we obtain

$\displaystyle\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}}$
$\displaystyle\frac{1}{L^{2}}\alpha_{i_{1},i_{2},j}(k_{1}^{\prime},k_{2}^{\prime},t)\alpha^{*}_{i_{1},i_{2},-j}(k_{1},k_{2},t-t^{\prime})\left\langle\left[\hat{A}_{k_{1}^{\prime},i_{1}}(t^{\prime})\hat{A}_{k_{2}^{\prime},i_{2}}(t^{\prime}),\hat{A}^{\dagger}_{k_{2},i_{2}}(0)\hat{A}^{\dagger}_{k_{1},i_{1}}(0)\right]\right\rangle_{0}=$
$\displaystyle=\frac{1}{(2\pi)^{2}}\bigg{\\{}-$
$\displaystyle\delta_{i_{1},i_{2}}\int_{-\infty}^{+\infty}dk_{1}\,e^{-iE_{A_{i_{1}}}(k_{1})t^{\prime}/\hbar}\frac{e^{\beta
E_{A_{i_{1}}}(k_{1})}}{1+e^{\beta E_{A_{i_{1}}}(k_{1})}}\cdot$
$\displaystyle\cdot$
$\displaystyle\int_{-\infty}^{+\infty}dk_{2}\,\alpha_{i_{1},i_{1},j}(k_{1},k_{2},t)\alpha^{*}_{i_{1},i_{1},-j}(k_{1},k_{2},t-t^{\prime})e^{-iE_{A_{i_{1}}}(k_{2})t^{\prime}/\hbar}\frac{e^{\beta
E_{A_{i_{1}}}(k_{2})}}{1+e^{\beta E_{A_{i_{1}}}(k_{2})}}+$ $\displaystyle+$
$\displaystyle\int_{-\infty}^{+\infty}dk_{1}\,e^{-iE_{A_{i_{1}}}(k_{1})t^{\prime}/\hbar}\frac{e^{\beta
E_{A_{i_{1}}}(k_{1})}}{1+e^{\beta E_{A_{i_{1}}}(k_{1})}}\cdot$
$\displaystyle\cdot$
$\displaystyle\int_{-\infty}^{+\infty}dk_{2}\,\alpha_{i_{1},i_{2},j}(k_{1},k_{2},t)\alpha^{*}_{i_{1},i_{2},-j}(k_{1},k_{2},t-t^{\prime})e^{-iE_{A_{i_{2}}}(k_{2})t^{\prime}/\hbar}\frac{e^{\beta
E_{A_{i_{2}}}(k_{2})}}{1+e^{\beta E_{A_{i_{2}}}(k_{2})}}\bigg{\\}}.$ (27)

The full expansion of Eq. (24) is cumbersome. However, Eqs. (24)-(27) are
useful to identify the typical structure of each term: a product of two
$\alpha$ coefficients - which, through the $\Gamma$s, contain all the details
specifying the tunnelling process - and two Green functions given by the two
integrals.

##  4 Results

###  4.1 Analytical results

The long and cumbersome calculation sketched in the previous section leads to
the following expression for the right/left current

$\displaystyle I^{r/l}(t)$
$\displaystyle=\mathcal{C}\,\text{Im}\bigg{\\{}e^{\mp
i(\omega_{J}t+\varphi^{0}_{r}-\varphi^{0}_{l})}\int_{0}^{+\infty}ds\,e^{\pm
is\tilde{V}}\left[A_{1}\cos{\left(\pi\frac{\phi}{\phi_{0}}\right)}+A_{2}\sin{\left(\frac{\pi}{2}\frac{\phi}{\phi_{0}}\right)}+A_{3}\right]\bigg{\\}}=$
$\displaystyle=\mathcal{C}\,\text{Im}\bigg{\\{}e^{\mp
i(\omega_{J}t+\varphi^{0}_{r}-\varphi^{0}_{l})}\left[\tilde{A}^{r/l}_{1}\cos{\left(\pi\frac{\phi}{\phi_{0}}\right)}+\tilde{A}_{2}^{r/l}\sin{\left(\frac{\pi}{2}\frac{\phi}{\phi_{0}}\right)}+\tilde{A}_{3}^{r/l}\right]\bigg{\\}}.$
(28)

Here, $\mathcal{C}\equiv(-2e\Delta\Gamma^{2})/(\pi^{2}\hbar^{3}v_{F}^{2})$ is
a constant. The dimensionless quantities $\tilde{V}=eV/\Delta$ and
$s=\frac{t^{\prime}\Delta}{\hbar}$, with $t^{\prime}$ a time variable have
been introduced. Notice that $A_{1},\,A_{2}$ and $A_{3}$ are three complex
factors that depend on all the parameters except for the flux $\phi$.
Likewise, $\tilde{A}^{r/l}_{1},\,\tilde{A}^{r/l}_{2},\,\tilde{A}^{r/l}_{3}$
stand for the three coefficients $A_{1},\,A_{2},\,A_{3}$ once the integration
over $s$ is done. More details can be found in Appendix B.

We draw the attention to the term proportional to
$\sin{\left(\frac{\pi}{2}\frac{\phi}{\phi_{0}}\right)}$. Indeed, it encodes a
$4\pi$-periodicity with respect to the magnetic flux. This term is absent in
the absence of forward scattering among the two edges ($f=0$). Moreover, in
order to have a nonzero $\tilde{A}^{r/l}_{2}$ one must additionally have
$\tilde{f}_{T}\neq 0$ provided by the strong spin-orbit coupling
characterising the structure. This is due to the fact that an odd number of
tunnellings within the constriction necessarily leads to a spin-flip of one
electron, that has to be counterbalanced by an additional spin-flip at the
SC-2DTI interface. What the analytical calculation shows is that the physical
processes related to the anomalous periodicity are the ones in which the CP
switches from being on one edge only to being delocalised between the two
edges. These processes are proportional to $\tilde{f}_{T}f_{C}$ and must
include one forward tunnelling in the constriction (or an odd number of them).

An example is depicted in Fig. 5, where a CP enters the junction with a DAR in
the upper edge and exits it with a CAR after one electron tunnels in the
constriction. The AB phases picked up by the two electrons sum up on the left
side of the constriction and cancel on the right side, while no meaningful
phase is picked along the constriction (making the precise location of the
tunnelling within the constriction irrelevant). The amplitude of the process
just described is proportional to $\Gamma_{11,1-1,1}\Gamma^{*}_{11,11,-1}$,
which gives

$\displaystyle\Gamma_{11,1-1,1}\Gamma^{*}_{11,11,-1}$
$\displaystyle=\left(\Gamma
f_{C}\,e^{-i\left(\varphi^{0}_{r}+\frac{1}{2}\omega_{J}t\right)}\right)\left(\Gamma\tilde{f}_{T}\,e^{i\left(\varphi^{0}_{l}-\frac{1}{2}\omega_{J}t-\frac{\pi\phi}{2\phi_{0}}\right)}\right)=$
$\displaystyle=\Gamma^{2}f_{C}\tilde{f}_{T}e^{-i\left[\left(\varphi^{0}_{r}-\varphi^{0}_{l}\right)+\omega_{J}t\right]}e^{-i\frac{\pi\phi}{2\phi_{0}}}.$

Notice that the flux dependence does correspond to what we commented before. A
spin-flip $\tilde{f}_{T}$ is needed due to the fact that in (1) and (5) in the
figure we have a singlet state and that within the constriction an odd number
(one) of inter-edge tunnellings results in a spin-flip of one electron.

Figure 5: A prototypical example of a process carrying a $4\pi$-periodic
dependence on the flux $\phi$: (1) a CP is injected from the left SC in the
upper edge; (2) during the injection, one of the electrons is spin-flipped,
and thus the CP enters the edge system in a spin-triplet state, acquiring a
$\tilde{f}_{T}$ factor accounting for the spin-flip. (3) Having enclosed a
flux $\phi/2$, in the constriction an electron tunnels to the lower edge; (4)
when the opposite end of the junction is reached, a final $f_{C}$ factor keeps
track of the CAR; (5) clearly, the CP reaches the right SC in a spin-singlet
state.

The anomalous periodicity hence represents a hallmark of the presence of
forward scattering and the coupling between the edges. Note that it is in
principle difficult to demonstrate the presence of such a coupling. Indeed, as
already discussed, in long constrictions the single particle backscattering is
only able to open a gap at the Dirac point. However such point is often hidden
in the valence band of the structure [53, 73, 74]. Achieving zero conductance
as a function of the gate voltage, that would strongly indicate the formation
of the constriction, is hence not always possible and was never experimentally
realised. At present, the signature that is related to the existence of the
constriction is a reduction of the conductance from $2e^{2}/h$ to $e^{2}/h$.
However, this behaviour is due to electronic interactions and is hence unclear
if it would persist in the presence of superconductors in close contact to the
constriction. A clearer signature on the formation of the constriction is thus
in order.

###  4.2 Quantitative analysis

In this subsection, in addition to the already introduced $\tilde{V}$, we
define the natural dimensionless quantities

$\displaystyle\tilde{L}=\frac{L\Delta}{\hbar v_{F}},\qquad\tilde{T}=\frac{\pi
k_{B}T}{\Delta},\qquad\tilde{\mu}=\frac{\mu}{\Delta},\qquad\tilde{f}=\frac{f}{\Delta}$

to better understand the relevant scales. Notice that our previous assumption
$E\ll\Delta$ requires $\tilde{V},\,\tilde{T}\ll 1$.

In the limit $\tilde{V}\rightarrow 0$, we inspect the critical current
$I_{c}$, given by the total current $I^{r}-I^{l}$ for
$\varphi^{0}_{r}-\varphi^{0}_{l}=\pi/2$. The anomalous periodicity is more
pronounced at high temperature $\tilde{T}$, while at low temperature, a
$2\pi$-periodicity is almost recovered, as shown in Fig. 6(a). There, the
current at different temperatures, normalised to its maximum value, is shown
as a function of the flux. This scaling allows to easily compare the
periodicity of the curves. In the picture, darker lines correspond to lower
temperatures. However, the magnitude of the current itself decreases as
temperature is raised, as shown in Fig. 6(b), where the plots are not scaled.

Figure 6: (a) $I_{c}$, normalised with respect to its maximum, for
$\tilde{L}=20$, $k_{F}L=6\pi$, $\tilde{f}_{T}=0.4$, $f_{C}=0.3$,
$\tilde{\mu}=10^{-4}$ and $\tilde{f}=0.3$. The plot is represented as a
function of $\phi$ (in units of $\phi_{0}$) and for $\tilde{V}=0$. We have
$\tilde{T}$ varying between 0.008 and 0.2 (see the red bar legend aside).
(b) Same but without the normalisation to the maximum.

In the finite $\tilde{V}$ regime the current becomes time dependent, in view
of the AC Josephson effect. As in Ref. [70], we hence analyse the Fourier
component of the current at the frequency $\omega_{J}$ given by the voltage.
Quantitatively, what we analyse is

$I_{\omega_{J}}^{r}=\left|\frac{1}{\text{T}}\int_{-\text{T}/2}^{\text{T}/2}e^{-i\omega_{J}t}I^{r}(t)\right|,$
(29)

with $\text{T}=2\pi/\omega_{J}$. The Fourier transform of the total current is
qualitatively similar. We observe that the applied bias increases the
visibility of the anomalous periodicity, as shown in Fig. 7. On top of that,
and differently from the zero bias case, the $4\pi$ component becomes more
pronounced for lower temperature (Fig. 8(a)). Moreover, as in the zero bias
limit, higher temperature makes the signal smaller. This fact is shown in Fig.
8(b), which represents the same plots as Fig. 8(a) but without scaling. The
zero bias and the finite bias regime, exhibiting different behaviours to
varying of the parameters, can hence be useful for different materials.

Figure 7: $I^{r}_{\omega_{J}}$, normalised with respect to its maximum, for
$\tilde{L}=20$, $k_{F}L=6\pi$, $\tilde{f}_{T}=0.4$, $f_{C}=0.3$,
$\tilde{\mu}=10^{-4}$, $\tilde{f}=0.3$ and $\tilde{T}=0.1$. The plot is
represented as a function of $\phi$ (in units of $\phi_{0}$) and for
$\tilde{V}$ varying between 0.008 and 0.2 (see the green bar legend aside).
Figure 8: (a) $I^{r}_{\omega_{J}}$, normalised with respect to its maximum,
for $\tilde{L}=20$, $k_{F}L=6\pi$, $\tilde{f}_{T}=0.4$, $f_{C}=0.3$,
$\tilde{\mu}=10^{-4}$, $\tilde{f}=0.3$ and $\tilde{V}=0.1$. The plot is
represented as a function of $\phi$ (in units of $\phi_{0}$) and for
$\tilde{T}$ varying between 0.008 and 0.2 (see the red bar legend aside).
(b) Same but without the normalisation to the maximum.

##  5 Discussion and conclusions

In this work, we have analysed a Josephson junction where the non-
superconducting element is a long constriction between helical edge states,
pierced by a magnetic flux. We have shown that the interplay between the
strong spin-orbit interaction characterising the system and the coupling
between the edges results in an anomalous $4\pi$-periodicity of the Josephson
current with respect to the flux. This behaviour is present both in the DC and
in the AC Josephson effect. It is robust with respect to temperature and no
fine tuning is necessary. The effect in our system needs the presence of three
properties: a DAR with spin-flipping, a tunnelling between the edges and a
CAR, as shown in Fig. 5. Throughout our discussion, we neglected the magnetic
flux enclosed in the constriction; if included, it would just add a weak
Fraunhofer-like decay to the pattern in our plots, without affecting
significantly our results.

The anomalous periodicity we found represents a hallmark of a constriction
between edges tunnel-coupled to superconductors. Its physical origin resides
in the possibility of switching the CP nature (localised on one
edge/delocalised on both edges) enabled by the single-electron tunnelling.

In the absence of inter-edge tunnelling, the fate of the CPs is known from the
very beginning: CPs entering the weak link in DAR on the left end
($x=-L/2,\,y=\pm W/2$) necessarily end up on the right end of the same edge
($x=L/2,\,y=\pm W/2$), picking a phase difference $\pm\pi\phi/\phi_{0}$;
analogously, CPs entering the link in CAR unavoidably leave it in CAR without
collecting any phase, as discussed in Subsec. (2.2). This scenario corresponds
to a standard SQUID pattern, similar to [70], and is plotted in Fig. 9.
On the other hand, the inter-edge tunnelling we included broadens the
possibilities, in particular allowing a CP to enter the junction in DAR and
leave it in CAR or viceversa. The single-electron tunnelling marks the
boundary between two kinds of CP, with different insights of the magnetic
flux. For the setup we sketched, the phase difference in this case amounts to
$\pm\pi\phi/2\phi_{0}$, leading to a $4\pi$-periodic contribution to the
Josephson current.
To summarise our finding: if $f=0$ (Fig. 9), there are a $\pi$ and a $2\pi$
components (the latter being usually known as even-odd effect, due to the
presence of CAR, as discussed for instance in [70, 75]); temperature and bias
allow to make the $2\pi$ term more or less prominent (as commented in [70]).
With a finite $f$ (Figs. 6-8), a novel $4\pi$ component arises (coexisting
with the other two that are more or less still present).

We argue that, whenever in a constriction there is the possibility of a swap
from a CP living on one side to a CP living on both sides, anomalous
periodicities can emerge. In this sense, they are more generally a phenomenon
related to single electron physics in the superconducting context. Applied to
the setup under investigation, this physical interpretation substantiate the
robustness of the anomalous periodicity with respect to slight spatial
variations of the parameters.

Figure 9: (a) $I^{r}_{\omega_{J}}$, normalised with respect to its maximum,
for $\tilde{L}=20$, $k_{F}L=6\pi$, $\tilde{f}_{T}=0.4$, $f_{C}=0.3$,
$\tilde{\mu}=10^{-4}$, $\tilde{f}=0$ and $\tilde{T}=0.1$. The plot is
represented as a function of $\phi$ (in units of $\phi_{0}$) and for
$\tilde{V}$ varying between 0.008 and 0.2 (see the green bar legend aside).
(b) Same parameters values, but now $\tilde{V}=0.1$ and $\tilde{T}$ varies
between 0.008 and 0.2 (see the red bar legend aside).

While our discussion is general and applies to all the quantum spin Hall
(candidate) systems, we argue that our results are readily observable in HgTe-
CdTe heterostructures. Indeed, with the realistic length scales $L\simeq
3\,\mu m$, $\ell\simeq 1\,\mu m$, $W\simeq 200\,nm$ and $w\simeq 30\,nm$, the
magnetic field necessary to observe the effect is of the order $B\simeq
10^{-3}\,T$. The temperature range is limited by the pairing potential
$\Delta\sim 0.40\,meV$ (a reasonable value for HgTe-based systems [76, 77]),
which determines an upper bound of $\sim 300\,mK$. Similarly, the bias should
not exceed $\sim 80\mu eV$. We mention in passing that the edge states can be
robust even in the presence of magnetic fields that are much stronger than the
ones needed for our proposal [78, 79, 80].

### Acknowledgments

This work was supported by the “Dipartimento di Eccellenza MIUR 2018-2022”, by
the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in
Quantum Matter (EXC2147, project-id 390858490) and by the DFG (SPP1666 and
SFB1170 “ToCoTronics”).

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## Appendix A: Hamiltonian of the SCs and tunnelling Hamiltonian

The $j^{th}$ superconductor Hamiltonian, written in the usual Nambu basis, is
given, in the absence of the magnetic field, by [85]

$\hat{H}_{S}^{j}=\frac{1}{2}\int
d\mathbf{r}\left(\hat{\Psi}^{\dagger}_{j,\uparrow}(\mathbf{r}),\hat{\Psi}^{\dagger}_{j,\downarrow}(\mathbf{r}),\hat{\Psi}_{j,\uparrow}(\mathbf{r}),\hat{\Psi}_{j,\downarrow}(\mathbf{r})\right)\mathcal{H}_{S}^{j}\begin{pmatrix}\hat{\Psi}_{j,\uparrow}(\mathbf{r})\\\
\hat{\Psi}_{j,\downarrow}(\mathbf{r})\\\
\hat{\Psi}^{\dagger}_{j,\uparrow}(\mathbf{r})\\\
\hat{\Psi}^{\dagger}_{j,\downarrow}(\mathbf{r})\end{pmatrix},$ (30)

with

$\mathcal{H}_{S}^{j}=\begin{pmatrix}-\frac{\hbar^{2}\nabla^{2}_{\mathbf{r}}}{2m}-\mu_{SC}&0&0&-\Delta
e^{-i\varphi_{j}^{0}}\\\
0&-\frac{\hbar^{2}\nabla^{2}_{\mathbf{r}}}{2m}-\mu_{SC}&\Delta
e^{-i\varphi_{j}^{0}}&0\\\ 0&\Delta
e^{i\varphi_{j}^{0}}&\frac{\hbar^{2}\nabla^{2}_{\mathbf{r}}}{2m}+\mu_{SC}&0\\\
-\Delta
e^{i\varphi_{j}^{0}}&0&0&\frac{\hbar^{2}\nabla^{2}_{\mathbf{r}}}{2m}+\mu_{SC}\end{pmatrix}.$
(31)

As mentioned in the main text, $\Delta$ is the superconducting pairing
amplitude, $\mu_{SC}$ the chemical potential - which we assume to be the same
in the two SCs, while $\varphi_{j}^{0}$ are the bare pairing phases that we
keep distinguished in principle. Moreover,
$\hat{\Psi}_{j,\uparrow/\downarrow}(\mathbf{r})$ is the annihilation operator
of an electron in the $j^{th}$ superconducting lead, with
$\uparrow/\downarrow$ spin polarisation (with respect to the same axis as the
spin polarisation defined in the edges) and at position $\mathbf{r}$. Eq. (30)
is nothing but the Hamiltonian of a standard BCS superconductor.

The coupling between the superconductors and the constriction is modeled by
means of the time-reversal invariant Hamiltonian [70]
$\hat{H}_{T}=\sum_{j}\hat{H}_{T}^{j}$, with

$\hat{H}_{T}^{j}=\int dx\int
d\mathbf{r}\left(\hat{\Psi}^{\dagger}_{j,\uparrow}(\mathbf{r}),\hat{\Psi}^{\dagger}_{j,\downarrow}(\mathbf{r})\right)\mathcal{T}^{j}(\mathbf{r},x)\begin{pmatrix}\hat{\psi}_{11}(x)\\\
\hat{\psi}_{-11}(x)\\\ \hat{\psi}_{-1-1}(x)\\\
\hat{\psi}_{1-1}(x)\end{pmatrix}+h.c.,$ (32)

where

$\displaystyle\mathcal{T}^{j}(\mathbf{r},x)=$
$\displaystyle\begin{pmatrix}e^{ik_{F}x}\delta(\mathbf{r}-\mathbf{r}_{j,1})&e^{-ik_{F}x}\delta(\mathbf{r}-\mathbf{r}_{j,1})(if_{T})&e^{-ik_{F}x}\delta(\mathbf{r}-\mathbf{r}_{j,-1})&e^{ik_{F}x}\delta(\mathbf{r}-\mathbf{r}_{j,-1})(if_{T})\\\
e^{ik_{F}x}\delta(\mathbf{r}-\mathbf{r}_{j,1})(if_{T})&e^{-ik_{F}x}\delta(\mathbf{r}-\mathbf{r}_{j,1})&e^{-ik_{F}x}\delta(\mathbf{r}-\mathbf{r}_{j,-1})(if_{T})&e^{ik_{F}x}\delta(\mathbf{r}-\mathbf{r}_{j,-1})\end{pmatrix}$
$\displaystyle\frac{\mathfrak{T}}{\sqrt{1+f_{T}^{2}}}\delta(x-j\frac{L}{2}).$
(33)

Here $k_{F}$ is the Fermi momentum in the constriction. Moreover
$\mathbf{r}_{j,\tau}=(jL/2,\tau W/2,0)^{T}$ are the contact points between the
$j$-th superconductor and the edges ($\mathbf{r}_{j,1}$ for the upper and
$\mathbf{r}_{j,-1}$ for the lower edge channel respectively). In Eq. (33) two
new parameters have been introduced: $f_{T}$ and $\mathfrak{T}$. The quantity
$f_{T}$ is the ratio of spin-reversing processes over the spin-conserving
ones. It is reasonable to include such a parameter in the model since it
allows to take into account the Rashba coupling in the material, which makes
spin flips possible. Typically, it is $f_{T}\ll 1$. The other parameter
$\mathfrak{T}$ is the tunnelling coefficient related to the opacity of the
barrier. These parameters are present in the $\Gamma_{\zeta_{1},\zeta_{2},j}$
coefficients as discussed in Sec. (2).

## Appendix B: Calculation of the current

In the following, we address the calculation of the current $I^{j}(t)$ in more
detail. Let’s look back at Eq. (27), and make use of the two definitions in
the main text

$\displaystyle\alpha_{i_{1},i_{2},j}(k_{1},k_{2})$
$\displaystyle\equiv\Gamma_{i_{1},i_{2},j}(k_{1},k_{2})-\Gamma_{i_{2},i_{1},j}(k_{2},k_{1})$
$\displaystyle\Gamma_{i_{1},i_{2},j}(k_{1},k_{2})$
$\displaystyle\equiv\sum_{\zeta_{1},\zeta_{2}}\Gamma_{\zeta_{1},\zeta_{2},j}a_{\zeta_{1},i_{1}}a_{\zeta_{2},i_{2}}e^{ik_{1}x_{j}^{-}}e^{ik_{2}x_{j}^{+}},$

where we remind that the possible values for $i_{1},i_{2}$ are

$\displaystyle 1,1\qquad 1,2\qquad 1,3\qquad 1,4\qquad 2,2\qquad 2,3\qquad
2,4\qquad 3,3\qquad 3,4\qquad 4,4$ (34)

and those for $\zeta_{1},\zeta_{2}$ are

$\displaystyle 11,11$ $\displaystyle 11,-11$ $\displaystyle 11,-1-1$
$\displaystyle 11,1-1$ $\displaystyle-11,-11$ $\displaystyle-11,-1-1$
$\displaystyle-11,1-1$ $\displaystyle-1-1,-1-1$ $\displaystyle-1-1,1-1$
$\displaystyle 1-1,1-1.$ (35)

We focus on the calculation of $I^{j}(t)$ for $j=r=1$. In this case, Eq. (27)
becomes

$\displaystyle\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}}\frac{1}{L^{2}}\alpha_{i_{1},i_{2},1}(k_{1}^{\prime},k_{2}^{\prime},t)\alpha^{*}_{i_{1},i_{2},-1}(k_{1},k_{2},t-t^{\prime})\left\langle\left[\hat{A}_{k_{1}^{\prime},i_{1}}(t^{\prime})\hat{A}_{k_{2}^{\prime},i_{2}}(t^{\prime}),\hat{A}^{\dagger}_{k_{2},i_{2}}(0)\hat{A}^{\dagger}_{k_{1},i_{1}}(0)\right]\right\rangle_{0}=$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}\bigg{\\{}-\delta_{i_{1},i_{2}}\sum_{\zeta_{1},\zeta_{2}}\Gamma_{\zeta_{1},\zeta_{2},1}\sum_{\zeta_{3},\zeta_{4}}\Gamma^{*}_{\zeta_{3},\zeta_{4},-1}\bigg{[}\int_{-\infty}^{+\infty}dk_{1}\,e^{-iE_{A_{i_{1}}}(k_{1})t^{\prime}/\hbar}\frac{e^{\beta
E_{A_{i_{1}}}(k_{1})}}{1+e^{\beta
E_{A_{i_{1}}}(k_{1})}}a_{\zeta_{1},i_{1}}a_{\zeta_{4},i_{1}}e^{ik_{1}\left[L-(\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right]}\cdot$
$\displaystyle\cdot\int_{-\infty}^{+\infty}dk_{2}\,e^{-iE_{A_{i_{1}}}(k_{2})t^{\prime}/\hbar}\frac{e^{\beta
E_{A_{i_{1}}}(k_{2})}}{1+e^{\beta
E_{A_{i_{1}}}(k_{2})}}a_{\zeta_{2},i_{1}}a_{\zeta_{3},i_{1}}e^{ik_{2}\left[L-(\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right]}\bigg{]}+$
$\displaystyle+\int_{-\infty}^{+\infty}dk_{1}\,e^{-iE_{A_{i_{1}}}(k_{1})t^{\prime}/\hbar}\frac{e^{\beta
E_{A_{i_{1}}}(k_{1})}}{1+e^{\beta
E_{A_{i_{1}}}(k_{1})}}\int_{-\infty}^{+\infty}dk_{2}\,e^{-iE_{A_{i_{2}}}(k_{2})t^{\prime}/\hbar}\frac{e^{\beta
E_{A_{i_{2}}}(k_{2})}}{1+e^{\beta E_{A_{i_{2}}}(k_{2})}}\cdot$
$\displaystyle\cdot\bigg{[}\sum_{\zeta_{1},\zeta_{2}}\bigg{(}\Gamma_{\zeta_{1},\zeta_{2},1}a_{\zeta_{1},i_{1}}a_{\zeta_{2},i_{2}}e^{ik_{1}\left[\frac{L}{2}-\delta_{\zeta_{1},\zeta_{2}}\frac{\xi}{2}\right]}e^{ik_{2}\left[\frac{L}{2}+\delta_{\zeta_{1},\zeta_{2}}\frac{\xi}{2}\right]}-\Gamma_{\zeta_{1},\zeta_{2},1}a_{\zeta_{1},i_{2}}a_{\zeta_{2},i_{1}}e^{ik_{2}\left[\frac{L}{2}-\delta_{\zeta_{1},\zeta_{2}}\frac{\xi}{2}\right]}e^{ik_{1}\left[\frac{L}{2}+\delta_{\zeta_{1},\zeta_{2}}\frac{\xi}{2}\right]}\bigg{)}\cdot$
$\displaystyle\cdot\sum_{\zeta_{3},\zeta_{4}}\bigg{(}\Gamma^{*}_{\zeta_{3},\zeta_{4},-1}a_{\zeta_{3},i_{1}}a_{\zeta_{4},i_{2}}e^{-ik_{1}\left[-\frac{L}{2}-\delta_{\zeta_{3},\zeta_{4}}\frac{\xi}{2}\right]}e^{-ik_{2}\left[-\frac{L}{2}+\delta_{\zeta_{3},\zeta_{4}}\frac{\xi}{2}\right]}-\Gamma^{*}_{\zeta_{3},\zeta_{4},-1}a_{\zeta_{3},i_{2}}a_{\zeta_{4},i_{1}}e^{-ik_{2}\left[-\frac{L}{2}-\delta_{\zeta_{3},\zeta_{4}}\frac{\xi}{2}\right]}e^{-ik_{1}\left[-\frac{L}{2}+\delta_{\zeta_{3},\zeta_{4}}\frac{\xi}{2}\right]}\bigg{)}\bigg{]}\bigg{\\}}.$
(36)

In the previous expression, the building block is represented by the integral

$\displaystyle\int_{-\infty}^{+\infty}dk\,e^{-iE_{A_{i_{1}/i_{2}}}(k)t^{\prime}/\hbar}\frac{e^{\beta
E_{A_{i_{1}/i_{2}}}(k)}}{1+e^{\beta
E_{A_{i_{1}/i_{2}}}(k)}}e^{ik\left[L+(\pm\delta_{\zeta_{1},\zeta_{2}}\pm\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right]},$
(37)

where all the combinations of signs $\pm$ are possibile. Let the generic
energy dispersion be written as
$E_{A_{i_{1}/i_{2}}}(k)=f_{i_{1}/i_{2}}+\rho_{i_{1}/i_{2}}\hbar v_{F}k-\mu$,
with $\rho_{i_{1}/i_{2}}=\pm 1$ and $f_{i_{1}/i_{2}}=\pm f$, then it can be
computed

$\displaystyle\int_{-\infty}^{+\infty}dk\,e^{-iE_{A_{i_{1}/i_{2}}}(k)t^{\prime}/\hbar}\frac{e^{\beta
E_{A_{i_{1}/i_{2}}}(k)}}{1+e^{\beta
E_{A_{i_{1}/i_{2}}}(k)}}e^{ik\left[L+(\pm\delta_{\zeta_{1},\zeta_{2}}\pm\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right]}=$
$\displaystyle=\exp\left\\{{\left[\left(L+(\pm\delta_{\zeta_{1},\zeta_{2}}\pm\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right)i-\rho_{i_{1}/i_{2}}\xi\right]\frac{(\mu-
f_{i_{1}/i_{2}})}{\rho_{i_{1}/i_{2}}\hbar
v_{F}}}\right\\}\frac{1}{\sinh{[\frac{\pi}{\hbar\beta
v_{F}}}(L\pm(\delta_{\zeta_{1},\zeta_{2}}\pm\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}-\rho_{i_{1}/i_{2}}v_{F}t^{\prime}+i\rho_{i_{1}/i_{2}}\xi)]}\cdot$
$\displaystyle\cdot\left(\frac{\pi i}{\rho_{i_{1}/i_{2}}\beta\hbar
v_{F}}\right).$ (38)

Inserting these results in Eq. (24) and distinguishing explicitly the cases
$i_{1}=i_{2}$ and $i_{1}\neq i_{2}$ in the sum over $i_{1}i_{2}$, the current
$I^{r}(t)$ reads as follows

$\displaystyle I^{r}(t)=$
$\displaystyle\frac{8e}{(2\pi\hbar)^{2}}\text{Im}\bigg{\\{}\int_{-\infty}^{+\infty}dt^{\prime}\Theta(t^{\prime})\sum_{i_{1},i_{2}}\sum_{\zeta_{1},\zeta_{2}}\sum_{\zeta_{3},\zeta_{4}}\Gamma_{\zeta_{1},\zeta_{2},+}\Gamma^{*}_{\zeta_{3},\zeta_{4},-}$
$\displaystyle\bigg{[}-\delta_{i_{1},i_{2}}\text{Im}\bigg{\\{}a_{\zeta_{1},i_{1}}a_{\zeta_{4},i_{1}}a_{\zeta_{2},i_{1}}a_{\zeta_{3},i_{1}}\exp{\left\\{\left[\left(L-(\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right)i-\rho_{i_{1}}\lambda\right]\frac{(\mu-
f_{i_{1}})}{\rho_{i_{1}}\hbar v_{F}}\right\\}}\cdot$
$\displaystyle\cdot\frac{1}{\sinh{[\frac{\pi}{\hbar\beta
v_{F}}}(L-(\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}-\rho_{i_{1}}v_{F}t^{\prime}+i\rho_{i_{1}}\lambda)]}\exp{\left\\{\left[\left(L+(\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right)i-\rho_{i_{1}}\lambda\right]\frac{(\mu-
f_{i_{1}})}{\rho_{i_{1}}\hbar v_{F}}\right\\}}\cdot$
$\displaystyle\cdot\frac{1}{\sinh{[\frac{\pi}{\hbar\beta
v_{F}}}(L+(\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}-\rho_{i_{1}}v_{F}t^{\prime}+i\rho_{i_{1}}\lambda)]}\left(\frac{\pi
i}{\rho_{i_{1}}\beta\hbar v_{F}}\right)^{2}\bigg{\\}}+$
$\displaystyle+\delta_{i_{1},i_{2}}\text{Im}\bigg{\\{}a_{\zeta_{1},i_{1}}a_{\zeta_{2},i_{2}}a_{\zeta_{3},i_{1}}a_{\zeta_{4},i_{2}}\exp{\left\\{\left[\left(L+(-\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right)i-\rho_{i_{1}}\lambda\right]\frac{(\mu-
f_{i_{1}})}{\rho_{i_{1}}\hbar v_{F}}\right\\}}\cdot$
$\displaystyle\cdot\frac{1}{\sinh{[\frac{\pi}{\hbar\beta
v_{F}}}(L+(-\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}-\rho_{i_{1}}v_{F}t^{\prime}+i\rho_{i_{1}}\lambda)]}\exp{\left\\{\left[\left(L+(\delta_{\zeta_{1},\zeta_{2}}-\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right)i-\rho_{i_{2}}\lambda\right]\frac{(\mu-
f_{i_{2}})}{\rho_{i_{2}}\hbar v_{F}}\right\\}}\cdot$
$\displaystyle\cdot\frac{1}{\sinh{[\frac{\pi}{\hbar\beta
v_{F}}}(L+(\delta_{\zeta_{1},\zeta_{2}}-\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}-\rho_{i_{2}}v_{F}t^{\prime}+i\rho_{i_{2}}\lambda)]}\left(\frac{\pi
i}{\rho_{i_{1}}\beta\hbar v_{F}}\right)\left(\frac{\pi
i}{\rho_{i_{2}}\beta\hbar v_{F}}\right)\bigg{\\}}+$
$\displaystyle+\delta_{i_{1},-i_{2}}\text{Im}\bigg{\\{}a_{\zeta_{1},i_{1}}a_{\zeta_{2},i_{2}}a_{\zeta_{3},i_{1}}a_{\zeta_{4},i_{2}}\exp{\left\\{\left[\left(L+(-\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right)i-\rho_{i_{1}}\lambda\right]\frac{(\mu-
f_{i_{1}})}{\rho_{i_{1}}\hbar v_{F}}\right\\}}\cdot$
$\displaystyle\cdot\frac{1}{\sinh{[\frac{\pi}{\hbar\beta
v_{F}}}(L+(-\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}-\rho_{i_{1}}v_{F}t^{\prime}+i\rho_{i_{1}}\lambda)]}\exp{\left\\{\left[\left(L+(\delta_{\zeta_{1},\zeta_{2}}-\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right)i-\rho_{i_{2}}\lambda\right]\frac{(\mu-
f_{i_{2}})}{\rho_{i_{2}}\hbar v_{F}}\right\\}}\cdot$
$\displaystyle\cdot\frac{1}{\sinh{[\frac{\pi}{\hbar\beta
v_{F}}}(L+(\delta_{\zeta_{1},\zeta_{2}}-\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}-\rho_{i_{2}}v_{F}t^{\prime}+i\rho_{i_{2}}\lambda)]}\left(\frac{\pi
i}{\rho_{i_{1}}\beta\hbar v_{F}}\right)\left(\frac{\pi
i}{\rho_{i_{2}}\beta\hbar v_{F}}\right)\bigg{\\}}+$
$\displaystyle-\delta_{i_{1},-i_{2}}\text{Im}\bigg{\\{}a_{\zeta_{1},i_{1}}a_{\zeta_{2},i_{2}}a_{\zeta_{3},i_{2}}a_{\zeta_{4},i_{1}}\exp{\left\\{\left[\left(L+(-\delta_{\zeta_{1},\zeta_{2}}-\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right)i-\rho_{i_{1}}\lambda\right]\frac{(\mu-
f_{i_{1}})}{\rho_{i_{1}}\hbar v_{F}}\right\\}}\cdot$
$\displaystyle\cdot\frac{1}{\sinh{[\frac{\pi}{\hbar\beta
v_{F}}}(L+(-\delta_{\zeta_{1},\zeta_{2}}-\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}-\rho_{i_{1}}v_{F}t^{\prime}+i\rho_{i_{1}}\lambda)]}\exp{\left\\{\left[\left(L+(\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right)i-\rho_{i_{2}}\lambda\right]\frac{(\mu-
f_{i_{2}})}{\rho_{i_{2}}\hbar v_{F}}\right\\}}\cdot$
$\displaystyle\cdot\frac{1}{\sinh{[\frac{\pi}{\hbar\beta
v_{F}}}(L+(\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}-\rho_{i_{2}}v_{F}t^{\prime}+i\rho_{i_{2}}\lambda)]}\left(\frac{\pi
i}{\rho_{i_{1}}\beta\hbar v_{F}}\right)\left(\frac{\pi
i}{\rho_{i_{2}}\beta\hbar v_{F}}\right)\bigg{\\}}+$
$\displaystyle-\delta_{i_{1},-i_{2}}\text{Im}\bigg{\\{}a_{\zeta_{1},i_{2}}a_{\zeta_{2},i_{1}}a_{\zeta_{3},i_{1}}a_{\zeta_{4},i_{2}}\exp{\left\\{\left[\left(L+(\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right)i-\rho_{i_{1}}\lambda\right]\frac{(\mu-
f_{i_{1}})}{\rho_{i_{1}}\hbar v_{F}}\right\\}}\cdot$
$\displaystyle\cdot\frac{1}{\sinh{[\frac{\pi}{\hbar\beta
v_{F}}}(L+(\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}-\rho_{i_{1}}v_{F}t^{\prime}+i\rho_{i_{1}}\lambda)]}\exp{\left\\{\left[\left(L+(-\delta_{\zeta_{1},\zeta_{2}}-\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right)i-\rho_{i_{2}}\lambda\right]\frac{(\mu-
f_{i_{2}})}{\rho_{i_{2}}\hbar v_{F}}\right\\}}\cdot$
$\displaystyle\cdot\frac{1}{\sinh{[\frac{\pi}{\hbar\beta
v_{F}}}(L+(-\delta_{\zeta_{1},\zeta_{2}}-\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}-\rho_{i_{2}}v_{F}t^{\prime}+i\rho_{i_{2}}\lambda)]}\left(\frac{\pi
i}{\rho_{i_{1}}\beta\hbar v_{F}}\right)\left(\frac{\pi
i}{\rho_{i_{2}}\beta\hbar v_{F}}\right)\bigg{\\}}+$
$\displaystyle+\delta_{i_{1},-i_{2}}\text{Im}\bigg{\\{}a_{\zeta_{1},i_{2}}a_{\zeta_{2},i_{1}}a_{\zeta_{3},i_{2}}a_{\zeta_{4},i_{1}}\exp{\left\\{\left[\left(L+(\delta_{\zeta_{1},\zeta_{2}}-\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right)i-\rho_{i_{1}}\lambda\right]\frac{(\mu-
f_{i_{1}})}{\rho_{i_{1}}\hbar v_{F}}\right\\}}\cdot$
$\displaystyle\cdot\frac{1}{\sinh{[\frac{\pi}{\hbar\beta
v_{F}}}(L+(\delta_{\zeta_{1},\zeta_{2}}-\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}-\rho_{i_{1}}v_{F}t^{\prime}+i\rho_{i_{1}}\lambda)]}\exp{\left\\{\left[\left(L+(-\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}\right)i-\rho_{i_{2}}\lambda\right]\frac{(\mu-
f_{i_{2}})}{\rho_{i_{2}}\hbar v_{F}}\right\\}}\cdot$
$\displaystyle\cdot\frac{1}{\sinh{[\frac{\pi}{\hbar\beta
v_{F}}}(L+(-\delta_{\zeta_{1},\zeta_{2}}+\delta_{\zeta_{3},\zeta_{4}})\frac{\xi}{2}-\rho_{i_{2}}v_{F}t^{\prime}+i\rho_{i_{2}}\lambda)]}\left(\frac{\pi
i}{\rho_{i_{1}}\beta\hbar v_{F}}\right)\left(\frac{\pi
i}{\rho_{i_{2}}\beta\hbar v_{F}}\right)\bigg{\\}}\bigg{]}\bigg{\\}}.$ (39)

The summation over $i_{1},i_{2}$ runs over 10 terms that can be divided into
three cases:

  1. 1.

$i_{1}=i_{2}$, including $i_{1},i_{2}=1,1\quad 2,2\quad 3,3\quad 4,4$;

  2. 2.

$i_{1}\neq i_{2}$, with $\hat{A}_{k_{1},i_{1}}$ and $\hat{A}_{k_{2},i_{2}}$
being both right-movers or both left-movers, including $i_{1},i_{2}=1,2\quad
3,4$;

  3. 3.

$i_{1}\neq i_{2}$, with $\hat{A}_{k_{1},i_{1}}$ and $\hat{A}_{k_{2},i_{2}}$
having opposite directions of propagation, including $i_{1},i_{2}=1,3\quad
1,4\quad 2,3\quad 2,4$.

Table 1 makes explicit the three cases, by looking at Eq. (8).

|  | $f_{i_{1}}$ | $f_{i_{2}}$ | $\rho_{i_{1}}$ | $\rho_{i_{2}}$
---|---|---|---|---|---
(i) | 1, 1 | $-f$ | $-f$ | -1 | -1
| 2, 2 | $f$ | $f$ | -1 | -1
| 3, 3 | $-f$ | $-f$ | 1 | 1
| 4, 4 | $f$ | $f$ | 1 | 1
(ii) | 1, 2 | $-f$ | $f$ | -1 | -1
| 3, 4 | $-f$ | $f$ | 1 | 1
(iii) | 1, 3 | $-f$ | $-f$ | -1 | 1
| 1, 4 | $-f$ | $f$ | -1 | 1
| 2, 3 | $f$ | $-f$ | -1 | 1
| 2, 4 | $f$ | $f$ | -1 | 1
Table 1: Details concerning the ten terms of the summation over $i_{1},i_{2}$
in Eq. (39).

The next step consists in developing the summations over $\zeta_{1},\zeta_{2}$
and $\zeta_{3},\zeta_{4}$, remembering that the possible values are only those
listed in (35). Since some of the $a_{\zeta,i}$ are zero, the corresponding
addends of the sum will not contribute to the current. The expression we
obtain, using the dimensionless quantities introduced in the main text, is

$\displaystyle
I^{r}(t)=\frac{-2e\Delta\tilde{T}^{2}\Gamma^{2}}{\pi^{2}\hbar^{3}v_{F}^{2}}\text{Im}\Bigg{\\{}e^{-i(\omega_{J}t+\varphi^{0}_{r}-\varphi^{0}_{l})}\int_{0}^{+\infty}dse^{is\tilde{V}}\Bigg{\\{}$
$\displaystyle\tilde{f}_{T}^{2}e^{-i2k_{F}L}\cos{\left(\pi\frac{\phi}{\phi_{0}}\right)}\text{Im}\Bigg{[}\frac{1}{2}\left(e^{\left(2\tilde{L}i+2\right)\left(-\tilde{\mu}-\tilde{f}\right)}+e^{\left(2\tilde{L}i+2\right)\left(-\tilde{\mu}+\tilde{f}\right)}\right)$
$\displaystyle\left(\frac{1}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}}-\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}-1+s-i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+1+s-i\right)\right]}}\right)+e^{-\left(2\tilde{L}i+2\right)\tilde{\mu}}\frac{1}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}}+$
$\displaystyle-\frac{1}{2}\left(e^{2i\tilde{f}}e^{-\left(2\tilde{L}i+2\right)\tilde{\mu}}+e^{-2i\tilde{f}}e^{-\left(2\tilde{L}i+2\right)\tilde{\mu}}\right)\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}+1+s-i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}-1+s-i\right)\right]}}\Bigg{]}+$
$\displaystyle+\tilde{f}_{T}^{2}e^{i2k_{F}L}\cos{\left(\pi\frac{\phi}{\phi_{0}}\right)}\text{Im}\Bigg{[}\frac{1}{2}\left(e^{\left(2\tilde{L}i-2\right)\left(\tilde{\mu}+\tilde{f}\right)}+e^{\left(2\tilde{L}i-2\right)\left(\tilde{\mu}-\tilde{f}\right)}\right)$
$\displaystyle\left(\frac{1}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}}-\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}-1-s+i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+1-s+i\right)\right]}}\right)+$
$\displaystyle+e^{\left(2\tilde{L}i-2\right)\tilde{\mu}}\frac{1}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}}-\frac{1}{2}e^{-2i\tilde{f}}e^{\left(2\tilde{L}i-2\right)\tilde{\mu}}\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}-1-s+i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+1-s+i\right)\right]}}+$
$\displaystyle-\frac{1}{2}e^{2i\tilde{f}}e^{\left(2\tilde{L}i-2\right)\tilde{\mu}}\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}-1-s+i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+1-s+i\right)\right]}}\Bigg{]}+$
$\displaystyle-\frac{1}{2}\cos{\left(\pi\frac{\phi}{\phi_{0}}\right)}\text{Im}\Bigg{[}\bigg{(}e^{\left(\tilde{L}i+1\right)\left(-\tilde{\mu}-\tilde{f}\right)}e^{\left(\tilde{L}i-1\right)\left(\tilde{\mu}+\tilde{f}\right)}+e^{\left(\tilde{L}i+1\right)\left(-\tilde{\mu}+\tilde{f}\right)}e^{\left(\tilde{L}i-1\right)\left(\tilde{\mu}-\tilde{f}\right)}+e^{\left(\tilde{L}i+1\right)\left(-\tilde{\mu}-\tilde{f}\right)}e^{\left(\tilde{L}i-1\right)\left(\tilde{\mu}-\tilde{f}\right)}+$
$\displaystyle+e^{\left(\tilde{L}i+1\right)\left(-\tilde{\mu}+\tilde{f}\right)}e^{\left(\tilde{L}i-1\right)\left(\tilde{\mu}+\tilde{f}\right)}\bigg{)}\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}}\Bigg{]}+$
$\displaystyle-2\tilde{f}_{T}f_{C}e^{i2k_{F}L}\sin{\left(\frac{\pi}{2}\frac{\phi}{\phi_{0}}\right)}\text{Im}\Bigg{[}\left(e^{i\tilde{f}}-e^{-i\tilde{f}}\right)e^{\left(2\tilde{L}i-2\right)\tilde{\mu}}\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}+\frac{1}{2}-s+i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}-\frac{1}{2}-s+i\right)\right]}}\Bigg{]}+$
$\displaystyle+2\tilde{f}_{T}f_{C}e^{-i2k_{F}L}\sin{\left(\frac{\pi}{2}\frac{\phi}{\phi_{0}}\right)}\text{Im}\Bigg{[}\left(e^{i\tilde{f}}-e^{-i\tilde{f}}\right)e^{-\left(2\tilde{L}i+2\right)\tilde{\mu}}\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}-\frac{1}{2}+s-i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+\frac{1}{2}+s-i\right)\right]}}\Bigg{]}+$
$\displaystyle+\tilde{f}_{T}^{2}e^{i(-k_{F}L2)}\text{Im}\Bigg{[}\frac{1}{2}\left(e^{\left(2\tilde{L}i+2\right)\left(-\tilde{\mu}-\tilde{f}\right)}+e^{\left(2\tilde{L}i+2\right)\left(-\tilde{\mu}+\tilde{f}\right)}\right)$
$\displaystyle\left(\frac{1}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}}-\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}-1+s-i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+1+s-i\right)\right]}}\right)-e^{-\left(2\tilde{L}i+2\right)\tilde{\mu}}\frac{1}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}}+$
$\displaystyle+\frac{1}{2}\left(e^{2i\tilde{f}}e^{-\left(2\tilde{L}i+2\right)\tilde{\mu}}+e^{-2i\tilde{f}}e^{-\left(2\tilde{L}i+2\right)\tilde{\mu}}\right)\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}+1+s-i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}-1+s-i\right)\right]}}\Bigg{]}+$
$\displaystyle+(f_{C}^{2}e^{i(-k_{F}L2)})\text{Im}\Bigg{[}e^{-\left(2\tilde{L}i+2\right)\tilde{\mu}}\frac{1}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}}\Bigg{]}+$
$\displaystyle+(\tilde{f}_{T}^{2}e^{i(k_{F}L2)})\text{Im}\Bigg{[}\frac{1}{2}\left(e^{\left(2\tilde{L}i-2\right)\left(\tilde{\mu}+\tilde{f}\right)}+e^{\left(2\tilde{L}i-2\right)\left(\tilde{\mu}-\tilde{f}\right)}\right)$
$\displaystyle\left(\frac{1}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}}-\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}-1-s+i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+1-s+i\right)\right]}}\right)+$
$\displaystyle-e^{\left(2\tilde{L}i-2\right)\tilde{\mu}}\frac{1}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}}+\frac{1}{2}e^{-2i\tilde{f}}e^{\left(2\tilde{L}i-2\right)\tilde{\mu}}\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}-1-s+i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+1-s+i\right)\right]}}+$
$\displaystyle+\frac{1}{2}e^{2i\tilde{f}}e^{\left(2\tilde{L}i-2\right)\tilde{\mu}}\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}-1-s+i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+1-s+i\right)\right]}}\Bigg{]}+$
$\displaystyle-(\tilde{f}_{T}^{2}f_{C}^{2})\text{Im}\Bigg{[}\left(e^{\left(\tilde{L}i+1\right)\left(-\tilde{\mu}-\tilde{f}\right)}e^{\left(\tilde{L}i-1\right)\left(\tilde{\mu}+\tilde{f}\right)}+e^{\left(\tilde{L}i+1\right)\left(-\tilde{\mu}+\tilde{f}\right)}e^{\left(\tilde{L}i-1\right)\left(\tilde{\mu}-\tilde{f}\right)}\right)\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}}\Bigg{]}+$
$\displaystyle-\frac{1}{2}\text{Im}\Bigg{[}\bigg{(}e^{\left(\tilde{L}i+1\right)\left(-\tilde{\mu}-\tilde{f}\right)}e^{\left(\tilde{L}i-1\right)\left(\tilde{\mu}+\tilde{f}\right)}+e^{\left(\tilde{L}i+1\right)\left(-\tilde{\mu}+\tilde{f}\right)}e^{\left(\tilde{L}i-1\right)\left(\tilde{\mu}-\tilde{f}\right)}-e^{\left(\tilde{L}i+1\right)\left(-\tilde{\mu}-\tilde{f}\right)}e^{\left(\tilde{L}i-1\right)\left(\tilde{\mu}-\tilde{f}\right)}+$
$\displaystyle+e^{\left(\tilde{L}i+1\right)\left(-\tilde{\mu}+\tilde{f}\right)}e^{\left(\tilde{L}i-1\right)\left(\tilde{\mu}+\tilde{f}\right)}\bigg{)}\frac{1}{\sinh{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}}\Bigg{]}+$
$\displaystyle+f_{C}^{2}e^{i2k_{F}L}\text{Im}\Bigg{[}e^{\left(2\tilde{L}i-2\right)\tilde{\mu}}\frac{1}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}}\Bigg{]}\Bigg{\\}}\Bigg{\\}}.$

At this stage, it is already possible to visualise the structure of Eq. (28).

The last step is the evaluation of the eighteen integrals in the previous
expression. In particular

$\displaystyle I_{1}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Im}\frac{\tilde{T}^{2}}{\sinh{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}},$
(40) $\displaystyle I_{2}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Re}\frac{\tilde{T}^{2}}{\sinh{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}},$
(41) $\displaystyle I_{3}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Im}\frac{\tilde{T}^{2}}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}},$
(42) $\displaystyle I_{4}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Re}\frac{\tilde{T}^{2}}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}},$
(43) $\displaystyle I_{5}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Im}\frac{\tilde{T}^{2}}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}},$
(44) $\displaystyle I_{6}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Re}\frac{\tilde{T}^{2}}{\sinh^{2}{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}},$
(45) $\displaystyle I_{7}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Im}\frac{\tilde{T}^{2}}{\sinh^{4}{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}},$
(46) $\displaystyle I_{8}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Re}\frac{\tilde{T}^{2}}{\sinh^{4}{\left[\tilde{T}\left(\tilde{L}-s+i\right)\right]}},$
(47) $\displaystyle I_{9}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Im}\frac{\tilde{T}^{2}}{\sinh^{4}{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}},$
(48) $\displaystyle I_{10}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Re}\frac{\tilde{T}^{2}}{\sinh^{4}{\left[\tilde{T}\left(\tilde{L}+s-i\right)\right]}},$
(49) $\displaystyle I_{11}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Im}\frac{\tilde{T}^{2}}{\sinh{\left[\tilde{T}\left(\tilde{L}-1-s+i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+1-s+i\right)\right]}},$
(50) $\displaystyle I_{12}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Re}\frac{\tilde{T}^{2}}{\sinh{\left[\tilde{T}\left(\tilde{L}-1-s+i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+1-s+i\right)\right]}},$
(51) $\displaystyle I_{13}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Im}\frac{\tilde{T}^{2}}{\sinh{\left[\tilde{T}\left(\tilde{L}-\frac{1}{2}-s+i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+\frac{1}{2}-s+i\right)\right]}},$
(52) $\displaystyle I_{14}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Re}\frac{\tilde{T}^{2}}{\sinh{\left[\tilde{T}\left(\tilde{L}-\frac{1}{2}-s+i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+\frac{1}{2}-s+i\right)\right]}},$
(53) $\displaystyle I_{15}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Im}\frac{\tilde{T}^{2}}{\sinh{\left[\tilde{T}\left(\tilde{L}-1+s-i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+1+s-i\right)\right]}},$
(54) $\displaystyle I_{16}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Re}\frac{\tilde{T}^{2}}{\sinh{\left[\tilde{T}\left(\tilde{L}-1+s-i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+1+s-i\right)\right]}},$
(55) $\displaystyle I_{17}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Im}\frac{\tilde{T}^{2}}{\sinh{\left[\tilde{T}\left(\tilde{L}-\frac{1}{2}+s-i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+\frac{1}{2}+s-i\right)\right]}},$
(56) $\displaystyle I_{18}$
$\displaystyle=\int_{0}^{+\infty}ds\,e^{i\tilde{V}s}\text{Re}\frac{\tilde{T}^{2}}{\sinh{\left[\tilde{T}\left(\tilde{L}-\frac{1}{2}+s-i\right)\right]}\sinh{\left[\tilde{T}\left(\tilde{L}+\frac{1}{2}+s-i\right)\right]}}.$
(57)

Only those in Eqs. (40)-(49) are independent. Indeed, up to the second order
in $\xi/L$, we have

$\displaystyle I_{11}$ $\displaystyle\approx I_{7}+I_{3},$ $\displaystyle
I_{12}$ $\displaystyle\approx I_{8}+I_{4},$ $\displaystyle I_{13}$
$\displaystyle\approx\frac{1}{4}I_{7}+I_{3},$ $\displaystyle I_{14}$
$\displaystyle\approx\frac{1}{4}I_{8}+I_{4},$ $\displaystyle I_{15}$
$\displaystyle\approx I_{9}+I_{5},$ $\displaystyle I_{16}$
$\displaystyle\approx I_{10}+I_{6},$ $\displaystyle I_{17}$
$\displaystyle\approx\frac{1}{4}I_{9}+I_{5},$ $\displaystyle I_{18}$
$\displaystyle\approx\frac{1}{4}I_{10}+I_{6}.$

Some of the independent integrals are already present in [70], the other ones
have been evaluated under the same assumptions and up to the same order in
$\xi$. We obtain

$\displaystyle I_{1}$
$\displaystyle\approx-\frac{\pi\tilde{T}e^{\left(i\tilde{V}\tilde{L}-\tilde{V}\right)}}{\sinh{\left(2\tilde{T}\tilde{L}\right)}},$
$\displaystyle I_{2}$
$\displaystyle\approx-i\frac{\pi\tilde{T}e^{\left(i\tilde{V}\tilde{L}-\tilde{V}\right)}}{\sinh{\left(2\tilde{T}\tilde{L}\right)}},$
$\displaystyle I_{3}$ $\displaystyle\approx
i\pi\tilde{V}e^{\left(i\tilde{V}\tilde{L}-\tilde{V}\right)},$ $\displaystyle
I_{4}$
$\displaystyle\approx-\pi\tilde{V}e^{\left(i\tilde{V}\tilde{L}-\tilde{V}\right)},$
$\displaystyle I_{5}$ $\displaystyle\approx 0,$ $\displaystyle I_{6}$
$\displaystyle\approx 0,$ $\displaystyle I_{7}$
$\displaystyle\approx-i\frac{\pi}{6}\tilde{V}\left(\tilde{V}^{2}+4\tilde{T}^{2}\right)e^{\left(i\tilde{V}\tilde{L}-\tilde{V}\right)},$
$\displaystyle I_{8}$
$\displaystyle\approx\frac{\pi}{6}\tilde{V}\left(\tilde{V}^{2}+4\tilde{T}^{2}\right)e^{\left(i\tilde{V}\tilde{L}-\tilde{V}\right)},$
$\displaystyle I_{9}$ $\displaystyle\approx 0,$ $\displaystyle I_{10}$
$\displaystyle\approx 0.$

Once the integration over $s$ is done, we obtain

$I^{r}(t)=\mathcal{C}\,\text{Im}\bigg{\\{}e^{-i(\omega_{J}t+\varphi^{0}_{r}-\varphi^{0}_{l})}\left[\tilde{A}^{r}_{1}\cos{\left(\pi\frac{\phi}{\phi_{0}}\right)}+\tilde{A}^{r}_{2}\sin{\left(\frac{\pi}{2}\frac{\phi}{\phi_{0}}\right)}+\tilde{A}^{r}_{3}\right]\bigg{\\}},$
(58)

with $\mathcal{C}=(-2e\Delta\Gamma^{2})/(\pi^{2}\hbar^{3}v_{F}^{2})$ as
defined in the main text and

$\displaystyle\tilde{A}^{r}_{1}=\frac{\pi}{6}\,\exp{\left[-2\tilde{\mu}+\tilde{V}(-1+i\tilde{L})\right]}\bigg{\\{}i\,\exp{\left[i\left(2k_{F}L+2\tilde{L}\tilde{\mu}\right)\right]}\,\tilde{f}_{T}^{2}\,\tilde{V}\bigg{[}6+\left(-6+4\,\tilde{T}^{2}+\tilde{V}^{2}\right)\cdot$
(59a)
$\displaystyle\cos{\left(2\tilde{f}\right)}+\left(4\,\tilde{T}^{2}+\tilde{V}^{2}\right)\cosh{\left(2\tilde{f}-i2\tilde{f}\tilde{L}\right)}\bigg{]}+\frac{12\,\tilde{T}}{\sinh{\left(2\tilde{L}\tilde{T}\right)}}\,\cosh{\left(\tilde{f}-i\tilde{f}\tilde{L}\right)}\,\cosh{\left(\tilde{f}+i\tilde{f}\tilde{L}\right)}\bigg{\\}},$
$\displaystyle\tilde{A}^{r}_{2}=-\frac{\pi
f_{C}\tilde{f}_{T}\tilde{V}}{6}\exp{\left[i2k_{F}L+\left(-1+i\tilde{L}\right)\left(\tilde{V}+2\tilde{\mu}\right)\right]}\left(-24+4\tilde{T}^{2}+\tilde{V}^{2}\right)\sin{\left(2\tilde{f}\right)},$
(59b)
$\displaystyle\tilde{A}^{r}_{3}=\frac{\pi}{6}\,\exp{\left[-2\tilde{\mu}+\tilde{V}(-1+i\tilde{L})\right]}\bigg{\\{}-i\,\exp{\left[2ik_{F}L+2i\tilde{L}\tilde{\mu}\right]}\,\tilde{V}\bigg{[}-6\left(f_{C}^{2}-\tilde{f}_{T}^{2}\right)+$
(59c)
$\displaystyle+\tilde{f}_{T}^{2}\left(-6+4\tilde{T}^{2}+\tilde{V}^{2}\right)\cos{\left(2\tilde{f}\right)}-\tilde{f}_{T}^{2}\left(4\tilde{T}^{2}+\tilde{V}^{2}\right)\cosh{\left(2\tilde{f}-2i\tilde{f}\tilde{L}\right)}\bigg{]}+$
$\displaystyle+6\frac{\tilde{T}}{\sinh{\left(2\tilde{L}\tilde{T}\right)}}\bigg{[}\left(-1+2f_{C}^{2}\tilde{f}_{T}^{2}\right)\cosh{\left(2\tilde{f}\right)}+\cosh{\left(-2i\tilde{f}\tilde{L}\right)}\bigg{]}\bigg{\\}}.$


1,000 [records]     1,000,000     1,000,000

4*Ours    Truncation     0.612     14.80     4.51

Inversion     0.0219     1.179     0.236

Square root     0.0147     0.608     0.136

Exponential     0.0355     1.017     0.237

Truncation     0.833     1.82     -

Inversion     0.0547     0.0567     -

Square root     0.0559     0.0574    -

Exponential     0.0350     0.0391    -

§ ADDITIONAL EXPERIMENTS WITH VARYING PARAMETERS.

In this section, we describe additional experiments, deferred from Section <ref>.
We measured the running time when varying parameters such as the batch size, the number of neurons,
the number of hidden layers, and the number of input records.
These numbers are useful to estimate the running time for different networks and data.

Varying Batch Sizes.
We show how the running time (for 1 epoch) depends on batch sizes when $n = 2, d_1 = d_2 = 128$ in Table <ref>.

Execution time for 1 epoch is almost in inverse proportion to batch size.
Therefore, increasing the batch size helps to implement faster learning.
This likely comes from the fact that the large batch size implies
fewer parameter updates.

Execution Time for each Batch Size

Batch Size     64     128     256     512

Execution Time [s]     $225$     $117$     $64$     $37$

Varying the Number of Neurons.
We show the running time (for 1 epoch) depends on the number of neurons when $n=2, m=128$ in Table <ref>.
Even for a large number of neurons, $256$, the execution time is only $198$ seconds.

Execution Time for each # of Neurons

# of Neurons     32     64     128     256

Execution Time     68     85     117     304

Varying the Number of Hidden Layers.
We show how the running time (for 1 epoch) depends on the number of hidden layers when $m=128, d_1=d_2=128$ in Table <ref>.
For 4 hidden layers, the execution time is less than $3$ minutes.
This result roughly shows that we need approximately 15 seconds more to have another hidden layer.
This helps us estimate execution time for deeper networks.

Execution Time for each # of Hidden Layers

# of Hidden Layers     1     2     3     4

Execution Time     102     117     133     149

Large Data: 100 Features $\times$ 10 Million Data Samples.

Table <ref> shows the execution time per epoch using 100 attributes $\times$ 10 million data samples.
The parameters $n = 2, d_0 = 100, d_1 = d_2 = 128, d_{3}=10$ are used.

Execution Time for 100 Attributes and 10 Million Data [s]

Batch Size    Our Protocol

$512$    $4,047$ [s] ($1.12$ [h])

$1024$    $2,724$ [s] ($0.76$ [h])