# Magnetic Breakdown and Chiral Magnetic Effect at
Weyl-Semimetal Tunnel Junctions

Adam Yanis Chaou  Vatsal Dwivedi  Maxim Breitkreiz<EMAIL_ADDRESS>berlin.de Dahlem Center for Complex Quantum Systems and Fachbereich Physik,
Freie Universität Berlin, 14195 Berlin, Germany

###### Abstract

We investigate magnetotransport across an interface between two Weyl
semimetals whose Weyl nodes project onto different interface momenta. Such an
interface generically hosts Fermi arcs that connect Weyl nodes of identical
chirality in different Weyl semimetals (homochiral connectivity) — in contrast
to surface Fermi arcs that connect opposite-chirality Weyl nodes within the
same Weyl semimetal (heterochiral connectivity). We show that electron
transport along the arcs with homochiral connectivity, in the presence of a
longitudinal magnetic field, leads to a universal longitudinal
magnetoconductance of $e^{2}/h$ per magnetic flux quantum. Furthermore, a weak
tunnel coupling can result in a close encounter of two homochiral-connectivity
Fermi arcs, enabling magnetic breakdown. Above the breakdown field the
interface Fermi arc connectivity is effectively heterochiral, leading to a
saturation of the conductance.

Introduction — Weyl semimetals (WSMs) are a class of three dimensional
semimetals characterized by pairs of opposite-chirality Weyl fermions
instantiated as topologically protected gapless points in the bulk Brillouin
zone (BZ) Wan  _et al._ (2011); Burkov and Balents (2011); Xu  _et al._ (2011,
2015a, 2015b); Lv  _et al._ (2015); Armitage  _et al._ (2018); Yan and Felser
(2017); Burkov (2018); Hasan  _et al._ (2021); Bernevig  _et al._ (2022).
Individual Weyl fermions exhibit the chiral anomaly Adler (1969); Bell and
Jackiw (1969) — a violation of particle-number conservation in presence of
parallel electric and magnetic fields. The chiral anomaly manifests as a
spectral flow on chiral zeroth Landau levels, which disperse either parallel
or anti-parallel to an applied magnetic field depending on chirality of the
Weyl fermion Nielsen and Ninomiya (1983). In a finite system, the reconnection
of this spectral flow necessitates the existence of gapless Fermi-arc surface
states, which connect Weyl nodes of opposite chirality.

One of the most striking transport phenomena associated with the chiral
anomaly is the chiral magnetic effect (CME), which leads to a positive
longitudinal magnetoconductance Son and Spivak (2013); Burkov (2018); Xiong
_et al._ (2015). In the ballistic (also called ultra-quantum) limit, where
transport is governed solely by the lowest (chiral) Landau level, the
longitudinal conductance is predicted to show a universal linear dependence pm
the magnetic field Altland and Bagrets (2016). Experimental evidence of the
chiral anomaly by way of the CME has, however, turned out to be challenging
because of various extrinsic effects dos Reis  _et al._ (2016); Lv  _et al._
(2021). Moreover, since the Weyl nodes often do not reside exactly at the
Fermi energy, the ballistic-limit CME is only achieved at large non-universal
field strengths.

In this work we show that in tunnel junctions between two WSMs both the CME
and the Fermi arcs combine to give alternative magnetoconductance signatures
of the chiral anomaly. We consider WSMs with Weyl nodes whose transverse
momenta are displaced with respect to one another. Previous work has focused
on the tunnel conductance across interfaces where the Fermi pockets of the two
WSMs overlap Kobayashi  _et al._ (2018); Sousa  _et al._ (2021); Tchoumakov
_et al._ (2021); Buccheri  _et al._ (2022). We instead consider non-
overlapping Fermi pockets, which, in the absence of further ingredients, would
simply result in a vanishing tunnel conductance. We, however, show that upon
adding a magnetic field normal to the interface, the chiral Landau levels can
transmit across the interface via the interface Fermi arcs, while higher
Landau levels are reflected. In contrast to bulk realisations, the ballistic-
limit CME across the tunnel junction (characterised by a universal, linear in
field conductance), occurs around zero magnetic field, irrespective of whether
the Weyl nodes are at exactly the Fermi energy.

We further show that magnetotransport across the tunnel-junction allows for
the exploration of the phenomenon of magnetic breakdown — magnetic-field
induced quantum tunneling between disjoint equienergy contours Cohen and
Falicov (1961); Blount (1962); Shoenberg (1984); Kaganov and Slutskin (1983).
In bulk materials, level separation is typically too large for magnetic
breakdown to manifest at realistic magnetic field strengths Shoenberg (1984);
van Delft  _et al._ (2018). In a tunnel junction however, the level repulsion
between Fermi arcs can be made small by weakening the coupling Sebastian  _et
al._ (2012) at the interface. The onset of magnetic breakdown causes an
effective switch between topologically distinct Fermi-arc connectivities at
the interface (see below), signified by a saturation of the magnetoconductance
above a characteristic mangnetic-breakdown field which is controlled by the
tunneling amplitude.

Tunnel-junction CME — We consider electron transport through a tunnel junction
of two Weyl semimetals in the presence of a magnetic field of magnitude $B$
normal to the interface. We assume that the projection onto the interface BZ
of the Fermi surfaces of all the Weyl nodes (on both sides of the interface)
are separated by lattice momentum much larger than the inverse magnetic length
$l^{-1}_{B}=\sqrt{eB/h}$. This ensures that the bulk Weyl nodes are not
coupled by the applied magnetic field.

In presence of the magnetic field, each Weyl node has an imbalance in the
number of left and right movers because of the $N(B)$–fold degenerate chiral
lowest Landau level, where $N(B)$ is the number of magnetic flux quanta
through the interface. As the spectral flow from the surplus chiral modes
cannot terminate at the interface, there must exist a continuous chain of
states (the interface Fermi arcs) that reconnects the interface projection of
one Weyl node to that of another, as illustrated in Fig. 1. This follows from
particle conservation and the observation that an infinitesimal field only
couples states that are infinitesimally close in transverse momenta.

Consequently, two such types of Fermi arcs are possible: those that connect
projections of opposite-chirality Weyl nodes in the same WSM and those which
connect same-chirality nodes in WSMs on opposite sides of the interface
Dwivedi (2018); Abdulla  _et al._ (2021); Mathur  _et al._ ; Kaushik  _et al._
. We term the two connectivities _heterochiral_ and _homochiral_ connectivity,
respectively. While the connectivity of the interface Fermi arcs is a robust
topological property (See the Supplemental Material (SM) for a topological
argument), their shape is non-universal and depends on system details, such as
boundary potentials and the tunneling amplitude.

We first consider the situation where the interface Fermi arcs are separated
by lattice momenta much larger than $\ell_{B}^{-1}$. In this case, a pair of
homochiral Fermi arcs perfectly transmit the incoming mode across the
interface, while a pair of heterochiral Fermi arcs are totally reflected.
Following a Landauer approach (see the SM for a detailed derivation), the
conductance is given by

$G=n_{\mathrm{ho}}N(B)\frac{e^{2}}{h},$ (1)

where $n_{\mathrm{ho}}$ is the number of homochiral Fermi arcs and $N(B)$ is
the number of incoming modes per unit area. Note that the conductance is
independent of the occupation of higher Landau levels, which are pefectly
reflected. This results in a universal conductance that is insensitive to
material-specific details such as the energies of the Weyl nodes and
velocities.

Figure 1:  Spectral flow at the interface between two Weyl semimetals with a
longitudinal magnetic field (a) Interface Fermi arcs with homochiral
connectivity (orange lines) lead to a full transmission of the chiral zeroth
Landau level across the interface, while the higher Landau levels are
reflected. (b) Fermi arcs of heterochiral connectivity (green lines) lead to
reflection of all Landau levels.

_Magnetic breakdown_ — The derivation of Eq. (1) breaks down once the magnetic
field is large enough to enable backscattering via the interface Fermi arcs.
This however, only happen once $\ell_{B}^{-1}\approx
0.004\,\text{\AA}^{-1}\sqrt{B[\mathrm{T}]}$ (with $B$ in Teslas) approaches
the reciprocal space separation between the two chiral modes. Realistic
magnetic fields can thus only couple modes whose separation is small compared
to the size of the surface BZ, resulting in the phenomena of magnetic
breakdown being considered a rather exotic Shoenberg (1984). However, in our
setup, a close encounter of interface Fermi arcs can be achieved by a weak
tunnel coupling of two WSMs whose Fermi arcs cross in the decoupled limit, as
illustrated in Fig. 2(a).

Near a close encounter of two Fermi arcs, the linearized interface Hamiltonian
reads

$H_{\mathrm{int}}({\bm{k}}_{\perp})=(v_{z}\Delta/2)\sigma_{x}+v_{y}k_{y}\sigma_{z}+v_{z}k_{z},$
(2)

where ${\bm{k}}_{\perp}$ is the transverse momentum measured from the mid-
point of the smallest separation between contours (see Fig. 2(b)), the Pauli
matrices correspond to the two Fermi arcs from the decoupled system, the
velocities $v_{y}$ and $v_{z}$ are fixed by the specific dispersion of the
Fermi arcs, and $\Delta$ quantifies the hybridization strength, which depends
on the tunneling amplitude. The shortest distance between the Fermi arcs is
then given by $|\Delta|$ and $\theta=\tan^{-1}(v_{z}/v_{y})$ is half the angle
of the Fermi arc intersection, as shown in Fig. 2(b).

The essential requirement for our setup is a crossing of interface Fermi arcs
in the decoupled limit. At finite tunneling, such a crossing turns into a
close encounter, unless protected by a symmetry deriving from the symmetries
of the WSMs. Explicitly, this must forbid mass terms proportional to both
$\sigma_{x}$ and $\sigma_{y}$ in Eq. (2). Such a protection, however, requires
a lattice symmetry; which, though it may hold for interfaces between a pair of
highly symmetric Weyl node configurations, would not hold for generic
interfaces. The close encounter described above is thus generic so long as the
decoupled Fermi arcs exhibit a crossing.

At a finite but small $\Delta$, a longitudinal magnetic field $B$ enables
quantum tunneling between the two Fermi arcs when
$\ell_{B}^{-1}\gtrsim\Delta$. To quantify this magnetic breakdown we deploy an
analytical description following the standard formalism of Refs Cohen and
Falicov (1961); Landau and Lifshitz (1977); Kaganov and Slutskin (1983);
Shoenberg (1984). Using the semiclassical wavefunctions of the Fermi arcs away
from the encounter as scattering states that move along the arcs in accordance
with the Lorentz force, we calculate the transition probability across the gap
by matching the scattering states with the exact solutions of
$H_{\mathrm{int}}$. The resulting probability of tunneling between the arcs is
given by

$P=e^{-B_{0}/B};\qquad B_{0}=\frac{\pi}{4}\Delta^{2}\left|\tan\theta\right|.$
(3)

A particle passing a single encounter will thus experience a heterochiral
connectivity — and is thus reflected — with probability $P$, so that the
probability of transmission across the interface is given by $1-P$. The
conductance is thus obtained by weighing Eq. (1) by a factor of $1-P$, so that

$G=N(B)\frac{e^{2}}{h}\sum_{i=1}^{n_{\mathrm{ho}}}\Big{(}1-e^{-B_{0,i}/B}\Big{)},$
(4)

where the sum runs over all homochiral connectivities and $B_{0,i}$ are the
corresponding breakdown fields. For $B\ll B_{0,i}$ Eq. (1) is recovered, while
for $B\gg B_{0,i}$ the conductance saturates at $(e^{2}/h)N(B_{0,i})$. In the
latter limit, the transmission probability approaches zero as $1/B$ (rendering
the connectivity effectively heterochiral) but as $N(B)$ is linear in $B$, the
conductance saturates to a constant value.

Figure 2: (a) Coupling the surfaces of two WSMs with crossing Fermi arcs,
whose hybridization leads to the interface Fermi arc connectivity switching
from heterochiral to homochiral. (b) The close encounter of interface Fermi
arcs for weak tunneling between the two WSMs.

If we further increase the magnetic field, it couples the Weyl nodes of the
same chirality and opposite sides of the interfaces. In this case, the modes
can transmit across the interface directly (skipping the Fermi arc), so that
$N(B)$ is no longer the upper bound for the transmission, and the transmission
probability becomes non-universal, depending on the number of occupied Landau
levels. If the field couples opposite chirality nodes, the WSM phase is
effectively destroyed Ramshaw  _et al._ (2018).

_Lattice simulation_ — We test the above predictions by numerical and semi-
analytical calculations on a WSM lattice model. We consider lattice models
described by Hamiltonians of the form Dwivedi and Chua (2016)

$\displaystyle\mathcal{H}({\bm{k}})=\mathcal{H}_{x}(k_{x})+\eta_{y}({\bm{k}}_{\perp})\sigma^{y}+\eta_{z}({\bm{k}}_{\perp})\sigma^{z},$
(5)

where $\mathcal{H}_{x}(k_{x})=\sin k_{x}\sigma^{x}+(1-\cos k_{x})\sigma^{z}$.
The hopping strength along $x$ and the lattice constant are set to one, and
$\eta_{y,z}({\bm{k}}_{\perp})$ are functions of the transverse momentum
${\bm{k}}_{\perp}\equiv(k_{y},k_{z})$. For $\eta_{z}({\bm{k}}_{\perp})>-2$,
the model hosts Weyl nodes in the $k_{x}=0$ plane at ${\bm{k}}_{\perp}$
satisfying $\eta_{y}({\bm{k}}_{\perp})=\eta_{z}({\bm{k}}_{\perp})=0$. At a
boundary normal to $x$, the Fermi arcs lie along
$\eta_{y}({\bm{k}}_{\perp})=0$ for $\eta_{z}({\bm{k}}_{\perp})<0$ Dwivedi and
Chua (2016).

Figure 3: Conductance in units of the saturation value $(e^{2}/h)N(B_{0})$ as
a function of the magnetic field in units of the breakdown field $B_{0}$. The
numerical computation is performed for model parameters $b=\pi/2$, energy
$E=0.1$ and various values of $\theta$ and $\kappa$. The analytical result
(black dashed curve) fits perfectly with the numerical results for weak
fields. The straight dashed lines indicate the predicted asymptotic low- and
high-field behavior. For the numerical data, the upper bound on the magnetic
field is set by the smallest magnetic length $\ell_{B}=3.1$ lattice units —
since smaller $\ell_{B}$ results in a trivial breakdown of WSM physics — while
the lower bound is set by computational limits to $13.8$ lattice units. Data
points corresponding to a smaller (larger) breakdown field $B_{0}$ [for
smaller (larger) $\kappa$] thus span a range of fields at higher (lower)
values of $B/B_{0}$. The inset shows a close-up at small fields; highlighting
the deviation from perfect transmission caused by the onset of magnetic
breakdown.

We consider an interface between two WSMs with

$\displaystyle\eta_{y}^{\pm}({\bm{k}}_{\perp})$ $\displaystyle=\frac{1}{\sin
b}\left[\pm\sin b_{y}\sin k_{y}+\sin b_{z}\sin k_{z}\right],$
$\displaystyle\eta^{\pm}_{z}({\bm{k}}_{\perp})$ $\displaystyle=\cos b_{y}+\cos
b_{z}-\cos k_{y}-\cos k_{z},$ (6)

where the superscript $\pm$ refers to the left/right WSM. The corresponding
Weyl nodes with chirality $\chi$ lie at ${\bm{k}}_{\perp}=\chi\mathbf{b}$,
where $\mathbf{b}=(b_{y},b_{z})=b(\sin\tilde{\theta},\cos\tilde{\theta})$. The
tunnel junction between these two WSMs is modelled by a reduced hopping
$0\leq\kappa\leq 1$ along $x$ at the interface. The conductance for this
lattice model for various values of $\theta$ , computed numerically using the
_Kwant_ package Groth  _et al._ (2014), are plotted in Fig. 3.

To compare the numerical results to the analytics, we derive the breakdown
field Eq. (3) from the lattice parameters. To this end, we need to determine
the intersection angle $\theta$ of the Fermi arcs in the decoupled limit
($\kappa=0$) as well as the minimum separation $\Delta$ between the hybridized
($\kappa\neq 0$) Fermi arcs [cf. Fig. 2(b)]. The Fermi arcs in the decoupled
limit are given by $\eta_{y}^{\pm}({\bm{k}}_{\perp})=0$, linearizing around
${\bm{k}}_{\perp}=0$ gives

$\tan\theta=\frac{\sin b_{y}}{\sin
b_{z}}=\frac{\sin(b\sin\tilde{\theta})}{\sin(b\cos\tilde{\theta})}.$ (7)

We next derive the interface Fermi arcs at a finite coupling $\kappa$ using a
transfer-matrix approach Dwivedi and Ramamurthy (2016); Dwivedi and Chua
(2016), as described in the SM. To leading order in $\kappa\ll 1$, the minimum
separation is given by

$\Delta=2\kappa\beta(2+\beta)\frac{\sin b}{\sin b_{y}},$ (8)

where $\beta=\cos b_{y}+\cos b_{z}-2$. The breakdown field can then be
determined by Eqs. (3), (7), and (8). The conductance computed using (4) with
the analytically computed breakdown field shows excellent agreement with the
numerical results so long as the magnetic length is much larger than the
lattice spacing, as shown in Fig. 3.

_Discussion and conclusion_ — In this article, we consider the
magnetoconductance across a tunnel barrier between two WSMs arranged such that
the projection of the Weyl node’s Fermi surfaces onto the interface BZ are
well separated and thereby aren’t coupled by a magnetic field. At generic
couplings of the WSMs the interface Fermi arcs, which come in two
topologically distinct connectivity types (heterochiral and homochiral), will
typically also be well separated. In this regime the system displays a
universal tunnel magnetoconductance of $(e^{2}/h)\,N(B)$ (where $N(B)$ is the
number of flux quanta through the interface) for each pair of homochiral-
connectivity Fermi arcs. While the CME of a bulk WSM in the ballistic limit is
also characterised by a universal magnetoconductance where the number of
Fermi-arc pairs is replaced by the number of Weyl-node pairs, it requires a
minimal magnetic-field strength, whose scale is set by the energy of the Weyl
nodes and by diffusion properties Altland and Bagrets (2016); Burkov (2018).
In contrast, the tunnel conductance considered here is independent of such
details because the interface is intransparent for higher Landau level and
thus acts like a filter. An alternative bulk system showing a similar effect
of the ballistic-limit CME extending down to zero magnetic field independent
of system details is achieved in Fermi-arc metals Breitkreiz and Brouwer ,
recently predicted by Brouwer and one of us.

Such interfaces can also be used to realize the phenomena of magnetic
breakdown. This requires a close encounter of two Fermi arcs so as to enable
magnetic field induced quantum tunneling between arcs. Such an encounter is
generic between WSMs whose interface Fermi arcs cross in the decoupled limit.
The magnetic breakdown leads to a suppression of the transmission probability
for field strengths above a characteristic breakdown field $B_{0}$ (set by the
coupling strength). Since the magnetic breakdown effectively turns a
homochiral connectivity into a heterochiral connectivity, one might expect the
conductance to drop to zero above $B_{0}$. The increased probability of
experiencing a heterochiral connectivity at a higher field is however balanced
by the increased degeneracy of transmitted modes, thus leading to the
saturation of the conductance at a finite value of $(e^{2}/h)\,N(B_{0})$.

Acknowledgments. We thank Piet W. Brouwer, Achim Rosch, Reinhold Egger, and
Francesco Buccheri for useful discussions. This research was supported by
projects A02 and A03 of the CRC-TR 183 “Entangled States of Matter” and Grant
No. 18688556 of the Deutsche Forschungsgemeinschaft (DFG, German Science
Foundation).

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## SUPPLEMENTAL MATERIAL

## Topological argument for homochiral interface Fermi arcs

Consider an interface between two WSMs whose Weyl nodes occur at different
transverse lattice momenta, as shown in Fig. S1. We begin by noting that the
the restriction of the lattice model to transverse momenta forming a loop in
the interface away from the projections of the Weyl nodes can be interpreted
as the band structure of two semi-infinite gapped 2d systems, which can
possess a Chern number. An interface between two gapped phases with Chern
numbers $C_{1}$ and $C_{2}$ hosts $\left|C_{1}-C_{2}\right|$ protected chiral
modes in the bulk gap. Thus, for a given loop, we can infer the number of
topologically protected zero modes crossing it, depending on the enclosed Weyl
nodes on either side of the interface. In particular, if the loop encloses a
single Weyl node of chirality $\chi$ from one of the WSMs and none from the
other, then the loop corresponds to an interface between gapped 2D phases with
Chern numbers $\chi$ and $0$, guaranteeing the existence of a chiral mode
crossing zero energy at some momentum on the loop. On the other hand, if the
loop encloses Weyl nodes of identical chirality $\chi$ from both WSMs, it
corresponds to an interface between two gapped phases with identical Chern
numbers, which does not host any topologically protected modes.

In terms of the interface between Fermi arcs, we thus conclude that a Fermi
arc emanate from each projection of a Weyl node, but given a loop enclosing
Weyl nodes of identical chiralities from the two sides, the Fermi arcs can
always be deformed to lie entirely inside it. Thus, the topologically
protected interface Fermi arcs must connect projections of Weyl nodes of
identical chiralities coming from the two WSMs.

## Tunnel conductance from Landauer approach

The Landauer formula $G=(e^{2}/h)T$ relates the conductance $G$ with the
transmission probability $T$, the sum of transmission probabilities over all
right- or leftmoving modes. We consider the transmission of a flat interface
where the modes (scattering states) are divided into two decoupled groups,
labeled $1$ and $2$, whereby in group $1$ the number of rightmovers is larger
by $N$ modes than the number of leftmovers, and vice versa for group $2$. The
scattering problem, illustrated in Fig. S2, with scattering matrices $S_{i}$
for the two groups $i=1,2$, reads

$\begin{pmatrix}\bm{a}^{-}_{i}\\\
\bm{b}^{+}_{i}\end{pmatrix}=\underbrace{\begin{pmatrix}r_{i}&t_{i}^{\prime}\\\
t_{i}&r_{i}^{\prime}\end{pmatrix}}_{\equiv
S_{i}}\begin{pmatrix}\bm{a}^{+}_{i}\\\ \bm{b}^{-}_{i}\end{pmatrix}.$ (S1)

Note that the reflection and transmission amplitude matrices have different
rectangular shapes, set by the size of the vectors of mode coefficients.

Figure S1:  (Left) Interface between two WSMs with Weyl nodes of
positive/negative chirality (red/blue spheres) and their projections onto the
interface from the upper WSM (filled circles) and lower WSM (empty circles).
Fermi arcs can either connect Weyl nodes of opposite chirality on the same
side of the interface (green lines) or same chirality on opposite sides of the
interface (orange lines), corresponding to heterochiral and homochiral
connectivities, respectively. The topology of interface Fermi arcs can be
inferred from considering the interface of 2D Chern insulators with Chern
number $C$, indicated as cylinders. (Right) Numerically computed interface
Fermi arcs for the lattice model Eq. (6) with $\tilde{\theta}=\pi/2$ for
$\kappa=0.1$ (top) and $\kappa=0.25$ (bottom).  Figure S2: Illustration of the
scattering problem, Eq. (S1), indicating the numbers of modes.

We will now show that the total transmission probability is bound from below
as

$T\equiv\sum_{i}\mathrm{Tr}\big{(}t^{\dagger}_{i}t^{\phantom{\dagger}}_{i}\big{)}\geq
N.$ (S2)

Heuristically one can understand from particle conservation that, considering
e.g., the modes to the left of the interface, only $a$ of the $a+N$
rightmovers in subsystem $1$ can be reflected since there are only $a$
leftmovers and, in subsystem 2, all $c$ rightmovers can be reflected since
there are $c+N$ leftmovers. Hence, in total, at least $N$ modes must be
transmitted. (The same conclusion can be reached considering the modes to the
right of the interface.) Formally, this derives from the unitarity of the
scattering matrices,
$t^{\dagger}_{1}t^{\phantom{\dagger}}_{1}+r^{\dagger}_{1}r^{\phantom{\dagger}}_{1}=\mathbb{1}_{(a+N)\times(a+N)}$
and
$t^{\prime}_{1}{t^{\prime}_{1}}^{\dagger}+r_{1}r^{\dagger}_{1}=\mathbb{1}_{a\times
a}$. Taking the trace of both equations and eliminating
$\mathrm{Tr}\big{(}r^{\dagger}_{1}r^{\phantom{\dagger}}_{1})$ and repeating
the same for the other subsystem, we obtain

$\displaystyle\mathrm{Tr}\big{(}t^{\dagger}_{1}t^{\phantom{\dagger}}_{1})=$
$\displaystyle\
\mathrm{Tr}\big{(}{t^{\prime}_{1}}^{\dagger}{t^{\prime}_{1}}^{\phantom{\dagger}}\big{)}+N$
(S3)
$\displaystyle\mathrm{Tr}\big{(}t^{\dagger}_{2}t^{\phantom{\dagger}}_{2})=$
$\displaystyle\
\mathrm{Tr}\big{(}{t^{\prime}_{2}}^{\dagger}{t^{\prime}_{2}}^{\phantom{\dagger}}\big{)}-N.$
(S4)

Equation (S2) follows by taking into account
$\mathrm{Tr}\big{(}{t^{(\prime)\dagger}_{i}}{t^{(\prime)}_{i}}^{\phantom{\dagger}}\big{)}\geq
0$.

We can apply this result to a pair of chiral Weyl Fermions in an applied
magnetic field directed perpendicular to the interface, at which the
homochiral Fermi arcs connect Weyl Fermions of the same chirality at opposite
sides of the interface. In this case, the difference in the number of right-
and leftmovers is the degeneracy of the chiral Landau level $N=N(B)$ and
subsystem $1$ and $2$ correspond to positive and negative chirality. The lower
bound of the conductance for a single homochiral Fermi arc is thus
$G=\frac{e^{2}}{h}T\geq\frac{e^{2}}{h}N(B)$. The upper bound is set by the
number of states at the smallest constriction. As argued in the main text, all
particles must pass the Fermi arc, which in the magnetic field becomes a
single $N(B)$-degenerate Landau level. It follows that

$G=n_{\mathrm{ho}}N(B)\frac{e^{2}}{h},$ (S5)

where we included the total number $n_{\mathrm{ho}}$ of homochiral
connectivities.

## Tunnel conductance from semiclassics

In an applied magnetic field directed perpendicular to the interface, the
Lorentz force moves a particle along the Fermi arc in time $dt$ by the amount
$dk=vdt/\ell_{B}^{2}$, where $v$ is the velocity and $\ell_{B}$ the magnetic
length. Particles contributing to transport are within the energy range $eU$,
set by the applied voltage $U$, which corresponds to the momentum $eU/\hbar
v$. Hence, the transported charge is

$dQ=e\frac{dk\,(eU/\hbar v)}{(2\pi)^{2}}{\cal A},$ (S6)

where ${\cal A}$ is the interface area. Inserting $dk=vdt/\ell_{B}^{2}$ and
with $N(B)={\cal A}/(2\pi\ell_{B}^{2})$ we obtain

$\frac{dQ}{Udt}=N(B)\frac{e^{2}}{h}.$ (S7)

This correctly corresponds to the tunneling conductance across the interface
via a single homochiral Fermi arc obtained from the scattering-matrix
formalism.

## Interface modes of a lattice model using transfer matrices

The generalized transfer matrix along $x$ for the two-band models lattice
model of Eq. (5) is given by Dwivedi and Ramamurthy (2016); Dwivedi and Chua
(2016)

$T(\varepsilon,{\bm{k}}_{\perp})=\frac{1}{1+\eta_{z}}\begin{pmatrix}\varepsilon^{2}-\Gamma^{2}&-\varepsilon+\eta_{y}\\\
\varepsilon+\eta_{y}&-1\end{pmatrix},$ (S8)

where $\Gamma\equiv\eta_{y}^{2}+(1+\eta_{z})^{2}$. For an interface along $x$
with tunneling strength $\kappa$, a necessary condition for the existence of a
mode localized at the interface is Dwivedi (2018)

$\det\left[T_{+},KT_{-}K\right]=0,$ (S9)

where $T_{\pm}$ denote the transfer matrices on the two sides of the interface
and $K=\text{diag}\left\\{(1/\kappa),\kappa\right\\}$. For the transfer matrix
Eq. (S8), this expression becomes

$\Lambda_{+}-\Lambda_{-}=\pm
2\sinh\lambda\sqrt{\Lambda_{+}\Lambda_{-}+4\cosh^{2}\lambda},$ (S10)

where $\lambda=\ln\kappa$ and

$\Lambda_{\pm}=\frac{1}{\eta_{y}^{\pm}}\left[1-\left(\eta_{y}^{\pm}\right)^{2}-\left(1+\eta_{z}^{\pm}\right)^{2}\right].$
(S11)

For $\Lambda_{+}=-\Lambda_{-}\equiv\Lambda$, Eq. (S10) simplifies to

$\Lambda=\pm 2\sinh\lambda=\pm\left(\kappa-\frac{1}{\kappa}\right).$ (S12)

This turns out to be the relevant limit for our setup.

For the interface between WSMs described by Eq. (6), linearizing around
${\bm{k}}_{\perp}=\mathbf{0}$ yields

$\displaystyle\eta^{\pm}_{y}({\bm{k}}_{\perp})$ $\displaystyle=\pm
v_{y}k_{y}+v_{z}k_{z},$ $\displaystyle\eta^{\pm}_{z}({\bm{k}}_{\perp})$
$\displaystyle=\beta+\frac{1}{2}\left(k_{y}^{2}+k_{z}^{2}\right),$ (S13)

where $v_{i}=\sin b_{i}/\sin b$ and $\beta=\cos(b_{y})+\cos(b_{z})-2$. Thus,
the Fermi arcs at the interface for $\kappa=0$ lie along $v_{z}k_{z}=\pm
v_{y}k_{y}$, leading to a crossing at ${\bm{k}}_{\perp}=0$. For $\kappa>0$,
the interface Fermi arc can be computed using Eqns. (S10) and (S11). In
particular, the minimum separation between the arcs lies along $k_{y}$.
Setting $k_{z}=0$, we get

$\Lambda_{\pm}=\pm\frac{1}{v_{y}k_{y}}\left[1-(v_{y}k_{y})^{2}-\left(1+\beta+\frac{k_{y}^{2}}{2}\right)^{2}\right]$
(S14)

so that $\Lambda_{+}=-\Lambda_{-}$. The zero crossings along $k_{z}=0$ and
thus the minimum separation between the interface Fermi arcs are given by Eq.
S12. For $\kappa\ll 1$, we only retain the terms at $\mathrm{O}(1/\kappa)$ and
$\mathrm{O}(1/k_{y})$, so that

$\frac{1}{\kappa}=-\frac{1}{v_{y}k_{y}}\left[1-\left(1+\beta\right)^{2}\right]=\frac{\beta(2+\beta)}{v_{y}k_{y}}.$
(S15)

The minimum separation between the interface Fermi arc at $\mathrm{O}(\kappa)$
is thus given by

$\Delta=2k_{y}=\frac{2\kappa\beta(2+\beta)}{v_{y}}.$ (S16)


# Proposal for detecting the $\pi-$shifted Cooper quartet supercurrent

Régis Mélin<EMAIL_ADDRESS>Univ. Grenoble-Alpes, CNRS, Grenoble
INP††thanks: Institute of Engineering Univ. Grenoble Alpes, Institut NEEL,
38000 Grenoble, France  Romain Danneau  Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, Karlsruhe D-76021, Germany
Clemens B. Winkelmann  Univ. Grenoble-Alpes, CNRS, Grenoble INP††thanks:
Institute of Engineering Univ. Grenoble Alpes, Institut NEEL, 38000 Grenoble,
France

###### Abstract

The multiterminal Josephson effect aroused considerable interest recently, in
connection with theoretical and experimental evidence for correlations among
Cooper pairs, that is, the so-called Cooper quartets. It was further predicted
that the spectrum of Andreev bound states in such devices could host Weyl-
point singularities. However, the relative phase between the Cooper pair and
quartet supercurrents has not yet been addressed experimentally. Here, we
propose an experiment involving four-terminal Josephson junctions with two
independent orthogonal supercurrents, and calculate the critical current
contours (CCCs) from a multiterminal Josephson junction circuit theory. We
predict a generically $\pi$-shifted contribution of both the local or nonlocal
second-order Josephson harmonics. Furthermore, we show that these lead to
marked nonconvex shapes for the CCCs in zero magnetic field, where the
dissipative state reenters into the superconducting one. Eventually, we
discuss distinctive features of the non-local Josephson processes in the
CCC’s. The experimental observation of the latter could allow providing firm
evidence of the $\pi$-shifted Cooper quartet current-phase relation.

##  I Introduction

Entanglement in electronic superconducting circuits is central to quantum
engineering, and prototypes of quantum processors were recently realized,
unveiling a variety of physical phenomena Google1 . Entanglement engines were
proposed in the early 2000s, with normal metal-superconductor-normal metal
($N$-$S$-$N$) hybrids as sources of entangled Einstein-Podolsky-Rosen pairs of
electrons theory-noise2 ; theory-CPBS1 ; theory-CPBS4 ; theory-CPBS6 ; theory-
CPBS7 ; theory-CPBS8 ; theory-CPBS11 ; Yeyati2007 ; theory-CPBS7 ; theory-
noise8 ; theory-noise9 ; theory-noise11 . A series of experiments addressed
nonlocality in the DC current response exp-CPBS1 ; exp-CPBS2 ; exp-CPBS3 ;
exp-CPBS6 ; exp-CPBS5 ; Tan2015 ; Borzenets2016a ; Danneau  and quantum noise
exp-CPBS8  as evidence for entangled split Cooper pairs theory-noise2 ;
theory-CPBS1 ; theory-CPBS4 ; theory-CPBS6 ; theory-CPBS7 ; theory-CPBS8 ;
theory-CPBS11 ; Yeyati2007 ; theory-CPBS7 ; theory-noise8 ; theory-noise9 ;
theory-noise11 . On the other hand, the emerging field of all-superconducting
multiterminal Josephson junctions 1979 ; Omelyanchouk1 ; Omelyanchouk2 ;
Omelyanchouk3 ; Omelyanchouk4  offers new perspectives such as exotic
transient quantum correlations among Cooper pairs, known as Cooper quartets
Cuevas-Pothier ; Freyn ; Melin1 ; Jonckheere ; Akh2 ; MF ; split-quartets .
While a series of experiments reported clear signatures of Cooper quartets
Lefloch ; Heiblum ; HGE ; Pribiag , these features were not observed by others
exp-multiter-supp ; multiterminal-exp2 ; multiterminal-exp3 ; multiterminal-
exp4 ; multiterminal-exp5 ; multiterminal-exp6 ; multiterminal-exp9 ;
multiterminal-exp10 , possibly due to delicate material and device fabrication
issues. In parallel, multiterminal Josephson junctions also focused strong
interest recently as a testbed of Floquet theory FWS ; engineering ; Berry ;
paperII ; paperIII ; ultralong ; Keleri ; Park , as well as a platform for the
emergence of energy level repulsion in Andreev molecules Pillet ; Pillet2 ;
Nazarov-PRR ; multiterminal-exp7 ; IBM ; Tarucha ; Tarucha2 , the production
of Weyl-point singularities in the Andreev spectrum vanHeck ; Padurariu ;
multiterminal-exp1 ; Nazarov1 ; Nazarov2 ; topo0 ; topo2 ; topo3 ; topo1-plus-
Floquet ; topo1 ; topo4 ; topo4-bis ; topo5 ; Berry ; Feinberg1 ; Feinberg2 ;
Levchenko1 ; Levchenko2 ; Akh2 ; Gavensky , and the multiterminal
superconducting diode effect SDE1 ; SDE2 .

In spite of intense experimental efforts for observing signatures of the
quartet state and its new physics beyond the standard Resistively Shunted
Josephson Junction model Lefloch ; HGE ; Heiblum , novel schemes are necessary
for ascertaining the Cooper quartets. When driving current between pairs of
contacts in a multiterminal Josephson junction with an even number $2n$ of
superconducting leads, $n$ equations of current conservation are imposed by
the external circuit. Those $n$ constraints (for a total of $2n$ phase
variables) allow for supercurrent inside a region in phase space parameterized
by $2n-n\equiv n$ independent variables. With four terminals, a DC
supercurrent is thus established within a two-dimensional region in the plane
of the bias currents, separated from the resistive state by a one-dimensional
critical current contour (CCC). In a recent work, Pankratova et al.
multiterminal-exp3  reported nonconvex shapes in the CCCs of four-terminal
semiconductor-superconductor Josephson junctions. However, these nontrivial
features appeared only at rather high magnetic fields, corresponding to about
half a flux quantum threading the central part of the device. The observation
of nonconvex CCCs was interpreted using Random Matrix Theory, assuming time-
reversal symmetry breaking, either due to an applied magnetic field or
preexisting in the normal state multiterminal-exp3 .

Here, we demonstrate that in the presence of at least one contact with an
intermediate transmission, another mechanism for the emergence of nonconvex
CCCs is possible, which does not require a magnetic field. Namely, we find
correspondence between the quartet physics and the emergence of nonconvex
sharp-angled points in the CCCs at zero magnetic field. This distinctive
signature stems from the interference between symmetric quartet channels,
which are dephased by a transverse supercurrent (see Fig. 1). In other words,
we demonstrate that macroscopic critical current measurements can probe the
microscopic internal structure of entangled split Cooper pairs theory-noise2 ;
theory-CPBS1 ; theory-CPBS4 ; theory-CPBS6 ; theory-CPBS7 ; theory-CPBS8 ;
theory-CPBS11 ; Yeyati2007 .

Figure 1: Sketch of the superconducting four-terminal device with either one
current and one phase bias (a), or two orthogonal current biases (b). The
superconductors $S_{L}$, $S_{R}$, $S_{T}$ and $S_{B}$ are connected to the
normal metallic region $N_{0}$. The four $N_{0}$-$S_{i}$ junctions consist of
tunable quantum point contacts, where the transmission of the $N_{0}$-$S_{T}$
interface is reduced by a scaling factor $\tau_{T}$. Panels c-e represent the
lowest-order Josephson processes occurring in a simplified toy-model. Panel c
shows the two-terminal DC-Josephson effect from $S_{B}$ to $S_{T}$, which is
insensitive to the horizontal contacts. Panels d and e show the Cooper quartet
processes, which take two Cooper pairs from $S_{B}$, exchange partners and
transmit the outgoing pairs into $(S_{T},\,S_{L})$ and $(S_{T},\,S_{R})$,
respectively. In presence of a horizontal phase drop, these two processes pick
up opposite phases, as shown in panels f and g. This leads to interfering
quartet supercurrent components within this simplified model, without and with
horizontal phase drop, respectively. Due to the $\pi$-shift, the critical
current along the vertical direction in panel f is reduced by the two quartet
processes. On panel g, a phase drop along the horizontal direction dephases
the negative contribution of both processes, resulting in an increased
critical current and thus a nonconvex CCC.

The article is organized as follows. The $\pi$-shifted quartets are introduced
in Sec. II. The device and the model are presented in Sec. III. The numerical
results, analytical and numerical, are presented and discussed in Sec. IV.
Concluding remarks are provided in Sec. V.

##  II $\pi$-shifted Cooper quartets

In this section, we provide physical arguments supporting the $\pi$-shifted
Cooper quartet current-phase relation. The key underlying concept can readily
be understood starting from a three-terminal configuration of Josephson
junctions in the DC superconducting state, connecting the leads $S$ with
respective indices $i,j,k$ Freyn ; Pillet , and biased with respective phases
$\varphi$. The corresponding spin-singlet wave-function of a split Cooper pair
for instance between $S_{i}$ and $S_{j}$ takes the form

$\psi=\frac{1}{\sqrt{2}}\left(c_{i,\uparrow}^{+}c_{j,\downarrow}^{+}-c_{i,\downarrow}^{+}c_{j,\uparrow}^{+}\right),$
(1)

where $c^{+}_{i,\sigma}$ creates a spin-$\sigma$ fermion in $S_{i}$. The
splitting event in Eq. (1) can come along with a second one. The resulting
composite four-fermion transient state, i.e. a Cooper quartet Freyn ; Lefloch
; Heiblum ; HGE ; Pribiag , ends up as two Cooper pairs transmitted into
$S_{i}$ and $S_{j}$, respectively, and described by

$\langle\psi^{2}\rangle=-\langle
c_{i,\uparrow}^{+}c_{i,\downarrow}^{+}\rangle\,\langle
c_{j,\uparrow}^{+}c_{j,\downarrow}^{+}\rangle,$ (2)

where $\langle\,...\,\rangle$ is a quantum mechanical expectation value
(details can be found in Appendix A). By probing the internal structure of
(double) split Cooper pairs, we mean providing experimental evidence for the
negative sign in Eq. (2), which is a direct consequence of both quantum
mechanical exchange and the split Cooper pair structure of Eq. (1).
Consequently, the relation between the quartet supercurrent $I_{q}$ and the
quartet phase $\varphi_{q}$ is inverted:

$\displaystyle I_{q}(\varphi_{q})$ $\displaystyle=$
$\displaystyle-|I^{c,q}|\sin\varphi_{q}$ (3) $\displaystyle\varphi_{q}$
$\displaystyle=$ $\displaystyle\varphi_{a}+\varphi_{b}-2\varphi_{c},$ (4)

where $\varphi_{a}$, $\varphi_{b}$ and $\varphi_{c}$ are the superconducting
phase variables of the leads $S_{a}$, $S_{b}$ and $S_{c}$ respectively. Eq.
(3) can be rewritten as $I_{q}(\varphi_{q})=|I^{c,q}|\sin(\varphi_{q}+\pi)$
and this $\pi$-shift is a macroscopic signature for the specific internal
structure of single split Cooper pairs, see Eq. (1).

Another simple perspective on the $\pi$-shift of the quartets readily follows
from considering a single two-terminal superconducting weak link with normal-
state transmission $\alpha$. Here, the energy-phase relation can be Fourier-
expanded as $E^{J}(\varphi)=E^{J}_{0}+E^{J}_{2e}\cos\varphi+E^{J}_{4e}\cos
2\varphi+...$. The $\cos\varphi$ term represents the Josephson Cooper-pair
energy, and is dominant in the limit of small transparency, while the $\cos
2\varphi$ one describes correlated tunneling of two Cooper pairs. We find
$E^{J}_{4e}/E^{J}_{2e}\approx-\alpha/16$ in the small-$\alpha$ limit and more
generally $E^{J}_{4e}/E^{J}_{2e}<0$ for all $\alpha<1$, see Appendix B. This
negative sign echoes the above current-phase relation of the quartets. More
generally, our work proposes a method to directly reveal these $\pi$-shifted
second-order Josephson harmonics, using a multiterminal configuration.

##  III The device and multiterminal Josephson circuit theory

In this section, we present the two types of devices and the approximations
sustaining multiterminal Josephson circuit theory. The proposed device
consists of four BCS superconducting leads $S_{L},\,S_{R},\,S_{B}$ and
$S_{T}$, with the respective superconducting phase variables
$\varphi_{L},\,\varphi_{R},\,\varphi_{B}$ and $\varphi_{T}$, and connected via
a square-shaped normal conductor $N_{0}$ as shown in Fig. 1. The external
circuit imposes current in orthogonal directions, that is, a vertical current
$I_{v}\equiv I_{T}=-I_{B}$ and a horizontal one $I_{h}\equiv I_{R}=-I_{L}$.
The absence of coupling between $I_{v}$ and $I_{h}$ produces a square or
rectangular CCC, while rounded CCCs are indicative of coupling.

Our main result is that assuming a single or two contacts with transparency
smaller than the others, nonconvex CCCs emerge in the $(I_{v},\,I_{h})$ plane
already under zero applied magnetic field. We thus find reentrance of the
dissipative state into the superconducting region as a distinctive signature
of the $\pi$-shifted contribution of second-order Josephson harmonics.
Furthermore, we show that the $\pi$-shifted Cooper quartet supercurrent
produces distinctive sharp reentrant sharp-angled points in the CCCs.

The four-terminal geometry is found by a straightforward generalization of
Josephson circuits, where now the $I_{v}$ and $I_{h}$ supercurrents result
from an interference between multipair processes involving the phases of more
than two terminals Freyn ; split-quartets . For instance, in a two-terminal
Josephson junction, the terms corresponding to Cooper pairs transmitted from
$S_{i}$ to $S_{j}$ couple to the difference
$\delta_{i,j}=\varphi_{i}-\varphi_{j}$. Similarly, with four terminals, the
relevant phase variables are then given by gauge-invariant combinations such
as $\delta_{i,j}+\delta_{k,l}$ split-quartets , which reduces to Eq. (4) for
three terminals Freyn .

In our multiterminal Josephson circuit model, we assume tunable contacts with
a few transmission modes connecting the four superconductors to a central
normal metal island (see Fig. 1), as was recently demonstrated in bilayer
graphene-based two-terminal Josephson devices Danneau2  and in multiterminal
semiconducting-superconducting quantum point contacts Pribiag . Considering
intermediate contact transparencies, although the DC-Josephson effect is
dominant, the next-order Cooper quartets still yield a sizable contribution,
while the even higher-order terms are smaller. This hierarchy justifies the
approach of the Letter, considering within a single four-terminal device all
the Josephson processes involving two, three and then four terminals. The
calculation involves two steps: our starting points are the approximate
analytical expressions of the current-phase relations discussed above, with
sign and amplitude as free parameters. This allows comparing the CCCs with
respectively positive or negative Cooper quartet contributions. From this we
will arrive to the conclusion that nonconvex CCCs in zero field carry the
unique signature of the microscopic $\pi$-shifted Cooper quartet current-phase
relation, and would be absent with a $0$-shift.

We consider intermediate transparency interfaces, with hopping amplitudes
$J_{L},\,J_{R},\,J_{B}$ and $J_{T}$ connecting respectively the four
superconducting leads $S_{L}$, $S_{R}$, $S_{B}$ and $S_{T}$ to a normal tight-
binding lattice $N_{0}$. The DC-Josephson supercurrent of Cooper pairs from
lead $S_{i}$ to lead $S_{j}$ is written as
$I_{P}=I^{c,P}_{i,j}\sin\delta_{i,j}$. The nonlocal DC-Josephson supercurrent
of the Cooper quartets involves, at the lowest order in tunneling, the
following three terms:

$\displaystyle I_{q}=I^{c,q}_{i,j,(k)}\sin(\delta_{i,k}+\delta_{j,k})$
$\displaystyle+$ $\displaystyle
I^{c,q}_{i,(j),k}\sin(\delta_{i,j}+\delta_{k,j})+I^{c,q}_{(i),j,k}\sin(\delta_{j,i}+\delta_{k,i}).$

Here, $I^{c,q}_{i,j,(k)}$ for instance represents the critical quartet current
of two pairs emitted by $S_{k}$ and recombining into $S_{i}$ and $S_{j}$. We
introduce the individual channel transmissions $\tau_{i}$ such that all
$J_{i}=\sqrt{\tau_{i}}J^{(0)}$, with $J^{(0)}$ a constant smaller than the
band-width $W$. The critical currents scale as follows:
$I^{c,P}_{i,j}=\tau_{i}\tau_{j}I^{c(0)}_{i,j}$ for the Cooper pairs, and
$I^{c,q}_{i,j,(k)}=\tau_{i}\tau_{j}\tau_{k}^{2}I^{c(0)}_{i,j,(k)}$,
$I^{c,q}_{i,(j),k}=\tau_{i}\tau_{j}^{2}\tau_{k}I^{c(0)}_{i,(j),k}$ and
$I^{c,q}_{(i),j,k}=\tau_{i}^{2}\tau_{j}\tau_{k}I^{c(0)}_{(i),j,k}$ for the
Cooper quartets, where the $I^{c(0)}$s do not scale with the transmissions.

##  IV Results

###  IV.1 Polarization with one current and one phase bias

In this subsection we present analytical results for the device polarized with
one current and one phase bias, see Fig. 1a. An external source drives a
supercurrent from $S_{B}$ to $S_{T}$ and an external loop fixes the phase
difference between $S_{L}$ and $S_{R}$. We additionally assume that the
$N_{0}$-$S_{T}$ link has a tunneling amplitude $J_{T}$ small compared to
$J_{L}=J_{R}=J_{B}\equiv J^{(0)}$, i.e.
$\tau_{T}\lesssim\tau_{L},\,\tau_{R},\,\tau_{B}\lesssim 1$. Then, we make a
perturbation expansion in tunneling of the Josephson circuit to the dominant
order $\tau_{T}$, neglecting the processes of order $\tau_{T}^{2}$ (see
Appendix C). In absence of the quartets, we find two types of processes: (i)
The direct two-terminal DC-Josephson effect of the Cooper pairs from $S_{B}$
to $S_{T}$ (see Fig. 1e), and (ii) The two-terminal DC-Josephson processes of
the Cooper pairs involving the lateral superconductors $S_{L}$ and $S_{R}$.
Adding now the quartets, we include all possible processes appearing at the
orders $\tau_{T}^{0}$ and $\tau_{T}$.

The cartoon shown in Fig. 1 illustrates the case where, at the order
$\tau_{T}$, the critical current $I^{c}_{v}$ from $S_{B}$ to $S_{T}$ results
from an interference between the amplitudes of the two-terminal DC Josephson
effect and both Cooper quartets (see Figs. 1f,g). Taking an opposite relative
sign of the two- and three-terminal contributions, respectively, leads to a
reduction of $I^{c}_{v}$ upon including the Cooper quartets. Notably, because
each quartet process picks up an opposite phase
$\varphi_{L}=-\varphi_{R}\equiv-\varphi/2$, their respective contributions are
dephased and the value of $I^{c}_{v}$ is restored upon applying a supercurrent
$I_{h}$ (or a phase gradient) in the transverse direction, as shown in Figs.
1f and g.

Figure 2: The figure shows the shape of the CCC, i.e. $F(\varphi/2\pi)$ as a
function of $\varphi/2\pi$, where $F(\varphi/2\pi)$ is proportional to the
critical current, see Eq. (8). Polarization is with one current and one phase
bias, see Fig. 1a. The notation $\varphi$ stands for the phase difference
between the $N_{0}-S_{R}$ and $N_{0}-S_{L}$ contacts. Two regimes are obtained
if $I_{q}^{c}<-I_{P}^{c}/14$ or $I_{q}^{c}>-I_{P}^{c}/14$, corresponding to
nonconvex or convex CCCs respectively.

Now, we evaluate the full set of microscopic two- and three-terminal processes
at the relevant orders (details in Appendix C). Using the notations
$\varphi_{L}=-\varphi_{R}=-\varphi/2$, we demonstrate in Appendix C that, at
small $\tau_{T}$ and quartet Josephson energy $E_{q}=(\hbar/2e)I^{c}_{q}$, the
critical current $I_{v}^{c}$ from $S_{B}$ to $S_{T}$ can be approximated as

$I_{v}^{c}\simeq\tau_{T}I_{P}^{c}\left\\{3+6\frac{I_{q}^{c}}{I_{P}^{c}}-\frac{\varphi^{2}}{4}\left[1+14\frac{I_{q}^{c}}{I_{P}^{c}}\right]\right\\},$
(6)

where $I_{P}^{c}$ and $I_{q}^{c}$ are proportional to the critical currents of
the two- and three-terminal Cooper pair and Cooper quartet processes
respectively. Eventually, Eq. (6) predicts nonconvex CCCs if the condition

$I_{q}^{c}<-\frac{I_{P}^{c}}{14}$ (7)

is fulfilled. In this case, the dissipative state reenters into the
superconducting one, as a result of the $\pi$-shifted Cooper quartet current-
phase relation coming from the spin-singlet minus signs in Eqs. (1) and (2).

We rewrite Eq. (6) as $I_{v}^{c}(\varphi)=\tau_{T}I_{P}^{c}F(\varphi/2\pi)$,
with

$F\left(\frac{\varphi}{2\pi}\right)=3+6\frac{I_{q}^{c}}{I_{P}^{c}}-\pi^{2}\left(\frac{\varphi}{2\pi}\right)^{2}\left[1+14\frac{I_{q}^{c}}{I_{P}^{c}}\right].$
(8)

The variations of $F(\varphi/2\pi)$ are shown in Fig. 2, confirming emergence
of nonconvex or convex CCC if $I_{q}^{c}<-I_{P}^{c}/14$ or
$I_{q}^{c}>-I_{P}^{c}/14$ respectively.

Figure 3: Critical current contours in the $(I_{v},I_{h})$ plane, with
$I_{q}^{c}/I_{P}^{c}=-0.2,\,0,\,0.2$ (on panels a, b, c respectively), and
with a single weak link. The contact transmission coefficients are such that
$\tau_{B}=\tau_{L}=\tau_{R}=1$ and $\tau_{T}=1$ (magenta), $\tau_{T}=0.5$
(green) and $\tau_{T}=0.25$ (blue). Each panel is rescaled to full size on the
$I_{v}$ and $I_{h}$-axis. Polarization is with two orthogonal current biases,
see Fig. 1b. Temperature is set to zero.

Figure 4: Critical current contours in the $(I_{v},I_{h})$ plane, with
$\tau_{B}=\tau_{R}=1$, and with two weak links. The values
($I_{q}^{c}/I_{P}^{c},I^{{}^{\prime}c}_{2T}/I_{P}^{c})=(-0.2,0)$,
$(-0.2,-0.2)$, $(0,-0.2)$ and $(0,0.2)$ are used on panels a-d respectively.
The panels are organized as a table, and the values of $\tau_{L}$, $\tau_{T}$
are indicated. Each panel is rescaled to full size on the $I_{v}$ and
$I_{h}$-axis. Polarization is with two orthogonal current biases, see Fig. 1b.
Temperature is set to zero.

Figure 5: Critical current contours in the $(I_{v},I_{h})$ plane, in the
presence of Josephson coupling to the four superconducting phase variables,
and with a single weak link. The contact transmission coefficients are such
that $\tau_{B}=\tau_{L}=\tau_{R}=1$ and $\tau_{T}=1$ (magenta), $\tau_{T}=0.5$
(green) and $\tau_{T}=0.25$ (blue). The different panels are organized like a
table, and the values of $I_{q}^{c}$ and $I^{{}^{\prime}c}_{q}$ are indicated
on the figure, both of those being normalized to $I_{P}^{c}=1$. Each panel is
rescaled to full size on the $I_{v}$ and $I_{h}$-axis. Polarization is with
two orthogonal current biases, see Fig. 1b. Temperature is set to zero.

###  IV.2 Polarization with two orthogonal current biases

In this subsection, we numerically solve a related model where we impose
current biases in both horizontal and vertical directions, such that
$I_{v}=I_{T}=-I_{B}$ and $I_{h}=I_{R}=-I_{L}$ (see Fig. 1b). The four
superconducting phase variables adjust accordingly. The numerical calculations
are based on evaluating convergence of the steepest descent algorithm for a
multiterminal Josephson junction. A dichotomic search was implemented, in
order to locate the CCCs to high accuracy. We use $I^{c}_{k,l}\equiv
I_{P}^{c}$ and $I^{c}_{k,l,(m)}\equiv I_{q}^{c}$ for the critical currents of
the processes coupling to two and three superconducting phase variables,
respectively. Fig. 3 shows the CCCs of a four-terminal device with the
transmission coefficient scaling factors $\tau_{B}=\tau_{L}=\tau_{R}=1$ and
different values of $\tau_{T}$. For positive values of $I_{q}^{c}/I_{P}^{c}$,
the CCCs have the shape of nested rounded rectangles. For sufficiently
negative $I_{q}^{c}/I_{P}^{c}$ however, the CCCs evolve from diamond-like to a
shape presenting nonconvex sharp-angled points when lowering $\tau_{T}$.
Notably, the CCCs with nonconvex sharp-angled points are only obtained for a
sufficiently negative Cooper quartet critical current (here
$I_{q}^{c}/I_{P}^{c}=-0.2$), which is in agreement with the preceding
analytical solution.

In Fig. 4a, we further implement two weak links with $\tau_{T},\,\tau_{L}\leq
1$, while maintaining $\tau_{B}=\tau_{R}=1$, and we use a negative Cooper
quartet critical current $I_{q}^{c}/I_{P}^{c}=-0.2$. Focusing on the panels on
the diagonal, i.e. $\tau_{T}=\tau_{L}=1/4,\,1/2,\,1$, we obtain an evolution
from diamond-like to square-like CCCs, as $\tau_{T}=\tau_{L}$ decreases. Since
a rectangular CCC is indicative of independent currents in orthogonal
directions, this evolution demonstrates a loss of quantum mechanical coupling
between $I_{v}$ and $I_{h}$ as the contact transmission coefficient scaling
factor decreases. The intermediate value $\tau_{T}=\tau_{L}=1/2$ yields
reentrance on both supercurrent axes, which originates from the underlying
diagonal mirror symmetry in the device. Considering now the off-diagonal
panels in Fig. 4a, we obtain shapes with nonconvex sharp-angled points on the
$I_{v}^{c}$ axis if $\tau_{T}=1/4,\,1/2$ and $\tau_{L}=1$, and the same on the
$I_{h}^{c}$ axis if $\tau_{T}=1$ and $\tau_{L}=1/4,\,1/2$. This is again in
qualitative agreement with the analytical model calculations presented in the
above Sec. IV.1.

In Fig. 4b, we introduce all possible higher-order two-terminal
$I^{{}^{\prime}c}_{2T}\sin(2(\varphi_{i}-\varphi_{j}))$ coupling terms, in
addition to the Cooper quartets. We observe the robustness of the reentrant
sharp-angled points with respect to addition of these. Qualitatively, this can
be interpreted as due to the fact that a smooth feature on top of a sharp cusp
does not alter the latter. Figs. 4c and d comparatively show the CCCs with
vanishingly small quartet critical current but with finite
$I^{{}^{\prime}c}_{2T}$ taking negative or positive values. The nonconvex
sharp-angled points are absent in the corresponding CCCs if $I^{c}_{q}=0$ and
$I^{{}^{\prime}c}_{2T}\neq 0$. Those nonconvex sharp-angled points are thus a
unique signature of the nonlocally $\pi$-shifted Cooper quartets.

Eventually, we demonstrate robustness of the reentrant pockets upon including
the DC-Josephson effect depending on all four superconducting phase variables
split-quartets . At the lowest order in tunneling, the corresponding Josephson
critical currents are denoted by $I^{{}^{\prime}c}_{q}$ and they scale like
$\tau_{L}\tau_{R}\tau_{B}\tau_{T}$. Fig. 5 provides the CCCs for variable
combinations of $I_{q}^{c}/I_{P}^{c}$ and $I^{{}^{\prime}c}_{q}/I_{P}^{c}$,
and with $\tau_{T}\lesssim 1$ and $\tau_{B}=\tau_{L}=\tau_{R}=1$. The data
with $I_{q}^{c}\gtrsim 0$ reveal smooth nonreentrant variations, contrasting
with the sharper reentrant-like variations on the other panels. We conclude
that reentrant features in CCCs at negative Cooper quartet critical current
$I_{q}^{c}$ are robust with respect to including higher-order Josephson terms.

##  V Conclusions

To conclude, it follows from basic theoretical arguments that the quartet
supercurrent contribution must be $\pi$-shifted with respect to the lowest
order Josephson Cooper pair supercurrent. We demonstrated that the nonconvex
two-dimensional critical current contours (CCCs) of a current-biased four-
terminal Josephson junction are generically due to a relative $\pi$-shift of
the higher-order terms in the current-phase relation. These can either
originate simply from the two-terminal Josephson current-phase relation, or,
more interestingly, from the Cooper quartets. Finally, we demonstrated that
nonconvex sharp-angled points in the CCCs are a distinctive signature of
negative Cooper critical current contributions. However, we note that too
small negative quartet critical currents will restore convex CCC, which sets
constraints on the transmissions for the observation of the characteristic
reentrance. A recent experiment multiterminal-exp3  reported the appearance of
nonconvex CCCs only under applied magnetic field. However, in contrast to our
assumptions, all contacts had large transparencies. Conclusive evidence for
the $\pi$-shifted quartet term could be realized with bilayer graphene- or
semiconducting-quantum point contacts Danneau2 ; Pribiag  with tunable contact
transparencies.

## Acknowledgements

The authors benefited from fruitful discussions with M. d’Astuto, S.
Collienne, T. Klein, F. Lévy-Bertrand, M.A. Méasson, P. Rodière, and A.
Silhanek. R.M. acknowledges a useful correspondence with V. E. Manucharyan.
R.M. thanks the Infrastructure de Calcul Intensif et de Données (GRICAD) for
use of the resources of the Mésocentre de Calcul Intensif de l’Université
Grenoble-Alpes (CIMENT). This work was supported by the International Research
Project SUPRADEVMAT between CNRS in Grenoble and KIT in Karlsruhe. This work
received support from the French National Research Agency (ANR) in the
framework of the Graphmon (ANR-19-CE47-0007) and JOSPEC (ANR-17-CE30-0030)
projects. This work was partly supported by Helmholtz Society through program
STN and the DFG via the Project No. DA 1280/7-1.

##  Appendix A Details on phenomenologically squaring the single Cooper pair
wave-function

In this Appendix, we detail how to deduce Eq. (2) from Eq. (1). Namely, we
square the wave-function of a Cooper pair split between $S_{i}$ and $S_{j}$:

$\displaystyle\psi^{2}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{\sqrt{2}}\left(c_{i,\uparrow}^{+}c_{j,\downarrow}^{+}-c_{i,\downarrow}^{+}c_{j,\uparrow}^{+}\right)\right]^{2}$
(9) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(c_{i,\uparrow}^{+}c_{j,\downarrow}^{+}-c_{i,\downarrow}^{+}c_{j,\uparrow}^{+}\right)\times\left(c_{i,\uparrow}^{+}c_{j,\downarrow}^{+}-c_{i,\downarrow}^{+}c_{j,\uparrow}^{+}\right)$
(10) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[c_{i,\uparrow}^{+}c_{j,\downarrow}^{+}c_{i,\uparrow}^{+}c_{j,\downarrow}^{+}-c_{i,\uparrow}^{+}c_{j,\downarrow}^{+}c_{i,\downarrow}^{+}c_{j,\uparrow}^{+}\right.$
$\displaystyle\left.-c_{i,\downarrow}^{+}c_{j,\uparrow}^{+}c_{i,\uparrow}^{+}c_{j,\downarrow}^{+}+c_{i,\downarrow}^{+}c_{j,\uparrow}^{+}c_{i,\downarrow}^{+}c_{j,\uparrow}^{+}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[-\left(c_{i,\uparrow}^{+}\right)^{2}\left(c_{j,\downarrow}^{+}\right)^{2}-\left(c_{i,\uparrow}^{+}c_{i,\downarrow}^{+}\right)\left(c_{j,\uparrow}^{+}c_{j,\downarrow}^{+}\right)\right.$
$\displaystyle\left.-\left(c_{i,\uparrow}^{+}c_{i,\downarrow}^{+}\right)\left(c_{j,\uparrow}^{+}c_{j,\downarrow}^{+}\right)-\left(c_{i,\downarrow}^{+}\right)^{2}\left(c_{j,\uparrow}^{+}\right)^{2}\right].$

Evaluating quantum mechanical expectation values in the final state leads to

$\langle\psi^{2}\rangle=-\langle
c_{i,\uparrow}^{+}c_{i,\downarrow}^{+}\rangle\times\langle
c_{j,\uparrow}^{+}c_{j,\downarrow}^{+}\rangle,$ (13)

where we used $\langle\left(c_{i,\uparrow}^{+}\right)^{2}\rangle=0$ because of
spin conservation. The above Eq. (13) matches the above Eq. (2).

##  Appendix B Details on a single superconducting weak link

In this Appendix, we evaluate the first- and second-order harmonics of the
Josephson current-phase relation for a single-channel superconducting weak
link having hopping amplitude $J_{0}$, and connecting the “left” and “right”
superconducting leads $S_{L}$ and $S_{R}$ with superconducting phases
$\varphi_{L}$ and $\varphi_{R}$ respectively. The Andreev Bound State (ABS)
energies take the following form:

$E_{\pm}(\varphi_{R}-\varphi_{L},\alpha)=\pm\Delta\sqrt{1-\alpha\sin^{2}\left(\frac{\varphi_{R}-\varphi_{L}}{2}\right)},$
(14)

where the dimensionless normal-state transmission coefficient $\alpha$ between
$0$ and $1$ is given by

$\alpha=\frac{4(J_{0}/W)^{2}}{\left[1+(J_{0}/W)^{2}\right]^{2}}.$ (15)

The ABS energies are expressed as

$\displaystyle
E_{\pm}(\varphi_{R}-\varphi_{L},\alpha)=\pm\Delta\sqrt{1-\frac{\alpha}{2}+\frac{\alpha}{2}\cos\left(\varphi_{R}-\varphi_{L}\right)}$
(16)
$\displaystyle=\pm\Delta\sqrt{1-\frac{\alpha}{2}}\sqrt{1+\frac{\alpha}{2-\alpha}\cos\left(\varphi_{R}-\varphi_{L}\right)}.$
(17)

Expanding to second order, we obtain

$\displaystyle E_{\pm}(\varphi_{R}-\varphi_{L},\alpha)$ $\displaystyle=$
$\displaystyle\pm\Delta\sqrt{1-\frac{\alpha}{2}}\left\\{1+\frac{\alpha}{2(2-\alpha)}\cos(\varphi_{R}-\varphi_{L})-\frac{1}{8}\left(\frac{\alpha}{2-\alpha}\right)^{2}\cos^{2}\left(\varphi_{R}-\varphi_{L}\right)\right\\}+...$
(18)

Using
$\cos^{2}\left(\varphi_{R}-\varphi_{L}\right)=\left[1+\cos\left(2(\varphi_{R}-\varphi_{L})\right)\right]/2$,
we obtain

$\displaystyle E_{\pm}(\varphi_{R}-\varphi_{L},\alpha)$ $\displaystyle=$
$\displaystyle\pm\left[E_{0}^{J}+E^{J}_{2e}\cos\left(\varphi_{R}-\varphi_{L}\right)\right.$
$\displaystyle+\left.E^{J}_{4e}\cos\left(2\left(\varphi_{R}-\varphi_{L}\right)\right)+...\right],$

with

$\frac{E^{J}_{4e}}{E^{J}_{2e}}=-\frac{\alpha}{16}$ (20)

in the small-$\alpha$ limit, where $E^{J}_{2e}>0$ is positive and
$E^{J}_{4e}<0$ is negative.

Now, we present supplemental numerical calculations for the amplitudes
$H_{1}(\alpha)$ and $H_{2}(\alpha)$ of the first and second Josephson
harmonics as a function of the dimensionless normal-state transmission
coefficient $\alpha$:

$\displaystyle H_{1}(\alpha)$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}\frac{d\varphi}{2\pi}E_{+}(\varphi,\alpha)\cos\varphi$
(21) $\displaystyle H_{2}(\alpha)$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}\frac{d\varphi}{2\pi}E_{+}(\varphi,\alpha)\cos(2\varphi).$
(22)

Figure 6 shows that $H_{2}(\alpha)/H_{1}(\alpha)<0$ is negative for all values
of $\alpha<1$. The limiting behavior $H_{2}(\alpha)/H_{1}(\alpha)=-\alpha/16$
is obtained at small $\alpha$, in agreement with the above Eq. (20).

Figure 6: The figure shows $H_{2}(\alpha)/H_{1}(\alpha)$ as a function of the
normal-state contact transparency $\alpha$ (magenta datapoints), where
$H_{1}(\alpha)$ and $H_{2}(\alpha)$ are the first and second harmonics of the
Josephson energy, see Eqs. (21) and (22). The green line shows a comparison to
$H_{2}(\alpha)/H_{1}(\alpha)=-\alpha/16$, see Eq. (20).

##  Appendix C Details on the device controlled with one current and one phase
bias

In this Appendix, we consider the multiterminal Josephson circuit shown on the
above figure 1a, consisting of the four superconducting leads $S_{L}$,
$S_{R}$, $S_{B}$ and $S_{T}$ with the phases $\varphi_{L}$, $\varphi_{R}$,
$\varphi_{B}$ and $\varphi_{T}$. The phase difference
$\varphi_{R}-\varphi_{L}=\Phi$ is controlled by the flux $\Phi$ in the loop,
and supercurrent $I_{v}=I_{B}=-I_{T}$ is forced to flow from the “bottom” to
the “top” superconducting leads. We use the notation $\varphi_{T}=\psi$. Given
those constraints, the supercurrent transmitted into $S_{T}$ is parameterized
by a single phase variable, for instance by the phase variable $\psi$, and the
critical current from bottom to top is obtained by maximizing the supercurrent
$I_{v}(\psi)$ over $\psi$.

The superconductor $S_{T}$ is assumed to be connected to the normal conductor
$N_{0}$ by hopping amplitude that is weaker than the others. A reduction
factor $\tau_{T}$ is applied to each Cooper pair crossing the $N_{0}-S_{T}$
interface.

Figure 7: The figure shows the six two-terminal processes transferring Cooper
pairs between the superconducting leads.

First considering vanishing small quartet Josephson energy $I^{c}_{q}=0$, we
obtain the following expression of the four-terminal Josephson junction energy
$E^{(0)}$:

$\displaystyle E^{(0)}$ $\displaystyle=$ $\displaystyle
E_{P}\tau_{T}\left\\{\cos\left(\varphi_{T}-\varphi_{L}\right)+\cos\left(\varphi_{T}-\varphi_{B}\right)\right.$
$\displaystyle+\left.\cos\left(\varphi_{T}-\varphi_{R}\right)\right\\}$
$\displaystyle+$ $\displaystyle
E_{P}\left\\{\cos\left(\varphi_{B}-\varphi_{L}\right)+\cos\left(\varphi_{B}-\varphi_{R}\right)\right.$
$\displaystyle+\left.\cos\left(\varphi_{R}-\varphi_{L}\right)\right\\},$

where $E_{P}$ is the Josephson energy associated to transferring Cooper pairs
between the leads. Each term entering Eq. (C) is schematically shown in figure
7. Using $\varphi_{L}=-\varphi_{R}=-\varphi/2$ and $\varphi_{T}=\psi$, we
obtain

$\displaystyle E^{(0)}$ $\displaystyle=$ $\displaystyle
E_{P}\tau_{T}\left\\{\cos\left(\psi+\frac{\varphi}{2}\right)+\cos\left(\psi-\varphi_{B}\right)+\cos\left(\psi-\frac{\varphi}{2}\right)\right\\}$
(24)
$\displaystyle+E_{P}\left\\{\cos\left(\varphi_{B}+\frac{\varphi}{2}\right)+\cos\left(\varphi_{B}-\frac{\varphi}{2}\right)+\cos\varphi\right\\}.$

Then:

$\displaystyle I_{T}$ $\displaystyle=$
$\displaystyle-\frac{2e}{\hbar}\frac{\partial E^{(0)}}{\partial\psi}$ (25)
$\displaystyle=$
$\displaystyle\frac{2eE_{P}\tau_{T}}{\hbar}\left\\{\sin\left(\psi+\frac{\varphi}{2}\right)+\sin\left(\psi-\varphi_{B}\right)\right.$
$\displaystyle\left.+\sin\left(\psi-\frac{\varphi}{2}\right)\right\\}$
$\displaystyle I_{B}$ $\displaystyle=$
$\displaystyle-\frac{2e}{\hbar}\frac{\partial E^{(0)}}{\partial\varphi_{B}}$
(27) $\displaystyle=$
$\displaystyle\frac{2eE_{P}\tau_{T}}{\hbar}\sin\left(\varphi_{B}-\psi\right)$
$\displaystyle+\frac{2E_{P}}{\hbar}\left\\{\sin\left(\varphi_{B}+\frac{\varphi}{2}\right)+\sin\left(\varphi_{B}-\frac{\varphi}{2}\right)\right\\}.$

Figure 8: The figure shows the nine three-terminal processes transferring
Cooper quartets between the superconducting leads, at the orders
$(\tau_{T})^{0}$ and $\tau_{T}$. The three higher-order processes of order
$(\tau_{T})^{2}$ are not shown on the figure.

The current source imposes

$I_{T}+I_{B}=-(2e/\hbar)\left[\partial E^{(0)}/\partial\psi+\partial
E^{(0)}/\partial\varphi_{B}\right]=0,$ (29)

which leads to the self-consistent $\varphi_{B}=\varphi_{B}^{*}$, with

$\sin\varphi_{B}^{*}=-\tau_{T}\sin\psi,$ (30)

showing that $\varphi_{B}^{*}$ is of order $\tau_{T}$. Then, at the order
$\tau_{T}$, we obtain

$I_{T}=-\frac{2e}{\hbar}\frac{\partial
E^{(0)}}{\partial\psi}\simeq\frac{2eE_{P}\tau_{T}}{\hbar}\left[1+2\cos\left(\frac{\varphi}{2}\right)\right]\sin\psi.$
(31)

Taking the maximum over $\psi$ and expanding in small $\varphi$ leads to
following expansion of the critical current flowing from bottom to top at
small $\varphi$, at the order $\tau_{T}$:

$I_{c}\simeq\frac{2eE_{P}\tau_{T}}{\hbar}\left[3-\frac{\varphi^{2}}{4}\right],$
(32)

leading to convex CCC in absence of the quartets. Eq. (32) is rewritten as

$I_{c}\simeq\tau_{T}I_{P}^{c}\left[3-\frac{\varphi^{2}}{4}\right],$ (33)

where $I_{P}^{c}=2eE_{P}/\hbar$ is related to the Cooper pair critical
current.

Now, we include finite but small quartet Josephson energy $E_{q}$. Those
processes are shown in figure 8, and, at the order $\tau_{T}$, they yield the
following correction $\delta E^{(0)}$ to the energy $E^{(0)}$ in Eq. (C):

$\displaystyle\delta E^{(0)}$ $\displaystyle=$ $\displaystyle
E_{q}\left\\{\tau_{T}\cos\left(\varphi_{L}+\varphi_{T}-2\varphi_{B}\right)+\tau_{T}\cos\left(\varphi_{R}+\varphi_{T}-2\varphi_{B}\right)+\cos\left(\varphi_{R}+\varphi_{L}-2\varphi_{B}\right)+\tau_{T}\cos\left(\varphi_{T}+\varphi_{R}-2\varphi_{L}\right)\right.$
$\displaystyle+\left.\tau_{T}\cos\left(\varphi_{T}+\varphi_{B}-2\varphi_{L}\right)+\cos\left(\varphi_{R}+\varphi_{B}-2\varphi_{L}\right)+\tau_{T}\cos\left(\varphi_{T}+\varphi_{L}-2\varphi_{R}\right)+\tau_{T}\cos\left(\varphi_{T}+\varphi_{B}-2\varphi_{R}\right)\right.$
$\displaystyle+\left.\cos\left(\varphi_{L}+\varphi_{B}-2\varphi_{R}\right)\right\\}.$

Then, we find

$\displaystyle-\frac{\partial\delta E^{(0)}}{\partial\varphi_{T}}$
$\displaystyle=$
$\displaystyle\tau_{T}E_{q}\left\\{\sin\left(\varphi_{L}+\varphi_{T}-2\varphi_{B}\right)+\sin\left(\varphi_{R}+\varphi_{T}-2\varphi_{B}\right)+\sin\left(\varphi_{T}+\varphi_{R}-2\varphi_{L}\right)+\sin\left(\varphi_{T}+\varphi_{B}-2\varphi_{L}\right)\right.$
$\displaystyle+\left.\sin\left(\varphi_{T}+\varphi_{L}-2\varphi_{R}\right)+\sin\left(\varphi_{T}+\varphi_{B}-2\varphi_{R}\right)\right\\}$
$\displaystyle=$
$\displaystyle\tau_{T}E_{q}\left\\{\sin\left(-\frac{\varphi}{2}+\psi-2\varphi_{B}\right)+\sin\left(\frac{\varphi}{2}+\psi-2\varphi_{B}\right)+\sin\left(\psi+\frac{3\varphi}{2}\right)+\sin\left(\psi+\varphi_{B}+\varphi\right)\right.$
$\displaystyle+\left.\sin\left(\psi-\frac{3\varphi}{2}\right)+\sin\left(\psi+\varphi_{B}-\varphi\right)\right\\}.$

Since $\varphi_{B}$ is of order $\tau_{T}$ [see the above Eq. (30)], we
replace $\varphi_{B}$ by $\varphi_{B}=0$ in the above Eqs. (C)-(C) to obtain
the order-$\tau_{T}$ contribution to $-\partial\delta
E^{(0)}/\partial\varphi_{T}$:

$\displaystyle-\frac{\partial\delta E^{(0)}}{\partial\varphi_{T}}$
$\displaystyle\simeq$
$\displaystyle\tau_{T}E_{q}\left\\{\sin\left(-\frac{\varphi}{2}+\psi\right)+\sin\left(\frac{\varphi}{2}+\psi\right)+\sin\left(\psi+\frac{3\varphi}{2}\right)+\sin\left(\psi+\varphi\right)+\sin\left(\psi-\frac{3\varphi}{2}\right)+\sin\left(\psi-\varphi\right)\right\\}$
(37) $\displaystyle=$ $\displaystyle
2\tau_{T}E_{q}\left\\{\cos\left(\frac{\varphi}{2}\right)+\cos\varphi+\cos\left(\frac{3\varphi}{2}\right)\right\\}\sin\psi.$
(38)

Considering now $-\partial\delta E^{(0)}/\partial\varphi_{B}$, we find

$\displaystyle-\frac{\partial\delta E^{(0)}}{\partial\varphi_{B}}$
$\displaystyle=$ $\displaystyle
E_{q}\left\\{2\tau_{T}\sin\left(2\varphi_{B}-\varphi_{L}-\varphi_{T}\right)+2\tau_{T}\sin\left(2\varphi_{B}-\varphi_{R}-\varphi_{T}\right)+2\sin\left(2\varphi_{B}-\varphi_{R}-\varphi_{L}\right)+\tau_{T}\sin\left(\varphi_{T}+\varphi_{B}-2\varphi_{L}\right)\right.$
$\displaystyle+\left.\sin\left(\varphi_{R}+\varphi_{B}-2\varphi_{L}\right)+\tau_{T}\sin\left(\varphi_{T}+\varphi_{B}-2\varphi_{R}\right)+\sin\left(\varphi_{L}+\varphi_{B}-2\varphi_{R}\right)\right\\}$
$\displaystyle=$ $\displaystyle
E_{q}\left\\{2\tau_{T}\sin\left(2\varphi_{B}+\frac{\varphi}{2}-\psi\right)+2\tau_{T}\sin\left(2\varphi_{B}-\frac{\varphi}{2}-\psi\right)+2\sin\left(2\varphi_{B}\right)+\tau_{T}\sin\left(\psi+\varphi_{B}+\varphi\right)\right.$
$\displaystyle+\left.\sin\left(\varphi_{B}+\frac{3\varphi}{2}\right)+\tau_{T}\sin\left(\psi+\varphi_{B}-\varphi\right)+\sin\left(\varphi_{B}-\frac{3\varphi}{2}\right)\right\\}.$

The phase variable $\varphi_{B}$ turns out to be linear in $\tau_{T}$ at
$E_{q}=0$ [see the above Eq. (30)]. At small $E_{q}$, both $-{\partial\delta
E^{(0)}}/{\partial\varphi_{T}}$ and $-{\partial\delta
E^{(0)}}/{\partial\varphi_{B}}$ are of order $E_{q}\tau_{T}$. Then,
$\varphi_{B}^{*}$ is linear in $\tau_{T}$ in the presence of small $E_{q}$, as
it was the case for $E_{q}=0$.

The supercurrent transmitted into the superconducting lead $S_{T}$ is then
given by the sum of Eqs. (31) and (38):

$\displaystyle I_{T}$ $\displaystyle=$
$\displaystyle-\frac{2e}{\hbar}\frac{\partial\left(E^{(0)}+\delta
E^{(0)}\right)}{\partial\psi}$ (41) $\displaystyle\simeq$
$\displaystyle\frac{2e\tau_{T}E_{P}}{\hbar}\left\\{3+6\frac{E_{q}}{E_{P}}-\frac{\varphi^{2}}{4}\left[1+14\frac{E_{q}}{E_{P}}\right]\right\\}\sin\psi.$
(42)

Taking the maximum over $\psi$ leads to the following expression of the
critical current:

$\displaystyle I_{v}^{c}$ $\displaystyle=$
$\displaystyle-\frac{2e}{\hbar}\frac{\partial\left(E^{(0)}+\delta
E^{(0)}\right)}{\partial\psi}$ (43) $\displaystyle\simeq$
$\displaystyle\frac{2e\tau_{T}E_{P}}{\hbar}\left\\{3+6\frac{E_{q}}{E_{P}}-\frac{\varphi^{2}}{4}\left[1+14\frac{E_{q}}{E_{P}}\right]\right\\}.$
(44)

Eq. (43) is rewritten as

$I_{v}^{c}=\simeq\tau_{T}I_{P}^{c}\left\\{3+6\frac{I_{q}^{c}}{I_{P}^{c}}-\frac{\varphi^{2}}{4}\left[1+14\frac{I_{q}^{c}}{I_{P}^{c}}\right]\right\\},$
(45)

where $I_{P}^{c}=2eE_{P}/\hbar$ and $I_{q}^{c}=2eE_{q}/\hbar$ are related to
the Cooper pair and Cooper quartet critical currents. This concludes the
demonstration of the above Eq. (6). Eq. (45) goes to Eq. (33) in the
$I_{q}^{c}\rightarrow 0$ limit of vanishingly small quartet energy. As a
consequence of Eq. (45), the CCC is nonconvex if

$I^{c}_{q}<-\frac{I^{c}_{P}}{14},$ (46)

thus necessarily being negative.

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