Multi-terminal Josephson junctions as topological matter

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### ARTICLE

Received 17 Sep 2015 | Accepted 26 Feb 2016 | Published 4 Apr 2016 DOI: 10.1038/ncomms11167 OPEN
## Multi-terminal Josephson junctions as topological matter

Roman-Pascal Riwar 1,2, Manuel Houzet 1,2, Julia S. Meyer 1,2 & Yuli V. Nazarov 3

Topological materials and their unusual transport properties are now at the focus of modern

experimental and theoretical research. Their topological properties arise from the

bandstructure determined by the atomic composition of a material and as such are difficult to
tune and naturally restricted to r3 dimensions. Here we demonstrate that n-terminal
Josephson junctions with conventional superconductors may provide novel realizations of

topology in n � 1 dimensions, which have similarities, but also marked differences with

existing 2D or 3D topological materials. For nZ4, the Andreev subgap spectrum of the

junction can accommodate Weyl singularities in the space of the n � 1 independent

superconducting phases, which play the role of bandstructure quasimomenta. The presence

of these Weyl singularities enables topological transitions that are manifested experimentally

as changes of the quantized transconductance between two voltage-biased leads, the
quantization unit being 4e 2 /h, where e is the electric charge and h is the Planck constant.

1 University of Grenoble Alpes, INAC-PHELIQS, F-38000 Grenoble, France. 2 CEA, INAC-PHELIQS, F-38000 Grenoble, France. 3 Kavli Institute of
NanoScience, Delft University of Technology, Lorentzweg 1, Delft NL-2628 CJ, The Netherlands. Correspondence and requests for materials should be
addressed to R.-P.R. (email: [riwar@hotmail.de).](mailto:riwar@hotmail.de)

osephson junctions created by coupling two superconductors
through a weak link have been studied extensively for many
years 1–3 . The current across a Josephson junction yields
# J
information about the Andreev bound states (ABS) forming at
the junction 4–7 . In turn, the ABS spectrum is determined by the
properties of the junction and the superconducting leads. For
instance, if the leads are topologically nontrivial, the Josephson
effect may be used to probe these topological properties. In
particular, the 4p-periodicity of the supercurrent indicates
the presence of topologically protected zero-energy Majorana
states 8–11, which may arise in one-dimensional spinless p-wave
superconductors, semiconductor nanowires with proximityinduced superconductivity or at the surface of bulk materials.

The rapidly growing field of three-dimensional (3D) Weyl
semimetals deals with a bandstructure that exhibits conical
energy gap closings: Weyl points 12–16 . Unlike the Dirac point in
graphene 17 that may be gapped out through an appropriate
coupling, isolated Weyl points are topologically protected. They
can be regarded as monopoles with a positive or negative charge.
A topological invariant—the Chern number—defined on a
surface in momentum space characterizes the total charge of
the monopoles it encloses. These monopoles give rise to many
unusual features, such as chiral edge states and associated surface
Fermi arcs 14 . Topological protection guarantees that the only way
to induce a gap is either to annihilate two Weyl points of opposite
charge by bringing them together, or to couple two cones
at a finite distance in momentum space through breaking of
momentum conservation 16 .

In this paper, we show that multi-terminal Josephson junctions
may be topologically nontrivial even if the superconducting leads
are topologically trivial and no exotic materials are used to make
the junction. Thus, the junction itself may be regarded as an
artificial topological material, which displays Weyl singularities,
when the energy of the lowest ABS goes to zero at certain values
of the superconducting phases such that the gap in the spectrum
closes. Below, we also show that their topological property can be
easily probed by the transconductance between two voltagebiased leads, which is proportional to the Chern number.

Results
The topology of the bound state spectrum. We consider a
junction with n superconducting leads connected through a
scattering region (Fig. 1a). The leads a ¼ {0, 1, y, n � 1} have the
same gap D, though they may differ in the phase of the superconducting order parameter, fa. Due to gauge invariance, only
n � 1 phases are independent, hence we may set f0 ¼ 0. Likewise
we choose to focus on a short scattering region, characterized by

an energy-independent scattering matrix ^ S in the basis of
N ¼ P a N a transport channels, N a being the number of channels

in contact a. In a longer scattering region described by an energydependent ^ SðEÞ, more bound states would appear at finite energy.
However, the presence of Weyl singularities at zero energy only
depends on ^ S - ^ Sð0Þ. We also assume time-reversal symmetry,
such that ^ S T ¼ ^ S, as well as spin-rotation symmetry. Note,
although, that our predictions are robust even if those symmetries
are broken, see Discussion section.

The junction hosts a set of spin-degenerate ABS, indexed by k,
whose energies EkZ0 are determined from the equation 4

with the Pauli matrices ^tx;y;z in the basis of the two degenerate
states corresponding to eigenvalue E ¼ 0. The fields hi depend
linearly on df ^ ¼f ^ - f ^ ð 0 Þ through the real matrix M^ .
The form of the Weyl Hamiltonian in equation (2) indicates
that we need at least three parameters to tune the system to the
degeneracy point, hi ¼ 0. Thus, for four terminals with three
independent phases, the Weyl singularities appear as points in the
3D phase space. For five terminals, the Weyl singularities occur in
general as one-dimensional curves in the four-dimensional (4D)
space of phases. This opens up the possibility to realize a
topological material in arbitrary dimensions. Note that such
multi-terminal Josephson junctions cannot be characterized
by means of the standard periodic table of topological
semimetals 18,19, due to the distinct behaviour of the
quasimomenta f^ under particle–hole symmetry. A proper
classification could be envisioned along the lines of ref. 20.

h i
det 1 � e � 2iw^ Se if^^ S � e � if^

¼ 0; ð1Þ

where w ¼ arccos(E/D), and e if^ is a diagonal matrix that assigns to
each channel the phase factor of the corresponding terminal. The
spin-degenerate ABS energies Ekðf ^ Þ (Fig. 1b) are periodic in all
phases with a period 2p. The total phase-dependent energy of
the junction reads E ¼ P ks (n ks � 1/2)E k , n ks ¼ 0, 1 being the

occupation of the state k with spin s.

Zero-energy states are most easily described by making use of
the mapping from states with spin s at energy EZ0 to states with
spin - s at energy - E. Then a gap closing at a certain value of
the phases, f ^ ð0Þ, corresponds to the crossing of two (singly
degenerate) states. Thus, the zero-energy state is doubly (spin-)
degenerate. We can then describe the lowest ABS band in the
vicinity of the gap closing by the two-by-two Weyl Hamiltonian
(Supplementary Note 1),

HW ¼ P

i¼x;y;z

hi^ti; hi ¼ P

dfaMai ; ð2Þ

0.8

0.6

0.4

0.2

- _�_ 0 _�_

_�_ 1

0.8

0.6

0.4

0.2

- _�_ 0 _�_

_�_ 1

_�_ 3

_�_ n–1

_�_ 1

_�_ 0 = 0

_�_ 2

_S_ ˆ

Figure 1 | General setup of the multi-terminal junction, and examples of typical ABS spectra. (a) The superconducting leads with phases fa, a ¼ 0, y,

n � 1, are connected through a scattering region described by the scattering matrix ^ S. (b) Generic ABS energy spectrum versus f1, away from a Weyl
singularity. (c) ABS energy spectrum versus f1, where the other phases are tuned to a Weyl singularity. Note the gap closing (red, dotted circle).

The topology of the junction can be characterized by a set of
Chern numbers. A Chern number may be defined in the twodimensional (2D) subspace of two phases fa and fb, through the
local Berry curvature of the ABS. The Berry curvature for the
bound state k with spin s is related to its wave function jjksi
through

- +

@jks

ab k �� 2Im

B ab k

@fa

the voltages, and the transconductance is defined by the Chern
number

�Ia¼G ab Vb with G ab ¼ � 4 h e2 Cab: ð6Þ

Equation (6) shows that multi-terminal junctions exhibit a d.c.
current response typical for the quantum Hall effect, although
based on different physics. The transconductance quantum is
four times bigger than in the quantum Hall effect, which can be
traced to the 2e charge of the superconducting Cooper pairs and
the presence of two spin bands. To extract the small d.c. signal,
the averaging time needs to be sufficiently long. The relevant time
scale is determined by the low-frequency current noise (see
Supplementary Note 3).

Generally, at low temperatures one expects relaxation processes
to bring the system to the ground state with nks ¼ 0. A peculiarity
of superconducting junctions is that internal relaxation processes
cannot change the parity of the quasiparticle number. Parity
changing processes require a quasiparticle from the bulk, and are
therefore rare as the concentration of such quasiparticles is
exponentially small at low temperatures. So one can say that a
superconducting junction can be in two different ground states,
with even or odd parity. The switching between parities occurs
on a long time scale: experiments with break junctions 21, for
example, yield switching times 40.1 ms, while for two other

�����

@jks

@fb

: ð3Þ

Note that B ab k

Note that B ab k does not depend on spin (Supplementary Note 2).

The total Berry curvature of the many-body superconducting
state then reads B ab ¼ P ks ð nks � 1=2 ÞB ab k , in analogy to the

ks ð nks � 1=2 ÞB ab k

state then reads B ab ¼ P ks ð nks � 1=2 ÞB ab k , in analogy to the

expression for the energy. For fixed occupations nks ¼ 0, 1, the
integral of the Berry curvature over the elementary cell yields an
integer, the Chern number

C ab ¼

k ab nks � 1 2

Ck ab

-
nks � 1 ð4Þ

- p dfadfbB k ab= ð 2p Þ.

with Ck ab¼

R p

- p

Since the Weyl singularities appear as points in the 3D phase
space, a third phase fg may be used to tune the system through
Weyl points, thus changing the Chern number. We see that a
given band k contributes to the total Chern number with - Ck ab

when it is empty, and with þ Ck ab when it is doubly occupied,k

when it is empty, and with þ Ck ab when it is doubly occupied,

whereas it gives a zero contribution if there is a single
quasiparticle in the band.

Probing Weyl points through quantized transconductance.
Importantly, the current response of the junction with slowly
varying phases reveals the Chern number. Biasing lead b with a
voltage eVb - D gives rise to the instantaneous current to
contact a (Supplementary Note 2 and Supplementary Fig. 1)

_�_ /2

_C_ 12

−1

−2

−3

- _�_  - _�_ /2 0

_�_ 3

_�_

IaðtÞ ¼ 2e

‘

@E
@fa

- 2ef _ bB ab ; ð5Þ

where f _ b¼2eVb=‘. The first term corresponds to the adiabatic
current and the second term is the first order correction in the
phase velocity. Let us now apply constant voltages to two leads.
For incommensurate voltages, the two phases uniformly sweep
the elementary cell. In the d.c. limit, the adiabatic current contribution then averages out, and the Berry curvature is replaced
by its average value. Thus, we find that the d.c. current is linear in

Figure 3 | Topological characterization of the 4-terminal junction for the
multi-channel case. Chern number as a function of f3 for a multi-channel
4-terminal junction, where the contacts 1, 2, 3, and 0 contain 12, 11, 10,
and 9 channels, respectively. In this particular example, the junction
hosts 36 Weyl points.

_�_

_C_ 12 = + 1

a b _G_ 12

_�_ 2

_�_ 4e 2 / _h_

_�_ 2

_G_ 12

- _�_ _�_

_�_

- _�_

−1 1

_�_

_�_ 3

- _�_

_�_ 4

- _�_

_C_ 12 = −1

_�_ 3

- _�_

- _�_ 0

_�_ 1

_�_

Figure 2 | Topological characterization of the 4-terminal junction for the
single-channel case. (a) Position of the four Weyl points in the space of
f1,2,3 of the single-channel 4-terminal junction, the colour code indicating
the respective charge. (b) The resulting transconductance G 12 indicating
the Chern number, as a function of f3 for the same single-channel
junction as in a.

Figure 4 | Topological characterization of a 5-terminal junction, each
contact having a single channel. The coloured areas display a nonzero
Chern number C 12 . The boundaries of these areas correspond to the
projection of the Weyl singularity lines to the (f3, f4)-plane. In a 3D
subspace the curves can be assigned a charge.

recent experimental setups 10 ms (ref. 22) and 1 min (ref. 23)
have been reported. We therefore expect a switching of the Chern
number and the associated quantized transconductance on this
time scale. In particular, for the situation we concentrate on, the
nontrivial Chern number C comes from the lowest band, and the
transconductance would switch between - (4e 2 /h)C for even
parity and zero for odd parity. If the current is averaged over time
intervals much longer than the switching scale, the resulting
transconductance will be proportional to the probability of
finding the junction in the even parity state.

Therefore, the topological signature is robust provided the
fermion parity is preserved, a fact which has also been pointed
out for other topological systems realized in superconductors 9 .
There are ways to control quasiparticle poisoning and reach
a desired even parity in the ABS occupation 7,24 . Note that
quantum Hall-like conductance quantization has also been
proposed in superconducting devices with finite charging
energy and hosting quantum phase slips 25 . Furthermore,
superconducting junctions with a gate-tunable charging energy
may realize topologically protected discrete charge pumping 26 .

We now focus on a four-terminal junction and investigate the
energy spectrum as a function of the three independent phases
f1,2,3. As mentioned above, such a 3D bandstructure may host
Weyl points with positive or negative topological charge. The
Nielsen-Ninomiya theorem 27 implies that the total topological
charge of the system is zero, such that the number of Weyl points
is always even. Furthermore, time-reversal invariance
corresponds to a mapping from f ^ to - f ^, hence a Weyl point
at f ^ ð0Þ has a counterpart at - f ^ ð0Þ with the same topological
charge. Thus, Weyl points come in groups of 4.

In the simplest case, where each contact contains only
one channel, the system may realize 0 or 4 Weyl points,
corresponding to a topologically trivial or nontrivial 3D material,
respectively. If a scattering matrix yielding Weyl points is found,
small changes in the scattering matrix only modify their position,
but cannot gap them. Namely, as the Weyl points carry a
topological charge, individual Weyl points are stable and
annihilation is possible only when two Weyl points with opposite
charges coincide.

A specific example is shown in Fig. 2. The position and charge
of the 4 Weyl points is shown in Fig. 2a. Without loss of
generality, we fix the phase f3 and compute the transconductance
G 12 between voltage-biased contacts 1 and 2. In Fig. 2b, one can
clearly see that the transconductance increases (decreases) by
4e 2 /h when f3 passes through a Weyl point with positive
(negative) charge. Interpreting f3 as a control parameter rather
than a quasimomentum, we thus see that the 2D bandstructure
of the system as a function of f1 and f2 undergoes a topological
transition when f3 passes through a Weyl point. The
transconductance directly measures the Chern number characterizing the corresponding 2D topological phase. Note that the
transconductance satisfies the relation G 12 ( � f3) ¼ - G 12 (f3)
due to time-reversal symmetry.

By randomly generating scattering matrices from the circular
orthogonal ensemble 28, we find that about 5% of scattering
matrices give rise to four Weyl points (Supplementary Note 4).
More Weyl points can be obtained in multi-channel junctions
where the maximal number of Weyl points is roughly
proportional to the number of channels, and the probability to
have no Weyl points is small (Supplementary Fig. 2). As a
consequence, a greater variety of 2D topological phases with
higher Chern numbers can be realized in that case. This is shown
in Fig. 3 for a multi-channel junction hosting 36 Weyl points,
where the maximal Chern number is 3. Recently realized
few-channel cross junctions 29 are promising to observe the
transconductance in a four-terminal junction.

We now turn to five-terminal junctions. In that case, the Weyl
singularities appear as closed loops in the 4D space of phases. The
simplest way to visualize them is to consider the additional phase
f4 as a tuning parameter of the 3D system described by the
phases (f1, f2, f3). Tuning f4 the Weyl points move, but remain
at zero energy. Note that in the 3D subspace, time-reversal
symmetry is effectively broken, as for a fixed nonzero f4, a Weyl
point at ðf ð 1 0Þ; fð 2 0Þ; fð 3 0ÞÞ does not have a counterpart at

�ðf ð 1 0Þ; fð 2 0Þ; fð 3 0ÞÞ anymore. The only constraint at a finite f4
is that the number of Weyl points is even. Once two Weyl points
with opposite charge meet, they annihilate. Thus their trajectories
describe closed loops in the 4D space of all phases.

As before, the presence of Weyl singularities may be probed by
the transconductance between two voltage-biased terminals, say
terminal 1 and 2. As a function of the two other phases f3 and
f4, we now obtain areas with different quantized values of the
transconductance, corresponding to different Chern numbers.
The boundaries of these areas are given by the projections of
the Weyl loops on the (f3, f4)-plane. An example is shown
in Fig. 4. Here time-reversal symmetry of the scattering matrix
manifests itself in the relation G 12 ( � f3, - f4) ¼ - G 12 (f3, f4).

Discussion
The above considerations are straightforwardly extended to a
larger number of terminals. Furthermore, as we saw for the
five-terminal junction, breaking time-reversal invariance does not
lift the topological protection of the Weyl points at zero energy,
as long as spin degeneracy is preserved. When spin degeneracy
is lifted due to a Zeeman field or due to spin–orbit interactions,
we expect the Weyl points to shift away from zero energy
while remaining stable (Supplementary Note 5 and Supplementary Fig. 3). The possibility of realizing zero-energy states in
three-terminal junctions with strong spin–orbit interaction was
studied in ref. 30.

To summarize, we predict the existence of topological Weyl
singularities in the ABS spectrum of n-terminal superconducting
junctions with nZ4. These Weyl singularities manifest themselves in a quantized transconductance between two voltagebiased contacts, when the remaining phases are tuned away from
the singularities. Tuning the system through a Weyl singularity,
the conductance displays a step, signalling a topological transition
of the 2D system described by the two phases of the voltagebiased contacts. This signature is robust provided the system
remains in its ground state. Thus, the quantized transconductance
should be accessible experimentally at low temperatures and
voltages, in multi-terminal junctions with, for example, 2D
electron gases, semiconducing crossed nanowires or graphene
(all systems through which a conventional Josephson effect has
already been measured).

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Acknowledgements
This work has been supported by the Nanosciences Foundation in Grenoble, in the frame
of its Chair of Excellence program. M.H. and J.M. acknowledge support through grants
No. ANR-11-JS04-003-01 and ANR-12-BS04-0016-03, and an EU-FP7 Marie Curie IRG.

Author contributions
M.H., J.S.M., and Y.V.N. formulated the model. R.-P.R. and Y.V.N. performed the
analytical derivations. R.-P.R. conducted the numerical calculations. All authors
contributed equally to design and writing of the manuscript.

Additional information
Supplementary Information accompanies this paper at [http://www.nature.com/](http://www.nature.com/naturecommunications)
[naturecommunications](http://www.nature.com/naturecommunications)

Competing financial interests: The authors declare no competing financial interests.

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How to cite this article: Riwar, R.-P. et al. Multi-terminal Josephson junctions as
topological matter. Nat. Commun. 7:11167 doi: 10.1038/ncomms11167 (2016).

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