Fermion-parity anomaly of the critical supercurrent in the quantum spin-Hall effect

0.65 MB·English

by  C. W. J. Beenakker

Fermion-parity anomaly of the critical supercurrent in the quantum spin-Hall effect
C. W. J. Beenakker,1 D. I. Pikulin,1 T. Hyart,1 H. Schomerus,2 and J. P. Dahlhaus1
1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
2Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom
(Dated: October 2012)
The helical edge state of a quantum spin-Hall insulator can carry a supercurrent in equilibrium
betweentwosuperconductingelectrodes(separationL,coherencelengthξ). Wecalculatethemax-
imum (critical) current I that can flow without dissipation along a single edge, going beyond the
c
short-junction restriction L (cid:28) ξ of earlier work, and find a dependence on the fermion parity of
2 the ground state when L becomes larger than ξ. Fermion-parity conservation doubles the critical
1 current in the low-temperature, long-junction limit, while for a short junction I is the same with
c
0 or without parity constraints. This provides a phase-insensitive, dc signature of the 4π-periodic
2
Josephson effect.

v
o PACSnumbers: 74.45.+c,71.10.Pm,74.78.Fk,74.78.Na
N

0 The quantum Hall effect and quantum spin-Hall effect uous spectrum above the gap can be neglected. As the
3 both refer to a two-dimensional semiconductor with an ratio L/ξ increases, the Andreev levels proliferate and
insulatingbulkandaconductingedge,andbothexhibita also the continuous spectrum starts to contribute to the

]
quantizedelectricalconductancebetweentwometalelec- supercurrent. Since σ is switched by changing the occu-
n
o trodes. If the electrodes are superconducting, a current pation of a single level, one might wonder whether a sig-
c canflowinequilibrium,inducedbyamagneticfluxwith- nificant parity dependence remains in the long-junction
- out any applied voltage. In the quantum Hall effect, the regime.
r
p edge states are chiral (propagating in a single direction Remarkably enough, the parity dependence becomes
u only) and two opposite edges are needed to carry a su- evenstronger. Whileinashortjunctionthetwobranches
s
percurrent [1–3]. Graphene is an ideal system to study I (φ) = −I (φ) differ only in sign, we find that in a
. + −
t this interplay of the Josephson effect and the quantum long junction they differ both in sign and in magnitude.
a
m Hall effect in a strong magnetic field [4–6]. In particular, the largest current that can flow without
- The interplay of the Josephson effect and the quan- dissipation is twice as large for I− than it is for I+. The
d tum spin-Hall effect, in zero magnetic field, has not yet differenceisillustratedinFig.1,inthezero-temperature
n been demonstrated experimentally but promises to be limit. Thebasicphysicscanbeexplainedinsimpleterms,
o strikingly different [7]. The quantum spin-Hall insulator as we will do first, and then we will present a complete
c
has helical edge states (propagating in both directions) theoryforfinitetemperatureandforarbitraryratioL/ξ.
[
that can carry a supercurrent along a single edge. The We set the stage by summarizing the findings of Fu

3 edgestatecouplesapairofMajoranazero-modes, allow- and Kane [7] in the short-junction regime. The spec-
v ing for the transmission of unpaired electrons with h/e trum of the Bogoliubov-De Gennes Hamiltonian HBdG
2
rather than h/2e periodic dependence on the magnetic is a ±ε symmetric combination of a discrete spectrum
1
flux [8, 9]. for |ε| < ∆ and a continuous spectrum for |ε| > ∆.
4
5 Anh/efluxperiodicitycorrespondstoa4π-periodicity Since backscattering along the quantum spin-Hall edge
. in terms of the superconducting phase difference φ, is forbidden by time-reversal symmetry [19], this is a
0
whichmeansthatthecurrent-phaserelationshiphastwo ballistic single-channel Josephson junction. In the limit
1
2 branchesI±(φ)andthesystemswitchesfromonebranch L/ξ →0thediscretespectrumconsistsofapairoflevels
1 totheotherwhenφisadvancedby2πatfixedtotalnum- at ε± =∓∆|cos(φ/2)|, while the continuous spectrum is
: ber N of electrons in the system. This is referred to as a φ-independent [20]. Quite generally, an eigenvalue ε(φ)
v
i fermion-parity anomaly, because the two branches have of HBdG contributes to the supercurrent an amount
X
different parity σ = ± of the number of electrons in the
ge d
r superconducting ground state [8]. I(φ)= ε(φ), (1)
a (cid:126) dφ
Josephsonjunctionscomeintwotypes[10],depending
onwhethertheseparationLofthesuperconductingelec- with g a factor that counts spin and other degeneracies
trodesissmallorlargecomparedtothecoherencelength [21]. There is no spin degeneracy at the quantum spin-
ξ =(cid:126)v/∆, or equivalently, whether the superconducting Hall edge (since spin is tied to the direction of motion),
gap ∆ is small or large compared to the Thouless energy so g =1 and the level ε contributes a supercurrent [7]
±
E =(cid:126)v/L. Existing literature [7–9, 11–18] has focused
T e∆
ontheshort-junctionregimeL(cid:28)ξ. Thesupercurrentis I (φ)=± sin(φ/2), |φ|<π. (2)
± 2(cid:126)
then determined entirely by the phase dependence of a
small number of Andreev levels in the gap, just one per To discuss the fermion-parity anomaly we assume, for
transverse mode. The phase dependence of the contin- definiteness, that the total number N of electrons in the
2
FIG. 1: Phase-dependent excitation spectrum of a Josephson junction along a quantum spin-Hall edge (left panels) and
corresponding zero-temperature supercurrent (right panels). The supercurrent I is 4π-periodic, with two branches I (blue
4π +
solid),I (redsolid)distinguishedbytheground-statefermionparityandwithaparityswitchatφ=±π. Thetoprowshows
−
the short-junction limit of Ref. 7, the bottom row the long-junction limit calculated here. (The jump in I at φ = 0 occurs
−
because of the change in slope indicated by the green arrows in the magnified central part of the spectrum.) The 2π-periodic
supercurrent I without parity constraints is also shown (green dashed). The critical current is the same for I and I in
2π 4π 2π
the short junction, but different by a factor of two in the long junction.
system is even. (A different choice amounts to a 2π supercurrent I is a 4π-periodic sawtooth with a slope
4π
phase shift, or equivalently, an interchange of I and ∆I /4π =eE /2π(cid:126).
+ − T
I .) Theground-statefermionparityσ isevenforφ=0 The corresponding parity-dependent supercurrents in
−
and switches to odd when φ crosses π. Since N is fixed, the reduced zone scheme are
thistopologicalphasetransitionmustbeaccompaniedby
eE eE
a switch between even and odd number of quasiparticle I = Tφ, I = T(φ−2πsignφ), |φ|<π. (3)
excitations. At zero temperature only the two levels ε + 2π(cid:126) − 2π(cid:126)
±
closesttotheFermilevel(ε=0)playarole,andthepar-
The 4π-periodic supercurrent I switches from I to
ity switch of σ means that a quasiparticle is transferred 4π +
I at φ = π, while I remains in the branch I by
from ε <0 to ε >0. It cannot relax back from ε to − 2π +
+ − − compensating the switch in ground-state fermion parity
ε at fixed parity of N.
+ σ by a switch in the parity of the electron number N.
The resulting current-phase relationship can be repre- These are the curves plotted in Fig. 1 (lower panels).
sented by a switch between 2π-periodic branches I (φ)
± The maximal supercurrent is reached near φ = 2π for
(reduced zone scheme), or equivalently as a 4π-periodic
I (withparityconstraint)andnearφ=πforI (with-
4π 2π
function I (φ) (extended zone scheme). Both represen-
4π outparityconstraint). Thereisafactoroftwodifference
tationsareshowninFig.1,upperpanels. Wealsoinclude
inmagnitudeofthesecriticalcurrentsinalongjunction,
the2π-periodiccurrentI thatresultsifthesystemcan
2π
relax to its lowest energy state without constraints on I =eE /(cid:126), I =eE /2(cid:126). (4)
4π,c T 2π,c T
the parity of N.
So much for the short-junction limit. An elementary Incontrast,forashortjunctionbotharethesame(equal
discussion of the long-junction regime (to be made rig- to e∆/2(cid:126)).
orous in just a moment) goes as follows. For L (cid:29) ξ we To determine the crossover from the short-junction
may assume [22–24] a local linear relation between the limit(2)tothelong-junctionlimit(3),includingthetem-
current density I and the phase gradient φ/L (cid:28) 1/ξ, of perature dependence, we adapt the scattering theory of
theformI =constant×evφ/L. ThelinearincreaseofI the Josephson effect [25] to include the fermion parity
−
is interrupted at φ=0 by a discontinuity ∆I =2ev/L. constraints. Inputisthescatteringmatrixs ofelectrons
− 0
Half of it results from the jump in the slope of the low- in the normal region and the Andreev reflection matrix
est occupied positive energy level ε = (π − |φ|)(cid:126)v/2L r at the normal-superconductor interfaces. These take
A
(green arrows in Fig. 1e). The jump in the slope of the a particularly simple 2×2 form at the quantum spin-
highest occupied negative energy level contributes the Hall edge, but our general formulas are applicable also
other half. In the extended zone scheme, the resulting to multi-channel topological superconductors.
3
The parity-dependent partition function is [12–14, 26] Eq. (12) contains a contribution from the supercon-
ducting electrodes,
(cid:32) (cid:33)(cid:32) (cid:33)
(cid:89) (cid:89) (cid:89)
Z = 1 eβε/2 (1+e−βε)± (1−e−βε)
± 2 ˆ
∞
ε>0 ε>0 ε>0
J = dερ (ε)lntanh(βε/2), (14)
(cid:18) (cid:19) S S
(cid:89)
= 1Z 1± tanh(βε/2) , (5) ∆
2 0
ε>0
with β = 1/k T and Z = (cid:81) 2cosh(βε/2) the par- which only plays a role at temperatures T (cid:38)∆/kB. The
B 0 ε>0 factoreJS canthereforebereplacedbyunityinthelong-
tition function without parity constraints. From the ex-
junction regime, when k T (cid:46)E (cid:28)∆.
pressionforZ onecanseethatthe±selectstermsthat B T
±
containaneven(+)oranodd(−)numberofquasiparti- Wenowspecifythesegeneralformulasforthequantum
cle excitation factors e−βε, as is dictated by the ground- spin-Hall edge, with Hamiltonian [30]
state fermion parity. The partition function Z gives the
free energy F and hence the supercurrent I [27],
(cid:18)vpσ +U(x) ∆∗(x)σ (cid:19)
H = z y . (15)
2edF BdG ∆(x)σ vpσ −U(x)
I = ±, F =−β−1lnZ , (6) y z
± (cid:126) dφ ± ±
2edF
I ≡I = 0, F =−β−1lnZ . (7) The edge runs along the x-axis, p = −i(cid:126)∂ is the mo-
2π 0 (cid:126) dφ 0 0 x
mentumoperator,andtheelectrostaticpotentialisU(x)
(measured relative to the Fermi level). The pair poten-
The density of states ρ(ε) contains both the discrete
tial ∆(x) vanishes in the normal region |x| < L/2. In
spectrum for |ε| < ∆ (a sum of delta functions at the
thetwosuperconductingregionsweset∆(x)=∆e±iφ/2,
Andreevlevels)andthecontinuousspectrumfor|ε|>∆,
with a step at x=±L/2. This socalled “rigid boundary
including also a contribution ρ from the superconduct-
S
condition” is justified for a single channel coupled to a
ingelectrodes. Scatteringtheorygivestheexpression[25]
bulk superconducting reservoir [10].
d
ρ(ε)=Im ν(ε+i0+)+ρ (ε), (8) A mode-matching calculation gives the scattering ma-
dε S trices
ν(ε)=−π−1lnDetX(ε), X =(1−M)M−1/2, (9)
M(ε)=rA∗(−ε)s∗0(−ε)rA(ε)s0(ε). (10) (cid:18) 0 eiχ(cid:19)
s = , χ(ε)=χ +ε/E , (16)
0 eiχ 0 0 T
The factor M−1/2 in the definition of X, as well as the
term ρ , give a φ-independent additive contribution to (cid:18)αeiφ/2 0 (cid:19) (cid:114) ε2 iε
S r = , α(ε)= 1− + ,
F0 without any effect on I0, but we need to retain these A 0 −αe−iφ/2 ∆2 ∆
terms here because they do enter into the parity con-
DetX(ε)=2cosφ+α2e2iε/ET +α−2e−2iε/ET. (17)
straint for I .
±
In the absence of parity constraints, Ref. 28 gives the
free energy
We discuss the various terms in these expressions. The
F =−β−1(cid:80)∞ lnDetX(iω ), (11) electron scattering matrix s0 is purely off-diagonal, be-
0 p=0 p cause of the absence of backscattering along the quan-
tum spin-Hall edge. The transmission phase χ depends
as a sum over fermionic Matsubara frequencies ω =
p
linearly on energy because of the linear dispersion. Elec-
(2p+1)π/β. A similar calculation [29] gives the parity-
trostatic potential fluctuations contribute only to the
dependence in the form ´
energy-independent offset χ = −((cid:126)v)−1 LUdx, which
(cid:20) 0 0
F =F −β−1ln1 1+σeJS(cid:112)DetX(0) drops out in Eq. (9). The Andreev reflection matrix rA
σ 0 2 (from electron to hole) is unitary below the gap. Above
the gap there is also propagation into the superconduc-
(cid:18) ∞ (cid:19)(cid:21)
(cid:88)
×exp (−1)plnDetX(iΩp/2) , (12) tor, so rA is sub-unitary. The same expression (16) for
r applies at all energies, evaluated at ε+i0+ to avoid
p=1 A
σ =sign(cid:2)Pf(r s −sTrT)(Detis )−1/2(cid:3) , (13) the branch cut of the square root.
A 0 0 A 0 ε=0
Putting all pieces together [29] we obtain the parity-
with bosonic Matsubara frequencies Ω = 2pπ/β. The dependent supercurrent for arbitary ratio ∆/E . In the
p T
ground-state fermion parity σ is given in terms of the short-junction limit ∆/E → 0 we recover the known
T
Pfaffian of the anti-symmetrized scattering matrix, eval- result (2), when the energy dependence of the scattering
uated at the Fermi energy. The sign ambiguity in the matrix and the phase sensitivity of the continuous spec-
square root is resolved by fixing σ =1 at φ=0. trum can both be ignored. In the opposite long-junction
4
effect (fractional Shapiro steps [9, 15–18]). A dc mea-
surement of the current-flux (I-Φ, φ = 2eΦ/(cid:126)) relation-
ship, on time scales large compared to the time τ (cid:39)µs
qp
forunpairedquasiparticlestotunnelintothesystem[35],
will measure the 2π-periodic I rather than I . Such a
2π 4π
phase-sensitive measurement (Fig. 2, upper inset) would
produce the critical current I without any signature
2π,c
of the parity anomaly. In contrast, a phase-insensitive
measurement of the critical current through the current-
voltage (I-V) characteristic (lower inset) will produce
I even on time scales (cid:29) τ , because the phase of
4π,c qp
a resistively shunted (overdamped) circuit can adjust to
a change in N on time scales much smaller than τ . A
FIG. 2: Phase dependence of the parity-constrained super- qp
change in the parity of N will be compensated by a 2π
current I (solid curves, in units of eE /(cid:126) ∝ 1/L), calcu-
4π T
phase shift, without a change in critical current [29]. In
lated by a numerical evaluation of the Matsubara sums. The
left panel shows the crossover from the short-junction to the ashortjunctionI2π,c andI4π,c arethesame,sothisdoes
long-junction regime in the zero-temperature limit (full in- not help, but in a long junction they differ by up to a
terval −2π < φ < 2π). The right panel shows the tempera- factor of two.
ture dependence in the long-junction limit (reduced interval In conclusion, we have presented a theory for the 4π-
0 < φ < 2π). The left panel also shows the supercurrent periodicJosephsoneffectonscaleslargecomparedtothe
I2π withoutparityconstraints(dashedcurves). Theinsetsin superconducting coherence length. A multitude of sub-
the right panel show current-biased superconducting circuits
gap states, as well as a continuum of states above the
that measure the I-V and I-Φ relationships of a Josephson
gap, contribute to the supercurrent for L (cid:29) ξ, but still
junction.
theparityanomalyresponsibleforthe4π-periodicityper-
sists. In fact, we have found that in a long junction the
anomaly manifests itself also in a phase-insensitive way,
limit ∆/E →∞ we find
T
throughadoublingofthecriticalcurrent. Thisopensup
I =I − 2e d ln(cid:2)1 +cos(φ/2)eS−π/2βET(cid:3), (18) new possibilities for the detection of this topological ef-
4π 0 (cid:126)βdφ 2 fect at the quantum spin-Hall edge [31–33], and possibly
∞ also in semiconductor nanowires [34, 36–38].
S =(cid:88)(−1)pln(cid:0)1+2e−Ωp/ETcosφ+e−2Ωp/ET(cid:1), (19) Discussions with A. R. Akhmerov are gratefully ac-
p=1 knowledged. This research was supported by the Dutch
∞ Science Foundation NWO/FOM and by an ERC Ad-
2e (cid:88)(cid:2) (cid:3)−1
I ≡I = sinφ cosφ+cosh(2ω /E ) . (20) vanced Investigator grant.
2π 0 (cid:126)β p T
p=0
The plot of the results in Fig. 2 shows that the crossover
fromasinetoasawtoothshapeoccursearly: alreadyfor
∆=E (soforL=ξ)themaximumofthecurrent-phase
T
relationship is close to φ = 2π. The sawtooth shape is
preserved with increasing temperature for k T (cid:46) 1E .
B 2 T
These are encouraging results for the experimental ac-
cessibility of the long-junction regime. The quantum
spin-Hall effect has been observed in HgTe/CdTe quan-
tumwells[31],andmorerecentlyinInAs/GaSbquantum
wells [32] — where also Andreev reflection from super-
conducting Nb electrodes was demonstrated [33]. For
a typical Fermi velocity of v (cid:39) 105m/s in a semicon-
ductor and superconducting gap ∆ (cid:39) 1meV in bulk
Nb, the coherence length is ξ = 70nm, so the Joseph-
son junction length L = 0.5µm from Ref. 33 is deep in
the long-junction regime. Since the long-junction regime
is already entered for L ≈ ξ, this would apply even
if the effective superconducting gap is well below the
bulk value of Nb. The corresponding Thouless energy
is E /k =1.5K, so at T =100mK one should be close
T B
to the low-temperature limit.
Intheongoingsearchforthe4π-periodicJosephsonef-
fect the first results have been reported [34] for the ac
5
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[21] It is a pervasive misunderstanding that the factor g =2
intherelation(1)betweensupercurrentandAndreevlev- Appendix A: Details of the calculation of the free
elsaccountsforthecharge2eoftheCooperpairs,rather energy
than counting the spin degeneracy. Because of this mis-
understanding, the results for the supercurrent carried
1. Transformation from the real to the imaginary
by Majorana zero-modes (which have g = 1) should be
energy axis
dividedbytwoinRef.9andmanyfollow-uppapers.The
origin of the misunderstanding is discussed in more de-
tailbyN.M.ChtchelkatchevandYu.V.Nazarov,Phys. AccordingtoEq.(5)thefreeenergyF =−β−1lnZ ,
σ σ
Rev. Lett. 90, 226806 (2003). with σ =±1the ground-statefermion parity, is givenby
[22] C. Ishii, Prog. Theor. Phys. (Kyoto) 44, 1525 (1970).
[23] J.BardeenandJ.L.Johnson,Phys.Rev.B5,72(1972). (cid:20) (cid:18) (cid:19)(cid:21)
(cid:88)
[24] A. V. Svidzinsky, T. N. Antsygina, and E. N. Bratus, J. F =F −β−1ln1 1+σexp lntanh(βε/2) ,
σ 0 2
Low Temp. Phys. 10, 131 (1973).
ε>0
[25] C. W. J. Beenakker, Phys. Rev. Lett. 67, 3836 (1991); (A1)
extended version at arXiv:cond-mat/0406127. (cid:88)
[26] M. T. Tuominen, J. M. Hergenrother, T. S. Tighe, and F0 =−β−1 ln[2cosh(βε/2)]. (A2)
M. Tinkham, Phys. Rev. Lett. 69, 1997 (1992). ε>0
[27] Thefactoroftwointherelation(6)betweensupercurrent
andfreeenergyaccountsfortheCooperpaircharge,with Here F is the free energy in the absence of parity con-
0
alldegeneraciesincludedinthepartitionfunction.Please straints. The infinite product over ε is defined in terms
6
of the density of states ρ by with ω = (2p + 1)π/β the fermionic Matsubara fre-
p
ˆ quency. For F the Matsubara sum contains bosonic
(cid:20) ∞ (cid:21) σ
(cid:89) Matsubara frequencies Ω =2pπ/β,
f(ε)=exp dερ(ε)lnf(ε) . (A3) p
ε>0 0
Integrals of the type (A3) can be done efficiently by
contour integration, made possible by the fact that the ˆ
(cid:88) 1 ∞ dν(ε+i0+)
scattering matrices are analytic in the upper half of the lntanh(βε/2)= dε
complex energy plane. This was worked out in Ref. 28 ε>0 2i −∞ dε
for F , and here we adapt that method to include the
0 ×ln|tanh(βε/2)|
parity constraint in F .
σ β ∞ 1
The function ν in the density of states (8) satisfies =− dεν(ε+i0+)
ν(ε)=ν∗(−ε),expressingtheelectron-holesymmetry. If 2i sinhβε
−∞
f(ε)isanevenfunctionofε,wemaythereforeconvertthe ∞
sum(cid:80) f(ε)overpositiveenergies(includingboththe =−π(cid:88)(−1)pν(iΩ /2)− 1πν(0). (A5)
ε>0 p 2
discrete and the continuous spectrum) into an integral p=1
along the entire real energy axis of fdν/dε, or νdf/dε
after a partial integration. Closing the contour in the
upperhalfofthecomplexenergyplanepicksupthepoles
of df/dε, which for the free energy are the Matsubara ffl
The notation indicates the Cauchy principal value of
frequencies on the imaginary axis.
the integral (which picks up half of the pole at ε = 0).
In Ref. 28 this transformation was carried out for F ,
0
We have made use of the identity
leading to
ˆ
∞
(cid:88)
ln[2cosh(βε/2)]= dερ(ε)ln[2cosh(βε/2)]
ε>0 ˆ 0 ˆ
1 ∞ dν(ε+i0+) df 1
dx ln|x|=− dxf(x) . (A6)
= dε ln[2cosh(βε/2)]
dx x
2i dε
−ˆ∞
β ∞
=− dεν(ε+i0+)tanh(βε/2)
4i
−∞
∞
(cid:88)
=−π ν(iω ), (A4)
p
p=0 For the free energy this gives
∞
(cid:88)
F =−β−1 lnDetX(iω ), (A7)
0 p
p=0
(cid:34) (cid:32) ∞ (cid:33)(cid:35)
(cid:112) (cid:88)
F =F −β−1ln1 1+σeJS DetX(0)exp (−1)plnDetX(iΩ /2) , (A8)
σ 0 2 p
p=1
(cid:112)
where we have substituted ν = −π−1lnDetX from σ DetX(0). We seek to express this quantity in terms
Eq. (9) and also included the factor eJS from the φ- of the normal-state scattering matrix s0(0). We will
independent density of states in the superconducting thenhaveobtainedascatteringmatrixexpressionforthe
electrodes. topological quantum number σ — equivalent to Kitaev’s
Hamiltonianexpression[8]butrequiringonlyFermi-level
information.
2. Scattering formula for the ground-state fermion
parity
The free energy (A8) incorporates the ground- We start from the definition of X = (1−M)M−1/2
state fermion parity dependence through the quantity and use that M(0)=U∗U with unitary U =r (0)s (0).
A 0
7
Since DetM =1, we have itconverges, andthentheremainingintegraloverδν be-
comes a converging sum over Matsubara frequencies.
DetX =Det(1−U∗U)=Det(1−U∗U)T More specifically, for the quantum spin-Hall edge we
Det(U −UT) [Pf(U −UT)]2 take
= = . (A9)
DetU DetU ν (ε)=−π−1ln(cid:2)(1−4ε2/∆2)e−2iε/ET(cid:3), (A12)
∞
The determinant of U is independent of φ,
in view of Eq. (17). The integral over ν can be evalu-
∞
DetU =Det[rA(0)s0(0)]=Det[is0(0)]. (A10) ated in closed form,

We may now identify β ∞ 1
J ≡− dεν (ε+i0+)
(cid:112) Pf(UT−U) ∞ 2i −∞ˆ ∞ sˆinhβε
σ DetX(0)= √ 1 ∞ x ∞ 1
DetU =− dx − dx
(cid:20)Pf(r s −sTrT)(cid:21) πβET −∞ sinhx β∆/2 sinhx
⇒σ =sign √A 0 0 A . (A11) π
Detis0 ε=0 =−2βE +lntanh(β∆/4). (A13)
T
Thetwobranchesofthesquarerootfunctionintroducea
sign ambiguity, which is resolved as follows. The square Theremainingintegraloverδν =ν−ν∞ thenbecomes
rootofDetX ≥0istakenontheprincipalbranch,while a convergent sum over Matsubara frequencies,
the branch of the square root of Detis is fixed by set-
0 β ∞ δν(ε+i0+)
ting σ = 1 at φ = 0. Once this sign is fixed, the phase − dε =S− 1πν(0), (A14)
dependence of the topological quantum number σ(φ) is 2i −∞ sinhβε 2
determined by Eq. (A11) entirely in terms of Fermi-level ∞
properties. S =−π(cid:88)(−1)p(cid:2)ν(iΩp/2)−ν∞(iΩp/2)(cid:3). (A15)
The scattering formula (A11) is applied in the main p=1
text to a single-channel Josephson junction, where we
This gives the regularized version of Eq. (A8),
know a priori that the topological phase transition oc-
curs at φ = π (mod 2π). More generally, in the multi-
channel case there may appear any even number of sign Fσ =F0−β−1ln21(cid:0)1+σeS+J∞+JS(cid:112)DetX(0)(cid:1). (A16)
changesofσbetweenφ=0andφ=π. Eq.(A11)isthen
particularly useful, since it permits the determination of
σ at a given φ without having to track the number of 4. Evaluation of the supercurrent along the
quantum spin-Hall edge
sign changes back to φ=0.
We conclude this subsection by placing Eq. (A11) in
the general context of topological states of matter [39]. We apply Eq. (A16) to the quantum spin-Hall edge.
The ground-state fermion parity σ is the Z topological According to Eq. (17), we have
2
quantum number of a system of dimensionality d = 0
tinionsysm) omreBtrDyIcl(awshseDn (swh=ensTs)0.hTashenodismymenmsieotnraylirteystzreirco- ν(iω)=−π1 ln(cid:2)2cosφ+ζ+(ω)+ζ−(ω)(cid:3), (A17)
refers to the fact that0this i0s a closed system. We may ζ (ω)=e±2ω/ET(cid:0)(cid:112)1+ω2/∆2±ω/∆(cid:1)2. (A18)
±
consider opening up the system, promoting it to d = 1,
by replacing one of the two superconducting contacts TheMatsubarasum(A15), withν givenbyEq.(A12),
∞
by a normal metal. The topological quantum number takes the form
Q then counts the number of Majorana zero-modes at
∞ (cid:20) (cid:21)
the normal-superconducting interface. It is given by the S =(cid:88)(−1)pln 2cosφ+ζ+(Ωp/2)+ζ−(Ωp/2)
dQet∈erZmi)na[4n0t]oofrtbhyeitrseflteractcieon(fomractlraisxs (BfoDrI,clwaistshDQ, ∈wiZth) p=1 (1+Ω2p/∆2)eΩp/ET
2 (A19)
[41].
The function J is given by Eq. (A13) and
∞
(cid:112)
σ DetX(0)=2cos(φ/2), (A20)
3. Regularization
in view of Eqs. (16) and (A11).
Thetransformationoftheintegral(A5)overrealener- Thesuperconductingelectrodescoupletothequantum
gies into a sum over imaginary frequencies requires that spin-Hall edge via a single transverse mode, over a total
ν(iω)goestozerofasterthan1/ω forω →∞. Toensure length L . The corresponding density of states is
S
this, we decompose ν = ν +δν, with ν the large-ω
∞ ∞
limitofν(iω). Itisconvenienttospecifyν (0)=0. The 2 |ε|
∞ ρ (ε)= √ , |ε|>∆. (A21)
integraloverν∞ isdonealongtherealenergyaxis,where S πES ε2−∆2
8
We have defined E = (cid:126)v /L . The superconducting In the zero-temperature limit J → 0 and we recover
S F S S
electrodes affect the parity-dependent free energy (A16) the result of Ref. 7,
through the factor eJS, with
ˆ 2e du e∆
J = ∞dερ (ε)lntanh(βε/2) Tli→m0I± =∓(cid:126)βdφ =±2(cid:126) sin(φ/2). (A30)
S S
∆ ˆ √ The parity dependence at finite temperature can be
2β ∞ ε2−∆2
=− dε . (A22) quantified by the difference δI = 12(I+ −I−), for which
πES ∆ sinhβε we find the compact expression
Collecting results, we arrive at the parity-dependent 4e τ 1 du
supercurrent δI =−(cid:126)β1−τ2sinh2udφ, τ =eJStanhu, (A31)
I =I − 2e d ln1(cid:2)1+σ|cos(φ/2)|eS+J∞+JS(cid:3), in agreement with Ref. 13.
σ 0 (cid:126)βdφ 2
(A23)
σ =sign[cos(φ/2)], (A24) 6. Zero-temperature limit in the long-junction
regime
with I the 2π-periodic supercurrent in the absence of
0
parity constraints, Another check on the formalism is provided by the
combinedzero-temperatureandlong-junctionlimits. We
∞
I = 4e sinφ(cid:88)[2cosφ+ζ (ω )+ζ (ω )]−1. (A25) again choose the interval |φ| < π. The Matsubara sums
0 (cid:126)β + p − p (A19) and (A25) are given in the long-junction limit by
p=0
∞
In the long-junction regime ∆ (cid:29) ET, kBT these results lim S =(cid:88)(−1)pln(cid:0)1+2e−Ωp/ETcosφ+e−2Ωp/ET(cid:1),
reducetotheequations(18)–(20)giveninthemaintext.
∆→∞
p=1
(A32)
5. Short-junction limit 2e (cid:88)∞ (cid:2) (cid:3)−1
lim I = sinφ cosφ+cosh(2ω /E ) .
∆→∞ 0 (cid:126)β p T
As a check on the consistency of the whole formalism, p=0
(A33)
we take the short-junction limit E → ∞ of the parity-
T
dependent supercurrent (A23) and see if we recover the
In the zero-temperature limit the sums may be con-
results of Fu and Kane [7]. We choose the phase interval
verted into integrals, with the results
|φ|<π and abbreviate
eE
u≡ 12cos(φ/2)β∆. (A26) S →−ln|2cos(φ/2)|, I0 → 2π(cid:126)T φ. (A34)
The Matsubara sums (A19) and (A25) can be evalu- The two terms J and J both vanish at T = 0. Sub-
∞ S
ated in closed form in the short-junction limit, stitution into Eq. (A23) gives I = I , in agreement
+ 0
with Eq. (3), while I remains undetermined. The zero-
lim S =(cid:88)∞ (−1)pln(cid:18)2cosφ+2+Ω2p/∆2(cid:19) temperature limit o−f I− depends on on higher order
ET→∞ p=1 1+Ω2p/∆2 terms in the low-temperature expansion of S that we
(cid:18) (cid:19) have not been able to calculate analytically. A numer-
tanhu
=ln , (A27) ical calculation (using the Pad´e approximant built into
2cos(φ/2)tanh(β∆/4) theNSum|AlternatingSignsroutineofMathematica)
4e (cid:88)∞ gives
lim I = sinφ [2cosφ+2+4ω2/∆2]−1
ET→∞ 0 (cid:126)β p=0 p 2 (cid:18) (cid:19)
lim lim ln 1 −|cos(φ/2)|eS+J∞ =|φ|−π,
= e∆sin(φ/2)tanhu. (A28) T→0∆→∞βET 2
2(cid:126) (A35)
resulting in a current I in agreement with Eq. (3).
−
InthesamelimitJ =lntanh(β∆/4);uponsubstitution
∞
into Eq. (A23) we arrive at
Appendix B: Circuits to measure the critical current
I =I − 2e d ln(cid:0)1 ± 1eJStanhu(cid:1) with and without parity constraints
± 0 (cid:126)βdφ 2 2
=−2e d ln(cid:0)coshu±eJSsinhu(cid:1). (A29) As explained in the main text, the two circuits shown
(cid:126)βdφ in Fig. 2 (inset) both measure the critical current of
9
E . We also wish to ensure that the Josephson junction
T
remains in its ground state during the phase relaxation,
which requires E τ /(cid:126)(cid:29)1. Together these three condi-
T J
tions are met if
(R/R )2τ (cid:28)RC (cid:28)τ , (B3)
Q J J
FIG. 3: Left panel a: Circuit of a current-biased, resistively
shunted Josephson junction, to measure the current-voltage
with R(cid:28)R .
characteristic. Right panel b: Circuit of an rf squid to mea- Q
At a fixed parity P, the bias current I = I +V/R
sure the current-phase relationship. P
drives the phase φ(t) according to
the Josephson junction, but their sensitivity to fermion- I τ dφ 1
= J + mod (φ+Pπ+π)−1. (B4)
parityconstraintsisfundamentallydifferent. Ameasure- I 2π dt 2π 4π
4π,c
ment of the current-phase relationship (upper circuit) is
insensitive to parity constraints when quasiparticles can
enter or leave the system on the time scale of the mea- A typical value ET/kB (cid:39)1K gives (cid:126)/ET (cid:39)10−11s. The
surement. This gives the critical current I . A mea- typical time scales for quasiparticle poisoning are in the
2π,c
surement of the current-voltage characteristic (lower cir- µs to ms range [35], so even if R (cid:28) RQ we can safely
cuit) remains governed by fermion-parity constraints as assume that τJ (cid:28) τqp and use Eq. (B4) to calculate the
longasthequasiparticletunnelingtimeτ islargecom- relaxationofthephaseinbetweentwoquasiparticletun-
qp
pared to the phase relaxation time τ of the resistively neling events.
J
shunted Josephson junction. This then gives I . Here The phase relaxation due to a quasiparticle tunneling
4π,c
we analyze these two circuits in some more detail. event at t = 0 (by which P (cid:55)→ −P) amounts to a 2π
In the zero-temperature, long-junction limit, we phase slip on a time scale τ ,
J
havethe4π-periodicsawtoothcurrent-phaserelationship
shown in Fig. 1 (lower panel). This plot is for an even φ(t)=φ(0)e−t/τJ +[φ(0)+2π](1−e−t/τJ). (B5)
numberofelectronsinthesystem,P ≡(−1)N =1,while
foroddparityP =−1thesawtoothisdisplacedhorizon-
tally by 2π. Both cases are contained in the formula Before and after the phase slip the junction is in a zero-
voltage state, for bias currents I (cid:46) I . During the
4π,c
I (φ)= I4π,cmod (φ+Pπ+π)−I . (B1) p´hase slip there is a voltage pulse of integrated area
P 2π 4π 4π,c V(t)dt=h/2e. The corresponding time-averaged volt-
age V¯ = h/2eτ is smaller by a factor τ /τ (cid:28) 1 than
qp J qp
The modulo function is defined by mod4π(φ)=φ−4πn, the voltage that develops for I (cid:38)I .
with n∈Z such that mod (φ)∈[0,4π). 4π,c
4π This shows that the current-biased circuit of Fig. 3a
We start by considering the current-biased circuit of
provides a dc measurement of the parity-constrained
Fig.3a. AvoltageV =((cid:126)/2e)dφ/dtdropsoveraresistor
criticalcurrentI . Incontrast,thecircuitofFig.3bis
RinparallelwiththeJosephsonjunction,ofcapacitance 4π,c
notsensitivetoparityconstraints. Thisrfsquidisphase-
C. The two characteristic time scales of the circuit are
biased for sufficiently small inductance L (cid:28) h/eI .
the RC time and the phase relaxation time 4π,c
Quasiparticle tunneling events have only a small effect
h 1 R (cid:126) on the phase, which remains fixed by the enclosed flux
τJ = 2eRI = 2RQE , (B2) Φ=((cid:126)/2e)φ≈LI.
4π,c T
At low temperatures the parity of the number of elec-
where R = h/e2 is the resistance quantum. The char- trons N in the system will equilibrate at the ground-
Q
acteristic energy scales of the Josephson junction are state fermion parity σ, which implies that N is even
the charging energy e2/C and the Josephson energy (P = 1) for mod (φ + π) < 2π and odd (P = −1)
4π
(cid:126)I /e=E . for mod (φ + π) > 2π. In either case the supercur-
4π,c T 4π
The capacitance should be suffiently small that the rent I given by Eq. (B1) cannot become larger than
P
phase dynamics is overdamped, RC (cid:28) τ , and suffi- I /2 = I — which is the critical current without
J 4π,c 2π,c
cientlylargethatthephasedynamicsisclassical,e2/C (cid:28) parity constraints.