Science - USA (2020-05-01)

the minimum ofDfn(Fig. 3C, magenta curves).
Accordingly, the ground state hasvs= 0 with
nfluxoids. The intersection of the branchDfn– 1
with the dashed line defines an excited state
withn–1 fluxoids and a large superfluid
velocityv′s. Expressed in terms ofY^2 e(equiv-
alentlyIc),thefreeenergyminimabecomethe
scalloped boundary (Fig. 3D, bold curves). AsI
is increased (along the Fig. 3D dashed arrow),
we encounter the excited state at a value ofI
[<Ic(B)] that varies withf,asinEq.1.
Last, we discuss an asymmetry exhibited by
these branches. In Fig. 3A, branches that flow
outward (toward increasingjBj) are observed,
whereas branches flowing inward are conspic-
uously absent. As shown in Fig. 4, the flow
direction is sensitive to the signs ofIandB.
Figure 4A, with outflowing branches, depicts
the situation already discussed (I> 0,B< 0).
When we reverse the sign ofB(keepingI> 0)

(Fig. 4B), the branches flow toward decreasing

jBjbut are less clearly resolved. Likewise, in

Fig. 4C (I< 0,B< 0), the flow is toward de-

creasingjBj. Last, withI<0andB>0(Fig.4D),

we recover the pattern in Fig. 4A. Empirically,

we conclude that the branches flow outward if

the productI·B< 0 (Fig. 4, A and D), whereas

they flow inward ifI·B>0(Fig.4,BandC).

The pattern favors one circulation ofvsover the

other (but respects time-reversal invariance).

These symmetry patterns, lying beyond the

scenario discussed above, require the role of

spin-orbit coupling and other topological prop-

erties of the edge modes to be better understood.

Aside from the symmetry breaking, the na-

ture of pairing of edge excitations—what pro-

tects the edge condensate against hybridization

with the bulk—and the role played by hinge

states ( 19 ) are issues that invite further inves-

tigation. More broadly, this method may be

extended to explore other topological super-

conductors ( 1 – 3 ) and chiral superconductors

( 20 , 21 ).

REFERENCES AND NOTES

1. L. Fu, C. L. Kane,Phys. Rev. Lett. 100 , 096407 (2008).


2. X.-L. Qi, T. L. Hughes, S. Raghu, S.-C. Zhang,Phys. Rev. Lett.
   102 , 187001 (2009).


3. L. Fu, E. Berg,Phys. Rev. Lett. 105 , 097001 (2010).


4. S. Uchida, T. Habe, Y. Asano,J. Phys. Soc. Jpn. 83 , 064711
   (2014).


5. H. Wei, S.-P. Chao, V. Aji,Phys. Rev. B Condens. Matter Mater. Phys.
   89 , 014506 (2014).


6. G. Bednik, A. A. Zyuzin, A. A. Burkov,Phys. Rev. B Condens.
   Matter Mater. Phys. 92 , 035153 (2015).


7. T.Zhou,Y.Gao,Z.D.Wang,Phys. Rev. B 93 , 094517
   (2016).


8. K. A. Madsen, E. J. Bergholtz, P. W. Brouwer,Phys. Rev. B 95 ,
   064511 (2017).


9. Y.Li,F.D.M.Haldane,Phys. Rev. Lett. 120 , 067003
   (2018).


10. A. A. Soluyanovet al.,Nature 527 , 495–498 (2015).


11. Z. Wanget al.,Phys. Rev. Lett. 117 , 056805 (2016).


12. Y. Qiet al.,Nat. Commun. 7 , 11038 (2016).


13. Materials and methods are available as supplementary
    materials.


14. H. B. Heersche, P. Jarillo-Herrero, J. B. Oostinga,
    L. M. K. Vandersypen, A. F. Morpurgo,Nature 446 , 56– 59
    (2007).


15. S. Hartet al.,Nat. Phys. 10 , 638–643 (2015).


16. V. S. Pribiaget al.,Nat. Nanotechnol. 10 , 593– 597
    (2015).


17. W. A. Little, R. D. Parks,Phys. Rev. Lett. 9 ,9–12 (1962).


18. M. Tinkham,Introduction to Superconductivity(Dover, 2004),
    chap. 4.


19. Z. Wang, B. J. Wieder, J. Li, B. Yan, B. A. Bernevig,Phys. Rev.
    Lett. 123 , 186401 (2019).


20. D. A. Ivanov,Phys. Rev. Lett. 86 , 268–271 (2001).


21. J. R. Kirtleyet al.,Phys. Rev. B Condens. Matter Mater. Phys.
    76 , 014526 (2007).


22. P. Ong, Data for observation of an edge supercurrent in the
    Weyl superconductor MoTe 2. Harvard Dataverse (2020);

ACKNOWLEDGMENTS

We thank B. A. Bernevig, B. I. Halperin, A. Yacoby, and A. Yazdani

for valuable discussions.Funding:The research was supported

by the U.S. Army Research Office (W911NF-16-1-0116). The dilution

refrigerator experiments were supported by the U.S. Department

of Energy (DE- SC0017863). N.P.O. and R.J.C. acknowledge

support from the Gordon and Betty Moore Foundation’s Emergent

Phenomena in Quantum Systems Initiative through Grants

GBMF4539 (N.P.O.) and GBMF-4412 (R.J.C.). The growth and

characterization of crystals were performed by F.A.C. and R.J.C.,

with support from the National Science Foundation (NSF MRSEC

grant DMR 1420541).Author contributions:W.W., S.K., and

M.L. jointly prepared the devices and performed all measurements.

The detailed analyses were carried out by W.W., S.K., and N.P.O.

The crystals were provided by F.A.C. and R.J.C. The manuscript

was written by N.P.O., W.W., and S.K.Competing interests:The

authors declare no competing interests.Data and materials

availability:All data are archived in Dataverse ( 22 ).

SUPPLEMENTARY MATERIALS

science.sciencemag.org/content/368/6490/534/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S10

Table S1

References ( 23 , 24 )

6 February 2019; resubmitted 22 August 2019

Accepted 27 March 2020

10.1126/science.aaw9270

SCIENCEsciencemag.org 1 MAY 2020•VOL 368 ISSUE 6490 537

Fig. 4. Symmetry breaking in the excitation branches.Shown are the data for sample S6. (A) The color
map is measured at 20 mK withB< 0 (inset) andI> 0 (flowing right to left). The observed excitation
branches flow to the left (increasing |B|). (B) IfBis reversed keepingI> 0 (inset), the excitation branches
flow left, toward decreasing |B|, although the pattern is less sharply resolved than in (A). (C) IfB< 0 and
I< 0, the color map is similar to that in (B) except for the reversal inI.(D) ForB> 0 andI< 0, we recover
the color map from (A). The symmetry breaking follows the sign of the productI·B. ForI·B> 0 [(B)
and (C)], the branches flow toward decreasingjBj, whereas forI·B< 0, the flow is toward increasingjBj
[(A) and (D)]. In (A) to (D), the patterns are nonhysteretic and independent of field-sweep direction. Sample
S2 shows a similar symmetry breaking ( 17 ).

RESEARCH | REPORTS