Gapless vortex bound states in superconducting topological semimetals
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Gapless vortex bound states in superconducting topological semimetals
Yi Zhang, ∗ Shengshan Qin, ∗ Kun Jiang, † and Jiangping Hu
2, 3, 1, ‡ Kavli Institute of Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Quantum Matter, Beijing, China (Dated: February 15, 2021)We find that the vortex bound states in superconducting topological semimetals are gapless owing to topolog-ical massless excitations in their normal states. We demonstrate this universal result in a variety of semimetalsincluding Dirac and Weyl semimetals, threefold degenerate spin-1 fermions, spin-3 / π phase superconductor-normal metal-superconductor (SNS) junctions. We further demonstrate that these gapless states are topologi-cally protected and can be derived from a topological pumping process. The topological semimetals (TSMs), such as Weyl andDirac semimetals where their low energy physics is gov-erned by topological gapless excitations protected by topol-ogy and symmetry [1, 2], have attracted much interest. Re-cently, by studying all space-group symmetries, new addi-tional types of TSMs have also been classified [2–7], in-cluding spin-1 excitation[8], spin-3 / ff erent indi ff erent types of TSMs. Until now, there is no single univer-sal property associated with all TSMs.Vortex is a topological object in real space and is a centralingredient for a type-II superconductor (SC) under an exter-nal magnetic field [29, 30]. For a conventional Fermi liq-uid with full gapped pairing, the SC pairing function formsa quantum well inside a vortex core to generate a bunchof gapped bound states, named as the Caroli–de Gennes-Matricon (CdGM) states [31, 32]. In this paper, we showthat the vortex bound states in superconducting topologicalsemimetals di ff er fundamentally from conventional Fermi liq-uids. There are gapless bound states in the superconductingTSM vortices, which is a universal property for TSMs be-yond Dirac and Weyl semimetals[33–36]. These states aretopologically protected due to the topological gapless exci-tations in their normal states. The formation of these statesis also closely related to the Andreev specular reflection andthe propagating Andreev modes in π phase superconductor-normal metal-superconductor (SNS) junctions.To illustrate the main idea, we consider superconductinggraphene as an example[37, 38]. In a conventional Fermi liq-uid, if we inject one electron into the superconductor (SC),one hole comes out, named the Andreev retro-reflection[39],as shown in Fig.1a. Since the reflected hole is from theconduction hole band, its velocity is opposite to the elec-tron velocity from the conduction electron band. However,in graphene, due to the topological gapless nature, the re- ∗ These two authors contributed equally † Electronic address: [email protected] ‡ Electronic address: [email protected] flected hole can sit in the valence hole band when Fermi en-ergy E F is around the Dirac point. Since the valence bandhas a di ff erent Fermi velocity compared with the conduc-tion band, the hole can retain its velocity along the reflectionplane, named the specular Andreev reflection[40], as shownin Fig.1b. The specular reflection leads to an unconventionalbehavior in the π phase SNS junction in graphene [41]. Con-ventionally, the localized Andreev bound states or Andreevlevel can be found in the normal metal π junctions, as illus-trated in Fig.1c [42]. On the contrary, owing to specular An-dreev reflection, the graphene π junctions show propagatingAndreev modes, as shown in Fig.1d. The physics has beengeneralized to the superconducting topological insulator sur-face states in the seminal Fu-Kane proposal [43], in which atri-junction is constructed for Majorana zero mode manipula-tion. This tri-junction can be viewed as a discrete analog ofthe superconducting topological insulator surface state vortex.The above physics can be extended to a superconductingvortex. In a continuum model for graphene, a supercon-ducting vortex has been shown to host zero-energy boundstates[44, 45]. For a vortex, the order parameter can be writ-ten as ∆ ( r ) e i θ , where θ is the phase angle centered at the vortexcore. The gap value changes its sign by 180 degree rotation.Therefore, we can slice the vortex into infinite numbers of π junctions, as illustrated in Fig.1e. The vortex is just the su-perposition of all the π junctions. Hence, we can conjecturethat if the propagating Andreev modes exist in the π junction,there should be exotic bound states in the graphene vortex.Furthermore, it is also reasonable to argue that the specularreflection should be a common feature for all other topologi-cal semimetals.To confirm this conjecture, we consider various new typesof fermionic semimetals in three dimensional (3D) solids. Be-sides the well-established spin-1 / / H = δ k · S (1)where δ k = k − k is the momentum deviation from the cross-ing point k and S stands for the matrices for the pseudospin e hSC yx a retro-reflection E F - ΔΔ c Andreev level e hSC specular reflectionE F - ΔΔ b Andreev mode . . .... .... .... . . .... Δ e i θ i Δ -i Δ Δ - Δ Δ e i π /4 −Δ e i π /4 Δ e i3 π /4 −Δ e i3 π /4 Δ e i δ −Δ e i δ −Δ e i( π−δ) Δ e i( π−δ) ed Normal Metal Graphene
FIG. 1: Schematic illustration of the Andreev reflection and its relation to vortex. a , retro-reflection in the conventional metal to SC interface,where one electron is injected with one hole with opposite velocity reflected. b , specular reflection in the graphene to SC interface, where thereflected hole retains its velocity along the reflection plane. c , Andreev level or Andreev bound state in SNS junction owing to the multipleretro-reflection at each interface. d , propagating Andreev mode in graphene SNS junction owing to specular reflections. e , by slicing thepairing function ∆ e i θ , the vortex can be mapped to infinite slices of π junctions by taking δ →
0. The π junctions along the x axis and y axisare highlighted. a Spin-1/2 Weyl Fermion Spin-1excitationC=1 C=2C=0Spin-3/2 RSW Fermion Double Weyl Fermion bc C=3C=1 C=1x2 -1 -0.5 0 0.5 1 -0.4-0.200.20.4 E /t k z / π -0.4-0.200.20.4 E /t -1 -0.5 0 0.5 1 k z / π - Δ Δ
FIG. 2: a Energy dispersions for multiple types of fermionic exci-tations including spin-1 / / b The π SNS junction spec-trum along the k z direction for Dirac semimetal at k y =
0. There aregapless dispersions around the Dirac points. c Vortex bound statespectrum for the Dirac semimetal. A similar gapless energy disper-sion is obtained. operators in each spin representation. From Fig.2a, we cansee that if the E F is close to the degenerate points, the spec-ular reflection always exists. To show this, we first take theDirac semimetal as an example.The lattice Hamiltonian for a Dirac semimetal can be writ-ten as H D ( k ) = ( m − t cos k x − t cos k y − t cos k z ) σ z + t sin k x σ x s z + t sin k z (cos k x − cos k y ) σ x s x − t sin k y σ y + t sin k z sin k x sin k y σ x s y , (2)where the Pauli matrices σ i and s i with i = x , y , z act in theorbital and spin spaces, respectively [33]. We set the hoppingparameters as { t , t , m } = { , . , . } . H D hosts two Dirac points at (0 , , ± k cz ) with k cz = arccos 0 .
4. With an s -wave su-perconductivity, the Hamiltonian can be written as H SC = H D ( k ) − µ ˆ ∆ ( r ) ˆ ∆ † ( r ) − H TD ( − k ) + µ ! , (3)in the basis Ψ k = (cid:16) c k ↑ , c k ↓ , c †− k ↑ , c †− k ↓ (cid:17) T , where µ is the chemicalpotential and ˆ ∆ ( r ) is the s -wave pairing function with ˆ ∆ ( r ) = ∆ ( r ) i σ s y .We consider a π junction structure with length L for theDirac semimetal along x-direction. The pairing function isdefined as ∆ ( r ) = ∆ , x ≤ L / − ∆ , x > L / . (4)For the Dirac semimetal, there are propagating modes alongthe y and z directions. As shown in Fig.2b, a 2D gapless phasefor Dirac semimetal π junction with gapless dispersion alongthe k z direction is obtained numerically [46].For the vortex configuration, we consider a flux along the(0,0,1) direction ( z -direction). The pairing function can bewritten as ∆ ( r ) = ∆ ( r ) e i θ , (5)where r = p x + y is the distance to the vortex line and θ isthe polar angle and ∆ ( r ) takes the form ∆ ( r ) = ∆ Θ ( r − R ) (6)with Θ ( r ) the step function and R is the size of the vortex.The qualitative results do not depend on the pairing function.If we pinch all the π junction slices into a vortex, the vortexbound states are dispersive along k z dispersion. As plotted inFig.2c, there is a gapless vortex bound state dispersion. Thisgapless feature has been obtained in Dirac semimetals[33–35]and Weyl semimetals[35, 36].There is also another large class of topological semimetalswith higher order massless fermions, named the chiral crys-tals like RhSi, CoSi, RhGe, CoGe etc[2, 23–25, 47]. For thechiral crystal, there is one long Fermi arc connecting C = = -2 degeneracy points, which has been experimentally con-firmed in CoSi [26, 27]. In order to study the physics of chiralcrystals, we apply a minimal eight-band tight-binding model,which is the prototype model for RhSi (details can be found inthe supplemental materials and Ref. [24]). The band structurefor this model is plotted in Fig.3a. From the band structure,we can see there are one Chern number C = Γ point and one C = -2 double WeylFermion around the R point (green dot) in the absence of SOC,which is the general feature for the chiral crystal. To study thevortex property, a magnetic flux along the (0,0,1) direction ofthe crystal is inserted. The Hamiltonian for this case can beobtained by changing the H D ( k ) to H W ( k ), as defined in thesupplementary material [48].As discussed above, it is interesting to see how the vor-tex bound states behave around these special gapless points.We first put the chemical potential exactly at the red spin-1 excitation point. In this case, the above condition for thespecular reflection is satisfied. Hence, there should be gaplessbound states inside the vortex, which is numerically calculatedand plotted in Fig.3b. There are two chiral vortex bands con-necting the upper and lower bounds of the SC gap around the k z =
0, as predicted. There are also another branch of chiralgapless vortex bound bands around the k z = π , which is relatedto the R point degeneracy as discussed later. If the chemicalpotential is moved to the green double Weyl points, similargapless bound states are also obtained, as shown in Fig.3c. In-terestingly, when the chemical potential sits between the spin-1 and double Weyl degeneracy point, we find the gapless vor-tex bound states always exist. And if chemical potential is inthe range of − . ≤ µ ≤ .
59, we can always find the vortexbound states have gapless chiral modes that disperse along the z -direction. Moving away from this region, a gapped vortex isobtained, similar to the normal SC cases.We can extend the above calculation to the spin-3 / / Γ point is split into afourfold degenerate point that describes a spin-3 / z -direction, we again obtain the gapless chiral vortexbound states dispersing along z -direction when the chemicalpotential is at the RSW degenerate point as shown in Fig.3d.Owing to SOC, the two spin degenerated vortex bound bandssplit into four bound bands around the Γ point. The gaplesschiral vortex bound states also exist as long as the chemicalpotential µ lies between µ c = -0.87 and µ c = Γ point and the Weyl fermions around R point result in along Fermi arc extending from the center to the corner in thesurface Brillouin Zone. The long Fermi arc also reflects thefact that, within each k z plane (0 < k z < π ) the band struc- -1012 -1 -0.5 0 0.5 1-0.100.1 μ=0 -1 -0.5 0 0.5 1-0.100.1 μ=−0.41 EE k z / π k z / πΓ X Γ M R X a bc E k z / π d -1 -0.5 0 0.5 1-0.100.1 μ=0 FIG. 3: a Band structure for the chiral crystal eight-band model with-out SOC. There are one spin-1 excitation (red dot) around the Γ pointand double Weyl fermions (green dot) around the R point. b-c Thevortex energy spectrum for a bands with µ = Γ point. Two double degeneratedchiral gapless vortex bound bands locate around the k z = k z = π . d The vortexspectrum for the chiral crystal in the presence of SOC with µ at theRSW degenerate point. The gapless chiral bands are robust while thedouble degenerated chiral bands in b are split due to SOC. tures of chiral crystal describe a 2D quantum Hall insulatorwith Chern number C =
2, and for each − k z plane the bandscarry Chern number C = −
2. The two Hamiltonians at k z and − k z are connected by the time reversal symmetry. Accord-ingly, there exist two chiral edge modes on the edge of each k z plane, and the chiral edge modes carry opposite chirality forthe − k z planes, as indicated in Fig.4a and calculated in Fig.4b.We can use topological flux pumping to show that there is aone-to-one correspondence between these chiral edge modesand the chiral vortex bound states in the weak pairing limit.For simplicity, we focus on one single chiral edge mode firstand generalize to multiple chiral modes in the end. We con-sider the vortex bound states problem in the cylinder geom-etry and the vortex line is parallel to the cylinder along the z -direction. Under this condition, in the absence of the vor-tex line, the chiral edge modes can be described by a seriesof states H chiral ( k z ) = v ( k z ) l z − µ , with l z = ± , ± , . . . the an-gular momentum owing to rotation symmetry and the sign of v ( k z ) characterizing the chirality. Then, we can switch on a π flux going through the vortex line. The chiral edge modesbecome H vortexchiral ( k z ) = v ( k z ) l ′ z − µ , with angular momentum in-creasing by one-half as l ′ z = l z + . Namely, each chiral edgemode is shifted by half of the minimal gap and the directionof the shift is determined by the chirality. And the chiral edgemodes on the edge of the k z plane and − k z plane carry oppositechirality due to time reversal symmetry. Therefore, H chiral ( k z )and H chiral ( − k z ) are both shifted half of the minimal gap but inopposite directions after the π flux is inserted.For a chiral crystal, we can always find one insulating k z a b k z k z =0 c d k z k z =0 k z0 -k z0 k z0 -k z0 -1 -0.5 0 0.5 1-0.500.5 E k y / π E l E l FIG. 4: a Chiral edge modes above the k z = k z = b Chiraledge modes for the chiral crystal eight-band model in the presenceof SOC with open boundary in the x -direction for a constant k z planewith k z = . π , where the mode is propagating along y with thechemical potential set at the 4-fold degenerate RSW point at Γ . c The chiral state (blue circles) energy at ± k z and their shift (greendashed circles) under topological pumping after the flux insertion.Notice that the chiral states at k z cross 0 under pumping. d Spectralflow for the vortex states. The spectrum for ± k z is related to eachother due to particle-hole symmetry. The bands connecting the ± k z with the k z = plane. And each chiral edge mode in the k z plane combinedwith its time reversal partner in the − k z plane contribute toa single state which goes across the Fermi energy after the π flux is inserted, as illustrated in Fig.4c. This pumping processcan be described by the following index N ( k z ) − N ( − k z ) = − sgn ( v ( k z )), where N ( k z ) is the number of states below theFermi energy at k z . For the superconducting state, an infinites-imal pairing can not change the above process, and the π fluxnot only pumps an electron but also pump a hole [48]. There-fore, in the superconducting state the above index is doubled,namely N chiral = N sc ( k z ) − N sc ( − k z ) = − sgn ( v ( k z )), where N sc ( k z ) is the number of the vortex bound states with negativeenergy at k z . Now, we can consider the spectral flow of thevortex bound states. Owing to the particle-hole symmetry, theenergy spectrum of bound states for k z is opposite to the − k z .Then, the energy spectrum for k z = k z =
0, as shown in Fig.4d. For k z , there are onepositive states and three negative energy states owing to fi-nite N chiral . Hence, there are two chiral vortex modes connect-ing the 0 to ± k z plane, as illustrated in Fig.4d. Obviously, N chiral is just the number of chiral vortex modes between k z and − k z . Considering the vorticity of the vortex and the num-ber of the chiral edge modes, the above topological index canbe generalized to N chiral = − η C ( k z ), with η the vorticity of the vortex and C ( k z ) the Chern number relating to the numberof chiral modes in the k z plane .Notice that in the above analysis, we assume an insulatinggap in the k z plane in the normal state. For the chiral crystal,for any chemical potential µ in between E Γ and E R , with E Γ and E R the energy of the degenerate point at Γ and R respec-tively, we can always find an insulating gap in some k z plane.Therefore, for any chemical potential satisfying E R < µ < E Γ ,there are always four chiral vortex modes near k z = k z = π . The topological pumping process can bedirectly generalized to the Weyl semimetals. As illustrated inFig.2a, the most important property for the Weyl semimetalsare the Weyl points containing the spin-1 / =
1. Similar to the chiral crystals,we can always find one C , k plane with chiraledge modes, where this plane sits between two Weyl pointswith opposite chirality. Based on the above analysis, the π fluxpumping gives rise to the chiral gapless vortex bound states inthe superconducting Weyl semimetal [35, 36].In summary, we study the gapless vortex bound states forthe superconducting topological semimetals with nontrivialfermionic excitations. Owing to their fermionic excitations,the topological semimetals always host the specular Andreevreflection, which gives rise to the propagating Andreev modesinside the TSM π phase SNS junctions. And the vortex can bemapped to the superposition of π SNS junctions. Therefore,a gapless vortex bound state can be obtained when the chem-ical potential sits at the degenerate points. Beyond the previ-ously studied Dirac and Weyl semimetals gapless vortex solu-tions, we generalize the above discussion to the chiral crystalsthat hold exotic gapless fermions such as quasi spin-1 Weylfermion and the spin-3 / [1] N. P. Armitage, E. J. Mele, and Ashvin Vishwanath, Rev. Mod.Phys. , 015001 (2018).[2] Barry Bradlyn, Jennifer Cano, Zhijun Wang, M.G. Vergniory,C. Felser, R. J. Cava, B. Andrei Bernevig, Science , 5037(2016).[3] B. J. Wieder, Y. Kim, A. M. Rappe, and C. L. Kane, Phys. Rev.Lett. , 186402 (2016).[4] H. Weng, C. Fang, Z. Fang, and X. Dai, Phys. Rev. B ,165201 (2016).[5] H. Weng, C. Fang, Z. Fang, and X. Dai, Phys. Rev. B ,241202 (2016).[6] Z. Zhu, G. W. Winkler, Q. S. Wu, J. Li, and A. A. Soluyanov,Phys. Rev. X , 031003 (2016).[7] B. Lv et al., Nature , 627 (2017).[8] J. L. Manes, Phys. Rev. B , 155118 (2012).[9] W. Rarita and J. Schwinger, Phys. Rev. , 61 (1941).[10] L. Liang and Y. Yu, Phys. Rev. B , 045113 (2016).[11] M. Ezawa, Phys. Rev. B , 195205 (2016).[12] Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H.Weng, X.Dai, and Z. Fang, Phys. Rev. B , 195320 (2012).[13] Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Phys. Rev. B , 125427 (2013).[14] P. Tang, Q. Zhou, G. Xu, and S.-C. Zhang, Nat. Phys. , 1100(2016).[15] Z. Liu et al., Nat. Mater. , 677 (2014).[16] Z. Liu et al., Science , 864 (2014).[17] S.-Y. Xu et al., Science , 294 (2015).[18] X. Wan, A. M. Turner, A. Vishwanath, S. Y. Savrasov, Phys.Rev. B , 205101 (2011).[19] H. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai, Phys.Rev. X , 011029 (2015).[20] B. Q. Lv et al., Phys. Rev. X , 031013 (2015).[21] S.-Y. Xu et al., Science , 613 (2015).[22] S.-M. Huang et al., Nat. Commun. , 7373 (2015).[23] Peizhe Tang, Quan Zhou, and Shou-Cheng Zhang, Phys. Rev.Lett. , 206402 (2017).[24] G. Chang et al., Phys. Rev. Lett. , 206401 (2017).[25] G. Chang et al, Nat. Mat. , 978 (2018).[26] Zhicheng Rao et al., Nature , 496 (2019). [27] Daniel S. Sanchez et al., Nature , 500(2019).[28] Hang Li et al., Nat. Commun. , 5505 (2019).[29] V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. , 1064(1950)[30] A. A. Abrikosov, Zh. Fksp. Teor. Fiz. , 1442 (1957) [Sov.Phys. JETP , 1174 (1957)][31] C. Caroli, P.G. de Gennes, and J. Matricon, Phys. Lett., , 556 (1969).[33] Shengshan Qin, Lunhui Hu, Congcong Le, Jinfeng Zeng, Fu-chun Zhang, Chen Fang, Jiangping Hu, Phys. Rev. Lett. ,027003 (2019).[34] Elio J. Konig and Piers Coleman, Phys. Rev. Lett. , 207001(2019).[35] Zhongbo Yan, Zhigang Wu, and Wen Huang, Phys. Rev. Lett. , 257001 (2020).[36] R. Giwa, P. Hosur, arXiv:2006.03613.[37] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,A. K. Geim, Rev. Mod. Phys. , 109 (2009).[38] K. S. Novoselov et al., Science , 666 (2004).[39] A. F. Andreev, Sov. Phys. JETP , 1228 (1964).[40] C. W. J. Beenakker, Phys. Rev. Lett. , 067007 (2006).[41] M. Titov and C. W. J. Beenakker, Phys. Rev. B , 041401(R)(2006); M. Titov, A. Ossipov, and C. W. J. Beenakker, ,045417 (2007).[42] I. O. Kulik, Sov. Phys. JETP , 841 (1966).[43] Liang Fu and C. L. Kane, Phys. Rev. Lett. , 096407 (2008).[44] I. M. Khaymovich, N. B. Kopnin, A. S. Melnikov, and I. A.Shereshevskii, Phys. Rev. B , 224506 (2009).[45] Doron L. Bergman and Karyn Le Hur, Phys. Rev. B , 184520(2009).[46] Actually there is a tiny gap at the junction spectrum in the nu-merical calculation owing to the breaking of rotation symmetry.[47] A. V. Tsvyashchenko, V. A. Sidorov, A. E. Petrova, L. N.Fomicheva, I. P. Zibrov, V. E. Dmitrienko, J. Alloys Comp.,686