Möbius topological superconductivity in UPt_3
aa r X i v : . [ c ond - m a t . s up r- c on ] J un M¨obius topological superconductivity in UPt Youichi Yanase ∗ and Ken Shiozaki Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan Department of Physics, University of Illinois at Urbana Champaign, Urbana, IL 61801, USA (Dated: June 30, 2017)Intensive studies for more than three decades have elucidated multiple superconducting phasesand odd-parity Cooper pairs in a heavy fermion superconductor UPt . We identify a time-reversalinvariant superconducting phase of UPt as a recently proposed topological nonsymmorphic super-conductivity. Combining the band structure of UPt , order parameter of E representation allowedby P /mmc space group symmetry, and topological classification by K -theory, we demonstrate thenontrivial Z -invariant of three-dimensional DIII class enriched by glide symmetry. Correspondingly,double Majorana cone surface states appear at the surface Brillouin zone boundary. Furthermore, weshow a variety of surface states and clarify the topological protection by crystal symmetry. Majoranaarcs corresponding to tunable Weyl points appear in the time-reversal symmetry broken B-phase.Majorana cone protected by mirror Chern number and Majorana flat band by glide-winding numberare also revealed. I. INTRODUCTION
Unconventional superconductivity in strongly corre-lated electron systems is attracting renewed interest be-cause it may be a platform of topological superconductiv-ity accompanied by Majorana surface/edge/vortex/endstates . Although previous studies focused on theproximity-induced topological superconductivity in s -wave superconductor (SC) heterostructures , natural s -wave SCs are mostly trivial from the viewpoint of topol-ogy. On the other hand, unconventional SCs may havetopologically nontrivial properties originating from non- s -wave Cooper pairing. In particular, time-reversal sym-metry (TRS) broken chiral SCs and odd-parity spin-triplet SCs are known to be candidates of topolog-ical superconductivity. However, from the viewpoint ofmaterials science, evidences for chiral and/or odd-paritysuperconductivity have been reported for only a fewmaterials, such as URu Si , SrPtAs , Sr RuO ,Cu x Bi Se , and ferromagnetic heavy fermion SCs .Superconductivity in UPt has been discovered in1980’s . Multiple superconducting phases illustrated inFig. 3 unambiguously exhibit exotic Cooper pairingwhich is probably categorized into the two-dimensional(2D) irreducible representation of point group D .After several theoretical proposals examined by exper-iments for more than three decades, the E representa-tion has been regarded as the most reasonable symmetryof superconducting order parameter . In particular,the multiple phase diagram in the temperature-magneticfield plane is naturally reproduced by assuming a weaksymmetry breaking term of hexagonal symmetry . Fur-thermore, a phase-sensitive measurement and the ob-servation of spontaneous TRS breaking in the low-temperature and low-magnetic field B-phase, which waspredicted in the E -state, support the E symmetryof superconductivity. The order parameter of E sym-metry represents odd-parity spin-triplet Cooper pairs.Therefore, topologically nontrivial superconductivity is expected in UPt . z=0z=1/2 r r r e e e FIG. 1. (Color online) Crystal structure of UPt . Uraniumions form AB stacked triangular lattice. 2D vectors, e i and r i , are shown by arrows. Furthermore, UPt has an intriguing feature in thecrystal structure, which is illustrated in Fig. 1. The sym-metry of the crystal is represented by nonsymmorphicspace group P /mmc ; glide and screw symmetries in-cluding a half translation along the c -axis are preservedin spite of broken mirror and rotation symmetries. Exoticproperties ensured by glide and/or screw symmetry areone of the central topics in the modern condensed matterphysics. This topic for SCs traces back to Norman’s workin 1995 for UPt ; a counterexample of Blount’s theo-rem . The line nodal excitation predicted by Normanhas been revisited by recent studies; group-theoreticalproof , microscopic origin , and topological protec-tion have been elucidated, and they have been con-firmed by a first principles-based calculation .Recent developments in the theory of nonsymmorphictopological states of matter have uncovered noveltopological insulators and SCs enriched by glide and/orscrew symmetry, which are distinct from those classi-fied by existing topological periodic table for symmor-phic systems . Since eigenvalues of glide and two-fold screw operators are 4 π -periodic, a M¨obius structureappears in the wave function and changes the topologi-cal classification. Although such topological nonsymmor-phic crystalline insulators have been proposed in KHgX(X = As, Sb, Bi) and CeNiSn , topological non-symmorphic crystalline superconductor (TNSC) has notbeen identified in materials. In this paper we show thetopological invariant specifying the TNSC by K -theory,and demonstrate its nontrivial value in UPt .Multiband structures give rise to both intriguing andcomplicated aspects of many heavy fermion systems.However, the band structure of UPt has been clarifiedto be rather simple . Fermi surfaces (FSs) are clas-sified into the two classes. The FSs of one class enclosethe A -point in the Brillouin zone (BZ) [band 1 and band2 in Ref. 54], while those of the other class are centeredon the Γ-point or K -point [bands 3, 4, and 5 in Ref. 54].The two classes are not hybridized in the surface stateon the (100)-direction where the glide symmetry is pre-served. Therefore, we can separately study the topolog-ical invariants and surface states arising from the multi-band FSs. The TNSC is attributed to the former FSs inSec. V. The latter FSs are also accompanied by varioustopological surface states, for which we identify topolog-ical invariant in Sec. VI.The paper is organized as follows. In Sec. II, we in-troduce a minimal two-sublattice model for nonsymmor-phic superconductivity in UPt . In Sec. IIB, Dirac nodallines protected by P /mmc space group symmetry areproved. In Sec. IIC, the order parameter of E sym-metry is explained. The calculated surface states onthe glide invariant (100)-surface are shown in Sec. III.In Sec. IV, three-dimensional (3D) TNSC of DIII classis classified on the basis of the K -theory. In Sec. V, weshow that the glide- Z invariant characterizing the TNSCis nontrivial in UPt A-phase. The underlying origin ofthe TNSC accompanied by double Majorana cone sur-face states is discussed. In Sec. VI, we characterize othertopological surface states by low-dimensional topologicalinvariants enriched by crystal mirror, glide, and rotationsymmetries. Constraints on these topological invariantsby nonsymmorphic space group symmetry are also re-vealed. Finally, a brief summary is given in Sec. VII.Ingredients giving rise to rich topological properties ofUPt are discussed. II. MODELA. Nonsymmorphic two-sublattice model
We study the superconducting state in UPt by ana-lyzing the Bogoliubov-de Gennes (BdG) Hamiltonian for nonsymmorphic two-sublattice model , H BdG = X k ,m,s ξ ( k ) c † k ms c k ms + X k ,s h a ( k ) c † k s c k s + h . c . i + X k ,m,s,s ′ α m g ( k ) · s ss ′ c † k ms c k ms ′ + 12 X k ,m,m ′ ,s,s ′ h ∆ mm ′ ss ′ ( k ) c † k ms c †− k m ′ s ′ + h . c i , (2.1)where k , m = 1 ,
2, and s = ↑ , ↓ are index of momen-tum, sublattice, and spin, respectively. The last termrepresents the gap function and others are normal partHamiltonian. Taking into account the crystal structureof UPt illustrated in Fig. 1, we adopt an intra-sublatticekinetic energy, ξ ( k ) = 2 t X i =1 , , cos k k · e i + 2 t z cos k z − µ, (2.2)and an inter-sublattice hopping term, a ( k ) = 2 t ′ cos k z X i =1 , , e i k k · r i , (2.3)with k k = ( k x , k y ). The basis translation vectors in twodimension are e = (1 , e = ( − , √ ), and e =( − , − √ ). The interlayer neighboring vectors projectedonto the basal plane are given by r = ( , √ ), r =( − , √ ), and r = (0 , − √ ). These 2D vectors areillustrated in Fig. 1.Although the crystal point group symmetry is cen-trosymmetric D , local point group symmetry at Ura-nium ions is D lacking inversion symmetry. Then,Kane-Mele spin-orbit coupling (SOC) with g -vector g ( k ) = ˆ z X i =1 , , sin k k · e i , (2.4)is allowed by symmetry. The coupling constant has to besublattice-dependent, ( α , α ) = ( α, − α ), so as to pre-serve the global D point group symmetry .Quantum oscillation measurements combined withband structure calculations have shown a pairof FSs centered at the A -point ( A -FSs) on the BZ face.Interestingly, the paired bands are degenerate on the A - L lines and form Dirac nodal lines , which have beenexperimentally observed by de Haas-van Alphen experi-ments . In the next subsection we show that the Diracnodal lines are protected by the nonsymmorphic spacegroup symmetry of P /mmc (No. 194) . Thus,the two A-FSs are naturally paired by the nonsymmor-phic crystal symmetry. By choosing a parameter set( t, t z , t ′ , α, µ ) = (1 , − , , ,
12) our two band model re-produces the paired A -FSs. In this paper we show thatthe peculiar band structure results in exotic supercon-ductivity in terms of symmetry and topology.First principles band structure calculations also pre-dict three FSs centered on the Γ-point (Γ-FSs), and twoFSs enclosing the K -point ( K -FSs) , although theexistence of K -FSs is experimentally under debates .We show that a variety of topological surface states mayarise from these bands. A parameter set ( t, t z , t ′ , α, µ ) =(1 , , , ,
16) reproduces one of the Γ-FSs, while anotherset ( t, t z , t ′ , α, µ ) = (1 , − , . , . , − .
2) is adopted forthe K -FSs. B. Dirac nodal lines in space group P /mmc The single particle states are four-fold degenerate onthe A - L lines [ k = (0 , k y , π ) and symmetric lines]. Inaddition to the usual Kramers degeneracy, the sublat-tice degree of freedom gives additional degeneracy. Thisfeature is reproduced in the normal part Hamiltonian,because the inter-sublattice hopping vanishes on the BZface ( k z = π ) and the SOC disappears on the A - L lines.Below we show that the existence of Dirac line nodes isensured by the space group symmetry.First, we show the additional degeneracy in the ab-sence of the SOC. In the SU(2) symmetric case, thetwo spin states are equivalent and naturally degenerate.Then, we can define the TRS, T = K , and screw symme-try S zπ ( k z ) in each spin sector, where K is the complexconjugate operator. At the BZ face, k z = π , we have S zπ ( π ) = iσ y where σ i is the Pauli matrix in the sublatticespace. Because the combined magnetic-screw symmetrysatisfies [ S zπ ( π ) T ] = −
1, the two-fold degeneracy in eachspin sector is proved by familiar Kramers theorem. Tak-ing into account the spin-degeneracy, we obtain four-folddegenerate bands on the entire BZ face.The four-fold degeneracy is partly lifted by the SOC.However, the degeneracy of two spinful bands is pro-tected on the A - L lines, that is proved as follows. Thelittle group on the A - L lines includes the rotation sym-metry IG xz ( k z ), mirror symmetry M yz , and magnetic-inversion symmetry IT . We here represent T = is y K , I = σ x , and M yz = is x , respectively. The glide symme-try is represented by G xz ( k z ) = s y σ y at k z = ± π while G xz ( k z ) = is y σ x at k z = 0. The nonsymmorphic prop-erty of rotation symmetry is emphasized by denoting as IG xz ( k z ). The symmetry operations satisfy the algebra[ IG xz ( π )] = − , (2.5) { IG xz ( π ) , IT } = 0 , (2.6) { IG xz ( π ) , M yz } = 0 . (2.7)The first relation ensures the sector decomposition to the λ = ± i eigenstates of rotation operator. Because the IT symmetry is preserved in each subsector, the Kramerstheorem holds. The anti-commutation relation of twounitary symmetries, Eq. (2.7), shows that a Kramerspair in one subsector has to be degenerate with anotherKramers pair in the other subsector. Therefore, the four- fold degeneracy on the A - L lines is protected by symme-try. -1.5-1-0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 ky kx k x k y k z = π FIG. 2. (Color online) FSs on the BZ face, k z = π . Thin redlines show the FSs in the presence of the SOC ( α = 1), whilethe thick black line is the overlapping FSs in the absenceof the SOC ( α = 0). Dashed lines show the A - L lines inthe first BZ. We set parameters ( t, t z , t ′ , µ ) = (1 , − , ,
12) toreproduce the A -FSs. Figure 2 shows the FSs in our model. It is illustratedthat the FSs completely overlap on the BZ face in theabsence of the SOC. Although the SOC splits the FSs,the degeneracy remains on the A - L lines. These featuresare consistent with above group theoretical analysis.The Dirac nodal lines in the P /mmc space groupare one of the typical examples of band degeneracy pro-tected by nonsymmorphic crystal symmetry . On theBZ face away from the A - L lines, the non-Kramers de-generacy is lifted purely by the SOC. Therefore, the SOCgives particularly significant effects on the BZ face. Thisis the underlying origin of the SOC-induced nodal loopin the superconducting gap . C. Order parameter of E representation The multiple superconducting phases in UPt havebeen reasonably attributed to two-component order pa-rameters in the E irreducible representation of D point group . The gap function is generally repre-sented by ˆ∆( k ) = η ˆΓ E + η ˆΓ E . (2.8)The two-component order parameters are parametrizedas ( η , η ) = ∆(1 , iη ) / p η , (2.9)with a real variable η . The basis functions ˆΓ E andˆΓ E are admixture of some harmonics. Adopting theneighboring Cooper pairs in the crystal lattice of U ions,we obtain the basis functionsˆΓ E = h δ { p x ( k ) s x − p y ( k ) s y } σ + f ( x − y ) z ( k ) s z σ x − d yz ( k ) s z σ y i is y , (2.10)ˆΓ E = h δ { p y ( k ) s x + p x ( k ) s y } σ + f xyz ( k ) s z σ x − d xz ( k ) s z σ y i is y , (2.11)which are composed of the p -wave, d -wave, and f -wavecomponents given by p x ( k ) = X i e xi sin k k · e i , (2.12) p y ( k ) = X i e yi sin k k · e i , (2.13) d xz ( k ) = −√ k z X i r xi e i k k · r i , (2.14) d yz ( k ) = −√ k z X i r yi e i k k · r i , (2.15) f xyz ( k ) = −√ k z X i r xi e i k k · r i , (2.16) f ( x − y ) z ( k ) = −√ k z X i r yi e i k k · r i . (2.17)Pauli matrix in the spin and sublattice space are denotedby s i and σ i , respectively.The purely f -wave state has been intensively investi-gated, and the phase diagram compatible with UPt hasbeen obtained . However, an admixture of a p -wavecomponent is allowed by symmetry and it changes thegap structure and topological properties . Thus, wehere take into account a small p -wave component with0 < | δ | ≪
1. The small p -wave component does not al-ter the phase diagram consistent with experiments. Onthe other hand, the dominantly p -wave state discussed inRef. 39 would fail to reproduce the phase diagram.Besides the p -wave component, a sublattice-singletspin-triplet d -wave component accompanies the f -wavecomponent as a result of the nonsymmorphic crystalstructure of UPt . The neighboring Cooper pairs on r i bonds give equivalent amplitude of d -wave and f -wavecomponents in Eqs. (2.10) and (2.11). The d -wave orderparameter plays a particularly important role on the su-perconducting gap at the BZ face, k z = π . Later we showthat the TNSC is induced by the d -wave component.Now we review the multiple superconducting phasesin UPt . Three thermodynamically distinguished super-conducting phases are illustrated in Fig. 3 . TheA-, B-, and C-phases are characterized by the ratio oftwo-component order parameters η = η /iη summarizedin Table. I. A pure imaginary ratio of η and η in theB-phase implies the chiral superconducting state which TH ACB
FIG. 3. (Color online) Multiple superconducting phases ofUPt in the magnetic field-temperature plane . The A-phase is identified as a TNSC. The shaded region shows theWeyl superconducting phase . Pair creation of Weyl nodesoccurs at the phase boundary. The dashed line indicates atopological phase transition, which is discussed in Sec. VI E. maximally gains the condensation energy. Owing to the p -wave component, the B-phase is non-unitary. It hasbeen considered that the A- and C-phases are stabilizedby weak symmetry breaking of hexagonal structure, pos-sibly induced by weak antiferromagnetic order .We here assume that the A-phase is the Γ state ( η = ∞ ),while the C-phase is the Γ state ( η = 0), and assumenon-negative η ≥ A-phase | η | = ∞ B-phase 0 ≤ | η | ≤ ∞ C-phase η = 0TABLE I. Range of the parameter η in the A-, B-, and C-phases of UPt . Contrary to the experimental indications for the E -pairing state mentioned before, a recent thermal conduc-tivity measurement has been interpreted in terms ofthe E symmetry of the orbital part of order parameter.However, this interpretation is not incompatible with the E symmetry of total order parameter. For instance,the basis functions, Eqs. (2.10) and (2.11), include com-ponents p x ( k ) s x − p y ( k ) s y and p y ( k ) s x + p x ( k ) s y , wherethe orbital part p x ( k ) and p y ( k ) belong to the E sym-metry. Although in Ref. 67 the superconducting statewith TRS has been discussed along with a theoreticalproposal , the spin part of order parameter can not bededuced from thermal conductivity measurements. Thus,we here assume the E -pairing state.For clarity of discussions for topological properties, wecarry out the unitary transformation for the BdG Hamil-tonian. When the model Eq. (2.1) is represented in theNambu space H BdG = 12 X k ˆ c † k ˆ H BdG ( k )ˆ c k , (2.18)withˆ c k = (cid:16) c k ↑ , c k ↑ , c k ↓ , c k ↓ , c †− k ↑ , c †− k ↑ , c †− k ↓ , c †− k ↓ , (cid:17) T , (2.19)the BdG matrix ˆ H BdG ( k ) in this form does not satisfythe periodicity compatible with the first BZ. To avoidthis difficulty, we represent the BdG Hamiltonian by˜ H BdG ( k ) = U ( k ) ˆ H BdG ( k ) U ( k ) † , (2.20)using the unitary matrix U ( k ) = (cid:18) e i k · τ (cid:19) σ ⊗ s ⊗ τ . (2.21)By choosing the translation vector, τ = (0 , − √ , ),˜ H BdG ( k ) is periodic with respect to the translation k → k + K for any reciprocal lattice vector K . Thetransformed BdG Hamiltonian has the same form asEq. (2.1), although the inter-sublattice components ac-quire the phase factor a ( k ) → ˜ a ( k ) ≡ a ( k ) e − i k · τ , (2.22) f i ( k ) → ˜ f i ( k ) ≡ f i ( k ) e − i k · τ , (2.23) d i ( k ) → ˜ d i ( k ) ≡ d i ( k ) e − i k · τ . (2.24) III. TOPOLOGICAL SURFACE STATES
We calculate the energy spectrum of quasiparti-cles with surface normal to the (100)-axis, E ( k sf ) = E ( k y , k z ), because the nonsymmorphic glide symmetryis preserved there. Both glide and screw symmetry arebroken in the other surface directions. Figures 4 and 5show results for the Γ-FS and A -FSs, respectively. Theblack regions represent the zero energy surface states. Itis revealed that a variety of zero energy surface states ap-pear on the (100)-surface in the A-, B-, and C-phases. Weclarify the topological protection of these surface statesbelow. Indeed, all of the zero energy surface states aretopologically protected. In Figs. 4 and 5, the topologicalsurface states discussed in Secs. V and VI A - VI E arelabeled by (V) and (A)-(E), respectively.The most panels of Figs. 4 and 5 show the results for α = 0 by neglecting the SOC. Most of the surface statesare indeed robust against the SOC. Exceptionally, thesurface states around k z = π are affected by the SOC,because the nodal bulk excitations may be induced bythe SOC . For our choice of parameters, the bulkexcitation gap remains finite at k z = π for α = 1 althoughthe gap may be suppressed for α = 2. Thus, we show the surface states for α = 2 in Fig. 5(f) for a comparison. Thegapless bulk excitations which are not shown for α = 0[Fig. 5(c)] are observed around the surface BZ boundary.One of the main results of this paper is a signatureof TNSC in UPt , that is indicated by the label (V) inFig. 5(e). This surface state is robust against the SOCunless the bulk excitation gap is closed. According to thefirst principles band structure calculation, the band split-ting by the SOC is tiny along the A - H lines and signif-icantly decreased by the mass renormalization factor , z ∼ /
100 in UPt . Thus, it is reasonable to assume asmall SOC leading to the gapped bulk excitations at theBZ face. This assumption is compatible with the recentfield-angle-dependent thermal conductivity measurementwhich has shown nodal lines/points lying away from theBZ face .In the next section, superconducting phases of 3D DIIIclass with additional glide symmetry are classified on thebasis of the K -theory, and the topological invariants arederived. In Sec. V, we show that a surface state labeledby (V) is protected by the strong topological index char-acterizing the TNSC. The topological protection of othersurface states is revealed in Sec. VI. IV. CLASSIFICATION OF CLASS DIIISUPERCONDUCTORS WITH GLIDESYMMETRY
Topological classification of TNSC is carried out forboth glide-even and glide-odd superconducting states ofDIII class. For simplicity, the cubic first BZ with vol-ume (2 π ) is assumed in this section. We do not rely onany specific model, and therefore, the results obtained inthis section are valid for all the superconducting statespreserving the glide symmetry and TRS. A. Glide-even superconductor
First, we study glide-even superconducting states. TheΓ -state (A-phase) of UPt corresponds to this case. Thesymmetries for the BdG Hamiltonian are summarized as C H ( k ) C − = −H ( − k ) , C = τ x K, (4.1) T H ( k ) T − = H ( − k ) , T = is y K, (4.2) G ( k ) H ( k ) G − ( k ) = H ( m y k ) , G ( m y k ) G ( k ) = − e − ik z , (4.3) T G ( k ) = G ( − k ) T, CG ( k ) = G ( − k ) C, (4.4)where m y k = ( k x , − k y , k z ) is the momentum flipped byglide operation, and K is the complex conjugate. Thestable classification of bulk superconductors is given bythe K -theory over the bulk 3D BZ torus with symmetries (a) η =0 (b) η =0.7 (c) η =1(d) η =1.5 (e) η =infty -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1ky kz 0 0.02 0.04 0.06 0.08 0.1 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 0.04 0.05-1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 0.04 0.05 0.06 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1ky kz 0 0.04 0.08 0.12 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 0.04 0.05 (B) (B) (E)(B) (C)(A) (A)(A) k z k y (C)(C) FIG. 4. (Color online) Energy of surface states on the (100)-surface. We impose open boundary condition along the[100]-direction and periodic boundary condition along the other directions. The lowest excitation energy of BdG quasipar-ticles [ ≡ min | E ( k sf ) | ] as a function of the surface momentum k sf = ( k y , k z ) is shown. Parameters ( t, t z , t ′ , α, µ, ∆ , δ ) =(1 , , , , , , .
02) are assumed so that the Γ-FS is reproduced. (a) C-phase ( η = 0), (b)-(d) B-phase ( η = 0 .
7, 1, and 1 . η = ∞ ). Arrows with characters (A), (B), (C) and (E) indicate surface states clarified in Secs. VI A, VI B,VI C, and VI E, respectively. The green circles show the projections of Weyl point nodes. (a) η =0 (b) η =0.6 (c) η =1(d) η =2 (e) η =infty -1.5 -1 -0.5 0 0.5 1 1.5 1.8 2 2.2 2.4 2.6 2.8 3 3.2ky kz 0 0.02 0.04 0.06 0.08 0.1 -1.5 -1 -0.5 0 0.5 1 1.5 1.8 2 2.2 2.4 2.6 2.8 3 3.2 0 0.01 0.02 0.03 0.04 -1.5 -1 -0.5 0 0.5 1 1.5 1.8 2 2.2 2.4 2.6 2.8 3 3.2 0 0.01 0.02 0.03 0.04-1.5 -1 -0.5 0 0.5 1 1.5 1.8 2 2.2 2.4 2.6 2.8 3 3.2 0 0.01 0.02 0.03 0.04 -1.5 -1 -0.5 0 0.5 1 1.5 1.8 2 2.2 2.4 2.6 2.8 3 3.2ky kz 0 0.02 0.04 0.06 0.08 (f) α=2, η =1 -1.5 -1 -0.5 0 0.5 1 1.5 1.8 2 2.2 2.4 2.6 2.8 3 3.2 0 0.01 0.02 0.03 0.04 (V) (C) (A) (A) (A) k z k y (C) (A) (A)(A) FIG. 5. (Color online) (a)-(e) Energy of surface states on the (100)-surface for parameters reproducing the paired A -FSs,( t, t z , t ′ , α, µ, ∆ , δ ) = (1 , − , , , , . , . η = 0), (b)-(d) B-phase ( η = 0 .
6, 1, and 2), and (e) A-phase( η = ∞ ). We choose α = 2 in (f) while the other parameters are the same as (c). Comparison between (c) and (f) revealsthe effect of the SOC. Arrows with characters (V), (A), and (C) indicate surface states discussed in Secs. V, VI A, and VI C,respectively. The green circles show the projections of Weyl point nodes. (4.1)-(4.4). From Ref. 42, the result is(The stable classification of bulk gapped SCs)= Z |{z} ( k x ,k y ,k z ) ⊕ Z ⊕ Z | {z } ( k x ,k z ) ⊕ Z |{z} ( k y ,k z ) ⊕ Z |{z} ( k z ) . (4.5)The bold style Z expresses an emergent topologicalphase which disappears if the glide symmetry is broken.Each underbrace represents the momentum dependenceof generating Hamiltonian. For instance, Z |{z} ( k x ,k z ) meansthat the generating Hamiltonian of the Z phase can be k y -independent, that is, the stacking of layered Hamil-tonians H y ( k x , k z ) in the xz -plane along the y -direction.We focus on the gapless states on the surfaces preserv-ing the glide symmetry, i.e. x = constant surface. Theclassification of the surface gapless states is given by asimilar K -theory over the surface 2D BZ torus underthe same symmetries (4.1)-(4.4) with the k x -direction ex-cluded. The bulk-boundary correspondence holds: the K -group for the surface gapless states is given by the di-rect summand of which generators are dependent on k x in Eq. (4.5). Thus, (cid:18) The classification of gapless stateson the x = constant surface (cid:19) = Z |{z} ( k x ,k y ,k z ) ⊕ Z ⊕ Z | {z } ( k x ,k z ) . (4.6)All the three Z invariants relevant to the surface gaplessstates are constructed on the k z = π plane . At k z = π ,the glide symmetry is reduced into the mirror symmetry G ( k x , k y , π ) H ( k x , k y , π ) G − ( k x , k y , π ) = H ( k x , − k y , π ) , (4.7) G ( k x , − k y , π ) G ( k x , k y , π ) = 1 , (4.8) T G ( k x , k y , π ) = G ( − k x , − k y , π ) T, (4.9) CG ( k x , k y , π ) = G ( − k x , − k y , π ) C. (4.10)On the k y = Γ y ≡ , π lines, since the TRS and theparticle-hole symmetry (PHS) commute with the glidesymmetry, we can define Z invariant ν (Γ y , ± ) ∈ { , } of one-dimensional (1D) class DIII SCs for each glide-subsectors G ( k x , Γ y , π ) = ± ν (Γ y , ± ) = iπ I π dk x X n D u ( I ) ± ,n ( k x , Γ y , π ) (cid:12)(cid:12)(cid:12) ∂ k x (cid:12)(cid:12)(cid:12) u ( I ) ± ,n ( k x , Γ y , π ) E (mod 2) , (Γ y = 0 , π ) , where u ( I ) ± ,n ( k x , Γ y , π ) represents one of the Kramerspair of occupied states in the glide-subsector G ( k x , Γ y , π ) = ±
1. Noticing that the combined symmetries
T G ( k x , k y , π ) and CG ( k x , k y , π ),[ T G ( k x , k y , π )] H ( k x , k y , π ) [ T G ( k x , k y , π )] − = H ( − k x , k y , π ) , (4.11) T G ( − k x , k y , π ) T G ( k x , k y , π ) = − , (4.12)[ CG ( k x , k y , π )] H ( k x , k y , π ) [ CG ( k x , k y , π )] − = −H ( − k x , k y , π ) , (4.13) CG ( − k x , k y , π ) CG ( k x , k y , π ) = 1 , (4.14)indicate the emergent class DIII symmetry for all k y , wehave a constraint ν (0 , +) + ν (0 , − ) = ν ( π, +) + ν ( π, − ) , (mod 2) . (4.15)Because of this emergent class DIII symmetry, all thesurface states on the k z = π plane show two-fold degen-eracy. The three kinds of surface states may be generatedby Z |{z} ( k x ,k y ,k z ) : (cid:0) ν (0 , +) , ν (0 , − ); ν ( π, +) , ν ( π, − ) (cid:1) = (1 ,
1; 0 ,
0) or (0 ,
0; 1 , , (4.16) Z |{z} ( k x ,k z ) : (cid:0) ν (0 , +) , ν (0 , − ); ν ( π, +) , ν ( π, − ) (cid:1) = (1 ,
0; 1 , , (1 ,
0; 0 , , (0 ,
1; 1 ,
0) or (0 ,
1; 0 , , (4.17) Z |{z} ( k x ,k z ) : (cid:0) ν (0 , +) , ν (0 , − ); ν ( π, +) , ν ( π, − ) (cid:1) = (1 ,
1; 1 , . (4.18)Here, Z |{z} ( k x ,k z ) shows the flat surface band on the k z = π plane. The Z |{z} ( k x ,k y ,k z ) is the strong index of TNSC, whichis denoted as glide- Z invariant, ν G ≡ Z |{z} ( k x ,k y ,k z ) . It isgiven by ν G = ν (0 , +) ν (0 , − ) − ν ( π, +) ν ( π, − ) (mod 2) . (4.19)Later, we show that the glide- Z invariant ν G is nontrivialin the A-phase of UPt . B. Glide-odd superconductor
Next, we study glide-odd superconducting states,which may be realized in the Γ -state of UPt (C-phase).Symmetries for the BdG Hamiltonian are T H ( k ) T − = H ( − k ) , T = is y K, (4.20) C H ( k ) C − = −H ( − k ) , C = τ x K, (4.21) G ( k ) H ( k ) G − ( k ) = H ( m y k ) , G ( m y k ) G ( k ) = − e − ik z , (4.22) T G ( k ) = G ( − k ) T, CG ( k ) = − G ( − k ) C. (4.23) From Ref. 42, the K -theory classification of the bulkreads(The stable classification of bulk gapped SCs)= Z ⊕ Z | {z } ( k x ,k y ,k z ) ⊕ Z |{z} ( k x ,k z ) ⊕ Z ⊕ Z | {z } ( k y ,k z ) ⊕ Z |{z} ( k z ) . (4.24)The bold-style indices express emergent topologicalphases which requires the glide symmetry. From thebulk-boundary correspondence, it holds that (cid:18) The classification of surface stateson the x = constant surface (cid:19) = Z ⊕ Z | {z } ( k x ,k y ,k z ) ⊕ Z |{z} ( k x ,k z ) . (4.25)The 3D Z |{z} ( k x ,k y ,k z ) index is the ordinary winding num-ber , N := 148 π Z trΓ( H − d H ) , Γ = iT C. (4.26)By imposing the glide symmetry, we have two Z in-variants θ (Γ y = 0 , π ) ∈ { , , , } on the glide invariant k y = 0 and π planes , θ (Γ y ) := 2 iπ h I π dk x tr A ( I )+ ( k x , Γ y , π ) + 12 Z π dk z I π dk x tr F + ( k x , Γ y , k z ) i (mod 4) , (Γ y = 0 , π ) . (4.27)Here, A ( I )+ ( k x , Γ y , π ) and F + ( k x , Γ y , k z ) are the Berryconnection of one of Kramers pair of occupied statesand the Berry curvature of the occupied states, respec-tively, with the positive glide eigenvalue G ( k x , Γ y , π ) = 1.In modulo 2, θ (Γ y ) is recast into the Z invariant at( k x , Γ y , k z = 0) lines as θ (Γ y ) := iπ I π dk x tr A + ( k x , Γ y ,
0) (mod 2) , (4.28) by the Stokes’ theorem.The three invariants { N, θ (0) , θ ( π ) } are not indepen-dent, since there is a constraint N + θ (0) + θ ( π ) = 0 (mod 2) , (4.29)which can be understood as follows. On the k z = 0 plane,the Z invariant ν = θ (0) + θ ( π ) (mod 2) is equivalent tothe 2D class DIII Z invariant. Since we can show thatthe existence of odd numbers of Majorana cones is al-lowed only on the k z = 0 plane, N (mod 2) is also equiv-alent to the Z invariant. Therefore, N = ν (mod 2),which implies Eq. (4.29). V. TOPOLOGICAL NONSYMMORPHICSUPERCONDUCTIVITY IN A-PHASE
Now we go back to the superconductivity in UPt .Let’s focus on the surface zero mode at k sf = (0 , π ) inthe TRS invariant A-phase. Naturally, the A -FSs areconsidered in this section. The surface states labeled by(V) in Fig. 5(e) have the spectrum shown in Fig. 6. As weproved in Sec. IV A, the quasiparticle states are two-folddegenerate on the k sf = ( k y , π ) line in the glide-even A-phase. Therefore, the spectrum of surface states showsdouble Majorana cone with four zero energy states at k sf = (0 , π ). In this section we show that the surfacedouble Majorana cone is protected by the strong Z in-dex ν G for glide-even TNSCs, which has been introducedin Eq. (4.19). -0.03-0.02-0.01 0 0.01 0.02 0.03 -0.03-0.02-0.01 0 0.01 0.02 0.03 k y k z E ( k s f ) -0.7 0.7 3.1323.151 FIG. 6. (Color online) Double Majorana cone in the A-phase.Energy spectrum of (100)-surface states around k sf = (0 , π )is shown. Parameters are the same as Fig. 5(e) for the paired A -FSs. The glide symmetry of P /mmc space group is G xz = { M xz | z } composed of mirror reflection and half transla-tion along the z -axis. Thus, the nonsymmorphic glideoperator is intrinsically k z -dependent. We have an op-erator for the normal part Hamiltonian, G xz ( k z ) = is y σ x V σ ( k z ), where V σ ( k z ) = (cid:18) e − ik z (cid:19) σ , (5.1)acts in the sublattice space. The superconducting statepreserves the glide symmetry in the TRS invariant A- andC-phases, although the glide symmetry is spontaneouslybroken in the B-phase. The glide operator in the Nambuspace depends on the glide-parity of the superconducting state; G xz BdG ( k z ) = G xz ( k z ) τ in the glide-even A-phasewhile G xz BdG ( k z ) = G xz ( k z ) τ z in the glide-odd C-phase.Then, the BdG Hamiltonian respects the glide symmetry G xz BdG ( k z ) ˜ H BdG ( k ) G xz BdG ( k z ) − = ˜ H BdG ( k x , − k y , k z ) . (5.2)The symmetries satisfy the algebra (4.1)-(4.4) and (4.20)-(4.23) in the A-phase and C-phase, respectively. A. Glide- Z invariant k x k y (K x +K y )/2(-K x +K y )/2(-K x -K y )/2 (K x -K y )/2 K x FIG. 7. (Color online) Unfolded BZ (solid line) and folded BZ(dashed line) projected onto a k z = constant plane. The latteris compatible with the surface BZ. K x = 2 π ˆ x and K y = π √ ˆ y are reciprocal lattice vectors of the folded BZ. As we showed in Sec. IV A, only the 2D plane at k z = π determines the topological properties of the glide-even A-phase. From Eq. (4.19), the glide- Z invariant ν G is givenby the 1D Z invariant of DIII class, ν (Γ y , ± ). Whenwe choose the rectangular BZ shown in Fig. 7, we haveΓ y = 0 , π/ √
3. For our choice of parameters, the FSsdo not cross a line k y = π/ √ ν ( π/ √ , ± ) is trivial, and the glide- Z invariant isobtained by evaluating ν (0 , ± ). Below, we show that ν (0 , ± ) = 1, and thus, the glide- Z invariant is nontriv-ial.First, the Hamiltonian is block-diagonalized by usingthe basis diagonal for G xz BdG ( π ) = s y σ y τ ,˜ H BdG ( k x , , π ) = ˜ H ( k x ) ⊕ ˜ H − ( k x ) , (5.3)on the 1D BZ k x ∈ [ − π, π ]. The glide-subsector witheigenvalue λ G = ± H ± ( k x ) = ˆ H (0) ± ( k x ) ˆ∆( k x )ˆ∆( k x ) † − ˆ H (0) ± ( − k x ) T ! , (5.4)with ˆ H (0) ± ( k x ) = (cid:18) ξ ( k x ) ∓ α g ( k x ) ∓ α g ( k x ) ξ ( k x ) (cid:19) , (5.5)ˆ∆( k x ) = i ∆ (cid:18) iδp x ( k x ) − d xz ( k x ) − d xz ( k x ) iδp x ( k x ) (cid:19) . (5.6)0We defined p x ( k x ) ≡ p x ( k x , , π ) = sin k x + sin k x and d xz ( k x ) ≡ d xz ( k x , , π ) = −√ k x . Thus, the glide-subsector is equivalent to the TRS invariant p -wave SC.It is easy to confirm that both TRS and PHS are pre-served in each glide-subsector as expected from Sec. IV A.In the A-phase we adopt time-reversal operator in theNambu space T BdG = iT τ z , since the gap function ischosen to be pure imaginary. Then, the commutationrelation [ C, T
BdG ] = 0 is satisfied.Although the inversion symmetry is broken in theglide-subsector by the SOC, we can adiabatically elim-inate the SOC as α →
0, unless the SOC is large enoughto suppress the superconducting gap . Then, theglide-subsector is reduced to the odd-parity spin-tripletSC, and the Z invariant is obtained by counting thenumber of Fermi points N ( λ G ) (per Kramers pairs) be-tween the time-reversal invariant momentum, k x = 0and 2 π . Since each glide-subsector represents a singleband model with N ( ±
1) = 1, the nontrivial Z invari-ant, ν (0 , ± ) = 1 (mod 2), is obtained from the formula( − ν (0 , ± ) = ( − N ( ± ) .Now we conclude that the glide- Z invariant is non-trivial, namely, ν G = 1, because (cid:16) ν (0 , +) , ν (0 , − ); ν ( π/ √ , +) , ν ( π/ √ , − ) (cid:17) = (1 ,
1; 0 , . (5.7)This is the strong topological index characterizing theTNSC with even glide-parity.It should be noticed that the paired FSs and thesublattice-singlet d -wave pairing are essential ingredients.Both of them are ensured by the nonsymmorphic spacegroup symmetry (see Secs. II B and II C). The pseudospindegree of freedom in the glide-subsector corresponds tothe pair of FSs. Although the f -wave component in theorder parameter disappears on the glide invariant plane k y = 0, the d -wave component induces the superconduct-ing gap and gives rise to 1D Z nontrivial superconduc-tivity.The topological surface state protected by the glide- Z invariant should appear as a signature of the TNSC. Be-cause the two glide-subsectors discussed above are TRSinvariant and Z nontrivial, two Majorana states per sub-sector, namely, four Majorana states in total, appear onthe glide invariant (100)-surface. Indeed, the double Ma-jorana cone centered at k sf = (0 , π ) (Fig. 6) is the charac-teristic topological surface states of the glide-even TNSC.For confirmation, we show the topological indices of1D Hamiltonian along the k = ( k x , ,
0) and ( k x , , π )lines and 2D Hamiltonian on the k z = 0 and π planesin Table II. For these low-dimensional Hamiltonian, theglide operator is momentum independent, and therefore,the topological classification can be carried out withouttaking care of the nonsymmorphic property. The (anti-)commutation relations of symmetry operators are sum-marized in Table II, and accordingly the topological in-dices are obtained on the basis of the periodic table forsymmorphic topological crystalline insulators and SCs . k z ( G xz BdG ) η T η C
1D invariant 2D invariantC-phase 0 -1 1 -1 Z Z ⊕ Z ( η = 0) π Z Z ( η = ∞ ) π Z ⊕ Z Z TABLE II. Classification of 1D and 2D BdG Hamiltonian inthe TRS invariant A- and C-phases. The low-dimensionalHamiltonian on the basal plane ( k z = 0) and BZ face( k z = π ) is classified. We show ( G xz BdG ) , η T , and η C . (Anti-)commutation relations with time-reversal and particle-holeoperators are represented as T BdG G xz BdG = η T G xz BdG T BdG and
C G xz BdG = η C G xz BdG C . The right two columns show the 1Dtopological index on the ( k y , k z ) = (0 ,
0) and (0 , π ) lines andthe 2D topological index on the k z = 0 and π planes. Indeed, in the A-phase we have Z ⊕ Z index for 1DHamiltonian on the k = ( k x , , π ) line, which are noth-ing but ν (0 , ± ). The Z index of 2D Hamiltonian on the k z = π plane is equivalent to the glide- Z invariant dis-cussed in this section. On the other hand, the k z = π plane is trivial in the glide-odd C-phase, consistent withthe absence of topological surface states in Fig. 5(a). B. Folded Brillouin zone
For the consistency with classification based on the K -theory in Sec. IV, we need to consider the folded Bril-louin zone compatible with the surface BZ. To be spe-cific, the translation symmetry along the [010]-axis ispartially broken on the (100)-surface. The basic transla-tion vectors on the surface are ( y, z ) = ( √ ,
0) and (0 , K y = √ π ˆ y and K z = 2 π ˆ z . Thus, the surface first BZ is a rectanglewith k y ∈ [ − π/ √ , π/ √
3) and k z ∈ [ − π, π ). We have al-ready adopted the bulk BZ compatible with the surfaceBZ. However, the periodicity with respect to K y is lostin the BdG Hamiltonian. To satisfy the periodicity, weequate k with k + K y , and accordingly, adopt the foldedBZ in Fig. 7.The folded BdG Hamiltonian is transformed b H BdG ( k ) = U sf ( k ) (cid:18) ˜ H BdG ( k ) 00 ˜ H BdG ( k + K y ) (cid:19) ρ U sf ( k ) † , (5.8)by the unitary matrix U sf ( k ) = 1 √ (cid:18) e i kτ ′ (cid:19) ρ (cid:18) − (cid:19) ρ , (5.9)with τ ′ = ( , √ , b H BdG ( k + K i ) = b H BdG ( k )with respect to K x = 2 π ˆ x , K y , and K z . The glidesymmetry is recast, b G xz BdG ( k ) b H BdG ( k ) b G xz BdG ( k ) − = b H BdG ( k x , − k y , k z ) , (5.10)1with using the glide operator for the folded BdG Hamil-tonian, b G xz BdG ( k ) = G xz BdG ( k z ) ⊗ (cid:18) e − i √ k y (cid:19) ρ . (5.11)The TRS and PHS are also preserved, T BdG b H BdG ( k ) T − = b H BdG ( − k ) , (5.12) C b H BdG ( k ) C − = − b H BdG ( − k ) . (5.13)The symmetry operators satisfy the following relations, T BdG C = C T
BdG , (5.14) b G xz BdG ( m k ) b G xz BdG ( k ) = − e − ik z , (5.15) T BdG b G xz BdG ( k ) = b G xz BdG ( − k ) T BdG , (5.16) C b G xz BdG ( k ) = ± b G xz BdG ( − k ) C. (5.17)The sign ± in Eq. (5.17) should be chosen in the A-phaseand C-phase, respectively. Thus, the algebra (4.1)-(4.4)and (4.20)-(4.23) are satisfied.The 1D Z invariants in the folded BZ are equivalentto those obtained in the unfolded BZ. This fact is sim-ply understood by looking at the surface states. Theenergy spectrum is not changed by the unitary transfor-mation, and we have obtained odd number of Majoranacone at k sf = (0 , π ) per glide-subsector. This fact indi-cates ν (0 , ± ) = 1 and ν ( π/ √ , ± ) = 0 (mod 2). There-fore, the glide- Z invariant is nontrivial, that is, ν G = 1. C. Deformation to M¨obius surface state
The glide- Z invariant ν G is the strong topological in-dex specifying the gapped TNSC. However, the A-phaseis actually gapless because of the point nodes of gap func-tion at the poles of 3D FSs. Figure 8 shows the surfacespectrum E (0 , k z ), and indeed, we observe gapless bulkexcitations away from the surface BZ boundary k z = π inaddition to the double Majorana cone at k z = π . There-fore, UPt A-phase does not realize the characteristic“M¨obius surface state” of topological nonsym-morphic insulators/superconductors.However, the nontrivial glide- Z invariant ensures thatthe gapped TNSC can be realized when the point nodesare removed by some perturbations preserving the sym-metry. Then, we obtain the M¨obius surface states withkeeping the nontrivial glide- Z invariant and the associ-ated double Majorana cone. In other words, the doubleMajorana cone around k sf = (0 , π ) is regarded as a remi-niscent of the M¨obius surface states of glide-even TNSC.A simple way is to deform the FS to be cylindricalso that the point nodes are removed. Then, the surfacespectrum in Fig. 9 is obtained. In Fig. 9(a), the surfacestates detached from bulk excitations show the M¨obiusstructure typical of glide-even TNSC. At k sf = 0, theKramers degeneracy is ensured by the TRS. The Kramers -0.3-0.2-0.1 0 0.1 0.2 0.3 2 2.5 3 3.5 4 4.5 E ( , kz ) kz k z E ( , k z) FIG. 8. (Color online) Energy spectrum on the (100)-surface in the A-phase. Parameters ( t, t z , t ′ , α, µ, ∆ , δ ) =(1 , − , , , , . , .
04) reproduce the paired A -FSs of UPt .Spectrum on the k sf = (0 , k z ) line is shown. Surface statesare highlighted by green lines. -0.4-0.2 0 0.2 0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 E ( , kz ) kz k z E ( , k z) (b) k z =0 (c) k z = π (a) k y =0 -0.4-0.2 0 0.2 0.4 -1.5 -1 -0.5 0 0.5 1 1.5 E ( ky , ) ky -0.6-0.4-0.2 0 0.2 0.4 0.6 -1.5 -1 -0.5 0 0.5 1 1.5 E ( ky , ) ky k y k y E ( k y , π ) E ( k y , ) FIG. 9. (Color online) M¨obius surface states in the 3Dglide-even TNSC. We choose parameters ( t, t z , t ′ , α, µ, ∆ , δ ) =(1 , , . , . , . , . , .
5) for 2D cylindrical FSs. Bulk andsurface states are shown by blue and green lines, respectively.(a) E (0 , k z ), (b) E ( k y , E ( k y , π ). The glide eigen-values are illustrated in (a). pair is formed by ± i glide eigenstates since the TRS andPHS are not preserved in the glide-subsector. When welook at the k sf = ( k y ,
0) line, Fig. 9(b) shows two helicalmodes protected by the mirror Chern number ν = 4,which is introduced in Sec. VI B. The nontrivial relation-ship between the mirror Chern number and the glide- Z . D. Broken glide symmetry by crystal distortion
Strictly speaking, the symmetry of crystal structurein UPt is still under debate, because a tiny crystal dis-tortion has been indicated by a x-ray diffraction mea-surement . The distortion leads to layer dimerizationthat breaks the glide and screw symmetry. Then, thespace group is reduced from nonsymmorphic P /mmc to symmorphic P ¯3 m
1. If the crystal distortion actuallyoccurs in UPt , the double Majorana cone protected bythe glide- Z invariant may be gapped. -0.3-0.2-0.1 0 0.1 0.2 0.3 2 2.5 3 3.5 4 4.5 E ( , kz ) kz k z E ( , k z) FIG. 10. (Color online) Energy spectrum on the (100)-surfacein the presence of layer dimerization that breaks the glidesymmetry. Parameters are the same as Fig. 8, while the inter-sublattice hybridization is replaced by Eq. (5.18) with d = 0 . The layer dimerization makes the inter-sublattice hy-bridization asymmetric between the + z and − z direc-tions. The asymmetry is taken into account by replacing, a ( k ) = 2 t ′ cos k z X i =1 , , e i k k · r i ⇒ t ′ h (1 + d ) e ik z / + (1 − d ) e − ik z / i X i =1 , , e i k k · r i . (5.18)The parameter d represents the strength of the layerdimerization. For a finite d , the double Majorana coneat k sf = (0 , π ) indeed acquires mass term. In Fig. 10,we show the surface spectrum gapped at k sf = (0 , π ). Inthe figure, a strong layer dimerization d = 0 . d is expected to be tinyeven if it is finite, because the crystal distortion reportedis small . Therefore, the gap in the double Majoranacone may be tiny, and a fingerprint of topological glide- Z superconductivity will appear even in the symmorphic P ¯3 m VI. OTHER TOPOLOGICAL SURFACE STATES
In contrast to toy models, the model specific for thereal material shows rich topological properties. In Figs. 4and 5, we have observed a variety of surface states otherthan the double Majorana cone discussed in Sec. V. Inthis section, we clarify the topological invariant protect-ing the surface states. In addition to the glide symmetry,we take the mirror symmetry into account. The Weylcharge, mirror Chern number, glide winding number, androtation winding number are discussed below.
A. Chiral Majorana arc in Weyl B-phase
The TRS broken B-phase identified as a Weyl super-conducting state hosts surface Majorana arcs, anal-ogous to Fermi arcs in Weyl semimetals . The ex-istence of Majorana arcs is ensured by the topologicalWeyl charge q i = 12 π I S d k ~F ( k ) , (6.1)which is nothing but the monopole of Berry flux, F i ( k ) = − iε ijk X E n ( k ) < ∂ k j h u n ( k ) | ∂ k k u n ( k ) i . (6.2)A wave function and energy of Bogoliubov quasiparticlesare denoted by | u n ( k ) i and E n ( k ), respectively. A non-trivial Weyl charge protects the Weyl point node in thebulk excitation spectrum. Indeed, the B-phase of UPt isa point nodal SC compatible with Blount’s theorem when the p -wave and d -wave order parameters are ap-propriately taken into account . Although the purely f -wave state has a nodal line at k z = 0 , it is an acciden-tal node removed by symmetry-preserving perturbation.In accordance with the bulk-boundary correspondence,the Majorana arcs appear on the surface and terminateat the projection of Weyl point nodes illustrated by greencircles in Figs. 4 and 5.Interestingly, the position of Weyl nodes is tunable. Inthe E scenario for UPt , the parameter η smoothlychanges from ∞ to 0 in the B-phase by decreasing thetemperature and/or increasing the magnetic field (seeFig. 3 and Table I). Then, the pair creation, pair annihi-lation, and coalescence of Weyl nodes occur as a con-sequence of the p - f mixing in the order parameter .Accordingly, the projection of Weyl nodes moves as il-lustrated in Figs. 4(b)-(d) and 5(b)-(d). The Majoranaarcs follow the Weyl nodes.In the generic E -state studied in this paper, the Weylnodes are purely protected by topology, and any crys-tal symmetry is not needed. Therefore, the positions ofWeyl nodes are not constrained by any symmetry. Al-though the Weyl nodes are pinned at the poles of FSin the purely f -wave E -state , that is an accidentalresult. In another candidate of Weyl SC URu Si , the33D d xz ± id yz -wave superconductivity has been revealedby experiments. Then, the Weyl nodes are pinnedand the traveling of Weyl nodes does not occur, in con-trast to UPt . (a) k z =1.98 [ ν (k z )=0] (b) k z =2.59 [ ν (k z )=4](c) k z =2.89 [ ν (k z )=8] (d) k z =3.09 [ ν (k z )=-4](e) k z = π [ ν (k z )=-4] (f) α =1, k z = π [ ν (k z )=-4] -0.06-0.04-0.02 0 0.02 0.04 0.06-1.5 -1 -0.5 0 0.5 1 1.5 E ( ky , kz ) ky -0.2-0.1 0 0.1 0.2-1.5 -1 -0.5 0 0.5 1 1.5 E ( ky , kz ) ky-0.06-0.04-0.02 0 0.02 0.04 0.06-1.5 -1 -0.5 0 0.5 1 1.5 E ( ky , kz ) ky -0.4-0.2 0 0.2 0.4-1.5 -1 -0.5 0 0.5 1 1.5 E ( ky , kz ) ky-0.6-0.4-0.2 0 0.2 0.4 0.6-1.5 -1 -0.5 0 0.5 1 1.5 E ( ky , kz ) ky -0.6-0.4-0.2 0 0.2 0.4 0.6-1.5 -1 -0.5 0 0.5 1 1.5 E ( ky , kz ) ky FIG. 11. (Color online) Energy spectrum on the (100)-surfacein the B-phase ( η = 0 . k z = constant planes are shown.The surface states are emphasized by green lines. The k z -dependent Chern number is shown in each panel. (a)-(e)Parameters are the same as Fig. 5(b). (f) α = 1 while theothers are the same as (e). The surface states are almosttwo-fold degenerate in (d) and (f) and four-fold degeneratein (e). Comparison of (e) and (f) reveals that the four-folddegeneracy in the absence of the SOC is lifted by the SOC. Here the number of Majorana arcs is verified by calcu-lating the Chern number of effective 2D models on k z = constant planes, ν ( k z ) = 12 π Z d k k F z ( k ) , (6.3)that is, a k z -dependent Chern number of class A. TheChern number indicates the number of chiral surfacemodes. In Weyl SCs, the Chern number may changeat a gapless k z = constant plane hosting Weyl nodes.Therefore, the zero energy surface states form arcs ter-minating at the projection of Weyl nodes. For parame-ters reproducing the A -FSs, the Chern number changes ν ( k z ) = 0 → → → − k z from 0 to π , while ν ( k z ) = 0 → → . Thebulk-boundary correspondence is confirmed by showingthe surface spectrum on the k z = constant lines inFig. 11. The number of chiral modes coincides with the k z -dependent Chern number. We also observe the signreversal of chirality in accordance with the sign changeof the Chern number.Finally, we discuss the Weyl superconducting phasein the phase diagram illustrated in Fig. 3. Because theTRS has to be broken in Weyl SCs, the A- and C-phasesare non-Weyl superconducting states. Furthermore, theB-phase in the vicinity of the A-B and B-C phase bound-aries is also non-Weyl state because the gap closing is re-quired for the topological transition. Therefore, the tran-sition from the non-Weyl state to the Weyl state occursin the B-phase. The shaded region in Fig. 3 schematicallyillustrates the Weyl superconducting phase. B. Majorana cone and mirror Chern number
Next we discuss the surface state around k sf = (0 , K -FS are considered in this subsection. k y k z E ( k s f ) -0.3 0.3 -0.10.1 -0.06-0.04-0.02 0 0.02 0.04 0.06 -0.06-0.04-0.02 0 0.02 0.04 0.06 FIG. 12. (Color online) Tilted Majorana cone at k sf =(0 ,
0) in the B-phase ( η = 0 . t, t z , t ′ , α, µ, ∆ , δ ) = (1 , , , , , , .
02) reproducing a Γ-FS.
We may understand the topological protection by im-plementing the crystal mirror reflection symmetry withrespect to the xy -plane. Mirror reflection operator forthe normal part Hamiltonian is, M xy ( k z ) = is z V σ ( k z ) . (6.4)The mirror reflection symmetry is equivalent to the prod-uct of inversion symmetry and screw symmetry, that4is, M xy ( k z ) = IS zπ ( k z ). The nonsymmorphic screwsymmetry S zπ = { R zπ | z } involves half translation alongthe z -axis, and therefore, the screw operator S zπ ( k z ) is k z -dependent. Thus, the mirror operator is also k z -dependent, and we have M xy ( π ) = is z σ z while M xy (0) = is z . This momentum dependence of M xy ( k z ) may yieldthe unusual line node in nonsymmorphic odd-paritySCs , a counterexample of Blount’s theorem.The normal part Hamiltonian is invariant under themirror reflection symmetry M xy ( k z ) ˜ H ( k ) M xy ( k z ) − = ˜ H ( k x , k y , − k z ) , (6.5)and the order parameter is mirror-odd irrespective of η , M xy ( k z ) ˜∆( k ) M xy ( − k z ) T = − ˜∆( k x , k y , − k z ) . (6.6)Thus, the BdG Hamiltonian respects mirror reflectionsymmetry, M xy BdG ( k z ) ˜ H BdG ( k ) M xy BdG ( k z ) − = ˜ H BdG ( k x , k y , − k z ) , (6.7)by defining the operator in the Nambu space, M xy BdG ( k z ) = (cid:18) M xy ( k z ) 00 − M xy ( − k z ) ∗ (cid:19) τ (6.8)= M xy ( k z ) ⊗ τ . (6.9)According to the K -theory for topological crystallineinsulators and SCs , the effective 2D Hamiltonian atmirror invariant planes, namely, k z = 0 and π , is specifiedby a topological index of class D, Z ⊕ Z , in the TRS brokenB-phase. This is ensured by the algebra [ M xy BdG (0)] =[ M xy BdG ( π )] = − { M xy BdG (0) , C } = { M xy BdG ( π ) , C } =0. One of the two integer topological invariants is nothingbut the Chern number ν ( k z ) introduced in Sec. VI A. Theother is the mirror Chern number, ν Γ z M ∈ Z (Γ z ≡ π ), which is defined below by using the mirror reflectionsymmetry . In the TRS invariant A- and C-phases,the Chern number must be zero, and the mirror Chernnumber is naturally the Z topological index of class DIIIappearing in Ref. 48.The commutation relation, h M xy BdG (Γ z ) , ˜ H BdG ( k k , Γ z ) i = 0, ensures that theBdG Hamiltonian is block-diagonalized at mirror in-variant planes on the basis diagonalizing M xy BdG (Γ z ). Inother words, the BdG Hamiltonian is decomposed intotwo mirror-subsectors with mirror eigenvalues ± i ,˜ H BdG ( k k , Γ z ) = ˜ H Γ z i ( k k ) ⊕ ˜ H Γ z − i ( k k ) . (6.10)The PHS is preserved in the mirror-subsector, becauseof [ M xy BdG (Γ z )] = − { M xy BdG (Γ z ) , C } = 0. On theother hand, the TRS is not preserved even in the TRS in-variant A- and C-phases since [ M xy BdG (Γ z ) , T ] = 0. Thus,the symmetry of the mirror-subsector is class D irre-spective of η , and the Chern number of mirror-subsectorHamiltonian given by ν Γ z ± i = 12 π Z d k k F Γ z z, ± i ( k k ) , (6.11) may be nontrivial. Here, F Γ z z, ± i ( k k ) is the Berry curvatureof ˜ H Γ z ± i ( k k ). The mirror Chern number is defined by ν Γ z M = ν Γ z + i − ν Γ z − i , (6.12)while the total Chern number is given by ν (Γ z ) = ν Γ z + i + ν Γ z − i .
1. Mirror Chern number at k z = 0 Later we show that the mirror Chern number at k z = π has to vanish owing to the constraint by glide symmetry.On the other hand, the mirror Chern number may benontrivial at k z = 0, and the surface states around k sf =(0 ,
0) are indeed protected by the mirror Chern number.Because we have M xy BdG (0) = is z σ τ , the mirror-subsector Hamiltonian ˜ H ± i ( k k ) is equivalent to the spinsector for s = ↑ and ↓ , respectively. Thus, we obtain˜ H ± i ( k k ) = (cid:18) ˆ h ± i ( k k ) ˆ∆ ± i ( k k )ˆ∆ ± i ( k k ) † − ˆ h ± i ( − k k ) T (cid:19) , (6.13)withˆ h ± i ( k k ) = (cid:18) ε ( k k ) ± αg ( k k ) ˜ a ( k k )˜ a ( k k ) ∗ ε ( k k ) ∓ αg ( k k ) (cid:19) , (6.14)andˆ∆ ± i ( k k ) = − ( η ± p (cid:2) p x ( k k ) ± ip y ( k k ) (cid:3) σ . (6.15)We denoted A ( k k ) = A ( k k ,
0) and ∆ p = δ ∆ / p η .It turns out that the mirror-subsector Hamiltonian isequivalent to the BdG Hamiltonian of a two-band chi-ral p -wave SC. In our model for the Γ-FS, only oneband crosses the Fermi level, and we obtain ν ± i = ± p -wave order parameter [see Eq. (6.15)]. Therefore, thetotal Chern number of the 2D BdG Hamiltonian is zero, ν (0) = ν i + ν − i = 0, even in the TRS broken B-phase.On the other hand, the mirror Chern number is nontriv-ial, ν = 2 . (6.16)We now understand that the (tilted-) Majorana conein Fig. 12 is the topological surface states ensured bythe bulk-boundary correspondence. Since the chirality ofMajorana modes corresponding to ν ± i = ± k z = 0, and the helical mode is gapped at k z = 0, implying the Majorana cone.Finally, we comment on the multiband effect. Al-though Eq. (6.16) is obtained for a hole Γ-FS, we ob-tain ν = − . Because the mirror Chern number is ad-ditive, we will obtain the mirror Chern number ν = − k sf = (0 ,
0) and two cones away fromthe Γ-point, k sf = ( ± k y , K -FSs havealso been predicted by band structure calculations ,the existence of them is still under debate . The K -FSsalso give nontrivial mirror Chern number ν = − ν = −
14 bytaking into account all the FSs. In any case, ν ∈ Z + 2indicates the existence of Majorana cone at k sf = (0 ,
2. Vanishing mirror Chern number at k z = π We here show that the mirror Chern number at k z = π must be trivial owing to the glide symmetry, namely, ν π M = 0 . (6.17)First, we consider the glide invariant A- and C-phases.The glide symmetry is also preserved in the mirror-subsectors at k z = π because of [ M xy BdG ( π ) , G xz BdG ( π )] = 0,although { M xy BdG (0) , G xz BdG (0) } = 0 indicates the brokenglide-symmetry in the mirror-subsectors at k z = 0. Then,we can prove the relation for Berry curvature, F πz, ± i ( k x , k y ) = − F πz, ± i ( k x , − k y ) . (6.18)Integration over the ( k x , k y ) plane ends up vanishingChern number, ν π ± i = 0, and thus, the mirror Chernnumber also vanishes. In the B-phase, the glide symme-try is spontaneously broken. However, considering themagnetic-glide symmetry T G xz BdG ( π ), we can show therelation F πz, ± i ( k x , k y ) = F πz, ∓ i ( − k x , k y ) , (6.19)which leads to ν πi = ν π − i . Therefore, the mirror Chernnumber at k z = π vanishes in the B-phase as well.The trivial mirror Chern number is confirmed in ourmodel as follows. Using the mirror reflection opera-tor M xy BdG ( π ) = is z σ z τ , we obtain the mirror-subsectorHamiltonian respecting the PHS,˜ H π ± i ( k k ) = (cid:18) ˆ h ± i ( k k ) ˆ∆ ± i ( k k )ˆ∆ ± i ( k k ) † − ˆ h ± i ( − k k ) T (cid:19) . (6.20)The normal part is given byˆ h ± i ( k k ) = (cid:2) ε ( k k ) ± αg ( k k ) (cid:3) σ . (6.21)For instance, the order parameter part isˆ∆ ± i ( k k ) = ∆ × (cid:18) ∓ δ (cid:2) p x ( k k ) ± ip y ( k k ) (cid:3) ˜ f ( x − y ) z ( k k ) + i ˜ d yz ( k k )˜ f ( x − y ) z ( k k ) ∗ − i ˜ d yz ( k k ) ∗ ± δ (cid:2) p x ( k k ) ∓ ip y ( k k ) (cid:3) (cid:19) , (6.22)in the C-phase. When the d + f -wave component is dom-inant as we assume in this paper, the p -wave compo-nent can be adiabatically reduced to zero without clos-ing the gap. Then, it turns out that the Chern number of mirror-subsectors is trivial because the phase windingof ˜ f ( x − y ) z ( k k ) ± i ˜ d yz ( k k ) along the FS is zero. Evenwhen the p -wave component is dominant, the Chernnumber vanishes because the chirality of gap function p x ( k k ) ± ip y ( k k ) is opposite between the pseudospin upand down Cooper pairs. Thus, we obtain ν π ± i = 0 and ν π M = 0 in the C-phase. It is straightforward to show ν π ± i = 0 in the A-phase as well. In the B-phase we haveobtained the nontrivial Chern number ν ( π ) = − A -FSs. However, the mirror Chern number remains triv-ial, because ν π ± i = −
2. We have numerically confirmedEq. (6.17) in the entire A-, B- and C-phases for all theFSs.
C. Majorana flat band and glide winding number
As shown in Figs. 4(d) and (e) and Figs. 5(d) and (e),the zero energy surface flat band appears in the A-phaseand in a “half” of the B-phase ( | η | >
1. A-phase
Let’s consider the glide invariant plane k y = 0 in the A-phase. The glide winding number is defined for 1D mod-els ˜ H BdG ( k x , , k z ) parametrized by k z . The 1D modelsdo not respect TRS and PHS unless (0 , , k z ) is a time-reversal invariant momentum. On the other hand, thecombined chiral symmetry, Γ = iT BdG C , is preserved.Thus, the winding number of 1D AIII class can be de-fined. However, it is obtained to be zero.The nontrivial winding number is obtained by imple-menting the glide symmetry, which has been representedby Eq. (5.2). The glide symmetry ensures the sector de-composition,˜ H BdG ( k x , , k z ) = ˜ H λ + ( k x , k z ) ⊕ ˜ H λ − ( k x , k z ) , (6.23)for eigenvalues λ ± = ± ie − ik z / of the glide operator.The chiral symmetry is preserved in the glide-subsectorHamiltonian ˜ H λ ± ( k x , k z ) because [Γ , G xz BdG ( k z )] = 0 inthe A-phase . Now we have two winding numbers ofAIII class, ω G (+ , k z ) and ω G ( − , k z ), which correspondto the Z ⊕ Z topological index of 1D AIII class with U + crystal symmetry .We here estimate the winding numbers by analyz-ing the original BdG Hamiltonian ˆ H BdG ( k ), instead of˜ H BdG ( k ). The periodicity along the k x -axis is satis-fied in ˆ H BdG ( k ), and the unitary transformation (2.20)does not alter the winding number, since [Γ , U ( k )] = 0.The glide operator for the original BdG Hamiltonianˆ H BdG ( k ) is G xz BdG = is y σ x τ e − ik z / in the A-phase while G xz BdG = is y σ x τ z e − ik z / in the C-phase.6In the A-phase, the glide-subsectors of ˆ H BdG ( k ) are,ˆ H λ ± ( k x , k z ) = ε k z ( k x ) σ τ z ± a k z ( k x ) σ z τ z + αg ( k x ) σ y τ − ∆ δp x ( k x ) σ τ x ± ∆ d k z xz ( k x ) σ y τ y . (6.24)The chiral symmetry is confirmed by n Γ s , ˆ H λ ± ( k x , k z ) o = 0, where Γ s = σ z τ y is thechiral operator in the subsector space. Thus, we obtainthe off-diagonal form U Γ s ˆ H λ ± ( k x , k z ) U † Γ s = (cid:18) q ± ( k x , k z )ˆ q †± ( k x , k z ) 0 (cid:19) , (6.25)by choosing the basis diagonalizing the chiral operator.From Eq. (6.24), we obtain q ± ( k x , k z ) = iε k z ( k x ) σ z ± ia k z ( k x ) σ + ∆ δp x ( k x ) σ + [ αg ( k x ) ± ∆ d k z xz ( k x )] σ y , (6.26)for the λ ± glide-subsector, respectively. We used abbre-viations, A k z ( k x ) = A ( k x , , k z ).Now the winding number of glide-subsectors given by ω G ( ± , k z ) = 14 πi Z π dk x Tr h ˆ q ± ( k x , k z ) − ∂ k x ˆ q ± ( k x , k z ) − ˆ q †± ( k x , k z ) − ∂ k x ˆ q †± ( k x , k z ) i , (6.27)is calculated. By adiabatically reducing αg ( k x ) → d k z xz ( k x ) → ω G ( ± , k z )= ( ∓ ε ( , k z ) + a ( , k z ) > > ε ( , k z ) − a ( , k z )]0 [otherwise] , (6.28)for t > t ′ > δ >
0. This means that the λ ± glide-subsectors of ˆ H BdG ( k ) [and equivalently the sub-sectors of the periodic BdG Hamiltonian ˜ H BdG ( k )] aretopologically characterized by the glide-winding number ω G ( ± , k z ) = ∓
1, when the condition ε ( , k z ) + a ( , k z ) > > ε ( , k z ) − a ( , k z ) is satisfied. This condition is equiv-alent to the number of FSs (per Kramers pairs) is odd.In Figs. 4(e) and 5(e), the flat band appears on the k y = 0 line of surface BZ where only one FS is pro-jected. The nontrivial glide-winding number demon-strated above protects this Majorana flat band. The zeroenergy states are two-fold degenerate in accordance withthe bulk-boundary correspondence. One comes from the λ + = ie − ik z / glide-subsector and the other comes fromthe λ − = − ie − ik z / glide-subsector. Note that the flatband is robust against the multiband effect. We find thatthe glide-winding number of K -FSs is zero. Taking into account three Γ-FSs, we will have glide-winding number ω G ( ± ,
0) = ∓
3, or ∓
1, or ±
1, or ±
2. B-phase
The glide-subsector is no longer well-defined in the B-phase, because the glide symmetry is spontaneously bro-ken. However, the glide-winding number is well-definedby the magnetic-glide symmetry G xz BdG T preserved in theB-phase. Then, the glide-winding number is given by ω G ( k z ) = i π Z π dk x Tr h Γ G ˜ H BdG ( k x , , k z ) − × ∂ k x ˜ H BdG ( k x , , k z ) i , (6.29)where Γ G = e iφ G xz BdG ( k z ) T BdG C is the glide-chiral oper-ator with Γ = 1.In the A-phase, Eq. (6.29) is reduced to ω G ( k z ) = ω G (+ , k z ) − ω G ( − , k z ) . (6.30)Thus, we obtain ω G ( k z ) = − | η | is large [see Fig. 13(a)]. When | η | is decreasedfrom infinity, the pair creation of Weyl nodes occurs inthe bulk BZ on the k y = 0 plane . Then, a part ofthe Majorana flat band disappears in between the pairof projected Weyl points [see Fig. 13(b)]. Therefore, theprojected Weyl points are end points not only of the Ma-jorana arc but also of the Majorana flat band. This fea-ture has been shown in Figs. 4(d) and 5(d). (a) (b) (c) FIG. 13. (Color online) Illustration of the Majorana flat band(a) in the A-phase and non-Weyl B-phase ( η > η c ), (b) in theWeyl B-phase ( η c > η > η = 1). Thick solid (purple) lines show the Majorana flatband. Thin lines illustrate the projection of a Γ-FS onto the(100)-surface BZ. The closed (blue) circles indicate projec-tions of Weyl point nodes. (a), (b), and (c) correspond to thenumerical results in Figs. 4(e), (d), and (c), respectively. At | η | = 1, a pair of Weyl nodes is annihilated onthe k = ( k x , ,
0) line, and other Weyl nodes coalesce onthe poles of FSs . Then, the Majorana flat band com-pletely disappears [Fig. 13(c)]. The fate of the Majorana7flat band in the B-phase is schematically illustrated inFig. 13, and shown in Figs. 4 and 5 by the numericaldiagonalization of the BdG Hamiltonian. D. Symmetry constraint on winding numbers
The crystal symmetries preserved on the (100)-surfaceare as follows. • Mirror symmetry M xy . • Glide symmetry G xz . • π -rotation symmetry R x .The π -rotation is given by the product of mirror and glideoperations.In addition to the glide-winding number studied inSec. VI C, we can define the mirror-winding number and the rotation-winding number in the same manner.They are given by ω Γ z M ( k y ) = i π Z π dk x Tr h Γ M (Γ z ) ˜ H BdG ( k x , k y , Γ z ) − × ∂ k x ˜ H BdG ( k x , k y , Γ z ) i , (6.31)and ω Γ z R = i π Z π dk x Tr h Γ R ˜ H BdG ( k x , , Γ z ) − × ∂ k x ˜ H BdG ( k x , , Γ z ) i . (6.32)Γ M (Γ z ) = e iθ M xy BdG (Γ z )Γ and Γ R = e iθ ′ R x BdG
Γ aremirror-chiral operator and rotation-chiral operator, re-spectively. The phase factors e iθ and e iθ ′ are chosen sothat Γ M (Γ z ) = Γ = 1. The mirror-winding number isdefined on the mirror invariant planes at k z = Γ z = 0 , π and k y -dependent. On the other hand, the rotation-winding number is defined on the rotation invariant lines.The mirror-winding number is defined only in the TRSinvariant A- and C-phases, since the mirror-chiral sym-metry is broken in the TRS broken B-phase.From the algebra of symmetry operations we can provethat most of the winding numbers vanish. The proofrelies on the fact that the winding number disappearswhen any unitary symmetry preserved on the surfaceanti-commutes with the chiral operator, { U, Γ V } = 0.This fact, ω V = 0, is understood by ω V = i π Z π dk x Tr h U Γ V ˜ H ( k x ) − ∂ k x ˜ H ( k x ) U † i = i π Z π dk x Tr h ( − Γ V ) ˜ H ( k x ) − ∂ k x ˜ H ( k x ) i = − ω V . (6.33) Furthermore, the TRS has to satisfy [ T, Γ V ] = 0 whenthe winding number is nontrivial. All of the mirror, glide,and rotation symmetries are preserved at the rotation in-variant lines in the A- and C-phases, although the glideand rotation symmetries are spontaneously broken in theB-phase. Thus, we obtain some constraints on the wind-ing numbers at k sf = (0 ,
0) and (0 , π ) in the A- andC-phases. Γ z c ( M xy , Γ M ) c ( G xz , Γ M ) c ( R x , Γ M ) c ( T, Γ M )A-phase 0 -1 -1 +1 -1 π -1 +1 -1 -1C-phase 0 -1 +1 -1 -1 π -1 -1 +1 -1TABLE III. Commutation (anti-commutation) relations ofthe mirror-chiral operator Γ M with the crystal symmetry andtime-reversal operators are represented by +1 ( − z c ( M xy , Γ G ) c ( G xz , Γ G ) c ( R x , Γ G ) c ( T, Γ G )A-phase 0 +1 +1 +1 +1 π -1 +1 -1 -1C-phase 0 +1 -1 -1 -1 π -1 -1 +1 +1TABLE IV. Commutation (anti-commutation) relations ofthe glide-chiral operator Γ G with the crystal symmetry andtime-reversal operators.Γ z c ( M xy , Γ R ) c ( G xz , Γ R ) c ( R x , Γ R ) c ( T, Γ R )A-phase 0 +1 -1 -1 -1 π -1 +1 -1 -1C-phase 0 +1 +1 +1 +1 π -1 -1 +1 +1TABLE V. Commutation (anti-commutation) relations of therotation-chiral operator Γ R with the crystal symmetry andtime-reversal operators. The commutation (anti-commutation) relations be-tween crystal symmetry operators M xy , G xz , R x andchiral operators Γ M , Γ G , and Γ R are summarized in Ta-bles III, IV, and V. From these algebra, we find that only ω G (0) and ω may be nontrivial. Interestingly, all thewinding numbers at k z = π vanish as a consequence ofthe nonsymmorphic glide symmetry. The mirror-windingnumber at k z = 0 also vanishes in both A- and C-phases.Furthermore, we see that the rotation-winding number ω disappears in the A-phase, while the glide-windingnumber ω G (0) disappears in the C-phase. These symme-try constraints are consistent with our numerical calcula-tions summarized in Table. VI, and also consistent withrecently obtained general rules for winding numbers .In addition to the glide-winding number ω G (0) dis-cussed in Sec. VI C, we may have a nontrivial rotation-winding number, which is introduced below for com-pleteness. Combining the π -rotation symmetry withTRS, we define the magnetic π -rotation symmetry by8 | η | > | η | < ω G (0) -2 0 ω T ′ = R xπ T = − is z σ x K . The BdG Hamiltonian is invari-ant T ′ BdG ˜ H BdG ( k ) T ′ − = ˜ H BdG ( − k x , k y , k z ) , (6.34)under the magnetic π -rotation in the Nambu space, T ′ BdG = (cid:18) T ′ T ′∗ (cid:19) τ = T ′ ⊗ τ z , (6.35)not only in the rotation invariant A- and C-phases butalso in the B-phase. According to the classification by K -theory , the 2D Hamiltonian of D class on the k z = 0 or π plane is specified by a Z ⊕ Z topological invariant byimplementing the magnetic π -rotation symmetry. Therelations ( T ′ BdG ) = 1 and [ T ′ BdG , C ] = 0 are used there.One of the integer topological numbers is the rotation-winding number given by Eq. (6.32), where the rotation-chiral operator is Γ R = T ′ BdG C = s z σ x τ y . E. Topological transition in B-phase
In Sec. VI D, symmetry constraints on the windingnumbers have been proved in the A- and C-phases. Inthis subsection the B-phase is discussed. We again seethat ω G ( π ) = ω π R = 0 owing to the mirror symmetry. Onthe other hand, we obtain ω G (0) = − | η | > ω = − | η | < | η | = 1 the jump of the winding numbers ω G (0) and ω indicates the gap closing. Equation (6.15) shows thatthe superconducting gap on the k z = 0 plane actuallydisappears at | η | = 1. This gap node has been reportedas unusual “quadratic line node” . In contrast to theusual linear line node with ∆( k ) ∝ | k z | , which appearsin the purely f -wave E -state , the line node of thegeneric E -state is accompanied by the quadratic be-havior, ∆( k ) ∝ | k z | . Such an unusual nodal structureat | η | = 1 has been attributed to the pair annihilationof Weyl nodes . It can also be viewed as a criticality oftopological phase transition specified by ω G (0) and ω .In contrast to the k z = 0 plane, all of the windingnumbers on the k z = π plane are zero irrespective of η .Thus, the gap closing enforced by the change of windingnumbers does not occur at k z = π . This is consistent withthe numerical result showing the finite superconductinggap on the k z = π plane. VII. SUMMARY AND DISCUSSIONS
We investigated topologically nontrivial superconduct-ing phases in UPt . Taking into account the FSs reportedby first principles band structure calculation and quan-tum oscillation experiments, we have calculated the topo-logical invariants specifying the superconducting statesand demonstrated topological surface states.Among a variety of topological properties in UPt , themost intriguing result is the nontrivial glide- Z invariantin the TRS invariant A-phase. By using the K -theory fortopological nonsymmorphic insulators/superconductors,we showed that the glide- Z invariant is the strong topo-logical index specifying the 3D glide-even superconduc-tivity of class DIII. Although UPt is a gapless SC inthe bulk, the glide- Z invariant is well-defined and non-trivial. Thus, the UPt A-phase can be reduced to a 3Dgapped TNSC with keeping double Majorana cone sur-face states, when the point nodes are removed by someperturbations. By these findings, UPt is identified as a3D gapless TNSC. At our best knowledge, this is the firstproposal for the material realization of emergent topolog-ical superconductivity enriched by nonsymmorphic spacegroup symmetry.Not only the A-phase but also the B- and C-phaseshave been identified as symmetry-enriched topologicalsuperconducting states. Combining the crystal symme-tries of UPt with the TRS and PHS, we find topologicalinvariants and surface states as follows. • Double Majorana cone protected by the glide- Z invariant in the A-phase • Chiral Majorana arcs in the Weyl B-phase • Majorana cone protected by the mirror Chern num-ber in the A-, B-, and C-phases • Majorana flat band protected by the glide-windingnumber in the A-phase and “half” of the B-phaseIt has been proved that the other mirror Chern num-ber and winding numbers must be trivial because of theconstraints by symmetry.From the results obtained in this paper, we notice richtopological properties of superconducting UPt . Under-lying origins of such topological superconducting phasesare as follows. (1) Spin-triplet odd-parity superconduc-tivity, which is often a platform of topological SC. (2) 2D E representation, which allows multiple superconduct-ing phases distinguished by symmetry. (3) Nonsymmor-phic space group symmetry P /mmc , which gives riseto following features distinct from symmorphic systems,1. Classification of topological insulators and SCschanges, and allows emergent topological phases.2. Dirac nodal lines yield the paired FSs which corre-spond to the pseudospin degree of freedom in glide-subsectors.93. The sublattice-singlet d -wave pairing naturally ad-mixes with the f -wave pairing, and leads to thenontrivial glide- Z invariant.4. Most mirror Chern numbers and winding numbersare forced to be zero, and do not support topolog-ical surface states.Thus, an old heavy fermion superconductor UPt is a pre-cious platform of topological superconductivity enrichedby nonsymmorphic space group symmetry. ACKNOWLEDGMENTS
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