Topological crystalline superconductors with linearly and projectively represented C n symmetry Chen Fang Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China B. Andrei Bernevig Department of Physics, Princeton University, Princeton NJ 08544 Matthew J. Gilbert 7 Micro and Nanotechnology Laboratory, University of Illinois, Urbana IL 61801 1 Department of Electrical and Computer Engineering, University of Illinois, Urbana IL 61801 and 0 Department of Electronics Engineering, University of Rome ”Tor Vergata”, Rome, Italy 00133 2 (Dated: January 10, 2017) n a We study superconductors with n-fold rotational invariance both in the presence and in the J absence of spin-orbit interactions. More specifically, we classify the non-interacting Hamiltonians by defining a series of Z-numbers for the Bogoliubov-de Gennes (BdG) symmetry classes of the 8 Altland-Zimbauer classification of random matrices in 1D, 2D, and 3D in the presence of discrete ] rotationalinvariance. Ouranalysisemphasizestheimportantroleplayedbytheangularmomentum n oftheCooperpairsinthesystem: forpairingsofnonzeroangularmomentum,therotationsymmetry o may be represented projectively, and a projective representation of rotation symmetry may have c anomalous properties, including the anti-commutation with the time-reversal symmetry. In 1D - r and 3D, we show how an n-fold axis enhances the topological classification and give additional p topological numbers; in 2D, we establish a relation between the Chern number (in class D and CI) u andtheeigenvaluesofrotationsymmetryathigh-symmetrypoints. Foreachnontrivialclassin3D, s we write down a minimal effective theory for the surface Majorana states. . t a m I. INTRODUCTION significantstepforwardwhentheHeuslerclassofmateri- - als were predicted as topological candidates29–32. While d the specific focus of the original work29,30 had been to n Condensed matter physics has, in recent years, been o partly focused on the search for new materials that har- exploretheexistenceofadditionalTRStopologicalband c insulatorcandidatesintheHeuslerclassofmaterials,the bor topological states. A topological state is a gapped [ Heusler compounds exhibit an extremely wide range of many-body state that cannot be adiabatically connected physicalphenomenasuchasferromagnetism33,whichex- 1 to the atomic limit while preserving a certain symme- v try group, and yet cannot be associated with any local presslybreaksTRS,andheavyfermionbehavior34. This 4 workprovidedearlyindicationsthatmultiplesymmetries order parameter. In place of order parameters, topologi- 4 can be present in materials and establish different topo- cal numbers, a global quantity contributed to by all the 9 logical orders. Fu35 was the first to provide an explicit electrons in system, distinguish a topological state from 1 proof that, in 2D and 3D band insulating systems, the 0 a trivial one. Initial efforts have focused on the eluci- presence of rotational symmetry in the underlying lat- . dation of topological states whose existence and global 1 tice structure, namely C , together with time-reversal properties are stabilized by the presence of time-reversal 4 0 symmetry give rise to a new Z classification for such 7 symmetry(TRS).Thissearchhasledtotheexamination 2 insulators even in the absence of spin-orbit coupling. 1 ofaplethoraof2D1–10and3Dnon-interactingbandinsu- Such materials which have topological states whose exis- : lating systems11–25 under a wide variety of experimental v tence is guaranteed by the presence of underlying crys- conditions seeking to explore the fundamental spin and Xi charge behavior of TRS topological band insulators. Yet talline symmetry are commonly referred to as topolog- ical crystalline insulators (TCI). From this early work, r the underlying principles of symmetry preservation re- a the search for topological materials beyond those with quired for the stability of a topological phase within a TRS has since been expanded. Predictions of other given host material are quite general in nature thereby types that can be generally classified as topological crys- makingtheTRSclassoftopologicalnon-interactingband talline systems have emerged, including that of inversion insulators but one in a long list of candidate topological symmetric topological materials36–38, and of rotation- materials. Therefore, while we understand a great deal ally invariant topological band insulators39. Topologi- about the physical nature of TRS topological band in- cal crystalline insulators protected by mirror reflection sulators, we are at but the beginning in the search for symmetry40–45 and glide reflection symmetries46–51 have topological materials26–28. beentheoreticallystudiedandsomehavebeenconfirmed From theoretical point of view, the discussion of topo- in experiments52–55. Most recently, predictions of topo- logical materials beyond those that preserve TRS took a 2 logical semimetals whose band crossings are protected their implications. byrotationalinvariance56–58 havealsoseenexperimental confirmation in Na Bi59 and in Cd As 60,61. 3 3 2 II. PRELIMINARIES One can further consider the case where particle num- ber is no longer conserved - that is, the case of su- perconductors, represented by a Bogoliubov-de Gennes A. Enhancement of the AZ Classification by Local Unitary Symmetries (BdG) Hamiltonian. Ref.[62–64] studied the topologi- cal classifications of fully gapped superconductors in all dimensions both in the presence and absence of time- The AZ classification ofsingle particle Hamiltonians76 reversalsymmetryandspin-rotationsymmetry. Thisini- is based on the transformation properties of the sin- tialworkhasbeenfollowedbyfurtherresearchintotopo- gle particle Hamiltonian under two local symmetries, logicalphasesinsuperconductorscontainingvarioussym- namely, particle-hole symmetry (PHS, or P) and time- metries such as: TRS superconductors65–67, reflection reversal symmetry (TRS, or T), and their composi- symmetric superconductors68,69, non-centrosymmetric tion which is called chiral symmetry. A non-interacting superconductors64, topological superfluid 3He-B65,70, Hamiltonian here refers to a Hamiltonian that only has and Weyl superconductors71,72. Nevertheless, the su- quadraticcouplingsamongthecreationandannihilation perconducting phase of materials with general rota- operators (in the second quantized form), and can be tional invariance has remained relatively unstudied (yet put into a matrix, H, in the orbital basis (or Nambu ba- see Ref.[73] for a discussion of superconductors with sis when charge is not conserved) in the first quantized twofold symmetries). Rotational symmetries are im- form. The two local antiunitary symmetries are, given a portant in understanding the superconducting behavior basis, represented by KT and KP, where T,P are uni- demonstrated in the Heusler alloys LaBiPt74 and, most tary matrices and K complex conjugation. A Hamilto- recently, in YPtBi75. Hence there is a need for theoreti- nian H is said to have T if and only if [KT,H] = 0 and cal elucidation of the possible corresponding topological haveP ifandonlyif{KP,H}=0. Weremarkthatthese nature of such superconducting systems. formulas apply in the first quantized form, where all op- erators are represented by matrices, while in the second Inthisworkwestudytopologicalsuperconductorsthat quantized form, where operators are expanded in terms possess C rotational symmetry with and without spin- n of fermion annihilation and creation operators, both P orbitalinteractions,andaskifthepresenceofC symme- n andTshouldcommutewithH,andPbecomesaunitary try can stabilize additional topological crystalline super- operator. Hereafter, we use hatted symbols for second conductors. WealsoexplorehowtheC symmetryplaces n quantized operators, and non-hatted ones for their first constraints on the invariants of the original BdG classes. quantizedforms.WhenbothP andT arepresent,wecan Weaskthisquestionbroadlybyconsideringthebehavior define S = KP ∗KT = P∗ ∗T such that {S,H} = 0, of each of the 4 distinct Bogoliubov- de Gennes (BdG) and we say the system has chiral symmetry S. However, symmetry classes, namely class C, D, CI and DIII, of there are cases where S is a symmetry, i.e., {S,H} = 0, random matrices from the Altland and Zimbauer (AZ)76 while neither KP nor KT is a symmetry. in 1D, 2D, and 3D. The answer to such a question has Now we consider a local unitary symmetry added to direct relevance to a wide range of unconventional su- thesystem,representedbysomeunitarymatrixL,which perconductingmaterialssuchascupratesandiron-based generically satisfies superconductors. In Section II, we discuss the necessary general background to understand the subsequent anal- [L,H]=0. (1) ysis. In Section III, we perform a complete classifica- tion of 1D superconductors with rotational symmetries. ThereforetheHamiltoniancanbeblock-diagonalizedinto In the classification, we find it important to distinguish sectors spanned by eigenvectors of L, namely, the case where the C symmetry is represented projec- n H =H ⊕H ⊕...+H , (2) tively from the case where it is linearly represented: in r1 r2 rl the former case, time-reversal symmetry anti-commutes where s is the number of eigenvalues of L, and r 1,...,l with the rotation symmetry when acting on a Bogoli- the eigenvalues; in sector r , the basis vectors are eigen- i ubov quasiparticle. Which case appears depends on the states of L with eigenvalue r . For each sector, we can i total angular momentum of the Cooper pair. In Section classify each H according to its transformation under ri IV, we consider Chern superconductors in 2D with ro- time-reversal,particle-holeandchiralsymmetries. Phys- tational symmetries, deriving explicit relations between ically, time-reversal and particle-hole symmetries com- the Chern number and the rotation eigenvalues of occu- mutewithallspatialsymmetries, andletusassumethat pied bands at high-symmetry points. In Section V, we [KT,L] = [KP,L] = 0. However, one is reminded that apply our knowledge of 1D superconductors to classify generally [KT,H ] (cid:54)= 0 and [KP,H ] (cid:54)= 0. This is be- ri ri both high-symmetry and generic lines within the Bril- causeifr ∈/ Real, KP orKT sendsthestatetoanother i louin zone (BZ) of 3D superconductors by determining sector with eigenvalue r∗. However the chiral symme- i not only the bulk invariants but also the effective sur- try, represented by a unitary matrix, preserves the block facetheory. InSectionVI,wesummarizeourresultsand structure of H. 3 Based on this discussion, we come to a simple con- of an electron. For our purposes, the term ‘spinless’ sim- clusion: for any r ∈ Real, the Hamiltonian H inherits ply refers to the unbroken SU(2) spin rotation symme- r theparticle-hole,time-reversalandchiralsymmetriesthe try, while the term ‘spinful’ indicates its absence. When system may have, thus having the same topological clas- the spin-orbit interaction is ignored, an SU(2) invariant sification as that found Ref.[62–64]. On the other hand, Hamiltonian can be block-diagonalized into two parts, if r ∈/ Real, H only inherits the chiral symmetry of the namely those of spin up and the spin down parts, while r system,shouldthesystempossesschiralsymmetry,while both the TRS and PHS can be composed with a spin- the time-reversal and particle-hole symmetries relate H rotation about a specific directional axis (say, y-axis) so r to H . Due to this relation, the topological invariants as to not change the spin state, thus acting within each r∗ for H and H will be shown to be either the same or block. This signifies that the symmetries of the whole r r∗ opposite, depending on S and the symmetry class of the systemcompletelypasstoeachoftherespectivespinsec- Hamiltonian. Therefore, a local unitary symmetry in tors, and it is therefore sufficient to study any one of the general enhances the topological classifications, as now two independent ‘spinless’ Hamiltonians to understand the full system is labeled by all the quantum numbers the properties of the system as a whole. fromeachsector(ifr ∈Real)andeachpairofsectors(if However, this is not quite the complete picture of the r ∈/ Real), ratherthanthesingleZ orZ numberforthe proper physics. One needs to take caution in that since 2 entire Hamiltonian H. PHS and TRS are now combined with a spin rotation, A point group, or specifically rotational, symmetry is their squares change sign, resulting in KT2 = −KP2 = ingeneralnot alocalsymmetry,asitchangestheposition 1. This becomes a vital distinction and is needed when of an electron. The only exceptions are mirror reflection we discuss rotational symmetries. A full rotation of an in a 2D system when then mirror plane and the system electron gives a −1 factor to the wavefunctions due to are coplanar, and the rotation in a 1D system when the the inherent π Berry’s phase. This points to the fact rotation axis coincides with the system. Nonetheless, for that for a general electronic system, we have Cn = −1 n single particle Hamiltonians with translational symme- where C is the rotation operator of an n-fold rotation. n try, there are always some sub-manifolds in the k-space When SU(2) is present, the rotation symmetry can be that are invariant under a point group symmetry. For redefined as a rotation of both spin and space followed example, in a 3D simple cubic lattice, the tight-binding by a spin rotation in the opposite direction. In such a Hamiltonian H(k ,k ,k ) in k-space is invariant under case, we have Cn = 1. Physically this means that in x y z n mirror reflection M : z → −z when k = 0 or k = π; the absence of SOC, a rotation symmetry only operates xy z z and it is invariant under the fourfold rotation about the on the spatial degrees of freedom, leaving the spin part z-axis C : (x,y,z) → (−y,x,z), if (k ,k ) = (0,0) or unchanged. Within the context of operators, the terms 4 x y (π,π), where the lattice constant is taken to be unity, ‘spinless’ and ‘spinful’ indicate Cn =KT2 =−KP2 =1 n a ≡ 1. For an invariant sub-manifold, the point group and −1, respectively. symmetry becomes a local symmetry, and hence, we can use the general scheme described above, i.e., clas- sifying the sectors labeled by the eigenvalues of L, to C. Rotational Symmetry of a BdG Hamiltonian findtheclassificationandthecorrespondinginvariantsof the Hamiltonian on the sub-manifold. The set of quan- Weareinterestedindiscussingthepropertiesofmean- tumnumbersdefinedonalltheseinvariantsub-manifolds field superconducting systems, and hence we must un- characterize a general topological crystalline insulator or derstand how the previously discussed rotational sym- superconductor. Itisthismethodthatwewillbeusingto metries manifest themselves in a Bogoliubov-de Gennes classify superconductors with rotation symmetry within (BdG) Hamiltonian. The second-quantized form of a the context of this work. We should note, however, that BdG Hamiltonian reads: this characterization is incomplete and there exist inter- estingexceptions35,77,78. Forexample,inRef.[35,77],the Hˆ =Hˆ0+∆ˆ +∆ˆ†, (3) authors show that 3D spinless systems host topological where numbers that are protected by rotation symmetry but yetcannotbedefinedonanyofthehigh-symmetrylines. Hˆ = H c†c +h.c., (4) 0 ab a b ∆ˆ = ∆ c c , ab a b B. Spinless Vs. Spinful Systems where a,b are composed indices labeling the site, orbital andspininalatticesystem. Inann-foldrotationinvari- Having clarified the role of rotational symmetries in ant system, we require that general Hamiltonians, it is important to mark the dif- [Cˆ ,Hˆ ]=0, (5) ference between spinless and spinful systems. As super- n 0 conductors are, naturally, electronic systems and their and constituent electrons are elementary particles with spin one-half, this distinction does not refer to the spin state Cˆ ∆ˆCˆ−1 =eiθ∆ˆ. (6) n n 4 Using the fact that Cˆn = ±1 for spinless and spinful If m = 0, Cˆ is the same as Cˆ and also commutes n n,m n systems respectively in conjunction with Eq.(6), we find with time-reversal. When m=n/2, which is only possi- einθ =1 ,or θ =2mπ/n where m=0,...,n−1. Consider bleifn∈even,thecommutationrelationbetweenCˆ n,n/2 a gauge transform Uˆ = exp(imπ/nQˆ), where Qˆ is the and Tˆ is total electric charge operator, such that {C ,KT}=0, (14) Uˆ c Uˆ−1 =e−imπ/nc , (7) n,n/2 m a m a where KT is the matrix representation of T where K is and if we combine Uˆ with Cˆ , from Eq. 5, we have complex conjugation. Eq.(14) indicates that the C , m n n,n/2 KT and KP form a projective representation of the [CˆnUˆm,Hˆ]=0. (8) group generated by T, Cn and P. We again defer the proof to Appendix B. Therefore, we may define Cˆ ≡ Cˆ Uˆ as a symmetry n,m n m ofthesystem. Inatranslationinvariantsystem,wehave III. CLASSIFICATION OF 1D Cn,mH(k)Cn−,1m =H(Cnk), (9) SUPERCONDUCTORS WITH ROTATIONAL SYMMETRIES where C is the first quantized matrix representation n,m of Cˆn,m in the Nambu basis. The significance of Eq.(9) We now proceed to 1D, and some quasi-1D systems, lies in that the spectrum of H(k) remains rotationally wheretherotationalonganaxisparalleltothesystemis invariant even when m (cid:54)= 0. Physically, m (cid:54)= 0 indi- a symmetry represented by a matrix C satisfying the n,m cates that the Cooper pair has total angular momentum commutation relation, m(cid:126) along the rotation axis (modulo n), because the pair gains a phase of ei2mπ/n after a rotation through 2π/n. [C ,H(k)]=0. (15) n,m Nevertheless, this change of phase is not reflected in the quasiparticle spectrum, which is still n-fold symmetric, It should be noted that in one dimension, there is no due to Eq.(9). This is because the phase can be com- crystallographic constraint on n, and therefore n ∈ Z+ pensatedbyaU(1)transformwhichleavesHˆ invariant. (positive integers). As we classify the respective gapped 0 Additionally, we should also point out that superconductorswithrotationalsymmetriesin1Dbelow, weseparatethediscussionsintotwodistinctsections: one Cn =(−1)mCˆn =(−1)m+F, (10) inwhichthetotalangularmomentumoftheCooperpair n,m n is zero (m=0) and one where the angular momentum is where F =0,1 for spinless and spinful fermions, respec- non-zero (m(cid:54)=0). tively. Therefore, when m ∈ odd, C is like a spinless n,m (spinful) rotation in a spinful (spinless) system. Math- ematically, when m ∈ odd, Eq.(10) indicates that C A. Pairing without Cooper Pair Angular n,m and KP form a projective representation of the group Momentum (m=0) generated by C and P, a fact that we rigorously prove n in Appendix A. Most generic of BdG Hamiltonians, which belong to class D in the AZ classification table, have preserved particle-hole symmetry, which is represented by KP in D. Time-reversal symmetry and projective theNambubasis, whereK isthecomplexconjugateand representation of the group generated by Cn and T P is a unitary matrix. Physically, we have Pˆ2 =1 for an electron, which leads to In this paper we also consider superconductors with TRS. Naturally, TRS implies that (KP)2 =PP∗ =1, (16) P =PT. Tˆ∆ˆTˆ−1 =∆ˆ. (11) SincePHSchangeselectronstoholesandviceversa,it Using Eq.(6), we have anti-commutes with the first quantized Hamiltonian and the momentum operator. This means that it sends one CˆnTˆ∆ˆTˆ−1Cˆn−1 =Cˆn∆ˆCˆn−1 =ei2mπ/n∆ˆ, (12) single particle state to another with opposite energy and momentum; symbolically we have and (KP)H(k)(KP)−1 =−H(−k), (17) TˆCˆ ∆ˆCˆ−1Tˆ−1 =Tˆei2mπ/n∆ˆTˆ−1 =e−i2mπ/n∆ˆ. (13) n n or However, since [Cˆ ,Tˆ] = 0, the m which satisfies both n Eq.(12) and Eq.(13) is m=0,n/2. PH(k)P† =−HT(−k). (18) 5 In addition to this, PHS commutes with the rotation also has Z classification, the Z number of H (k) is 2 2 −1 symmetry as the same as the Z number of the full Hamiltonian. 2 In addition to PHS, which is shared by all supercon- [KP,Cn]=0 → PCnP† =Cn∗. (19) ductors, we consider the presence of TRS, corresponding toclassDIIIintheAZclassification. IntheNambubasis, In the basis spanned by eigenstates of Cn, (φ1,φ2,...)T, TRS is represented by KT, where T is a unitary matrix. we have For spinful electron, we have (cid:88) Cn = rIdr, (20) (KT)2 =−1→T =−TT. (24) ⊕r TheactionofTRSreversesthemomentumofanelectron where Idr is a dr-by-dr identity matrix and dr is the without changing its energy, or symbolically, degeneracy of the eigenvalue r. Using Eq.(19) and Eq.(20), we obtain the expression for P in the basis (KT)H(k)(KT)−1 =H(−k), (25) (φ1,φ2,...,φ∗1,φ∗2,...)T as those spanned by the eigen- TH(k)T† =HT(−k). states. TRS also commutes with all spatial symmetries. Specif- (cid:18) (cid:19) (cid:88) (cid:88) 0 Q P = P ⊕ r , (21) ically, r QT 0 r ⊕r∈real ⊕Im[r]>0 [KT,C ]=0 → TC T† =C∗. (26) n n n where P and Q are unitary matrices. This indicates r r In the basis spanned by the eigenvectors of C , we have that the PHS operator leaves unchanged the eigenspace n the following block-diagonalization of T of the rotation matrix, C , with a real eigenvalue, but n maps the eigenspace with a complex eigenvalue to its (cid:18) (cid:19) (cid:88) (cid:88) 0 R complex conjugate. T = Tr⊕ −RT 0r , (27) UsingEq.(16)andEq.(21), wethenhaveforr ∈Real, ⊕Im[r]=0 ⊕Im[r]>0 r PrHr(k)Pr† =−HrT(−k), (22) where Tr and Rr are unitary matrices and Tr is anti- symmetric from Eq.(24). Using Eq.(25) and Eq.(27), we and for r ∈/ Real, we have have for each r ∈Real, (cid:18) 0 Qr(cid:19)(cid:18)Hr(k) 0 (cid:19)(cid:18) 0 Q∗r(cid:19) (23) TrHr(k)Tr† =HrT(−k), (28) QT 0 0 H (k) Q† 0 r r∗ r and for each r ∈/ Real (cid:18)HT(−k) 0 (cid:19) = − r 0 HrT∗(−k) (cid:18) 0 Rr(cid:19)(cid:18)Hr(k) 0 (cid:19)(cid:18) 0 −Rr∗(cid:19) (29) −RT 0 0 H (k) R† 0 r r∗ r According to the AZ classification, Hr(k) belongs to (cid:18)HT(−k) 0 (cid:19) class D if r ∈ Real and to class A (i.e., no symme- = r 0 HT (−k) . try because PHS relates r to r∗) if r ∈/ Real, while r∗ Hr(k)⊕Hr∗(k) again belongs to class D. In 1D, class D FromEq.(28),weunderstandthateachsectorwithrealr has a Z2 classification whereas class A is trivial. There- has both TRS and PHS and hence belongs to class DIII, fore, each Hr∈Real possesses its own Z2-index. There- which in 1D has a Z2 number. For r ∈/ Real, by utilizing fore, the question that remains to be answered is can a combination of Eq.(23) and Eq.(29), we obtain H (k) ⊕ H (k) be Z nontrivial? We argue that it r r∗ 2 is impossible by examination of a simple contradiction. (cid:18)Q∗RT 0 (cid:19)(cid:18)H (k) 0 (cid:19)(cid:18)R†QT 0 (cid:19) r r r r r (30) Should Hr(k) ⊕ Hr∗(k) be Z2 nontrivial, then for an 0 Q†rRr 0 Hr∗(k) 0 Rr†Qr open chain there must be a single Majorana mode at (cid:18) (cid:19) H (k) 0 each end79. Due to C -symmetry, the Majorana mode =− r . n 0 H (k) must either have a rotation eigenvalue of r or r∗, but r∗ eitherchoicebreakstheinherentPHS.Basedonthisdis- If we define S = Q∗RT and S = Q†R for each r ∈/ cussion, we find that the topological classification of a r r r r∗ r r Real, Eq.(30) leads to C -invariant1Dsuperconductorwithoutadditionalsym- n metries is given by a set of Z2 numbers from each Hr(k) {Sr,Hr(k)}=0, (31) with r ∈ Real. Using rn = −1, it is obvious that for n = even all eigenvalues are complex, thus, the classifi- which indicates that H (k) belongs to the chiral r∈/Real cation is always trivial. Meanwhile for n = odd, r = −1 class AIII. Although a real sector H also has chi- r∈Real istheonlyrealeigenvalueofC ,andtheonlytopological ral symmetry defined as S = P T , one cannot use this n r r r number is the Z number of H (k). It must be noted symmetry for classification, as the topological invariants 2 −1 thatsincethefullHamiltonianbelongstoclassD,which protectedbyS areconstrainedtocertainnumbers, zero r 6 in this case, by the individual PHS or TRS symmetry. by C as the rotation symmetry of the system. In the n,m Class AIII in 1D has a Z number, so each sector with Nambu basis, it is represented by complex r has a Z number denoted by z(r). In fact, we can further argue that z(r) =−z(r∗). The chiral symme- Cn,m =Cneiτzmnπ, (34) try is the composition of TRS and PHS, thus satisfying S2 = P2T2 = −1. This indicates that Sr has eigen- where τz is the Pauli matrix in the particle-hole indices values ±i. Any class AIII Hamiltonian having z(r) > 0 due to Um. We understand that PHS commutes with (z(r) <0) means that there are |z(r)| edge states at each Cn,m because(i)PHScommuteswithCn and(ii)itanti- end of an open system that are eigenstates of S with commutes with both i and τz, or eigenvalue +i (−i). But under TRS, an edge state hav- ingCneigenvaluerandSr eigenvalue+imapstoanother [KP,Cn,m]=0 → PCn,mP† =Cn∗,m. (35) edgestatehavingC eigenvaluer∗ andS eigenvalue−i, n r ComparingEq.(35)andEq.(19),weseethatallpreceding implying that understandingobtainedintheprevioussectionwherewe z(r) =−z(r∗), (32) ignored the angular momentum of the Cooper pair also applies to the system with both PHS and C symme- n,m as a result of which the number of independent Z num- try. Therefore, we simply apply the results and arrive bers is determined by one-half the number of complex at the following conclusions: (i) For m,n ∈ even, the eigenvaluesofCn. Eq.(32)alsoimpliesthatforr ∈Real, classification of the system is trivial, as we have already TRS sets this topological number to zero, as r =r∗. For stated. (ii) For m ∈ odd and n ∈ even, r may take the n ∈ even, the full classification is given by n/2 integers value r = ±1, and H (k) belongs to class D and gives ±1 (as all Cn eigenvalues appear in complex pairs), and if two Z numbers, z(±1). (iii) For m∈even and n∈odd, n ∈ odd, it is given by one Z number and (n − 1)/2 2 2 2 r maytake−1butnot+1,andH (k)hasaZ number −1 2 integers (for all eigenvalues except −1 appear in pairs). z(−1). (iv) Finally, for m,n ∈ odd, r may take the value Again,theZ numberofthefullHamiltonianisgivenby 2 thesameast2heZ2 numberofH−1(k)whenn∈odd,and of +1 but not −1, and H1(k) gives a Z2 number z2(1). is trivial if n∈even. The Z2-index for the full Hamiltonian, neglecting rota- Finally, let us consider spinless electrons, or equiva- tion symmetry, is the same as the sum of the Z2-indices lently, adding spin-SU(2) symmetry. We can follow all corresponding to each sector with a real eigenvalue of thestepsabovetofindtheclassifications,keepinginmind Cn,m. the distinction that for spinless electrons, we have Now we consider adding TRS to the system. In Sec.IID we have shown that the only nonzero m that T = TT, (33) is compatible with TRS is m = n/2 when n ∈ even. In P = −PT, this case, Cn,n/2 and KT anti-commute, i.e., Cn = 1. n TC T† =−C∗ . (36) n,n/2 n,n/2 Using nearly identical calculations to those presented in this section, we may derive the following additional re- Eq.(36)indicatesthatTRSmapsastatewithC eigen- n,m sults: (i) Without TRS, any H (r) with real r and any value r to a state with eigenvalue −r∗. We note that r H (k)⊕H (k) with complex r belong to class C and in this case time-reversal operator anti-commutes with r r∗ have a trivial classification. (ii) Additionally, without C from Eq.(34) due to the fact that: (i) TRS com- n,m TRS, any H (k) with complex r belongs to class A and mutes with C yet anti-commutes with imaginary unit r n alsohasatrivialclassification. (iii)WithTRS,anyH (r) i and (ii) it commutes with τ for it does not inter- r z with real r and any H (k)⊕H (k) with complex r be- change particles and holes. Therefore, for the case when r r∗ long to class CI, which has a trivial classification in 1D m = n/2 and we have n ∈ even, we separately discuss (iv) In the presence of TRS, any H (k) with complex thefollowingtwoconstraintsontherotationalsymmetry: r r belongs to class AIII, having a Z classification, under (i) m ∈ even ⇔ n = 4k and (ii) m ∈ odd ⇔ n = 4k−2. the constraint as outlined in Eq.(32). Therefore in class When n = 4k (k being a non-negative integer) and the CI, for n ∈ even, there are (n−2)/2 integers to specify general eigenvalue of C is r = ei2π(s+1/2)/n (s being n,m s the topological state and for n∈odd there are (n−1)/2 a non-negative integer), then under PHS the r -sector s integers, corresponding to the number of conjugate pairs and the r -sector are mapped to each other while, n−s−1 of complex eigenvalues of C . at the same time, under TRS, the r -sector and the n,m s r -sector are mapped to each other. Therefore, n/2−s−1 H ⊕H (note that r = −r in this case) be- rs rn/2+s n/2+s s B. Pairing with Cooper Pair Angular Momentum longs to class AIII , thereby having a Z-index. (Since (m(cid:54)=0) P∗T mapss→n−s−1→n/2−(n−s−1)−1=n/2+s, the direct sum is invariant under the composite symme- When the angular momentum of the Cooper pair is try.) However, this Z-number must vanish and this can considered, namely when m (cid:54)= 0, C must be replaced be shown by contradiction. Suppose this Z number is n 7 z >0, then on the edge there are z states that are eigen- n∈odd n=4k n=4k−2 vectors of P ∗T having eigenvalue +i, such that P ∗T C 0 0 0 in the Hilbert space spanned by the zero modes is iI, D, m∈even 0 Z Z 2 2 where I is the identity matrix the dimension of which D, m∈odd Z2 Z Z is the number of zero modes. Then we assume that the 2 2 2 CI, m=0 Z(n−1)/2 Z(n−2)/2 Z(n−2)/2 rotation symmetry C be represented by some matrix n,m R, and we have [R,P ∗T] = 0 contradicting Eq.(35,36). CI, m=n/2 0 0 0 Considering the second case, n = 4k−2, we know that DIII, m=0 Z ×Z(n−1)/2 Zn/2 Zn/2 2 the general eigenvalue of Cn,n/2 is rs = e2sπ/n. Under DIII, (m=n/2) 0 0 Z2 PHS, the r -sector and the r sectors are mapped to s n−s each other with the exceptions of s = n/2 and s = n, where r =−1 and r =+1, respectively. As in the previ- TABLE I: Complete classification of 1D gapped supercon- ductors with rotation symmetry, C . Within the table, ‘0’ ous case, under TRS, the r -sector and the r -sector n,m s n/2−s indicatesthatforthegivensystemtheclassificationistrivial. are mapped to each other. Therefore, for s (cid:54)= n/2, the Hamiltonian H ⊕H belongs to class AIII, while rs rn/2+s H ⊕ H belongs to class DIII. Based on above ar- +1 −1 gument, the Z-index of the class AIII component must IV. 2D SUPERCONDUCTORS WITH vanish, leaving to an overall classification to be Z . The ROTATION SYMMETRIES 2 Z -index for the full Hamiltonian, again without consid- 2 ering rotation symmetry, is the same as the Z2-index for With our discussion of gapped 1D superconductors H+1⊕H−1. with rotation symmetries complete, we focus our atten- Finally, we consider adding SU(2) symmetry. With tion in this section on the study of 2D gapped supercon- PHS and SU(2) the full Hamiltonian is in class C, hav- ductorswithoutTRS(classDandclassC)inthepresence ing (KP)2 = −1 as the only symmetry. In this case, ofC invariance,wheretherotationaxisisassumedto n,m all sectors of the Hamiltonian belong either to class C be the axis perpendicular to the system. Lattice peri- or to class A, both being trivial. When we include TRS odicity is compatible with rotation symmetry only when along with PHS and SU(2), we again need to separately n=2,3,4,680 . InBZ,thereexisthigh-symmetrypoints consider the two cases above with regards to the ro- that are invariant under C , where n˜ is a factor of n, n˜ tational symmetry of the given system. We first con- denoted by K . For example, in a C -invariant system, n˜ 4 sider the case for n = 4k in which a generic eigenvalue X = (π,0) and Y = (0,π) are points that are C in- 2 of C is r = e2sπ/n. Under PHS, the r -sector variant; and in a C -invariant system, K and K(cid:48) are n,n/2 s s 6 and the r sectors are mapped to each other with C -invariant. At K , each energy eigenstate is also an n−s 3 n˜ the exceptions of s = n/2 and s = n. Further, under eigenstate of C (it being understood that m is a mod n˜,m the application of TRS, the r -sector and the r - n˜ number). For each eigenvalue r of C , we count at s n/2−s n˜,m sector are mapped to each other, with the exceptions of K the number of occupied energy eigenstates that are n˜ s = n/4 and s = 3n/4, where r -sector is mapped to also eigenstates of C with eigenvalue r and denote it s n˜m itself. Therefore, for s (cid:54)= n/4,n/2,3n/4,n, the Hamilto- by N (K ). We show that these numbers are related to r n˜ nian H ⊕H belongs to class AIII and is char- the Chern numbers in superconductors. Physically, the (rs) (rn/2+s) acterized by a vanishing Z-index. Similarly, the sectors Chernnumberofasuperconductorisdeterminedbyboth corresponding to H ⊕H and H ⊕H be- thebandstructureofthenormalstateandthesymmetry (+1) (−1) (+i) (−i) long to class CI, and possess only a trivial classifica- of the pairing amplitude on the Fermi surface. The for- tion resulting in an overall classification for the system mer contribution is related to the C eigenvalues of the n that is trivial. For the case corresponding to rotational occupied bands at high symmetry points in the normal symmetries satisfying n = 4k − 2, we have eigenvalue states39,78,81,82, while the latter contribution is related of r = ei2π(s+1/2)/n. As before, under PHS, the r - tom,namely,theangularmomentumoftheCooperpair s s sector and the r -sector are mapped to each other, modulon. Inthissection,wefocusonhowthesenumbers n−s−1 and the application of TRS maps the r -sector and the relate to the Chern number in gapped 2D superconduc- s r -sectortooneanother,withtheexceptionofs= tors. n/2−s−1 (n/2−1)/2. Therefore, for s(cid:54)=(n/2−1)/2,(n/2+1)/2, theHamiltonianH ⊕H belongstoclassAIII,hav- rs rn/2+s ing a Z-classification but with vanishing Z-index, while H ⊕H belongs to class CI, having trivial classi- A. Continuum Limit (n=∞) (+i) (−i) fication. Accordingly, the overall classification is again trivial. To begin our analysis, let us first consider the contin- In Table I, We summarize the classification of all uum limit with full SO(2) symmetry. In this limit, the gapped 1D superconductors within four BdG classes (C, angular momentum of the Cooper pair, m, can take any D, CI and DIII) of the AZ classification enhanced by integer. We choose to work in an orbital basis in which C -symmetry. the generator of the rotation operator Jˆis diagonalized. n,m 8 Therefore, in the Nambu basis, we have general relation between the Chern number and all j ’s m at k=0 and k=∞: J˜=τ ⊗diag{j ,j ,...,j }, (37) z 1 2 Norb N(cid:88)orb C = [ji (0)−ji (∞)]. (47) where τ is the Pauli matrix acting on the particle-hole m m z index and j is the angular momentum of the α-th elec- i=1 α tronic orbital. The second quantized form of Jˆis given If one considers the gapped BdG Hamiltonian to be the by sameasthatofaninsulatorwithaccidentalparticle-hole symmetry, Eq.(47) simply means that its Chern number Jˆ= (cid:88) jiφ†iφi, (38) equalsthetotalangularmomentum(alongz-axis),where i=1,...,Norb Jˆm is the angular momentum operator, of all occupied states. To heuristically observe this, we notice that for where φ is the annihilation operator of angular momen- i any occupied state, |ψ(k)(cid:105) at a generic k (cid:54)= 0,∞, the tum j . Under rotation through θ via the application of the roitation operator, we have state eiJˆmθ|ψ(k)(cid:105) must also be an occupied state with momentum R(θ)k. One can always construct |ψ (cid:105) = n eiJˆθ∆ˆe−iJˆθ =eimθ∆ˆ, (39) (cid:82)02πdθeinθeiJˆmθ|ψ(k)(cid:105)foranyintegernand,therefore,all contributiontothetotalangularmomentumfromgeneric or its infinitesimal version k’s cancel each other, leaving the only contribution from k=0,∞. We then recall that any rotation about k=0 [∆ˆ,Jˆ]=−m∆ˆ. (40) is equivalent to an inverse rotation about k=∞, so the total angular momentum is the difference, not the sum, Furthermore, we also know of ji ’s at 0 and ∞. m Now examine the weak pairing limit, where we may [∆ˆ,Qˆ]=−2∆ˆ, (41) separate the contribution due to the normal state band structure from that of the pairing on the Fermi surface. where Qˆ(cid:80)c†c is the total charge. Using Eq. (41) in Intheweakcouplinglimit,ateachk,theoccupiedbands α α conjunction with Eq. (40), we can prove that intheBdGHamiltonianconsistoftwodistinctparts: the occupiedbandsofthenon-superconductingHamiltonian, [Jˆ ,∆ˆ]=0, (42) andtheparticle-holepartnerofalltheunoccupiedbands. m Keepinginmindthataholestatehasoppositechargeand where Jˆ ≡Jˆ− m(cid:81) (1−2c†c ). In the Nambu basis, angular momentum compared with an electron state, we m 2 α α α Jˆ is represented by have m J˜m =J˜− m2 τz⊗INorb. (43) N(cid:88)orbjmi (K) = j1(K)+...+jNocc(K)(K) (48) i=1 SinceJˆcommuteswiththenormalpart,Hˆ oftheHamil- − (jNocc(K)+1(K)+...+jNorb(K)) 0 tonian, using Eq.(42), we know that Jˆ commutes with m m − [N (K)−N (K)], the full Hamiltonian 2 occ unocc [Jˆ ,Hˆ]=0, (44) where Nocc and Nunocc are the number of occupied and m unoccupied bands, respectively. Substituting Eq.(48) to Eq.(47), we obtain a simple formula or, in the presence of translational symmetry H(k eiθ,k e−iθ)=exp(i(J˜− mτ )θ)× C =2[J(0)−J(∞)]−m[Nocc(0)−Nocc(∞)]. (49) + − 2 z m (45) ThephysicalmeaningofEq.(49)isclearasthefirstterm H(k+,k−)exp(−i(J˜− 2 τz)θ), is simply two times the total angular momentum of the normal state, where the factor of two is because of the wherek =k ±ik . Inthe2Dcontinuumk-space,k=0 Fermion doubling in the Nambu basis. The second term ± x y and k = ∞ are the only two points that are invariant isthetotalangularmomentumofthepairingonallFermi under rotation. At these points we have surfaces. To see this, we need to notice two separate facts: (i) Eq.(39) indicates that m is the total angular [J˜ ,H(0)]=[J˜ ,H(∞)]=0, (46) momentum of a Cooper pair, and (ii) N (0)−N (∞) m m occ occ is the difference in the occupation numbers at k = 0 where we have implicitly assumed that H(∞) is well de- and k = ∞. Suppose N (0) > N , and by traversing occ occ fined. Each state of H(0) or H(∞) is also an eigenstate any path from 0 to ∞ one crosses N electron-like Fermi e of J˜ of eigenvalue ji (i denoting the occupied bands surfaces and N hole-like Fermi surfaces, then we have m m h in the BdG Hamiltonian). In Appendix C, we prove a N −N =N (0)−N (∞). e h occ occ 9 B. Finite Rotational Symmetry C (n=2,3,4,6) where ζ ,θ ,ξ ,η are the product of all eigenvalues n m m m m of C , C , C and C at corresponding high- 2,m 3,m 4,m 6,m When considering the more realistic case of a 2D lat- symmetry points on the lower half BdG bands, respec- tice, the continuous rotation symmetry C breaks down tively. In the weak coupling limit, they again reduce to ∞ toCn=2,3,4,6. Inourapproachhere, wecloselyfollowour expressions that only involve the eigenvalues of Cn˜, the previous work39, in order to obtain the Chern number occupationnumberateachhigh-symmetrypointandthe up to a multiple of n in terms of the eigenvalues of C angular momentum of the Cooper pair (mod n). To be n˜,m at k-points invariant under C where n˜ divides n. For specific, n˜ C we find: n=2,3,4,6 ei2πC/2 =ζ (Γ)ζ (M )ζ (M )ζ (M ), (50) m m 1 m 2 m 3 ei2πC/3 =(−1)Norb(m+1)θm(Γ)θm(K)θm(K(cid:48)), ei2πC/4 =(−1)Norb(m+1)ξm(Γ)ξm(M)ζm(X), ei2πC/6 =(−1)Norb(m+1)ηm(Γ)θm(K)ζm(M), ei2πC/2 = exp[i(m+1)π(N (Γ)+N (M )+N (M )+N (M )], (51) occ occ 1 occ 2 occ 3 θ2(Γ) 2mπ ei2πC/3 = exp[i (2N (Γ)−N (K)−N (K(cid:48)))], θ(K)θ(K(cid:48)) 3 occ occ occ ξ2(Γ) 2mπ ei2πC/4 = exp(−i )(N (Γ)+N (M)−2N (X)), ξ2(M) 4 occ occ occ η2(Γ)ζ(M) 2mπ ei2πC/6 = exp[−i (N (Γ)+2N (K)−3N (M))]. θ(K) 6 occ occ occ and the definition of high-symmetry points is given in If m=0, then we have (since C2 =−1) 2 Fig.1. Let us derive the n=2 case here in detail. In the weakcouplinglimit,eachoccupiedstateofC eigenvalue 2 ζ at Γ is an eigenstate of C2,m with eigenvalue ζeim2π, and each unoccupied state with eigenvalue ζ at Γ is an j eigenstate of C2,m with eigenvalue ζj∗e−im2π after PHS ( (cid:89) ζ )2 =(−1)Nocc(Γ), (53) transform. Therefore, the total product of eigenvalues of i C at Γ is i∈occ 2,m ζm(Γ) = ( (cid:89) ζieim2π)( (cid:89) ζj∗ei−m2π) (52) i∈occ j∈unocc = ( (cid:89) ζi)2( (cid:89) ζn∗)eimπNocc(Γ)e−im2πNorb. so i∈occ n∈orb (cid:89) (−1)C = (−1)Nocc(Γ)+Nocc(M1)+Nocc(M2)+Nocc(M3)( ζ∗)4 (54) n n∈orb = (−1)Nocc(Γ)+Nocc(M1)+Nocc(M2)+Nocc(M3) If m=1, then so (cid:89) ( ζ )2 =1, (55) i i∈occ 10 (cid:89) (−1)C = (−1)2Nocc(Γ)+2Nocc(M1)+2Nocc(M2)+2Nocc(M3)( ζ∗)4e−i2πNorb (56) n n∈orb = 1. In Eq.(51), the contribution to the Chern number either T, P or Cn/2, this momentum is mapped to −k¯, n,m again decomposes into two parts as promised, but the but under P(cid:48) or T(cid:48) the momentum is mapped back to physical meaning is not as transparent as in Eq.(49), be- k¯, while sending the momentum along the line to its op- causeheretheangularmomentumiswell-definedonlyup posite value. Using the commutation relations similar in to a multiple of n, and states at high-symmetry points form to those used in Eq. (35) and Eq. (14) , we have other than Γ contribute to the total angular momentum in different ways. We hope our eigenvalue formulas for (KP(cid:48))2 = (−1)m+F(KP)2 =(−1)m+1, (57) projector Chern numbers can be useful in the search of (KT(cid:48))2 = (−1)mn/2+m+F(KT)2 =(−1)F(KT)2 =1. topological chiral superconductors. Eq.(57) makes the class of generic lines different from the class of the whole system, depending on the parity V. 3D SUPERCONDUCTORS WITH of m. Let us examine each of the subsequent possibili- ROTATIONAL SYMMETRIES ties of lines embedded in the 3D BZ in turn: Case (i): class C and m even, then Eq.(57) states that (KP(cid:48))2 = A. Bulk Invariants (KP)2 =−1, thus H(k¯) belongs to class C and has triv- ial classification. Case (ii): class C and m odd, then Eq.(57) states that (KP(cid:48))2 = −(KP)2 = 1, so H(k¯) We move on to discuss the partial topological classifi- belongs to class D, which gives it a Z classification. cationof3Dgappedsuperconductorswithrotationsym- 2 Case (iii): class D and m even, then Eq.(57) states that metries. In a 3D lattice, crystallographic constraint dic- (KP(cid:48))2 = −(KP)2 = −1 indicating that H(k¯) belongs tates that n=2,3,4,6 and within a C invariant lattice n toclassCandhasatrivialclassification. Case(iv): class system, there are discrete lines in the 3D BZ that are D and m odd, Eq.(57) states that (KP(cid:48))2 =(KP)2 =1, invariantunderC wheren˜ isafactorofn. Therefore, n˜>1 so H(k¯) belongs to class D, having Z classification. in order to classify 3D gapped superconducting systems, 2 Case (v): class CI and m even and Eq.(57) states that we can apply the classification of 1D superconductors (KP(cid:48))2 = (KP)2 = −1 and (KT(cid:48))2 = (KT)2 = 1, so with C invariance to these sub-manifolds, and the set n˜ H(k¯) belongs to class CI, having trivial classification. of quantum numbers of all C -invariant lines gives the n˜ Case (vi): class CI and m odd thus Eq.(57) states that enhancedclassificationofthe3Dsystem. Wenoticethat (KP(cid:48))2 = −(KP)2 = 1 and (KT(cid:48))2 = (KT)2 = 1, for n˜ = 2, 4, and 6 all three local symmetries we have indicating that H(k¯) belongs to class BDI, which has discussed in the text, namely time-reversal, particle-hole a Z classification. Case (vii): class DIII and m even and spin rotation, of the 3D system are also preserved so Eq.(57) states that (KP(cid:48))2 = −(KP)2 = −1 and on the C invariant lines. The same applies to the line n˜ (KT(cid:48))2 = −(KT)2 = 1 indicating that H(k¯) belongs that includes Γ when n = 3. In Fig.1, we schemati- toclassCI,havingtrivialclassification. Case(viii): class cally represent these special lines by their projections DIII and m odd, Eq.(57) states that (KP(cid:48))2 =(KP)2 = onto the surface BZ, where the surface is perpendicu- 1 and (KT(cid:48))2 = −(KT)2 = 1, so H(k¯) belongs to class lar to the rotation axis. For the outset, we show that BDI and has a Z classification. special treatment is needed for the following lines: (i) a We now consider a C -invariant line that does not generic vertical (parallel to the rotation axis) line in BZ 3 include Γ. If the system under consideration has C - in a system with C -symmetry and (ii) a C -invariant 3 2,4,6 3 invariance but not C -invariance, then the line does not line that does not include Γ. We must treat these lines 6 possessPHS.TheHamiltonianonthelinehencebelongs specially because, while TRS and PHS are not symme- to class A and accordingly possesses a trivial classifica- tries, compositions such as C ∗ P and C ∗ T, P ∗ T 2 2 tion. However, if the system also contains TRS , the might be symmetries. Due to the lack of PHS on these line possesses the combined symmetry KP ∗KT. Since genericlines,theclassificationderivedinSec.IIIdoesnot both KT and KP commute with C , each sector of the apply for these lines. In a system with C -symmetry, 3 2,4,6 occupied state has the chiral symmetry or, a generic vertical line does not have TRS or PHS, but it may have the following symmetries: KP(cid:48) ≡ KPCn/2 n,m {H ,S }=0, (58) r r and KT(cid:48) ≡KTCn/2, where we should note that TRS is n,m only possible for m = 0 and m = n/2. To see this, we where the eigenvalues are r ∈ (−1)F{1,ω ≡ ei2π/3,ω¯ ≡ note that each generic vertical line is labeled by its mo- e−iπ/3}. S hereisthematrixrepresentationofthechiral r mentumperpendiculartotherotationaxis,k¯,andunder symmetry S = P ∗T projected to the r-sector. Eq.(58)