Can deeply underdoped superconducting cuprates be topological superconductors?
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Can deeply underdoped superconducting cuprates be topological superconductors?
Yuan-Ming Lu,
1, 2
Tao Xiang, and Dung-Hai Lee
1, 2 Department of Physics, University of California, Berkeley, CA 94720 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China. (Dated: August 3, 2018)The nodal d x − y superconducting gap is a hallmark of the cuprate high T c superconductors.Surprisingly recent angle-resolved photoemission spectroscopy of deeply underdoped cuprates re-vealed a nodeless energy gap which is adhered to the Fermi surface. Importantly this phenomenonis observed for compounds across several different cuprate families. In this letter we propose anexciting possibility, namely the fully gapped state is a topological superconductor. PACS numbers:
The nodal d x − y gap function is a defining prop-erty of the copper-oxide (cuprate) high temperature su-perconductors. Many low temperature properties areaffected because of the existence of gap nodes. Ac-cording to topological arguments[1–4] (see the supple-mentary material) the nodes should be stable againstperturbations. Thus it is very surprising that recentangle-resolved photoemission spectroscopy (ARPES) ex-periments done on deeply underdoped cuprate sam-ples (samples which are at the border between the an-tiferromagnetic (AF) phase and the superconducting(SC) phase) have revealed a full particle-hole symmetricgap for Bi Sr CaCu O δ (Bi2212)[5, 6], La − x Sr x CuO (LSCO)[7, 8], Bi Sr − x La x CuO δ (Bi2201)[9] andCa − x Na x CuO Cl (NaCCOC)[10]. The fact that thereis a non-zero energy gap along the diagonal direction ofthe Brillouin zone (where the d x − y gap nodes usuallysit) is referred to as the “nodal gap” phenomenon. In-terestingly the nodal gap has been observed in systemswhose magnetic and transport properties range fromAF insulator[8, 9] to superconductor[6, 8]. For Bi2212Ref. [6] proposes a phase diagram with a new supercon-ducting (SC) phase appearing at the underdoping endof the SC dome. In contrast the samples showing the“nodal gap” in Ref. [9] are insulating and antiferromag-netic. However despite the difference in transport proper-ties, relatively sharp coherence peaks (i.e. spectral peaksat the nodal gap edge) were observed in both systemsso long as the doping concentration is not too low[6, 9].(Given the fact that samples at such a low doping levelcan be phase separated[11], it is possible that the nodalgapped phase observed in Ref. [9]) lies in disconnect SCislands embedded in an insulating background.) In theliterature possible cause of the nodal gap ranges from dis-order induced Coulomb gap[12] to the polaron effect[9].Motivated by Ref. [6, 8, 9] and the fact that non-superconducting gaps usually do not adhere to the Fermisurface, we assert that the state in question is a fullygapped SC state. Moreover we shall assume that it isnot due to extrinsic effects such as disorder. Moreover,because samples exhibiting the nodal gap are exclusively found at the border between AF and SC we need to con-sider the possibility that such SC state coexists with theAF order. In the rest of the paper we first list all possi-ble fully gapped SC states (with or without AF order),and organize them according to symmetry and topology(TABLE I). This information will be combined with ex-plicit effective theory calculations which determine theleading and subleading SC instabilities under differentconditions. The combination of theses two approachesallows us to pin down the most likely candidate for thenodal gapped state, namely, a topological superconduc-tor.Start with the maximally symmetric SC state, we sys-tematically break down the SU(2) spin × T symmetry. (Be-cause of superconductivity there is no U(1) charge symme-try). For each residual symmetry we use the method ofRef. [13, 16] to classify the possible fully gapped (twodimensional) SC states into topological classes. The re-sult is TABLE I, which contains eight cases in row 2 -9. We group these cases according to whether supercon-ductivity coexists with the AF order or not. (In view ofthe fact that the samples are likely disordered we do notconsider crystal symmetries such as lattice translations.The only exception is that we do regard the system ashaving inversion symmetry, at least on average, so thatspin singlet and spin triplet pairing will not mix. In theend we will briefly comment on the effects of inversionsymmetry breaking.)In the absence of Neel order there is spin SU(2) symme-try (in this paper we assume there is no spin-orbit inter-action, which is a good approximation for the cuprates).In the singlet pairing case there are two classes of fullygapped SC state - the s -wave pairing (row 2) and d ± i d pairing (row 3). The latter is a topological SC state withchiral (complex) fermion edge modes. In fact Ref. [8]proposed d ± i d as an explanation of the nodal gap inLSCO. In the triplet pairing case there are three classesof fully gapped SC state. They are listed in row 4-6 ofTABLE I. The ( p ± ip ) ↑↓ SC state in the 4th row breaksthe time-reversal symmetry but preserve U(1) spin ro-tation around, say, the z-axis. It is a representative
Neel order Symmetry Generators Classification AZ class ExamplesNo SU (2) spin × T { T e i πS x , T e i πS y , T e i πS z } π ( R ) = 0 CI s-waveNo SU (2) spin { e i πS x , e i πS y } π ( R ) = Z C d ± i d No U (1) spin { e i πS z } π ( C ) = Z A ( p ± i p ) ↑↓ No T T π ( R ) = Z DIII ( p ± i p ) ↑↑ + ( p ∓ i p ) ↓↓ No None N/A π ( R ) = Z D α ( p ± i p ) ↑↑ + β ( p ± η i p ) ↓↓ Yes U (1) spin × T e i πS x { T e i πS x , T e i πS y } π ( R ) = 0 AI s -waveYes U (1) spin { e i πS z } π ( C ) = Z A ( d ± i d ) ; ( p ± i p ) ↑↓ Yes None N/A π ( R ) = Z D α ( p ± i p ) ↑↑ + β ( p ± η i p ) ↓↓ TABLE I: Symmetry and topological classification of fully gapped SC phases in two spatial dimensions. We assume there is nospin-orbit interaction. The 2nd column lists the symmetry group whose generators are given in the 3rd column. The 4th columngives the Abelian group whose element each represents a topological class of SC states[13, 14] (for details see the supplementarymaterial). 0 means no topological superconductors, Z means there exists one type of topological superconductor in addition tothe trivial (s-wave) superconductor. Z represents the existence of infinite number of different topological superconductors eachwith protected gapless edge modes. The 5th column locates the symmetry class of each row in the Altland-Zirnbauer[15] 10-foldway[13, 16]. The last column provides examples of gapped superconducting states in each symmetry class. Here ( p + i p ) ↑↓ denotes the p + i p pairing between the spin up and spin down electrons, and ( p + i p ) ↑↑ , ( p + i p ) ↓↓ represents p + i p pairingamong spin-up and and/or spin-down electrons. In row 6 and 9 α and β denote generic complex numbers. of a family of degenerate triplet pairing state given bycos θ ( p ± i p ) ↑↓ + sin θe iφ ( p ± ip ) ↑↑ − sin θe − iφ ( p ± ip ) ↓↓ ,where ~d = (sin θ cos φ, sin θ sin φ, cos θ ) is the directionof the axis around which the U(1) spin symmetry is pre-served. These states span a manifold which is isompor-phic to S = { (sin θ cos φ, sin θ sin φ, cos θ ); θ ∈ [0 , π ] , φ ∈ [0 , π ) } . The SC states in this class possess chiral (com-plex) fermion edge modes hence are topologically non-trivial. In row 5 the ( p ± ip ) ↑↑ + ( p ∓ ip ) ↓↓ SC state pre-serves the time-reversal symmetry but completely breaksthe spin SU(2) symmetry. It has a pair of counter prop-agating Majorana modes along each edge. They are pro-tected from back scattering by the time-reversal symme-try, hence ( p ± ip ) ↑↑ +( p ∓ ip ) ↓↓ is a time-reversal invarianttopological superconductor. The SC state in the 6th rowof TABLE I has no residual symmetry. In all but the η = − η = − η = − Z .The cases where the fully gapped SC state coexistswith Neel order are listed in the last three rows of TA-BLE I. Here, without loss of generality, we can assumethe staggered magnetic moments to point in the ± z -direction. The pairing states in the 7th row are all topo-logically trivial, they are exemplified by the s -wave pair-ing. In contrast the d ± i d and the ( p ± i p ) ↑↓ SC states inthe 8th row both give rise to chiral topological supercon-ductors with chiral (complex) fermion edge modes. Wenote that the residual symmetry of the ( p ± i p ) ↑↓ super- conductor in row 8 is exactly the same as that in row4. However, unlike row 4, there is no continuous degen-eracy and associated Goldstone modes anymore becausethe SU(2) spin is already broken down to U(1) spin by theformation of the Neel order. The SC states in the 9th roware analogous to that given in the 6th row. Again, in allcases but η = − α ( p ± i p ) ↑↑ + β ( p ± η i p ) ↓↓ aretopologically non-trivial.Having surveyed all possible fully gapped SC states,the next step is to determine which class of TABLE Idoes the experimentally observed fully gapped SC statebelong to. To achieve that we use the effective theory ofRef. [17] H eff = X k ′ X σ ǫ ( k ) ψ + σ k ψ σ k + J X h i,j i S i · S j . (1)It has been demonstrated that the Eq. (1) is capable ofcapturing all experimentally observed electronic orders ofthe cuprates. Specifically the leading and subleading in-stabilities of Eq. (1) in the particle-particle (Cooper pair-ing) and particle-hole (density wave and Fermi surfacedistortion) channels capture the d-wave superconductiv-ity as well as the spin/charge density wave, nematicityand the Q = 0 magnetic order[17]. Encouraged by suchsuccess we use it to predict the leading and subleadingSC instabilities in the presence/absence of AF order. (Seesupplementary material for details.)The normal state Fermi surface with and withoutAF order are shown in Fig. 1. In the AF phase theband dispersion and Bloch wavefunctions are the eigen-values and eigenvectors of the following 2 × ǫ ( k ) σ mσ m ǫ ( k + Q ) ! . Here Q = ( π, π ) is the Neelwavevector, m is the staggered magnetization, and σ = ± ǫ ( k ), the paramagnetic normal H a L H b L FIG. 1: The Fermi surface of cuprates without (a) and with(b) AF order. The dashed line enclose the AF Brillouin zonewhoses vertices are ( ± π/a, ± π/a ). The staggered momentused to construct panel (b) is m = 0 . state dispersion, is given by µ − t (cos k x + cos k y ) + t cos k x cos k y − t (cos 2 k x + cos 2 k y ) with t = 1 , t =0 . , t = 0 . µ = 0 .
14. In the rest of the paper weshall set the value of m to 0 .
1. With this value of m thereare electron pockets centered around ( ± π,
0) and (0 , ± π )and hole pockets centered around ( ± π/ , ± π/ J is turned upfrom zero. (See the supplementary material.) Here acomment is in order. As mentioned earlier, the sampleswhere the nodal gap is observed can be phase separated.However in our calculation translation symmetry is as-sumed. Therefore one should interpret our results as thelocal pairing instabilities. (i) Cooper pairing in the absence of AF order. Fig. 2(a) and (b) illustrate the leading and subleading SCinstabilities in the absence of AF order[17]. The d x − y pairing in panel (a) has four nodes, hence can not beresponsible for the fully gapped state observed in exper-iments. Panel (b) illustrates the subleading extended s-wave pairing instability. It, too, has nodes. These nodes,like those of the d x − y pairing are topologically stableagainst perturbations. Although they are not requiredby the point group symmetry, it requires, e.g. strong dis-order, to get rid of them (the same for the d x − y nodes).Given the fact that the d x − y nodes have been observedin very disordered samples, we regard it unlikely that theextended s-wave pairing instability is responsible for thenodal gap. In the triplet pairing channel Eq. (1) has noSC instability (as long as J > d xy of d x − y + i d xy ) of the gap function isnot among the top (i.e. leading or subleading) pairinginstabilities in Fig. 2(a,b). (ii) Cooper pairing in the presence of AF or-der. In this case the leading and subleading pairing in-stabilities occurs in the S z = 0 channel and are shownin Fig. 2(c)-(e). The leading paring symmetry is again d x − y , which can not account for the presence of nodal H a L H b L H c LH d L H e L H f L FIG. 2: The leading superconducting instabilities of Eq. (1)in the absence (a)-(b) and presence (d)-(f) of Neel order. (a)singlet d x − y symmetry, (b) extended s symmetry, (c) singlet d x − y symmetry, (d) p x + y and (e) p x − y symmetries. Herethe hatch size is proportional to the magnitude of the gap andthe color indicates the sign (red: negative, blue: positive). (f)The energy gap correspond to | ∆ d ( k ) + i∆ e ( k ) | . Here Blackmeans positive and the hatch size is proportional to the gapmagnitude. gap. The subleading pairing symmetries, p x + y and p x − y in Fig. 2(d,e), are degenerate. However although theyeach has nodes, the linear combination ( p ± i p ) ↑↓ can giverise to a fully gapped chiral topological superconductor.This superconductor belong to the topological class of the8th row of TABLE I Therefore combining TABLE I withexplicit calculations we conclude that the best candidatesfor the experimentally observed fully gapped state is the( p ± i p ) ↑↓ SC coexists with AF order (the 8th row ofTABLE I).According to Ref. [6, 8, 9] the nodal gap magnitudesincreases as k moves away from the diagonal direc-tion. This is qualitatively consistent with the behaviorof | ∆ d ( k ) + i∆ e ( k ) | (see Fig. 2(f)) where ∆ d,e ( k ) are thegap functions of Fig. 2(d). In Fig. 3 we show the edgespectrum of the SC state discussed above. Explicit wave-function calculation shows the left/right moving in-gapmodes are localized on opposite edges. However despitethe presence of edge states, we do not expect the super-conducting vortex to harbor zero modes. This is becausein one dimension (the dimension of a loop surroundingthe vortex) the symmetry class of row 8 of TABLE I onlyhas trivial states (see supplementary materials). Discussions:
A natural question one might ask is whydoes triplet pairing instability exist in the AF state butnot in the paramagnetic state? It turns out that this is re-lated to the unit cell doubling in the AF state. After suchdoubling Q = ( π, π ) becomes a reciprocal lattice vector.Hence center-of-mass (COM) momentum (0 ,
0) and ( π, π )Cooper pairing can coexist. In the AF state although theAF exchange interaction in Eq. (1) favors singlet pairingfor COM momentum (0 , - - k x (cid:144) Π - - - E FIG. 3: The edge spectrum of the ( p + i p ) ↑↓ -AF coexistingstate. Here due to the Brillouin zone folding k x = − π/ k x = π/ COM momentum is ( π, π ). One might also ask “underwhat condition will the triplet pairing in Fig. 2(d,e) be-come the leading instability?” It turns out that this canbe achieved by increasing the staggered moment m , orby slight modifying the bandstructure so that in the AFstate there is only hole pockets. For example by using t = 1 , t = 0 . , t = 0 , µ = 0 . m = 0 . V with V /J & . ~d ∈ S ) in the 4th row of TABLE I is reduced to the twofold degeneracy ( ~d k ± ˆ z ) in the 8th row of TABLE I. (Infact the overall sign of ~d can be absorbed by the chargeU(1) phase of superconductivity, hence there is really nodegeneracy left). The preceding fact implies the directionof the Neel order parameter pins the ~d vector of tripletpairing. If so it is reasonable to expect the orientationfluctuations of the Neel order parameter can impede (oreven destroy) the SC coherence. This reasoning suggeststhat strong fluctuation of the Neel order can stretch theregime of fluctuation SC (hence the “pseudogap region”)to a much wider temperature interval. This might berelated to the observation of pseudogap above the SCtransition[6]. Lastly we comment on the effects of inver-sion symmetry breaking. Without inversion symmetrysinglet (Fig. 2(c)) and triplet (Fig. 2(d,e)) pairing chan-nels can mix. This can occur, e.g., locally due to disorderor phase separation. We have checked that the supercon- ductor with αd x − y + β ( p ± i p ) ↑↓ can be fully gapped.In addition there is a phase transition from a topologi-cally trivial to non-trivial phase as | β/α | increases. Thedetails of this investigation will be published in futurepublication. Conclusion:
We propose that deeply underdopedcuprates might be a topological superconductor. Oneway to experimentally test our prediction is to use STMto image the edge states. Given the likelihood that thesample is phase separated, STM is a particularly valuableprobe for the signature of the topological superconduc-tivity locally.We thank Yu He and Makoto Hashimoto for useful dis-cussions. This work is supported by DOE Office of Ba-sic Energy Sciences, Division of Materials Science, grantDE-AC02-05CH11231 (YML,DHL). [1] G. E. Volovik,
The Universe in a Helium Droplet , The In-ternational Series of Monographs on Physics, 117 (Book117) (Oxford University Press, USA, 2003).[2] P. Horava, Phys. Rev. Lett. , 016405 (2005), URL http://link.aps.org/doi/10.1103/PhysRevLett.95.016405 .[3] F. Wang and D.-H. Lee, Phys.Rev. B , 094512 (2012), URL http://link.aps.org/doi/10.1103/PhysRevB.86.094512 .[4] S. Matsuura, P.-Y. Chang, A. P. Schny-der, and S. Ryu, New Journal of Physics , 065001 (2013), ISSN 1367-2630, URL http://stacks.iop.org/1367-2630/15/i=6/a=065001 .[5] K. Tanaka, W. S. Lee, D. H. Lu, A. Fujimori, T. Fujii,Risdiana, I. Terasaki, D. J. Scalapino, T. P. Dev-ereaux, Z. Hussain, et al., Science .[6] I. M. Vishik, M. Hashimoto, R.-H. He, W.-S.Lee, F. Schmitt, D. Lu, R. G. Moore, C. Zhang,W. Meevasana, T. Sasagawa, et al., Proceedings ofthe National Academy of Sciences .[7] A. Ino, C. Kim, M. Nakamura, T. Yoshida, T. Mizokawa,Z.-X. Shen, A. Fujimori, T. Kakeshita, H. Eisaki,and S. Uchida, Phys. Rev. B , 4137 (2000), URL http://link.aps.org/doi/10.1103/PhysRevB.62.4137 .[8] E. Razzoli, G. Drachuck, A. Keren, M. Radovic, N. C.Plumb, J. Chang, Y.-B. Huang, H. Ding, J. Mesot,and M. Shi, Phys. Rev. Lett. , 047004 (2013), URL http://link.aps.org/doi/10.1103/PhysRevLett.110.047004 .[9] Y. Peng, J. Meng, D. Mou, J. He, L. Zhao, Y. Wu, G. Liu,X. Dong, S. He, J. Zhang, et al., Nature Communications , 2459 (2013), 1302.3017.[10] K. M. Shen, T. Yoshida, D. H. Lu, F. Ron-ning, N. P. Armitage, W. S. Lee, X. J. Zhou,A. Damascelli, D. L. Feng, N. J. C. Ingle,et al., Phys. Rev. B , 054503 (2004), URL http://link.aps.org/doi/10.1103/PhysRevB.69.054503 .[11] K. Fujita, A. R. Schmidt, E.-A. Kim, M. J. Lawler, D. H.Lee, J. C. Davis, H. Eisaki, and S. ichi Uchida, Journal of the Physical Society of Japan , 011005 (2012), URL http://jpsj.ipap.jp/link?JPSJ/81/011005/ .[12] W. Chen, G. Khaliullin, and O.P. Sushkov, Phys. Rev. B , 094519 (2009).[13] A. Kitaev, AIP Conf. Proc. , 22 (2009), URL http://link.aip.org/link/?APC/1134/22/1 .[14] X.-G. Wen, Phys. Rev. B , 085103 (2012), URL http://link.aps.org/doi/10.1103/PhysRevB.85.085103 .[15] A. Altland and M. R. Zirnbauer, Phys. Rev. B , 1142 (1997), URL http://link.aps.org/doi/10.1103/PhysRevB.55.1142 .[16] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W.Ludwig, Phys. Rev. B , 195125 (2008), URL http://link.aps.org/doi/10.1103/PhysRevB.78.195125 .[17] J. C. S´eamus Davis and D.-H. Lee, Proceedings of theNational Academy of Sciences , 17623 (2013), URL . arXiv1309.2719. r X i v : . [ c ond - m a t . s up r- c on ] N ov Supplementary materials
Yuan-Ming Lu,
1, 2
Tao Xiang, and Dung-Hai Lee
1, 2 Department of Physics, University of California, Berkeley, CA 94720 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China. (Dated: August 3, 2018)
PACS numbers:
STABILITY OF NODAL QUASIPARTICLES
In a singlet superconductor with no magnetic orders, the system preserves SU (2) spin rotational symmetry and timereversal symmetry T . A generic nodal particle in this 2+1-D time-reversal-invariant (TRI) singlet superconductorcan be described as H Q + k = k x γ + k y γ . (1)where we have normalized the Fermi velocities to unity without loss of generality. k is a small momentum expandedaround the nodal point Q . Usually the nodal Hamiltonian H Q + k is written in complex fermion basis Ψ = ( c, c † ) T ,but we can also represent it in the Majorana basis[1] η through relation c = η + i η . In the Majorana fermion basis γ , are real symmetric matrices which square to be +1[2]. Time reversal symmetry for spin=1 / T η i T − = T i,j η j , T = − . (2)where T is again a real matrix. Meanwhile spin rotational symmetries are e i πS α η i e − i πS α = (cid:0) Σ α (cid:1) i,j η j , (Σ α ) = − . (3)where S α , α = x, y, z are the total spin operators. Note that [ T, Σ α ] = 0. A symmetric nodal Hamiltonian H Q + k obey { γ i , T } = [ γ i , Σ α ] = 0 . (4)The symmetry group SU (2) s × T can be generated by the following 3 generators T e i πS α → T α ≡ T Σ α , ( T α ) = 1 , { T α , γ i } = 0 . (5)Notice that matrices { γ i , T α } form Clifford algebra The problem of classifying different nodal Hamiltonians H Q + k reduces to the following mathematical question: what are the distinct classes of real matrix γ (with γ and T α fixed)satisfying condition (5)? This question is related to (real) Clifford algebra Cl , and the associated classifying spacefor γ is R = (cid:0) Sp ( k + m ) /Sp ( k ) × Sp ( m ) (cid:1) × Z in notation of Ref. 1. Therefore the symmetry classification of nodalHamiltonian H Q + k with symmetry (5) is given by π ( R ) = Z . An intuitive way to understand this Z classificationis the following[3]: each node also appears on the surface of a 3+1-D TRI singlet superconductor. As a result thesymmetry classification of distinct 2+1-D nodal Hamiltonians is the same as symmetry classification of 3+1-D gappedTRI singlet superconductors (class CI[1, 4]) with protected surface states.Now we show how to compute this integer “winding number” for a given nodal Hamiltonian. In the presence of SU (2) s × T symmetry, the quadratic electron Hamiltonian near the node at Q can always be written as H Q = P k (cid:18) c † Q + k, ↑ c − Q + k, ↓ (cid:19) T h k (cid:18) c Q + k, ↑ c †− Q + k, ↓ (cid:19) + (cid:18) c † Q + k, ↓ c − Q + k, ↑ (cid:19) T τ y h k τ y (cid:18) c Q + k, ↓ c †− Q + k, ↑ (cid:19) (6)where τ x,y,z represent Pauli matrices in the particle-hole channel. Let’s label the filled bands (negative energyeigenstate) of Hermitian matrix h k by | k , α i , then the integer winding number w for nodal point Q is given by w = i2 π I Q d k Tr (cid:2) ∂ k ˆ P k (cid:3) , ˆ P k ≡ X α filled | k , α ih k , α | . (7)It’s straightforward to check[5] that w = ± d x − y -wave cuprate superconductor. CLASSIFICATION OF GAPPED MAGNETIC SUPERCONDUCTORS
In this section we derive the symmetry classification presented in TABLE I of the main text. Following Ref. 1,as far as topological properties are concerned, a gapped phase in two spatial dimensions can always be obtained byadding a mass matrix M to the Dirac Hamiltonian H = i ∂ x γ + i ∂ y γ . (8)In the Majorana basis γ , are real symmetric matrices which square to be +1. The mass terms correspond toreal antisymmetric matrices M satisfying { M, γ i } = 0. If the mass term M , matrices γ , and all symmetry groupgenerators form real clifford algebra Cl p, , the associated classifying space[1, 2] for mass M is R − p mod 8 .In the presence of SU (2) spin rotational symmetry and time reversal symmetry T , the symmetry group for non-interacting fermions is generated by time reversal followed by π -spin-rotations along the 3 axes: T α = T e i πS α . Inthe Majorana basis the T α are real symmetric matrices satisfying { T α , T β } = 2 δ α,β and { T α , γ i } = 0. Therefore thesymmetry-allowed M form the real Clifford algebra Cl , with γ i and T α and hence belong to classifying space R .Since π ( R ) = 0 there is only trivial mass, hence there are no topological superconductors with SU (2) × T symmetryin two space dimensions. This corresponds to Altland-Zirnbauer class[6, 7] CI in the 10-fold way of topologicalinsulators/superconductors[4].When time reversal is spontaneously broken but spin SU (2) symmetry is preserved, the generators of symmetry are { e i πS x , e i πS z } . In the Majorana basis the two generators are again real anti-symmetric matrices Σ x,z both of whichsquare to −
1. Without loss of generality, we can choose the two generators to be Σ x,z = σ x,z ⊗ (i σ y ) ⊗ N × N ( ~σ arethe Pauli matrices and N is an arbitrary integer), and the mass matrix M must have the following form M = σ ⊗ σ ⊗ H + i σ y ⊗ σ x ⊗ H x + σ ⊗ (i σ y ) ⊗ H y + i σ y ⊗ σ z ⊗ H z ,H T = − H , H Tα = H α ( α = x, y, z ) . so that [Σ x,z , M ] = 0. It’s straightforward to show that M = − H ) − X α = x,y,z ( H α ) = − N × N , { H , H α } = 0 , [ H α , H β ] = 0 . (9)Such a mass term M can be mapped into a quaternion matrix[2] H = H + i H x + j H y + k H z , H = − N × N . (10)where i , j , k satisfy the quaternion algebra. Meanwhile notice that the presence of 4 anti-commuting symmetrygenerators (squaring to be +1) also lead to the classifying space of a quaternion matrixΓ , = σ x,z ⊗ σ ⊗ σ ⊗ N × N , Γ , = σ y ⊗ σ y ⊗ σ x,z ⊗ N × N , { M, Γ i } = 0 = ⇒ M = 1 ⊗ H + i ⊗ H x + j ⊗ H y + k ⊗ H z , (11)1 ≡ i σ y ⊗ σ y ⊗ σ y , i ≡ i σ y ⊗ σ x ⊗ σ , j ≡ i σ y ⊗ σ z ⊗ σ , k ≡ σ y ⊗ σ ⊗ σ y . After taking the two Dirac matrices γ , into account ( γ , = 1), we can see all matrices { M, γ , , Γ i } form Cliffordalgebra Cl , , which leads to classifying space R = Sp ( l + m ) Sp ( l ) × Sp ( m ) × Z for SU (2) spin -symmetric superconductors in 2+1-D. Since π ( R ) = Z , distinct singlet topological superconductors are labeled by an integer i.e. their spin quantumHall conductance[8]. This corresponds to AZ class C.When spin rotational symmetry is spontaneously broken while preserving time reversal symmetry ( T = − T satisfying T = −
1. Therefore the matrices { γ , , T, M } form Clifford algebra Cl , ,which leads to classifying space[1, 2] R = O ( n ). Since π ( R ) = Z , there is only one type of topologically nontrivialsuperconductor with time reversal symmetry ( T = −
1) in addition to the topologically trivial ones. This correspondsto AZ class DIII.When AF Neel order (along z-axis) coexists with superconductivity, the symmetry group reduces to U (1) spinrotation along z-axis and time reversal combined with an in-plane π -spin-rotation. This symmetry group for non-interacting fermions is generated by T e i πS x and T e i πS y (since e i πS z = T e i πS x T e i πS y , we don’t need to write down e i πS z additionally). Again in Majorana basis these two generators are real symmetric matrices T α with { T α .T β } =2 δ α,β . Therefore the real Cliford algebra formed by mass M and { γ , , T x,y } is Cl , and the symmetry-allowed mass M belong to classifying space R . Since π ( R ) = 0 there are no topologically nontrivial superconductors protectedby this symmetry. It corresponds to AZ class AI.When the symmetry further breaks down to just U (1) spin rotation along the axis of collinear AF Neel order, saythe z -axis, the symmetry group is generated by π -spin-rotation along z -axis. In the Nambu basis ψ = ( c ↑ , c †↓ ) thequadratic gapped Hamiltonian has the following form: H = i ∂ x γ + i ∂ y γ + M,γ i = 1 , M = +1 , { M, γ i } = 0 . (12)where we’ve normalized the fermi velocities and the size of mass term M . The classifying space for M is C = U ( m + n ) U ( m ) × U ( n ) × Z and we have π ( C ) = Z . This means there is an infinite number of distinct topological superconductorswith S z conservation in 2+1-D and they are labeled by an integer. This integer is nothing but the Chern number[9] ofquadratic Hamiltonian H , associated to the number of chiral (complex) fermion modes on the edge. This correspondsto AZ class A.When this U (1) spin rotation is further broken, there is no symmetry in the system. The mass term M together withDirac matrices γ , forms clifford algebra Cl , , and the classifying space for mass term M is R = O ( m + n ) O ( m ) × O ( n ) × Z .Since π ( R ) = Z there are an infinite number of topological superconductors in two space dimensions with nosymmetry. They are labeled by an integer, which equals the number of chiral Majorana mode on the edge. Thiscorresponds to AZ pclass D.In the end we comment on vortex bound states in these superconductors. Notice that a superconducting vortexalways breaks time reversal symmetry, Meanwhile the existence of protected in-gap bound state at a point defect (thevortex core) is determined by the symmetry classification in one space dimension less[2, 10]. Since the topologicalclassification is always trivial for class A in one spatial dimension, the topological ( p + i p ) ↑ , ↓ superconductors withchiral edge modes don’t support zero-energy vortex bound state. EFFECTIVE THEORY CALCULATIONS
We follow the procedures of Ref. 11 to determine the top SC instabilities from the effective Hamiltonian. Startingwith H eff = X k ′ X σ ǫ ( k ) ψ + k σ ψ k σ + J X h i,j i S i · S j . (13)In Eq. (13) P k ′ stands for sum within a thin shell around the Fermi surface (so that | ǫ ( k ) | is less than an energycutoff). Moreover ψ σ k annihilates an electron with spin σ in band eigenstate at momentum k within the momentumshell. Note in ψ σ k we did not keep the band indices. This is because with the restriction to a thin energy shell,momentum actually fixes the band index.First we rewrite the last term of Eq. (13) in momentum space. In view of the fact that we also need to treat pairingin the AF ordered state, we use the folded, i.e. the magnetic, Brillouin zone (MBZ). (We use MBZ even when treatingthe paramagnetic state). V int = J X h i,j i S i · S j → N X ′ k , p , q ∈ MBZ X a,b,c,d n U bacd ( q ) C + k + q ,a C k ,b C + p − q ,c C p ,d + U bacd ( q + Q ) C + k + q + Q ,a C k ,b C + p − q − Q ,c C p ,d + U bacd ( q ) C + k + q + Q ,a C k + Q ,b C + p − q ,c C p ,d + U bacd ( q + Q ) C + k + q ,a C k + Q ,b C + p − q − Q ,c C p ,d + U bacd ( q ) C + k + q ,a C k ,b C + p + Q − q ,c C p + Q ,d + U bacd ( q + Q ) C + k + q + Q ,a C k ,b C + p − q ,c C p + Q ,d + U bacd ( q ) C + k + q + Q ,a C k + Q ,b C + p + Q − q ,c C p + Q ,d + U bacd ( q + Q ) C + k + q ,a C k + Q ,b C + p − q ,c C p + Q ,d o (14)In Eq. (14) N is the total number of unit cells, Q = ( π, π ) is the AF ordering wavevector, a,b,c,d are spin induces.The U bacd ( q ) in Eq. (14) is given by U bacd ( q ) = ( ~σ ab · ~σ cd ) J ( q ) , J ( q ) = J (cos q x + cos q y ) , (15)where ~σ are the spin Pauli matrices. The operator ψ k in Eq. (13) is related to C k σ in Eq. (14) by ψ k σ = φ k ,σ,n ( k ) , ( k ) C k ,σ + φ k ,σ,n ( k ) , ( k ) C k + Q ,σ . (16)Here φ Tn,σ = ( φ n,σ, , φ n,σ, ) us the n th ( n = 1 ,
2) eigenvector of H σ ( k ) = (cid:18) ǫ ( k ) σ mσ m ǫ ( k + Q ) (cid:19) (17)given in the main text. In Eq. (16) n ( k ) in the subscript of φ denotes is the index of the band that is closest to thefermi energy at momentum k . After some algebra Eq. (14) can be reduced to V eff = 1 N X ′ k , p , q ∈ MBZ X a,b,c,d n J ( q ) K baba ( k , p ; q ) ψ + k + q ,a ψ + p − q ,b ψ p ,a ψ k ,b − J ( q ) K bbaa ( k , p ; q ) ψ + k + q ,b ψ + p − q ,a ψ p ,a ψ k ,b o , (18)where K bacd ( k , p ; q ) = n φ ∗ k + q ,a, φ k ,b, φ ∗ p − q ,c, φ p ,d, − φ ∗ k + q ,a, φ k ,b, φ ∗ p − q ,c, φ p ,d, + φ ∗ k + q ,a, φ k ,b, φ ∗ p − q ,c, φ p ,d, − φ ∗ k + q ,a, φ k ,b, φ ∗ p − q ,c, φ p ,d, + φ ∗ k + q ,a, φ k ,b, φ ∗ p − q ,c, φ p ,d, − φ ∗ k + q ,a, φ k ,b, φ ∗ p − q ,c, φ p ,d, + φ ∗ k + q ,a, φ k ,b, φ ∗ p − q ,c, φ p ,d, − φ ∗ k + q ,a, φ k ,b, φ ∗ p − q ,c, φ p ,d, o . (19)In Eq. (19) we have omitted n ( k ) in the subscript of φ because in the momentum thin shell k fixes n ( k ).In the Cooper pairing channel the surviving term of Eq. (19) are V eff → − N X ′ k , p , q ∈ MBZ X a,b,c,d Λ abab ( k ; p ) ψ + − k ,b ψ + k ,a ψ p ,a ψ − p ,b , (20)where Λ abab ( k ; p ) = 2 J ( p + k ) K baba ( − p , p ; p + k ) + J ( p − k ) K bbaa ( − p , p ; p − k ) . (21)Upon SC mean-field factorization Eq. (20) becomes V eff → − X ′ k , p ∈ MBZ X a,b,c,d Λ abab ( k ; p ) n ψ + − k ,b ψ + k ,a ∆ ab ( p ) + ∆ ∗ ab ( k ) ψ p ,a ψ − p ,b − ∆ ∗ ab ( k )∆ ab ( p ) o . (22)where due to fermion anticommutation relation ∆ ab ( k ) = − ∆ ba ( − k ). Thus∆( k ) = (cid:18) ∆ t, ( k ) ∆ s ( k ) + ∆ t, ( k ) − ∆ s ( k ) + ∆ t, ( k ) ∆ t, ( k ) (cid:19) . (23)where ∆ s ( k ) is even in k and ∆ t,j ( k ) are odd in k . In the literature Eq. (23) is often written as∆( k ) = [ ψ ( k ) σ + ~d ( k ) · ~σ ](i σ y ) , (24)where ψ ( k ) = ∆ s ( k ) , d x ( k ) = [ − ∆ t, ( k ) + ∆ t, ( k )] , d y ( k ) = i [∆ t, ( k ) + ∆ t, ( k )] , d z ( k ) = ∆ t, ( k ).We then “integrate out” the electrons and keep up to the quadratic terms in ∆’s. The result is the following freeenergy form F = 1 N X ′ k , p ∈ MBZ X a,b ∆ ∗ ab ( k ) K ab ( k , p ; T )∆ ab ( p ) , (25)where K ab ( k , p ; T ) = Λ abab ( k ; p ) − N X ′ q ∈ MBZ Λ abab ( k ; q ) χ ( q ; T )Λ abab ( q ; p ) (26)where the temperature ( T )-dependent free fermion pair susceptibility is given by χ ( k ; T ) ∝ − f ( ǫ ( k )) ǫ ( k ) . (27)Here the proportionality constant is un-important for our purposes as it can be absorbed into J (see below).The leading (sub-leading) gap functions are the eigenfucntions of χ ( q ; T )Λ abab ( q ; p ) with the largest (second largest)eigenvalue among all a, b (The proportionality constant in χ T changes all eigenvalues by the same multiplicativeconstant but not the eigenfunctions.) These are the order parameters which will first (second) become unstableas J increases (at a temperature T much less than the thickness of the energy shell). These eigenfunctions areobtained numerically after discretizing the momentum space enclosed by the energy shell (under such discretization χ ( q ; T )Λ abab ( q ; p ) becomes a matrix for each pair of ( a, b )). We diagonalize each matrix then average the eigenfunctionsalong the direction perpendicular to the fermi surface. This leads to the results presented in the text. [1] A. Kitaev, AIP Conf. Proc. , 22 (2009), URL http://link.aip.org/link/?APC/1134/22/1 .[2] X.-G. Wen, Phys. Rev. B , 085103 (2012), URL http://link.aps.org/doi/10.1103/PhysRevB.85.085103 .[3] S. Matsuura, P.-Y. Chang, A. P. Schnyder, and S. Ryu, New Journal of Physics , 065001 (2013), ISSN 1367-2630, URL http://stacks.iop.org/1367-2630/15/i=6/a=065001 .[4] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B , 195125 (2008), URL http://link.aps.org/doi/10.1103/PhysRevB.78.195125 .[5] F. Wang and D.-H. Lee, Phys. Rev. B , 094512 (2012), URL http://link.aps.org/doi/10.1103/PhysRevB.86.094512 .[6] M. R. Zirnbauer, Journal of Mathematical Physics , 4986 (1996), URL http://scitation.aip.org/content/aip/journal/jmp/37/10/10.1063/1.531675 .[7] A. Altland and M. R. Zirnbauer, Phys. Rev. B , 1142 (1997), URL http://link.aps.org/doi/10.1103/PhysRevB.55.1142 .[8] T. Senthil, J. B. Marston, and M. P. A. Fisher, Phys. Rev. B , 4245 (1999), URL http://link.aps.org/doi/10.1103/PhysRevB.60.4245 .[9] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. , 405 (1982), URL http://link.aps.org/doi/10.1103/PhysRevLett.49.405 .[10] J. C. Y. Teo and C. L. Kane, Phys. Rev. B , 115120 (2010), URL http://link.aps.org/doi/10.1103/PhysRevB.82.115120 .[11] J. C. S´eamus Davis and D.-H. Lee, Proceedings of the National Academy of Sciences , 17623 (2013), URL