Z 2 index theorem for Majorana zero modes in a class D topological superconductor
aa r X i v : . [ c ond - m a t . s up r- c on ] O c t Z index theorem for Majorana zero modes in a class D topological superconductor Takahiro Fukui and Takanori Fujiwara
Department of Physics, Ibaraki University, Mito 310-8512, Japan (Dated: August 10, 2018)We propose a Z index theorem for a generic topological superconductor in class D. Introducing aparticle-hole symmetry breaking term depending on a parameter and regarding it as a coordinate ofan extra dimension, we define the index of the zero modes and corresponding topological invariantfor such an extended Hamiltonian. It is shown that these are related with the number of the zeromodes of the original Hamiltonian modulo two. PACS numbers: 71.10.Pm, 73.43.-f, 74.45.+c, 11.15.Tk
Topological invariants are useful tools in various fieldsin physics. In particle physics, interesting phenomenasuch as chiral and gauge anomalies, instantons, vortices,Skyrmions and many other nonperturbative effects arerelated to the topological invariants [1, 2]. In condensedmatter physics, they also play a crucial role in the classi-fication of various kinds of phases of matter [3–7]. Somekinds of defects [8] or solitons [2] are also classified bythe use of topological numbers.Recently, index theorems [9, 10] for fermions coupledwith a Higgs field in a monopole or vortex background[11, 12] have been attracting renewed interest in con-densed matter physics. It is due to zero-energy Majo-rana bound states obeying non-Abelian statistics, pio-neered by Read and Green [13] in a p -wave supercon-ductor and by Kitaev [14, 15] in quantum computations.Extensive studies have recently predicted them in vari-ous kinds of superfluids and superconductors [15–39]. Ina special case in which the systems have enhanced sym-metry (chiral symmetry), it has been shown [36] that theindex theorem ensures the existence of such zero modes.However, for generic topological superconductors, thereare no index theorems which relate zero modes with atopological invariant.In this paper, we propose a Z index theorem for Ma-jorana zero modes in a vortex of a generic topologicalsuperconductor in class D. We first introduce a particle-hole symmetry breaking term that depends continuouslyon a parameter. We next regard it as a coordinate ofan extra dimension and extend the d -dimensional Hamil-tonian to ( d + 1)-dimensional one with chiral symmetry.This enables us to define the index of the extended Hamil-tonian and also the topological invariant correspondingto it. Since the index is equal to the number of zeromodes of the original d -dimensional Hamiltonian mod-ulo two , we thus have a Z index theorem for the orig-inal system, implying that the Majorana zero mode intopological superconductors in class D is also protectedtopologically. We will give concrete calculations in twodimensional models, but the extension to an arbitraryhigher dimensions is straightforward.We investigate the Hamiltonian proposed by Fu andKane [21] for the surface states of a topological insulator with a proximity effect of an s -wave superconductor, H = iγ j ∂ j − µ ⊗ σ + γ a +2 φ a , (1)where γ j and γ a +2 ( j, a = 1 ,
2) form a set of γ matrices γ µ in four dimensions satisfying { γ µ , γ ν } = 2 δ µν . We ex-plicitly employ γ j = σ j ⊗ σ and γ a +2 = 1 ⊗ σ a , where,in the tensor product of the Pauli matrices, the former(latter) describes the spin (Nambu) space. The pairingpotential is included as φ = (Re∆( x ) , Im∆( x )). Pro-vided that φ φ µ = 0, the Hamiltonian (1) belongs toclass D, since it has only particle-hole symmetry C H C − = −H , where charge conjugation operator C is defined by C = iγ γ K with K being complex conjugation [6, 7, 40].This symmetry ensures that all states with nonzero ener-gies appear as paired states with ± energies. Let ε n be aneigenvalue of the Hamiltonian (1) labeled by an integer n .Then, it is natural to label the paired state with the op-posite energy by − n , and hence, particle-hole symmetrycan be characterized by the relation, ε − n = − ε n . E ε (a) (b) ε -1 ε ε -2 FIG. 1: The spectrum of the Hamiltonian in the case of q = 3vorticity. (a) Three Majorana zero modes, marked by an oval,appear when µ = 0. (b) Two of these get finite ± energies ifa nonzero chemical potential is switched on, but an unpairedstate is protected from it, sitting exactly at the zero energydue to particle-hole symmetry. Let us assume that there is a vortex at the origin whichis described by φ = ∆( r )(cos Θ( θ ) , sin Θ( θ )), where ( r, θ )is the polar coordinates in two dimensions and the phasesatisfies Θ(2 π ) = Θ(0) + 2 πq with q being an integer. Weassume that ∆(0) = 0 and ∆( ∞ ) = ∆ >
0. Then, it hasbeen argued [19, 20] that a Majorana zero mode appearsfor odd q , whereas for even q no zero modes are allowed.This can be understood from the perturbation theorybased on a simpler model with µ = 0 which belongs toclass BDI due to additional chiral symmetry [6, 7, 40]. Inthis case of µ = 0, the exact q zero modes wave functionscan be obtained similarly to [12]. We can also apply theindex theorem to this special model [10, 36], and showthat the index computed by the exact zero modes men-tioned above and the topological invariant, i.e., the wind-ing number of the pairing potential, indeed coincide, bothof which are given by − q . This result matches the classi-fication scheme due to Teo and Kane [41]. On the otherhand, in the case of µ = 0, index theorems cannot applyany longer due to the absence of chiral symmetry. In-stead, perturbative arguments strongly suggest that evennumber of zero modes disappear in pairs with nonzero ± energy, whereas an unpaired state is protected to stay atthe zero energy due to the particle-hole symmetry. Theseimply that the number of Majorana zero modes in classD is classified by Z , even or odd q [19, 20, 36, 41]. Thesefeatures are illustrated in Fig. 1.Beyond perturbative arguments, we propose an indextheorem valid for class D superconductors without chiralsymmetry. To this end, we define, in the former part ofthe paper, an analytical index for a generic model (1)with a nonzero chemical potential. Firstly, we introducesymmetry breaking term [28, 41] to the Hamiltonian (1)such that H ( τ ) = iγ j ∂ j − µ ⊗ σ + γ a +2 φ a − λ ( τ ) γ b , (2)where γ b could be any hermitian matrix with Cγ b C − = γ b , which may be several products of the γ matrices.One of simpler choices is γ b = γ ≡ ( − i ) γ γ γ γ . Forsimplicity and for convenience, we will restrict our dis-cussions only to this case, γ b = γ . As for λ ( τ ), weassume that it is an odd function of τ , λ ( − τ ) = − λ ( τ ),and hence λ (0) = 0. We also assume that λ ( ∞ ) = λ isfinite. Then, particle-hole symmetry can be generalizedto C H ( τ ) C − = −H ( − τ ) . Provided that the spectrum of H ( τ ) is a smooth functionof τ , it can be labeled by the same quantum number n for the τ = 0 Hamiltonian such that H ( τ ) ϕ n ( τ, x ) = ε n ( τ ) ϕ n ( τ, x ), where ε n (0) = ε n . From the point of viewof the spectrum, the generalized particle-hole symmetryleads to the new relationship, ε − n ( − τ ) = − ε n ( τ ) . (3)In this sense, the states labeled by ± n can be still re-garded as the paired states. The spectral flow as a func-tion of τ is shown in Fig. 2. E (a) ε -1 (τ) ε (τ)τ ε (τ)τ τ - E (b) ε -1 (τ) ε (τ)τ ε (τ) FIG. 2: The spectral flow of the Hamiltonian for a genericcase of µ = 0. The n = 0 state crosses the E = 0 line atleast τ = 0. (a) If any other eigenvalue crosses it at a finite τ , say, ε n ( τ ) = 0, the spectral symmetry (3) ensures that ε − n ( − τ ) = 0. (b) If the model parameters are changed, theabove ± n modes may come not to cross the zero energy. Evenin this case, the index changes by even integers because of thesymmetry (3). Although the extended Hamiltonian (2) does not havechiral symmetry, we can employ the spectral flow for theindex theorem. To see this, we introduce a kinetic termfor the parameter τ and define H (3) = iσ ⊗ ∂ τ + σ ⊗ H ( τ ) ≡ i Γ j ∂ j + Γ a +3 φ a + i Γ Γ Γ µ, (4)where j, a = 1 , ,
3. Newly defined Γ-matrices obey { Γ µ , Γ ν } = 2 δ µν . H (3) can be regarded as a Hamiltoniandefined in three dimensions spanned by the coordinates x , x and x ≡ τ . We have also introduced φ = λ and regarded it as a component of a generalized orderparameter φ = (Re∆ , Im∆ , λ ). Note that the extendedHamiltonian (4) has chiral symmetry Γ H (3) = −H (3) Γ ,where Γ = ( − i ) Γ · · · Γ . Therefore, if the Hamiltonian(4) has zero modes, they have definite chirality.The eigenvalue equation for the zero modes is H (3) Φ =0, which is given by H ( τ ) as follows: ∂ τ Φ = σ ⊗ H ( τ )Φ . (5)To solve this equation, let us set Φ n ( τ, x ) = f n ( τ ) ϕ n ( τ, x ). Then, ∂ τ Φ n = ( ∂ τ f n ) ϕ n + f n ( ∂ τ λ ) ∂ λ ϕ n ,since the Hamiltonian H ( τ ) depends on τ only through λ ( τ ). Provided that ∂ τ λ can be neglected in the adia-batic approximation and that ϕ n ( τ, x ) is normalizable, itturns out that f n is given by f ± n ( τ ) = e ± R τ dτ ′ ε n ( τ ′ ) f ± c , where f c is a constant spinor with Γ f ± c = ± f ± c . If agiven state for n = 0 satisfies ε n (+ ∞ ) > ε n ( −∞ ) <
0, which is the case of n = 1 and n = − n and − n are normalizable zeromodes with chirality −
1. Likewise, if ε n (+ ∞ ) < ε n ( −∞ ) > ± n states are normalizable zero modes withchirality +1. Now, define the index of the Hamiltonian H (3) by ind H (3) = N + − N − . (6)Then, these paired zero modes contribute to the indexfor H (3) always by two. On the other hand, if ε n ( ±∞ )has the same sign, such ± n states cannot be zero modes,since the wave functions are not normalizable. Therefore,these states give no contribution to the index. It thusturns out that n = 0 modes can change the index byeven integers. Contrary to these modes, the n = 0 mode,which is always a zero mode of H (3) unless ε ( ∞ ) = 0,determines whether the index is even or odd. Thus, itturns out that the number of zero modes of H in Eq. (1),which we denote as N ( H ), is given by the index of H (3) in Eq. (6) as N ( H ) = ind H (3) mod 2 . (7)So far we have discussed the index of the extendedHamiltonian (4) and its modulo-two relationship withthe number of zero modes of the original Hamiltonian(1). The rest of the present paper is devoted to calcu-lations of the corresponding topological invariant. For the Hamiltonian with chiral symmetry such as H (3) , it ispossible to define the topological invariant equal to theindex and to claim the index theorem [9, 10, 36] suchthat ind H (3) = − Z dS j J j ( x, , ∞ ) , (8)where dS j is an infinitesimal surface elements at theboundary ( x → ∞ ) of the Euclidean space R , and J i isthe axial vector current defined by J i ( x, m, M )= lim y → x tr Γ Γ i (cid:18) m − i H (3) − M − i H (3) (cid:19) δ ( x − y ) . The first term in the above parentheses becomes the in-dex in the limit m → M → ∞ after the calculations. When we calculate ther.h.s. of Eq. (8), it may be easy to use the plane wavebasis. Possible terms contributing to the index are J i ( x, , ∞ ) = − Z d k (2 π ) G tr Γ Γ i (cid:0) − i Γ j k j − i Γ a +3 φ a + Γ Γ Γ µ (cid:1) ( K − µ Γ)Λ( K − µ Γ)Λ( K − µ Γ) , where G − = "(cid:18)q k + k + λ − µ (cid:19) + k + ∆ k + k + λ + µ (cid:19) + k + ∆ , (9) K = k j + φ a + µ , Γ = 2 i (cid:0) Γ Γ k + Γ Γ k + Γ Γ λ (cid:1) , Λ = i Γ j Γ a +3 ∂ j φ a . The first step of the calculations is to take the traceof the Γ-matrices by the use of tr Γ Γ µ Γ ν · · · Γ µ Γ ν =(2 i ) ǫ µ ν ··· µ ν . Lengthy but straightforward calcula-tions lead to J i = 2 ǫ ijk ǫ abc φ a ∂ j φ b ∂ k φ c Z d k (2 π ) G K − δ i µ ǫ ab λ∂ φ a ∂ φ b Z d k (2 π ) G . As a next step, we carry out the integration over the mo-mentum k j in the above and over the space coordinates x j in Eq. (8). In particular in the latter integration,since the generalized order parameter φ depends cylin-drically on the coordinates ( r, θ, τ ), it may be natural toregard the boundary of R as a cylinder illustrated inFig. 3. For convenience, let us divide it into two piecesI ( r → ∞ ) and II ( τ → ±∞ ). The calculations on these two regions are quite similar to those in [42] when µ = 0.However, in the case of µ = 0, we have to choose ap-propriate λ ≡ λ ( ∞ ) to make the calculations valid. Tosee this, let us set ∆ → ∆ and λ → λ , respectively, inthe regions I and II. Then, we see that in the region I, G in Eq. (9) is always finite because of finite ∆ . Onthe other hand, it becomes singular in the region II if | λ | < | µ | . Indeed, in this case, G − in Eq. (9) van-ishes at some momentum at the core of the vortex r = 0,where ∆(0) = 0. Therefore, we restrict our discussionsto the case of | λ | ≥ | µ | . After some straightforwardcalculations, we arrive atind H (3) = 12 µ ( | λ − µ | − | λ + µ | ) 12 π I dθǫ ab ˆ φ a ∂ θ ˆ φ b = − sgn( λ ) q, (10) x x x =τ IIII I
FIG. 3: The cylindrical surface of the integration in Eq. (8).“I” denotes the side of the cylinder r → ∞ , whereas “II”denote two discs at τ → ±∞ . where ˆ φ a ≡ φ a / ∆ , the subscript a, b are restricted to1 ,
2, and the θ -integration is over S at r → ∞ in theregion II, which becomes q . It turns out that the indexis basically determined only by the vorticity q , and is thesame as the the index of H when µ = 0, which has beencalculated in ref. [36] as ind H = − q : The artificially in-troduced parameter λ just change the sign of the index.It follows that the index is topological: It is indeed pro-tected from infinitesimal changes of the three parameters µ , q , and λ . Eqs. (10) and (7) are Z index theorem fora generic model in class D.As we have mentioned, the topological invariant (10) isinvalid if | λ | < | µ | . In this case, the zero mode equation(5) may not have normalizable solutions. This can bechecked by the use of perturbations to this equation. Letus consider two possibilities of small parameters, µ and λ . If one regards µ as a small parameter and appliesthe first order perturbation to µ = 0 solutions, one hasindeed one normalizable solution. On the other hand,in the case of small λ , one can obtain the first orderperturbative energy eigenvalue, but the first order wavefunction cannot be normalizable. This implies that inthe case | λ | ≪ | µ | , the relationship between the spectralflow of H ( τ ) and the zero mode of H (3) does not hold.Therefore, we conclude, from these observations, togetherwith the calculations of the topological invariant, thatthe present formulation of the Z index theorem requires | λ | ≥ | µ | .The authors would like to thank H. Suzuki and M.Nitta for valuable discussions. This work was supportedin part by Grants-in-Aid for Scientific Research (GrantsNo. 20340098 and No. 21540378). [1] S. B. Treiman et. al. eds, Current algebra and anoma-lies (World Scientific Publishing Co Pte Ltd, Singapore,1985).[2] N. Manton and P. Sutcliffe,
Topological Solitons (Cam-bridge University Press, Cambridge, 2004).[3] D.J. Thouless, M. Kohmoto, M. P. Nightingale, and M.den Nijs, Phys. Rev. Lett. , 405 (1982). [4] M. Kohmoto, Ann. Phys. , 355 (1985).[5] G. E. Volovik, The Universe in a Helium Droplet (OxfordUniversity Press, Oxford, 2003).[6] A. Schnyder, S. Ryu, A. Furusaki, and A. Ludwig, Phys.Rev. B , 195125 (2008); AIP Conf. Proc. , 10(2009).[7] A. Kitaev, Proceedings of the L.D.Landau Memo-rial Conference “Advances in Theoretical Physics”,Chernogolovka, Moscow region, Russia, 22-26 June 2008(unpublished).[8] For a review, see N. D. Mermin, Rev. Mod. Phys. ,591 (1979).[9] C. Callias, Commun. Math. Phys. , 213 (1978).[10] E. J. Weinberg, Phys. Rev. D , 2669 (1981).[11] R. Jackiw and C. Rebbi, Phys. Rev. D , 3398 (1976).[12] R. Jackiw and P. Rossi, Nucl. Phys. B190 , 681 (1981).[13] N. Read and D. Green, Phys. Rev. B , 10267 (2000).[14] A. Kitaev, Proceedings of the Mesoscopic and StronglyCorrelated Electron Systems Conference, Chernogolovka,Moscow Region, Russia, 9-16 July 2000 (unpublished),(arXiv:cond-mat/0010440).[15] A. Kitaev, Ann. Phys. , 2 (2006).[16] D. A. Ivanov, Phys. Rev. Lett. , 268 (2001).[17] A. Stern, F. von Oppen, and E. Mariani, Phys. Rev. B , 205338 (2004).[18] S. Das Sarma, C. Nayak, and S. Tewari, Phys. Rev. B , 220502 (2006).[19] S. Tewari, S. Das Sarma, and D.-H. Lee, Phys. Rev. Lett. , 037001 (2007).[20] V. Gurarie and L. Radzihovsky, Phys. Rev. B , 212509(2007).[21] L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407(2008).[22] X.-L Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang, Phys.Rev. Lett. , 187001 (2009).[23] M. Sato, Y. Takahashi, and S. Fujimoto, Phys. Rev. Lett. , 020401 (2009).[24] Y. Tanaka, T. Yokoyama, and N. Nagaosa, Phys. Rev.Lett. , 107002 (2009).[25] G. E. Volovik, Pis’ma ZhETF , 639 (2009).(arXiv:0909.3084)[26] D. L. Bergman and K. Le Hur, Phys. Rev. B , 184520(2009).[27] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma,Phys. Rev. Lett. , 040502 (2010).[28] J. C. Y. Teo and C. L. Kane, Phys. Rev. Lett. ,046401 (2010).[29] P. A. Lee, arXiv:09072681.[30] J. Alicea, Phys. Rev. B , 125318 (2010).[31] I. F. Herbut, Phys. Rev. Lett. , 066404 (2010).[32] J. Linder, Y. Tanaka, T. Yokoyama, A. Sudbo, and N.Nagaosa, Phys. Rev. Lett. , 067001 (2010).[33] I. F. Herbut, Phys. Rev. B , 205429 (2010).[34] S. Tewari, J. D. Sau, and A. Das Sarma, Annals Phys. , 219 (2010).[35] L. Santos, T. Neupert, C. Chamon, and C. Mudry, Phys.Rev. B , 184502 (2010).[36] T. Fukui and T. Fujiwara, J. Phys. Soc. Jpn. , 033701(2010).[37] R. Roy, arXiv:1001.2571.[38] Y. Nishida, Phys. Rev. D , 074004 (2010).[39] S. Yasui, K. Itakura, and M. Nitta, Phys. Rev. D ,105003 (2010).[40] A. Altland and M. Zirnbauer, Phys. Rev. B , 1142 (1997).[41] J. C. Y. Teo and C. L. Kane, arXiv:1006.0690. [42] T. Fukui, Phys. Rev. B81