Topology of Andreev bound state ISSP, The University of Tokyo, Masatoshi Sato

In collaboration with Satoshi Fujimoto, Kyoto University Yoshiro Takahashi, Kyoto University Yukio Tanaka, Nagoya University Keiji Yada, Nagoya University Akihiro Ii, Nagoya University Takehito Yokoyama, Tokyo Institute for Technology

Outline Andreev bound state “Edge (or Surface) state” of superconductors Part I. Andreev bound state as Majorana fermions Part II. Topology of Andreev bound states with flat dispersion

Part I. Andreev bound state as Majorana fermions

What is Majorana fermion Dirac fermion with Majorana condition Dirac Hamiltonian Majorana condition particle = antiparticle Originally, elementary particles. But now, it can be realized in superconductors.

chiral p+ip–wave SC [Read-Green (00), Ivanov (01)] analogues to quantum Hall state = Dirac fermion on the edge [Volovik (97), Goryo-Ishikawa(99),Furusaki et al. (01)] chiral edge state B 1dim (gapless) Dirac fermion Majorana condition is imposed by superconductivity TKNN # = 1

Majorana zero mode in a vortex creation = annihilation ? We need a pair of the vortices to define creation op. vortex 2 vortex 1 non-Abelian anyon topological quantum computer

uniqueness of chiral p-wave superconductor spin-triplet Cooper pair full gap unconventional superconductor no time-reversal symmetry Question: Which property is essential for Majorana fermion ? Answer: None of the above .

Spin-orbit interaction is indispensable ! Majorana fermion is possible in spin singlet superconductor MS, Physics Letters B (03), Fu-Kane PRL (08), MS-Takahashi-Fujimoto PRL (09) PRB (10), J.Sau et al PRL (10), Alicea PRB(10) .. Majorana fermion is possible in nodal superconductor MS-Fujimoto PRL (10) Time-reversal invariant Majorana fermion Tanaka-Mizuno-Yokoyama-Yada-MS, PRL (10) MS-Tanaka-Yada-Yokoyama, PRB (11) Spin-orbit interaction is indispensable !

Majorana fermion in spin-singlet SC 2+1 dim odd # of Dirac fermions + s-wave Cooper pair Majorana zero mode on a vortex [MS (03)] Non-Abelian statistics of Axion string [MS (03)] On the surface of topological insulator [Fu-Kane (08)]  Bi1-xSbx Bi2Se3 Spin-orbit interaction => topological insulator

Majorana fermion in spin-singlet SC (contd.) s-wave SC with Rashba spin-orbit interaction [MS, Takahashi, Fujimoto (09,10)] Rashba SO p-wave gap is induced by Rashba SO int.

a single chiral gapless edge state appears like p-wave SC ! Gapless edge states x y Majorana fermion For a single chiral gapless edge state appears like p-wave SC ! Chern number Similar to quantum Hall state nonzero Chern number

strong magnetic field is needed a) s-wave superfluid with laser generated Rashba SO coupling [Sato-Takahashi-Fujimoto PRL(09)] b) semiconductor-superconductor interface [J.Sau et al. PRL(10) J. Alicea, PRB(10)] c) semiconductor nanowire on superconductors ….

Majorana fermion in nodal superconductor [MS, Fujimoto (10)] Model: 2d Rashba d-wave superconductor Rashba SO Zeeman dx2-y2 –wave gap function dxy –wave gap function

Edge state dx2-y2 –wave gap function x y dxy –wave gap function

Majorana zero mode on a vortex Zero mode satisfies Majorana condition! Non-Abelian anyon The zero mode is stable against nodal excitations 1 zero mode on a vortex 4 gapless mode from gap-node From the particle-hole symmetry, the modes become massive in pair. Thus at least one Majorana zero mode survives on a vortex

The non-Abelian topological phase in nodal SCs is characterized by the parity of the Chern number There exist an odd number of gapless Majorana fermions There exist an even number of gapless Majorana fermions + nodal excitation + nodal excitation No stable Majorana fermion Topologically stable Majorana fermion

How to realize our model ? 2dim seminconductor on high-Tc Sc (a) Side View (b) Top View dxy-wave SC Zeeman field Semi Conductor d-wave SC dx2-y2-wave SC

Time-reversal invariant Majorana fermion [Tanaka-Mizuno-Yokoyama-Yada-MS PRL(10) Yada-MS-Tanaka-Yokoyama PRB(10) MS-Tanaka-Yada-Yokoyama PRB (11)] Edge state time-reversal invariance time-reversal invariance

dxy+p-wave Rashba superconductor Majorana fermion [Yada et al. (10) ] The spin-orbit interaction is indispensable No Majorana fermion

Summary (Part I) With SO interaction, various superconductors become topological superconductors Majorana fermion in spin singlet superconductor Majorana fermion in nodal superconductor Time-reversal invariant Majorana fermion

Part II. Topology of Andreev bound state

Bulk-edge correspondence Gapless state on boundary should correspond to bulk topological number Chern # (=TKNN #) Chiral Edge state

different type ABS = different topological # chiral helical Cone Chern # (TKNN (82)) Z2 number (Kane-Mele (06)) 3D winding number (Schnyder et al (08)) Sr2RuO4 Noncentosymmetric SC (MS-Fujimto(09)) 3He B

Which topological # is responsible for Majorana fermion with flat band ?

The Majorana fermion preserves the time-reversal invariance, but without Kramers degeneracy Chern number = 0 Z2 number = trivial 3D winding number = 0 All of these topological number cannot explain the Majorana fermion with flat dispersion !

Symmetry of the system Particle-hole symmetry Time-reversal symmetry Nambu rep. of quasiparticle Time-reversal symmetry

Combining PHS and TRS, one obtains Chiral symmetry c.f.) chiral symmetry of Dirac operator

The chiral symmetry is very suggestive The chiral symmetry is very suggestive. For Dirac operators, its zero modes can be explained by the well-known index theorem. Number of zero mode with chirality +1 Number of zero mode with chirality -1 2nd Chern #= instanton #

Indeed , for ABS, we obtain the generalized index theorem Number of flat ABS with chirality +1 Superconductor Number of flat ABS with chirality -1 ABS Generalized index theorem [MS et al (11)]

Atiya-Singer index theorem Our generalized index theorem Dirac operator General BdG Hamiltonian with TRS Topology in the coordinate space Topology in the momentum space Zero mode localized on soliton in the bulk Zero mode localized on boundary

Topological number Integral along the momentum perpendicular to the surface Periodicity of Brillouin zone

To consider the boundary, we introduce a confining potential V(x) vacuum Superconductor

Introduce an adiabatic parameter 𝑎 in the Planck’s constant Strategy Introduce an adiabatic parameter 𝑎 in the Planck’s constant original value of Planck’s constant Prove the index theorem in the semiclassical limit Adiabatically increase the parameter 𝑎 as 𝑎→1

In the classical limit 𝑎=0, vacuum Superconductor Gap closing point => zero energy ABS

Around the gap closing point, Replacing 𝑘 𝑥 with −𝑖ℏ 𝜕 𝑥 in the above, we can perform the semi-classical quantization , and construct the zero energy ABS explicitly. From the explicit form of the obtained solution, we can determine its chirality as

We also calculate the contribution of the gap-closing point to topological # , The total contribution of such gap-closing points should be the same as the topological number 𝑊( 𝒌 ∥ ) , Because each zero energy ABS has the chirality Γ=sgn[ 𝜂 𝑖 ] Index theorem (but ℏ≪1)

Now we adiabatically increase 𝑎 →1 (ℏ→ ℏ 0 ) 𝑛 + − 𝑛 − is adiabatic invariance Non-zero mode should be paired Thus, the index theorem holds exactly

dxy+p-wave SC Thus, the existence of Majorana fermion with flat dispersion is ensured by the index theorem

remark It is well known that dxy-wave SC has similar ABSs with flat dispersion. S.Kashiwaya, Y.Tanaka (00) In this case, we can show that 𝑊 𝑘 𝑦 =±2. Thus, it can be explain by the generalized index theorem, but it is not a single Majorana fermion

Summary Majorana fermions are possible in various superconductors other than chiral spin-triplet SC if we take into accout the spin-orbit interctions. Generalized index theorem, from which ABS with flat dispersion can be expalined, is proved. Our strategy to prove the index theorem is general, and it gives a general framework to prove the bulk-edge correspondence.

Reference Non-Abelian statistics of axion strings, by MS, Phys. Lett. B575, 126(2003), Topological Phases of Noncentrosymmetric Superconductors: Edge States, Majorana Fermions, and the Non-Abelian statistics, by MS, S. Fujimoto, PRB79, 094504 (2009), Non-Abelian Topological Order in s-wave Superfluids of Ultracold Fermionic Atoms, by MS, Y. Takahashi, S. Fujimoto, PRL 103, 020401 (2009), Non-Abelian Topological Orders and Majorna Fermions in Spin-Singlet Superconductors, by MS, Y. Takahashi, S.Fujimoto, PRB 82, 134521 (2010) (Editor’s suggestion) Existence of Majorana fermions and topological order in nodal superconductors with spin-orbit interactions in external magnetic field, PRL105,217001 (2010) Anomalous Andreev bound state in Noncentrosymmetric superconductors, by Y. Tanaka, Mizuno, T. Yokoyama, K. Yada, MS, PRL105, 097002 (2010) Surface density of states and topological edge states in noncentrosymmetric superconductors by K. Yada, MS, Y. Tanaka, T. Yokoyama, PRB83, 064505 (2011) Topology of Andreev bound state with flat dispersion, MS, Y. Tanaka, K. Yada, T. Yokoyama, PRB 83, 224511 (2011)

Thank you !

The parity of the Chern number is well-defined although the Chern number itself is not Formally, it seems that the Chern number can be defined after removing the gap node by perturbation perturbation However, the resultant Chern number depends on the perturbation. The Chern number

On the other hand, the parity of the Chern number does not depend on the perturbation particle-hole symmetry T-invariant momentum all states contribute

Non-centrosymmetric Superconductors (Possible candidate of helical superconductor) CePt3Si LaAlO3/SrTiO3 interface Bauer-Sigrist et al. Space-inversion Mixture of spin singlet and triplet pairings Possible helical superconductivity M. Reyren et al 2007