Topological Cooper-pairing based on spin-orbit interactions 753 - PowerPoint PPT Presentation

Frontiers of Nanoscience International Center for Theoretical Physics Trieste, August 23 – September 1, 2015 Topological Cooper-pairing based on spin-orbit interactions

753 ¡ Topological Superconductors or Superfluids Why are they of interest? possible candidates for quantum computing reason : excitations (particles), topolog. protected against decoherence Majorana fermions , non-Abelian statistics Different types of topological superfluids : solids or cold atoms (a) superconductivity induced by proximity effect in - topolog. semiconductors - convent. semimetals or metals with large SO interaction (b) intrinsic topolog. superfluids , e.g., due to spin-orbit interactions 3 He B-phase , Cu x Bi 2 Se 3

749a ¡ prerequisite topolog. insulator or superconductor : integer Chern number ≠ 0 topol. order parameter class of Hamiltonians with the same Chern number , effective field theory analogue G − L ( ) = e i kr u m k r ( ) ψ m k r Berry phase: Bloch state m = band index closed path C ( ) = u m start ( ) e [ ] i γ m C u m end Berry-­‑flux ¡ [ ] = i ( ) ∇ k u m k ( ) ( ) ∫ ∫ γ m C = i A m k u m k d k d k C C ( ) dS ∫ = i curl A m k n m = 1 ( ) dS ( ) = curl A m k ( ) ∫ ; „magnet. flux“ F m k F m k 2 π BZ occ ∑ N = n m Chern number m

748a ¡ Edge or surface modes of topological insulators and superconductors when C ≠ 0 edge modes ; well known example : QH effect time-reversal symm. plays important role no TR symm : chiral edge states , e.g. QH effect , classified by integer when SC edge states give raise to Majorana fermions with TR symm : helical edge states , e.g. SQH effect , classified by  2

742a ¡ Special features of topol. superconductors ⎛ ⎞ Δ ⎛ ⎞ H 0 ( ) H = 1 c ∑ c + c ⎜ ⎟ Bogoliubov – de Gennes Hamiltonian ⎜ ⎟ ⎜ ⎟ Δ * − H 0 c + ⎝ ⎠ 2 ⎝ ⎠ k for Δ = 0 two copies of H 0 ¡ ¡ particle-hole symm. ( ) Ξ − 1 = − H − k ( ) Ξ = τ x K ⇒ Ξ H k bound states : C ¡= ¡0 ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡C ¡= ¡1 ¡ Zero energy mode ( ) E QP ( ) E = 0 ( ) E = 0 ≡ γ + = QP QP ( ) − E = QP ( ) E + QP Majorana bound states stored nonlocally at the end of a chain single state ( ) E ( ) − E + or in a vortex associated with and ! QP QP

754 ¡ Majorana Fermions particle is own antiparticle 1 electron = 2 majoranas fractionalization c + = 1 c + = 1 ( ) ( ) γ A − i γ B γ A + i γ B + + γ A = γ A γ B = γ B ; , 2 2 { } = δ ij γ i , γ j C ≠ 0 prerequisite : Chern number , e.g. 3 He-B phase , Cu x Bi 2 Se 3 how to generate topological superconductor? proximity effect 1D 2D Majorana end states Majorana vertex (L. Fu + C. Kane) (J. Sau et al.)

728b ¡ Superconductors with Rashba-type spin-orbit interaction e.g., monolayer of Pb or ultracold atoms or via proximity effect ( ) ∑ H s0 = α v F0 e z × k σ Rashba k W 2 ( ) = ε 0 k ( ) + λα v F0 k W 1 ε k λ ; h = 0 d intraband vs inter-band pairing è intraband pairing : spin singlet + triplet G. Zwicknagl inter-band pairing : spin triplet

751 ¡ inter-band pairing : finite pairing momentum c k + q 2 , λ c − k + q 2 , λ k SO interaction is pair breaking - k degeneracy q ∑ Δ = Δ ν e i q ν r ν intraband pairing : time-reversed pairing topolog. trivial

733a ¡ Large spin-orbit interaction only intraband pairing left ( ) ( ) = ε 0 k ( ) − α k x σ y − k y σ x H = H 0 + H int Hamiltonian : H 0 k electron-phonon interaction : ∑ Δ λ = λλ ' c − k ' λ ' c k ' λ ' mean field : v 0 ∑ ∑ + c − k λ + c − k ' λ ' c k ' λ ' H int = λλ ' c k λ v 0 k ' λ ' kx ≥ 0 k λ k ' λ ' kx ≥ 0 k ' x ≥ 0 Δ + = −Δ − = Δ with G. Zwicknagl ¡ topological superfluids (C. Zhang et al. (2008), H ze = σ z h 2D : out of plane magnetic field Sato et al. (2009)) ⎛ ⎞ E E ⎛ ⎞ H 0 k + q − i Δ σ y ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 2 ( ) = ⎜ ⎟ µ 1 H k ⎜ ⎟ ⎛ ⎞ * − k + q i Δ * σ y − H 0 ⎜ ⎜ ⎟ ⎟ ⎝ ⎠ ⎝ 2 ⎠ µ 2 k k q = pairing momentum

747a ¡ Topological edge states external magnetic field perpendicular to plane two Fermi surfaces one Fermi surface Zeeman field  H  z 2 + µ 2 2 + µ 2 µ B H > Δ 0 µ B H = 0 µ B H = Δ 0 Sato, Takahashi, Fujimoto

746aEN ¡ A. Yazdani, A. Bernevig et al. (2014) chain of Fe atoms on Pb (in a trench) strong SO interaction induction of SC on Fe chain (proximity effect) STM along chain

752 ¡ modified version (J. Alicea) : splitting of SO bands by in-plane field prerequisite : semiconduct. grown along (110) direction, e.g., InSb Rashba + Dresselhaus SO interaction Dresselhaus : favors spin alignment normal to plane Rashba : spins within plane are favored ⎡ ⎤ ⎛ ⎞ 2 + 1 1 ∫ H 0 + H 0 = d 2 r ψ + − ∂ x ∂ y ⎟ − µ − i α D σ z ∂ x ψ ⎢ ⎥ 2 ⎜ ⎝ ⎠ ⎢ 2m x 2m y ⎥ ⎣ ⎦ H z e = g µ B ( ) ψ r ψ + α x σ x ∂ y − α y σ y ∂ x ∫ ∫ H R = − i d 2 r ψ + σ y ψ 2 h y d 2 SC via proximity effect in-plane field opens gap when µ < g µ B 2 h y

725 ¡ 2D case : magnet. field in plane ; h = µ B H / Δ 0 ∑ ( ) = α = 0.1 for h > 0.85 Δ λ q λλ ' c v 0 2 λ ' c − k ' + q k ' + q 2 λ ' k ' λ ' kx > 0 inhomogeneous sc. state ! corresponds to type II superconductors interband pair scattering is still taking place h > h cr Cooper pairs break up ( ) ( ) N E N E N 0 N 0 N 0 = N - + N + 1 1 h > 0.85 h  1 E E with G. Zwicknagl Δ < Δ 0 Δ 0 0 0

745a ¡ as h increases more and more 2. phase transition is expected Cooper pairs with pairing momentum Q 1 = 2q 1 Q 2 = -2q 1 q 1 = q 2 = h v F character of 2. phase transition is unknown details rather different Q = 0 Q 1 ≠ 0 Q 1 , Q 2 ≠ 0 h for thin films and for 0 h c  Δ 0 ultracold atoms due to h c different energy scales! (W. Zhang + W. Yi (2013) Y. Cao at al. (2014), C. Qu et al. (2014))

734a ¡ h x + h ⊥ Two applied magnetic fields : finite pairing momentum ( ) Δ = Δ 0 e iqy ; q = 0 , q , 0 no charge current t-FFLO state gapless t-sc? polarized fermionic system like paired spinless special case fermions similar to Fe chain no spin current k 0 ¡

750 ¡ Conclusion • Topological superfluidity is feasible in systems with sufficiently large SO interaction in thin films (absence of inversion symmetry) or in optical lattice with ultracold atoms • SC may be induced by proximity effects • Magnetic fields perpendicular and parallel to the film play an important role in order to generate topological superconductivity • Topological superconductors will have Majorana bound states in vortices