Multichannel Majorana Wires

Transcription

1 Multichannel Majorana Wires Piet Brouwer Frascati, 2014 Dahlem Center for Complex Quantum Systems Physics Department Inanc Adagideli Freie Universität Berlin Mathias Duckheim Dganit Meidan Graham Kells Felix von Oppen Maria-Theresa Rieder Alessandro Romito

2 Excitations in normal metals Sommerfeld model: Electrons can be described as free fermions ε Eigenvalue equation for single-particle energies: ε F =0 Fermi energy Hψ = εψ Ground state: Single-particle states with energy ε < ε F are occupied. wikimedia.org Arnold Sommerfeld

3 Excitations in normal metals Sommerfeld model: Electrons can be described as free fermions ε Eigenvalue equation for single-particle energies: ε F = 0 Hψ = εψ Ground state: Single-particle states with energy ε < ε F are occupied. electron-like excitation hole-like excitation wikimedia.org Arnold Sommerfeld

4 Excitations in normal metals Excitation spectrum: Same as excitations of free fermions ε Eigenvalue equation? ε F = 0 electron-like excitations hole-like excitations Lev Landau

5 Excitations in normal metals Excitation spectrum: Same as excitations of free fermions ε Eigenvalue equation? ε F = 0 electron-like excitations hole-like excitations Electron-like and hole-like excitations both have positive excitation energy

6 Excitations in normal metals Excitation spectrum: Same as excitations of free fermions ε Eigenvalue equation? ε F = 0 Combined spectrum of electron-like and hole-like excitations

7 Excitations in normal metals Excitation spectrum: Same as excitations of free fermions ε Eigenvalue equation? ε F = 0 H 0 0 H electron * u v u = ε v hole Combined spectrum of electron-like and hole-like excitations

8 Excitations in normal metals Excitation spectrum: Same as excitations of free fermions Eigenvalue equation? ε particle-hole conjugation u v* ε F = 0 H 0 0 H electron * u v u = ε v hole Combined spectrum of electron-like and hole-like excitations particle-hole symmetry: eigenvalue spectrum is +/- symmetric one fermionic excitation one pair of eigenvalues ±ε

9 Excitations in superconductors Cooper-pair Superconductor Superconductor = (Macroscopic) superconductors with different numbers of Cooper pairs cannot be distinguished. Leon Cooper Brian Josephson

10 Excitations in superconductors In superconductors one cannot distinguish between electron-like and hole-like excitations Superconductor = Superconductor

11 Excitations in superconductors Excitation spectrum Eigenvalue equation: H Δ u u = ε Δ* H * v v ε F = 0 u: electron v: hole superconducting order parameter Bogoliubov-de Gennes equation particle-hole symmetry: eigenvalue spectrum is +/- symmetric one fermionic excitation two solutions of BdG equation ε particle-hole conjugation u v*

12 Topological superconductors Excitation spectrum Eigenvalue equation: H Δ u u = ε Δ* H * v v ε F = 0 Spectra with and without single level at ε = 0 are topologically distinct. particle-hole symmetry: eigenvalue spectrum is +/- symmetric one fermionic excitation two solutions of BdG equation ε ε particle-hole conjugation u v*

13 Topological superconductors Excitation spectrum Eigenvalue equation: H Δ* Δ H * u v u = ε v ε ε particle-hole conjugation u v* Spectra with and without single level at ε = 0 are topologically distinct. Excitation at ε = 0 is particle-hole symmetric: Majorana state one fermionic excitation two solutions of BdG equation

14 Topological superconductors Excitation spectrum Eigenvalue equation: H Δ* Δ H * u v u = ε v ε ε particle-hole conjugation u v* Spectra with and without single level at ε = 0 are topologically distinct. Excitation at ε = 0 is particle-hole symmetric: Majorana state Excitation at ε = 0 corresponds to ½ fermion: non-abelian statistics

15 Topological superconductors In nature, there are only whole fermions. Majorana states always come in pairs. ε ε particle-hole conjugation u v* In a topological superconductor pairs of Majorana states are spatially well separated. Excitation at ε = 0 is particle-hole symmetric: Majorana state Excitation at ε = 0 corresponds to ½ fermion: non-abelian statistics

16 Overview Spinless superconductors as a habitat for Majorana fermions Disordered multichannel spinless superconducting wires ε -ε

17 Particle-hole symmetric excitation Can one have a particle-hole symmetric excitation in a spinfull superconductor? Superconductor Superconductor =

18 Particle-hole symmetric excitation Can one have a particle-hole symmetric excitation in a spinfull superconductor? Superconductor Superconductor =

19 Particle-hole symmetric excitations Existence of a single particle-hole symmetric excitation: One needs a spinless (or spinpolarized) superconductor. Superconductor Superconductor

20 Particle-hole symmetric excitations Existence of a single particle-hole symmetric excitation: One needs a spinless (or spinpolarized) superconductor. H Δ* Δ H * u v u = ε v Δ is an antisymmetric operator. Without spin: Δ must be an odd function of momentum. p-wave:

21 Spinless p-wave superconductors superconducting order parameter has the form one-dimensional spinless p-wave superconductor spinless p-wave superconductor bulk excitation gap: Δ = Δ p F Majorana fermion end states Kitaev (2001) N p -p r he r eh Δ(p)e iφ(p) S Andreev reflection at NS interface: scattering amplitudes and Andreev (1964) *

22 Spinless p-wave superconductors superconducting order parameter has the form one-dimensional spinless p-wave superconductor spinless p-wave superconductor bulk excitation gap: Δ = Δ p F Majorana fermion end states Kitaev (2001) e iη e -iη N p -p r he r eh Δ(p)e iφ(p) S Bohr-Sommerfeld: Majorana state (i.e., bound state at ε = 0) if Always satisfied if r he =1.

23 Spinless p-wave superconductors superconducting order parameter has the form one-dimensional spinless p-wave superconductor spinless p-wave superconductor bulk excitation gap: Δ = Δ p F Majorana fermion end states Kitaev (2001) e h S ξ = hv F /Δ Argument does not depend on length of normal-metal stub

24 Proposed physical realizations fractional quantum Hall effect at ν=5/2 Moore, Read (1991) unconventional superconductor Sr 2 RuO 4 Das Sarma, Nayak, Tewari (2006) Fermionic atoms near Feshbach resonance Proximity structures with s-wave superconductors, and topological insulators semiconductor quantum well Gurarie, Radzihovsky, Andreev (2005) Cheng and Yip (2005) Fu and Kane (2008) Sau, Lutchyn, Tewari, Das Sarma (2009) Alicea (2010) Lutchyn, Sau, Das Sarma (2010) Oreg, von Oppen, Refael (2010) ferromagnet metal surface states Duckheim, Brouwer (2011) Chung, Zhang, Qi, Zhang (2011) Choy, Edge, Akhmerov, Beenakker (2011) Martin, Morpurgo (2011) Kjaergaard, Woelms, Flensberg (2011) Weng, Xu, Zhang, Zhang, Dai, Fang (2011) Potter, Lee (2010) (and more)

25 Disordered spinless superconductor spinless p-wave superconductor x=0 without disorder: x Majorana end states ε 0 µ = F 2 pf 2m p = k = F mv F

26 Brouwer, Duckheim, Romito, Von Oppen (2011) Disordered spinless superconductor spinless p-wave superconductor x=0 without disorder: with disorder:? x Motrunich, Damle, Huse (2001) Gruzberg, Read, Vishveshwara(2005) µ = F 2 pf 2m p = k = F mv F

27 Disordered spinless superconductor Disordered normal metal spinless p-wave superconductor x=0 without disorder: with disorder:? x Transfer matrix: has eigenvalues

28 Disordered spinless superconductor Disordered normal metal spinless p-wave superconductor x=0 without disorder: with disorder: x Transfer matrix: has eigenvalues topological phase persists for

29 Disordered spinless superconductor spinless p-wave superconductor x=0 without disorder: with disorder: x topological phase persists for

30 Rieder, Brouwer, Adagideli (2013) Multichannel wire with disorder? p+ip? W x=0 x bulk gap: coherence length

31 Multichannel wire with disorder? p+ip? W x=0 x Series of N topological phase transitions at n=1,2,,n 0 disorder strength

32 Multichannel spinless p-wave wire? p+ip? W L induced superconductivity is weak: and Without Δ p y : effective time-reversal symmetry, τ 3 Hτ 3 = H* Combine with particle-hole symmetry: chiral symmetry, H anticommutes with τ 2 Tewari, Sau (2012)

33 Periodic Multichannel table of spinless topological p-wave insulators wire? p+ip? IQHE W induced superconductivity is weak: Without Δ p y : effective time-reversal symmetry, τ 3 Hτ 3 = H* Combine with particle-hole symmetry: chiral symmetry, L bulk gap: coherence length 3DTI and QSHE Θ: Time-reversal symmetry Ξ: Particle-hole symmetry Π = ΘΞ: Chiral symmetry H anticommutes with τ 2 Schnyder, Ryu, Furusaki, Ludwig (2008) Tewari, Kitaev Sau (2012) (2009)

34 Periodic Multichannel table of spinless topological p-wave insulators wire? p+ip? IQHE W induced superconductivity is weak: L bulk gap: coherence length 3DTI and QSHE Without Δ p y : effective time-reversal symmetry, τ 3 Hτ 3 = H* Combine with particle-hole symmetry: chiral symmetry, Θ: Time-reversal symmetry Ξ: Particle-hole symmetry Π = ΘΞ: Chiral symmetry H anticommutes with τ 2 Schnyder, Ryu, Furusaki, Ludwig (2008) Tewari, Kitaev Sau (2012) (2009)

35 Multichannel wire with disorder? p+ip? W x=0 x Without Δ y and without disorder: N Majorana end states ψ sin( nπy W ) x / ξ e n=1 n=2 n=3

36 Multichannel wire with disorder Disordered normal metal with N channels? p+ip? W x=0 x Without For N channels, Δ y and without wavefunctions disorder: ψn n increase Majorana exponentially end states at N different rates ψ sin( nπy W ) x / ξ e n=1 n=2 n=3

37 Multichannel wire with disorder Disordered normal metal with N channels? p+ip? W x=0 x Without For N channels, Δ y but with wavefunctions disorder: ψ n increase exponentially at N different rates

38 Multichannel wire with disorder? p+ip? W x=0 x Without Δ y but with disorder: n = N, N-1, N-2,,1 0 N N-1 N-2 N-3 number of Majorana end states disorder strength

39 Series of topological phase transitions? p+ip? W x=0 x # Majorana end states ξ/(n+1)l disorder strength

40 Series of topological phase transitions? p+ip? W x=0 x With Δ y and with disorder: = Topological phase transitions at n = N, N-1, N-2,,1 Δ y /Δ x = 0 disorder strength (N+1)l /ξ disorder strength

41 Series of topological phase transitions? p+ip? W x=0 x With Δ y and with disorder: = Topological phase transitions at n = N, N-1, N-2,,1 Δ y /Δ x = 0 disorder strength (N+1)l /ξ disorder strength

42 Summary One-dimensional superconducting wires come in two topologically distinct classes: with or without a Majorana state at each end. Multiple Majoranas may coexist in the presence of an effective timereversal symmetry. Majorana states may persist in the presence of disorder and with multiple channels. For multichannel p-wave superconductors there is a sequence of disorder-induced topological phase transitions. The last phase transition takes place at l=ξ/(n+1). 0 disorder strength