Subgap States in Majorana Wires

1. Subgap States in Majorana Wires Piet Brouwer Dahlem Center for Complex Quantum Systems Physics Department FreieUniversität Berlin InancAdagideli Mathias Duckheim DganitMeidan Graham Kells Felix von Oppen Maria-Theresa Rieder Alessandro Romito Aachen, 2013

2. excitations in superconductors D: antisymmetric operator ue, ve: solution of Bogoliubov-de Gennes equation: Eigenvalues of HBdGcome in pairs±e, with (particle-hole symmetry) e onefermionic excitation → two solutions of BdG equation Eigenstate of HBdG at e = 0: Majoranafermion, -e

3. Overview • Spinless superconductors as a habitat for Majorana fermions • Semiconductor nanowires as a spinless superconductor • Disordered spinlesssupeconducting wires • Multichannel spinless superconducting wires • Disordered multichannel superconducting wires e -e

4. superconductor proximity effect ideal interface: S S S N N rhe(e) = ae-if Deif reh(e) = aeif h e a = e-iarccos(e/D) x1 x2 x3 n/nN S I(nA) S N x1 x2 x3 e (mV) Guéronet al. (1996) Mur et al. (1996)

5. spinless superconductors are topological e S scattering matrix for Andreev reflection: h e S is unitary 2x2 matrix scattering matrix for point contact to S particle-hole symmetry: if e = 0 combine with unitarity: Andreev reflection is either perfect or absent Béri, Kupferschmidt, Beenakker, Brouwer (2009)

6. spinlessp-wave superconductors superconducting order parameter has the form one-dimensional spinlessp-wave superconductor spinlessp-wave superconductor bulk excitation gap Majoranafermion end states Kitaev (2001) p rhe S N D(p)eif(p) -p reh Andreev reflection at NS interface p-wave: * Andreev (1964)

7. spinlessp-wave superconductors superconducting order parameter has the form one-dimensional spinlessp-wave superconductor spinlessp-wave superconductor bulk excitation gap Majoranafermion end states Kitaev (2001) eih p rhe S N e-ih D(p)eif(p) -p reh Bohr-Sommerfeld: Majorana bound state if * Always satisfied if |rhe|=1.

8. Proposed physical realizations • • fractional quantum Hall effect at ν=5/2 • • unconventional superconductor Sr2RuO4 • • Fermionic atoms near Feshbach resonance • Proximity structures with s-wave superconductors, and • topological insulators • semiconductor quantum well • ferromagnet • metal surface states Moore, Read (1991) Das Sarma, Nayak, Tewari (2006) 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) 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)

9. Semiconductor proposal S spin-orbit coupling N B and SOI Zeeman field proximity coupling to superconductor Semiconducting wire with spin-orbit coupling, magnetic field Sau, Lutchyn, Tewari, Das Sarma (2009) Alicea (2010) Lutchyn, Sau, Das Sarma (2010) Oreg, von Oppen, Refael (2010)

10. Semiconductor proposal S spin-orbit coupling N B and SOI Zeeman field proximity coupling to superconductor e p

11. Semiconductor proposal S N B and SOI e e e B -pF pF p p

12. Semiconductor proposal S N B and SOI e e e B B -pF pF -pF pF p p p

13. Semiconductor proposal S N B and SOI e e e e B B -pF pF -pF pF p p p p

14. Semiconductor proposal S N B and SOI e e e e B B -pF pF -pF pF D p p p p spinlessp-wave superconductor

15. Semiconductor proposal S N B and SOI e e e e B B -pF pF -pF pF D p p p p B spinlessp-wave superconductor -pF Mouriket al. (2012) p

16. Spinless superconductor S Effective description as a spinless superconductor N B and SOI e e e e B B -pF pF -pF pF D p p p p spinlessp-wave superconductor

17. Spinless superconductor spinlessp-wave superconductor e 0

18. Disorder-induced subgap states spinlessp-wave superconductor with disorder: topological phase persists for e density of subgap states: at critical disorder strength: Motrunich, Damle, Huse (2001)

19. Disorder-induced subgap states Power-law tail for density of states: n n n -1 if 2lel > x 2lel > x 2lel < x ~ ~ ~ e e e 2lel » x weak disorder almost critical beyond critical Disorder-induced subgap states are localized in the bulk of the wire. Localization length in topological regime:

20. Disorder-induced subgap states spinlessp-wave superconductor L Finite L: discrete energy eigenvalues fermionicsubgap states algebraically small energy for large L Majorana state exponentially small energy for large L e0,max: log-normal distribution e1,min: distribution has algebraic tail near zero energy

21. beyond one dimension semiconductor model: 2 3 3 2 1 If BD, ap: semiconductor model can be mapped to p+ip model 1 projection onto “spinless” transverse channels Tewari, Stanescu, Sau, Das Sarma (2012) Rieder, Kells, Duckheim, Meidan, Brouwer (2012)

22. Multichannel spinless p-wave superconducting wire p+ip ? W ? L bulk gap: coherence length induced superconductivity is weak: and D

23. Multichannel spinless p-wave superconducting wire p+ip ? W ? L bulk gap: coherence length induced superconductivity is weak: and Without D’py:chiral symmetry … Majorana end-states → D inclusion of py: effective Hamiltonian Hmn for end-states Hmn is antisymmetric: One zero eigenvalue if N is odd, no zero eigenvalue if N is even. 0

24. Multichannel wire with disorder p+ip ? W ? L bulk gap: coherence length

25. Multichannel wire with disorder p+ip ? W ? L Series of N topological phase transitions at n=1,2,…,N 0 disorder strength

26. Multichannel wire with disorder p+ip ? W ? L Without Dy’: chiral symmetry (Hanticommutes with ty) With Dy’: Topological number Q= ±1 Topological number Qchiral . Qchiral is number of Majorana states at each end of the wire. Without disorder Qchiral = N.

27. Scattering theory p+ip ? N S L Fulga, Hassler, Akhmerov, Beenakker (2011) Without Dy’: chiral symmetry (Hanticommutes with ty) With Dy’: Topological number Q= ±1 Topological number Qchiral . Qchiral is number of Majorana states at each end of the wire. Without disorder Qchiral = N.

28. Chiral limit p+ip ? N S L Basis transformation:

29. Chiral limit p+ip ? N S L Basis transformation: imaginary gauge field if and only if

30. Chiral limit p+ip ? N S L Basis transformation: imaginary gauge field if and only if

31. Chiral limit p+ip ? N S L Basis transformation: imaginary gauge field if and only if “gauge transformation”

32. Chiral limit p+ip ? N S L Basis transformation: imaginary gauge field if and only if “gauge transformation”

33. Chiral limit p+ip ? N S L Basis transformation: N, with disorder L “gauge transformation”

34. Chiral limit p+ip ? N S L Basis transformation: N, with disorder L “gauge transformation”

35. Chiral limit p+ip ? N S L N, with disorder L : eigenvalues of

36. Chiral limit p+ip ? N S L N, with disorder L Distribution of transmission eigenvalues is known: : eigenvalues of with , self-averaging in limit L→∞

37. Series of topological phase transitions p+ip ? W ? L Without Dy’: chiral limit Qchiral topological phase transitions at n=1,2,…,N x/(N+1)l disorder strength

38. Series of topological phase transitions p+ip ? W ? L Without Dy’: chiral limit With Dy’: topological phase transitions at n=1,2,…,N

39. Series of topological phase transitions p+ip ? W ? L With Dy’: Dy’/Dx’ topological phase transitions at n=1,2,…,N (N+1)l/x disorder strength

40. Series of topological phase transitions p+ip ? W ? L With Dy’: Dy’/Dx’ topological phase transitions at n=1,2,…,N (N+1)l/x disorder strength

41. Conclusions • Majorana fermions may persist in the presence of disorder and with multiple channels • Disorder leads to fermionicsubgap states in the bulk; Density of states has power-law singularity near zero energy. • Multiple channels may lead to fermionicsubgap states at the wire ends. • For multichannel p-wave superconductors there is a sequence of disorder-induced topological phase transitions. The last phase transition takes place at l=x/(N+1). 0 disorder strength