Multichannel Majorana Wires

1. Multichannel MajoranaWires 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 Capri, 2014

2. Excitations in superconductors Excitation spectrum Eigenvalue equation: particle-hole conjugation u ↔ v* e eF= 0 u: “electron” v: “hole” superconducting order parameter Bogoliubov-de Gennes equation particle-hole symmetry: eigenvalue spectrum is +/- symmetric onefermionic excitation → two solutions of BdG equation

3. Topological superconductors Excitation spectrum Eigenvalue equation: particle-hole conjugation u ↔ v* e e eF= 0 Spectra with and without single level at e = 0 are topologically distinct. particle-hole symmetry: eigenvalue spectrum is +/- symmetric onefermionic excitation → two solutions of BdG equation

4. Topological superconductors Excitation spectrum Eigenvalue equation: particle-hole conjugation u ↔ v* e e Spectra with and without single level at e = 0 are topologically distinct. Excitation at e = 0 is particle-hole symmetric: “Majorana state” onefermionic excitation → two solutions of BdG equation

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

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

7. Overview • Spinless superconductors as a habitat for Majorana fermions • Multichannel spinless superconducting wires • Disordered multichannel superconducting wires • Interacting multichannel spinless superconducting wires e -e

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

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

10. Particle-hole symmetric excitations Existence of a single particle-hole symmetric excitation: Superconductor • One needs a spinless (or spin-polarized) superconductor. Superconductor

11. Particle-hole symmetric excitations Existence of a single particle-hole symmetric excitation: • One needs a spinless (or spin-polarized) superconductor. • D is an antisymmetric operator. • Without spin: D must be an odd function of momentum. • p-wave:

12. 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 Law, Lee, Ng (2009) Béri, Kupferschmidt, Beenakker, Brouwer (2009)

13. 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: |rhe| = 1: “topologically nontrivial” |rhe| = 0: “topologically trivial”

14. 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: Q = detS = -1: “topologically nontrivial” Q = detS = 1: “topologically trivial” Fulga, Hassler, Akhmerov, Beenakker (2011)

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

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

17. Spinlessp-wave superconductors superconducting order parameter has the form one-dimensional spinlessp-wave superconductor spinlessp-wave superconductor bulk excitation gap: D = D’pF Majoranafermion end states Kitaev (2001) e S h x = hvF/D Argument does not depend on length of normal-metal stub

18. 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)

19. Kells, Meidan, Brouwer (2012) Multichannel spinless p-wave wire p+ip ? W ? L bulk gap: coherence length induced superconductivity is weak: and

20. Kells, Meidan, Brouwer (2012) Multichannel spinless p-wave wire p+ip ? W ? L bulk gap: coherence length induced superconductivity is weak: and Without superconductivity:transverse modes n = 1,2,3,… n=1 n=2 n=3

21. Multichannel spinless p-wave wire p+ip ? W ? L bulk gap: coherence length induced superconductivity is weak: and With D’px, but without D’py:transverse modes decouple … Majorana end-states → D N 0

22. Multichannel spinless p-wave wire p+ip ? W ? L bulk gap: coherence length induced superconductivity is weak: and With D’px, but without D’py:transverse modes decouple … Majorana end-states → D With D’py: effective Hamiltonian Hmn for end-states Hmn is antisymmetric: Zero eigenvalue (= Majorana state) if and only if N is odd. 0

23. Multichannel spinless p-wave wire p+ip ? W ? L bulk gap: coherence length induced superconductivity is weak: and Black: bulk spectrum Red: end states D Majorana if N odd

24. Multichannel spinless p-wave wire p+ip ? W ? L bulk gap: coherence length induced superconductivity is weak: and t3Ht3 = H* Without D’py:effective “time-reversal symmetry”, Combine with particle-hole symmetry:chiral symmetry, Hanticommutes with t2 Tewari, Sau (2012)

25. “Periodic table of topological insulators” Multichannel spinless p-wave wire p+ip ? W ? IQHE L bulk gap: coherence length 3DTI induced superconductivity is weak: and QSHE t3Ht3 = H* Without D’py:effective “time-reversal symmetry”, Combine with particle-hole symmetry:chiral symmetry, Hanticommutes with t2 Q: Time-reversal symmetry X: Particle-hole symmetry P = QX: Chiral symmetry Schnyder, Ryu, Furusaki, Ludwig (2008) Kitaev (2009) Tewari, Sau (2012)

26. “Periodic table of topological insulators” Multichannel spinless p-wave wire p+ip ? W ? IQHE L bulk gap: coherence length 3DTI induced superconductivity is weak: and QSHE t3Ht3 = H* Without D’py:effective “time-reversal symmetry”, Combine with particle-hole symmetry:chiral symmetry, Hanticommutes with t2 Q: Time-reversal symmetry X: Particle-hole symmetry P = QX: Chiral symmetry Schnyder, Ryu, Furusaki, Ludwig (2008) Kitaev (2009) Tewari, Sau (2012)

27. Multichannel spinless p-wave wire p+ip ? W ? L bulk gap: coherence length induced superconductivity is weak: and As long asD’pyremains a small perturbation, it is possible in principle that there are multiple Majorana states at each end, even in the presence of disorder. Tewari, Sau (2012) Rieder, Kells, Duckheim, Meidan, Brouwer (2012)

28. Multichannel spinless p-wave wire p+ip ? W ? L bulk gap: coherence length induced superconductivity is weak: and Without D’py:chiral symmetry, Hanticommutes with t2 : integer number Fulga, Hassler, Akhmerov, Beenakker (2011)

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

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

31. Multichannel wire with disorder p+ip ? W ? x x=0 Without Dy’ and without disorder: NMajorana end states

32. Multichannel wire with disorder Disordered normal metal with N channels p+ip ? W ? x x x=0 x=0 For N channels, wavefunctionsynincrease exponentially at N different rates Without Dy’ and without disorder: NMajorana end states

33. Multichannel wire with disorder Disordered normal metal with N channels p+ip ? W ? x x x=0 x=0 For N channels, wavefunctionsynincrease exponentially at N different rates WithoutDy’ but with disorder:

34. Multichannel wire with disorder p+ip ? W ? x x=0 WithoutDy’ but with disorder: n = N, N-1, N-2, …,1 N N-1 N-2 N-3 number of Majorana end states 0 disorder strength

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

36. Rieder, Brouwer, Adagideli (2013) 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.

37. Scattering theory p+ip ? N S L Basis transformation:

38. Scattering theory p+ip ? N S L Basis transformation: imaginary gauge field if and only if

39. Scattering theory p+ip ? N S L Basis transformation: imaginary gauge field if and only if

40. Scattering theory p+ip ? N S L Basis transformation: imaginary gauge field if and only if “gauge transformation”

41. Scattering theory p+ip ? N S L Basis transformation: imaginary gauge field if and only if “gauge transformation”

42. Scattering theory p+ip ? N S L Basis transformation: N, with disorder L “gauge transformation”

43. Scattering theory p+ip ? N S L Basis transformation: N, with disorder L “gauge transformation”

44. Scattering theory p+ip ? N S L N, with disorder L : eigenvalues of

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

46. Series of topological phase transitions p+ip ? W ? x x=0 = = WithDy’ and with disorder: Topological phase transitions at Dy’/Dx’ n = N, N-1, N-2, …,1 (N+1)l/x 0 disorder strength disorder strength

47. Series of topological phase transitions p+ip ? W ? x x=0 = = WithDy’ and with disorder: Topological phase transitions at Dy’/Dx’ n = N, N-1, N-2, …,1 (N+1)l/x 0 disorder strength disorder strength

48. Interacting multichannel Majorana wires p+ip ? W ? t3Ht3 = H* Without D’py:effective “time-reversal symmetry”,

49. Interacting multichannel Majorana wires Lattice model: a: channel index j: site index HS is real: effective “time-reversal symmetry”, Topological number Qchiral . Qchiral is number of Majorana states at each end of the wire, counted with sign. With interactions: Topological number Qint 8 Fidkowski and Kitaev (2010)

50. Interacting multichannel Majorana wires Qchiral= -4 Qchiral= -3 Qchiral= -2 Qchiral= -1 Qchiral= 0 a: channel index j: site index Qchiral= 1 Qchiral= 2 Qchiral= 3 Qchiral= 4 Topological number Qchiral . Qchiral is number of Majorana states at each end of the wire, counted with sign. With interactions: Topological number Qint 8 Fidkowski and Kitaev (2010)