1. Subgap States in Majorana Wires
Piet Brouwer
Dahlem Center for Complex Quantum Systems
Physics Department
Freie Universität Berlin
Inanc Adagideli
Mathias Duckheim
Dganit Meidan
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 HBdG come in pairs ±e, with
(particle-hole symmetry) e
one fermionic excitation → two solutions of BdG equation
Eigenstate of HBdG at e = 0: Majorana fermion, -e

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

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

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

7. spinless p-wave superconductors
superconducting order parameter has the form
one-dimensional spinless p-wave superconductor
spinless p-wave superconductor
bulk excitation gap
Majorana fermion 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
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
spin-orbit coupling N B
and SOI
Zeeman field
proximity coupling to superconductor e p

11. Semiconductor proposalB
and SOI e e e B
-pF pF p p

12. Semiconductor proposalB
and SOI e e e B B
-pF pF
-pF pF p p p

13. Semiconductor proposalB
and SOI e e e e B B
-pF pF
-pF pF p p p p

14. Semiconductor proposalB
and SOI e e e e B B
-pF pF
-pF pF D p p p p
spinless p-wave superconductor

15. Semiconductor proposalB p
-pF
Mourik et al. (2012) S N B
and SOI e e e e B B
-pF pF
-pF pF D p p p p
spinless p-wave superconductor

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

17. Spinless superconductor
spinless p-wave superconductor e

18. Disorder-induced subgap states
spinless p-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:
if 2lel > x ~ n n n e e e
2lel » x
2lel > x ~
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: -1

20. Disorder-induced subgap states
spinless p-wave superconductor L
Finite L: discrete energy eigenvalues
fermionic subgap 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:1 2 3 3 2 1
If B D, ap: semiconductor model can be mapped to p+ip model
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.

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
disorder strength

26. Multichannel wire with disorder?
p+ip ? W L
Without Dy’: chiral symmetry
(H anticommutes 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 ? N p+ip S L Without Dy’: chiral symmetry
Fulga, Hassler, Akhmerov, Beenakker (2011)
Without Dy’: chiral symmetry
(H anticommutes 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 ? N
p+ip S L
Basis transformation:

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

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

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

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

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

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

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

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

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

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 fermionic subgap states in the bulk; Density of states has power-law singularity near zero energy.
Multiple channels may lead to fermionic subgap 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).
disorder strength