1. Multichannel 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
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
one fermionic 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
one fermionic 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”
one fermionic 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 symmetric excitation
Can one have a particle-hole symmetric excitation in a spinfull superconductor?
Superconductor
Superconductor =

9. Particle-hole symmetric excitation
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
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
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
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 = det S = -1: “topologically nontrivial”
Q = det S = 1: “topologically trivial”
Fulga, Hassler, Akhmerov, Beenakker (2011)

15. Spinless p-wave superconductors
superconducting order parameter has the form
one-dimensional spinless p-wave superconductor
spinless p-wave superconductor
bulk excitation gap: D = D’ pF
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)

16. Spinless p-wave superconductors
superconducting order parameter has the form
one-dimensional spinless p-wave superconductor
spinless p-wave superconductor
bulk excitation gap: D = D’ pF
Majorana fermion 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. Spinless p-wave superconductors
superconducting order parameter has the form
one-dimensional spinless p-wave superconductor
spinless p-wave superconductor
bulk excitation gap: D = D’ pF
Majorana fermion 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. Multichannel spinless p-wave wire
Kells, Meidan, Brouwer (2012)
Multichannel spinless p-wave wire ?
p+ip ? W L
bulk gap:
coherence length
induced superconductivity is weak:
and

20. Multichannel spinless p-wave wire
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

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.

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
Combine with particle-hole symmetry: chiral symmetry,
H anticommutes with t2
Without D’py : effective “time-reversal symmetry”,
t3Ht3 = H*
Tewari, Sau (2012)

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

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

27. Multichannel spinless p-wave wire?
p+ip ? W L
bulk gap:
coherence length
induced superconductivity is weak:
and
As long as D’py remains 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,
H anticommutes with t2
: integer number
Fulga, Hassler, Akhmerov, Beenakker (2011)

29. Multichannel wire with disorder
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
disorder strength

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

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

33. Multichannel wire with disorder
Disordered normal metal with N channels x
x=0
For N channels, wavefunctions yn increase exponentially at N different rates ?
p+ip ? W x
x=0
Without Dy’ but with disorder:

34. Multichannel wire with disorder?
p+ip ? W x
x=0
Without Dy’ but with disorder:
n = N, N-1, N-2, …,1 N
N-1
N-2
N-3
number of Majorana end states
disorder strength

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

36. Scattering theory ? N p+ip S L Without Dy’: chiral symmetry
Rieder, Brouwer, Adagideli (2013)
Scattering theory ? N
p+ip S L
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.

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

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

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

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

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

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

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

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

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

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

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

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

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
Qchiral = 1
a: channel index
j: site index
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)

51. Interacting multichannel Majorana wires
Qchiral = -4
Qchiral = -3
Qchiral = -2
Qchiral = -1 ~
Qchiral = 0
Qchiral = 1
a: channel index
j: site index
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)

52. Interacting multichannel Majorana wires
ideal normal lead
a: channel index
j: site index
With interactions?
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)

53. Interacting multichannel Majorana wires
Meidan, Romito, Brouwer (2014)
Interacting multichannel Majorana wires S
ideal normal lead
Qchiral = -i tr reh
a: channel index
j: site index
With interactions?
Qchiral = -4
Qchiral = -3
Qchiral = -2
Qint = 0
, ±1
, ±2
, ±3
Qchiral = -1
S well defined;
Qchiral = 0
Qchiral = 1
Qint = -i tr reh
Qchiral = 2
Qchiral = 3
Qchiral = 4

54. The case Q = 4 S Low-energy subspace Kondo! -i tr reh = 4
a: channel index
j: site index
Low-energy subspace
Kondo!
Low-energy Fermi liquid fixed point:
2fold degenerate excited state
tunneling to/from normal lead
→ S well defined;
2fold degenerate ground state
-i tr reh = 4

55. The case Q = -4 S Low-energy subspace Kondo! -i tr reh = -4
a: channel index
j: site index
Low-energy subspace
Kondo!
Low-energy Fermi liquid fixed point:
2fold degenerate excited state
tunneling to/from normal lead
→ S well defined;
2fold degenerate ground state
-i tr reh = -4

56. The case Q = ±4 S Hint(q) = Hint,1 sinq + Hint,2 cosq
Interpolation between Q = 4 and Q = -4:
Hint(q) = Hint,1 sinq + Hint,2 cosq
Low-energy subspace
2fold degenerate ground state
1-4 e
transitions:
tunneling
to/from leads 1-4
q ≈ 0

57. The case Q = ±4 S Hint(q) = Hint,1 sinq + Hint,2 cosq
Interpolation between Q = 4 and Q = -4:
Hint(q) = Hint,1 sinq + Hint,2 cosq
Low-energy subspace
2fold degenerate ground state
9-12 e
transitions:
tunneling
to/from leads 9-12
q ≈ p/2

58. The case Q = ±4 S Hint(q) = Hint,1 sinq + Hint,2 cosq
Interpolation between Q = 4 and Q = -4:
Hint(q) = Hint,1 sinq + Hint,2 cosq
Low-energy subspace
2fold degenerate ground state
1-4
5-8
9-12 e
transitions:
tunneling
to/from leads 1-4, 5-8, or 9-12
3-channel Kondo!
Low-energy Fermi liquid fixed point for generic q, separated by Non-Fermi liquid point.
p/2 q
-i tr reh = 4
-i tr reh = -4
generic q

59. 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 time-reversal 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=x/(N+1).
An interacting multichannel Majorana wire can be mapped to an effective Kondo problem if coupled to a normal-metal lead.
disorder strength