### Multichannel Majorana Wires

```Multichannel Majorana Wires
Piet Brouwer
Dahlem Center for Complex Quantum Systems
Physics Department
Freie Universität Berlin
Mathias Duckheim
Capri, 2014
Dganit Meidan
Graham Kells
Felix von Oppen
Maria-Theresa Rieder
Alessandro Romito
Excitations in superconductors
Excitation spectrum
Eigenvalue equation:
e
  u  u 
 H

    e  
   *  H *  v   v 
u: “electron”
particle-hole
conjugation
u ↔ v*
eF = 0
v: “hole”
superconducting order parameter
Bogoliubov-de Gennes equation
particle-hole symmetry: eigenvalue spectrum is +/- symmetric
one fermionic excitation → two solutions of BdG equation
Topological superconductors
Excitation spectrum
Eigenvalue equation:
  u  u 
 H

    e  
   *  H *  v   v 
e
e
particle-hole
conjugation
u ↔ v*
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
Topological superconductors
Excitation spectrum
Eigenvalue equation:
  u  u 
 H

    e  
   *  H *  v   v 
e
e
particle-hole
conjugation
u ↔ v*
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
Topological superconductors
Excitation spectrum
Eigenvalue equation:
  u  u 
 H

    e  
   *  H *  v   v 
e
e
particle-hole
conjugation
u ↔ v*
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
Topological superconductors
In nature, there are only whole
fermions.
→Majorana states always come
in pairs.
e
e
particle-hole
conjugation
u ↔ v*
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
Overview
• Spinless superconductors as a habitat for Majorana fermions
• Multichannel spinless superconducting wires
• Disordered multichannel superconducting wires
• Interacting multichannel spinless superconducting wires
e
e
Particle-hole symmetric excitation
Can one have a particle-hole symmetric
excitation in a spinfull superconductor?
Superconductor
Superconductor
=
Particle-hole symmetric excitation
Can one have a particle-hole symmetric
excitation in a spinfull superconductor?
Superconductor
Superconductor
=
Particle-hole symmetric excitations
Existence of a single particle-hole symmetric
excitation:
• One needs a spinless (or spinpolarized) superconductor.
Superconductor
Superconductor
Particle-hole symmetric excitations
Existence of a single particle-hole symmetric
excitation:
• One needs a spinless (or spinpolarized) superconductor.
  u  u 
 H

    e  
   *  H *  v   v 
•  is an antisymmetric
operator.
• Without spin:  must be an
odd function of momentum.
p-wave:
Spinless superconductors are topological
S
e
scattering matrix for Andreev reflection:
h
e
scattering matrix for point
contact to S
S is unitary 2x2 matrix
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)
Spinless superconductors are topological
S
e
scattering matrix for Andreev reflection:
h
e
scattering matrix for point
contact to S
S is unitary 2x2 matrix
particle-hole symmetry:
if e = 0
combine with unitarity:
|rhe| = 1: “topologically nontrivial”
|rhe| = 0: “topologically trivial”
Spinless superconductors are topological
S
e
scattering matrix for Andreev reflection:
h
e
scattering matrix for point
contact to S
S is unitary 2x2 matrix
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)
Spinless p-wave superconductors
superconducting order parameter has the form
one-dimensional spinless p-wave superconductor
spinless p-wave superconductor
bulk excitation gap:  = ’ pF
Majorana fermion end states
N
p
-p
rhe
reh
Kitaev (2001)
S
(p)eif(p)
Andreev reflection at NS interface
Andreev (1964)
p-wave:
*
Spinless p-wave superconductors
superconducting order parameter has the form
one-dimensional spinless p-wave superconductor
spinless p-wave superconductor
bulk excitation gap:  = ’ pF
Majorana fermion end states
e ih
e-ih
N
p
-p
rhe
reh
Kitaev (2001)
S
(p)eif(p)
Bohr-Sommerfeld: Majorana state if
*
Always satisfied if |rhe|=1.
Spinless p-wave superconductors
superconducting order parameter has the form
one-dimensional spinless p-wave superconductor
spinless p-wave superconductor
bulk excitation gap:  = ’ pF
Majorana fermion end states
Kitaev (2001)
e
S
h
x  hvF/
Argument does not depend on length of normal-metal stub
Proposed physical realizations
• fractional quantum Hall effect at ν=5/2
• unconventional superconductor Sr2RuO4
• Fermionic atoms near Feshbach resonance
Das Sarma, Nayak, Tewari (2006)
Cheng and Yip (2005)
• Proximity structures with s-wave superconductors, and
topological insulators
semiconductor quantum well
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
(and more)
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)
Kells, Meidan, Brouwer (2012)
Multichannel spinless p-wave wire
?
p+ip
?
L
bulk gap:
coherence length
induced superconductivity is weak:
and
W
Kells, Meidan, Brouwer (2012)
Multichannel spinless p-wave wire
p+ip
?
?
L
bulk gap:
coherence length
induced superconductivity is weak:
Without superconductivity:
ny
W
  sin( )e
 n 
2
2

  k x  kF
W 
2
ikx x
n = 1,2,3,… n=1
and
transverse modes
n=2
n=3
W
Multichannel spinless p-wave wire
?
p+ip
?
W
L
bulk gap:
coherence length
induced superconductivity is weak:
and
With ’px, but without ’py : transverse modes decouple
…
Majorana end-states
→

N
0
Multichannel spinless p-wave wire
?
p+ip
?
W
L
bulk gap:
coherence length
induced superconductivity is weak:
and
With ’px, but without ’py : transverse modes decouple
…
Majorana end-states
→
With ’py: effective Hamiltonian Hmn for end-states
Hmn is antisymmetric: Zero eigenvalue
(= Majorana state) if and only if N is odd.

0
Multichannel spinless p-wave wire
?
p+ip
?
W
L
bulk gap:
coherence length
induced superconductivity is weak:
Black: bulk spectrum
Red: end states
and

Majorana if N odd
Multichannel spinless p-wave wire
?
p+ip
?
W
L
bulk gap:
coherence length
induced superconductivity is weak:
and
Without ’py : effective “time-reversal symmetry”, t3Ht3 = H*
Combine with particle-hole symmetry: chiral symmetry,
H anticommutes with t2
Tewari, Sau (2012)
“Periodic
Multichannel
table ofspinless
topological
p-wave
insulators”
wire
?
p+ip
IQHE
?
W
L
bulk gap:
coherence length
3DTI
induced superconductivity is weak:
and
QSHE
Without ’py : effective “time-reversal symmetry”, t3Ht3 = H*
Combine with particle-hole symmetry: chiral symmetry,
H anticommutes with t2
Q: Time-reversal symmetry
X: Particle-hole symmetry
P = QX: Chiral symmetry
Schnyder, Ryu, Furusaki, Ludwig (2008)
(2009)
Tewari,Kitaev
Sau (2012)
“Periodic
Multichannel
table ofspinless
topological
p-wave
insulators”
wire
?
p+ip
IQHE
?
W
L
bulk gap:
coherence length
3DTI
induced superconductivity is weak:
and
QSHE
Without ’py : effective “time-reversal symmetry”, t3Ht3 = H*
Combine with particle-hole symmetry: chiral symmetry,
H anticommutes with t2
Q: Time-reversal symmetry
X: Particle-hole symmetry
P = QX: Chiral symmetry
Schnyder, Ryu, Furusaki, Ludwig (2008)
(2009)
Tewari,Kitaev
Sau (2012)
Multichannel spinless p-wave wire
?
p+ip
?
W
L
bulk gap:
coherence length
induced superconductivity is weak:
and
As long as ’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)
Multichannel spinless p-wave wire
?
p+ip
?
L
bulk gap:
coherence length
induced superconductivity is weak:
and
Without ’py : chiral symmetry, H anticommutes with t2
: integer number
Fulga, Hassler, Akhmerov, Beenakker (2011)
W
Multichannel wire with disorder
?
x=0
p+ip
?
x
bulk gap:
coherence length
W
Multichannel wire with disorder
p+ip
?
?
W
x
x=0
Series of N topological phase transitions at
n=1,2,…,N
0
disorder strength
Multichannel wire with disorder
?
x=0
p+ip
?
x
Without y’ and without disorder: N Majorana end states
W
Multichannel
Disordered
normalwire
metalwith
withdisorder
N channels
?
x=0
x=0
p+ip
?
xx
For N channels,
wavefunctions
exponentially
n increase
Without
y’ and without
disorder:N
Majorana
end states at
N different rates
W
Multichannel
Disordered
normalwire
metalwith
withdisorder
N channels
?
x=0
x=0
p+ip
?
xx
For N channels,
wavefunctions
Without
y’ but with
disorder: n increase exponentially at
N different rates
W
Multichannel wire with disorder
p+ip
?
?
W
x
x=0
Without y’ but with disorder:
n = N, N1, N2, …,1
N N-1 N-2 N-3
0
number of Majorana end states
disorder strength
Series of topological phase transitions
?
x=0
p+ip
?
W
x
# Majorana end states
x/(N+1)l
disorder strength
Scattering theory
?
p+ip
N
S
L
Fulga, Hassler, Akhmerov, Beenakker (2011)
Without y’: chiral symmetry
(H anticommutes with ty)
Topological number Qchiral
Qchiral is number of Majorana
states at each end of the wire.
Without disorder Qchiral = N.
With y’:
Topological number Q = ±1
.
Scattering theory
?
p+ip
N
L
Basis transformation:
S
Scattering theory
?
p+ip
N
S
L
Basis transformation:
imaginary gauge field
if and only if
Scattering theory
?
p+ip
N
S
L
Basis transformation:
imaginary gauge field
if and only if
Scattering theory
?
p+ip
N
S
L
Basis transformation:
imaginary gauge field
if and only if
“gauge transformation”
Scattering theory
?
p+ip
N
S
L
Basis transformation:
imaginary gauge field
if and only if
“gauge transformation”
Scattering theory
?
p+ip
N
S
L
Basis transformation:
“gauge transformation”
N, with disorder
L
Scattering theory
?
N
p+ip
S
L
Basis transformation:
“gauge transformation”
N, with disorder
L
Scattering theory
?
N
p+ip
S
L
: eigenvalues of
N, with disorder
L
Scattering theory
?
p+ip
N
S
L
: eigenvalues of
N, with disorder
Distribution of transmission eigenvalues is known:
with
L
, self-averaging in limit L →∞
Series of topological phase transitions
p+ip
?
?
W
x
x=0
=
Topological phase transitions at
n = N, N1, N2, …,1
y’/x’
With y’ and with disorder:
=
0
disorder strength
(N+1)l /x
disorder strength
Series of topological phase transitions
p+ip
?
?
W
x
x=0
=
Topological phase transitions at
n = N, N1, N2, …,1
y’/x’
With y’ and with disorder:
=
0
disorder strength
(N+1)l /x
disorder strength
Interacting multichannel Majorana wires
?
p+ip
?
W
Without ’py : effective “time-reversal symmetry”, t3Ht3 = H*
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)
Interacting multichannel Majorana wires
Qchiral = -4
Qchiral = -3
Qchiral = -2
Qchiral = -1
Qchiral = 0
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.
a: channel index
j: site index
.
With interactions:
Topological number Qint
8
Fidkowski and Kitaev (2010)
Interacting multichannel Majorana wires
~
Qchiral = -4
Qchiral = -3
Qchiral = -2
Qchiral = -1
Qchiral = 0
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.
a: channel index
j: site index
.
With interactions:
Topological number Qint
8
Fidkowski and Kitaev (2010)
Interacting multichannel Majorana wires
S
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)
Meidan, Romito, Brouwer (2014)
Interacting multichannel Majorana wires
S
a: channel index
Qchiral = -i tr reh
j: site index
With interactions?
Qint = 0, ±1, ±2, ±3
S well defined;
Qint = -i tr reh
Qchiral = -4
Qchiral = -3
Qchiral = -2
Qchiral = -1
Qchiral = 0
Qchiral = 1
Qchiral = 2
Qchiral = 3
Qchiral = 4
The case Q = 4
S
a: channel index
j: site index
Low-energy subspace
2fold degenerate excited state
2fold degenerate ground state
Kondo!
Low-energy Fermi
liquid fixed point:
→ S well defined;
i tr reh = 4
The case Q = 4
S
a: channel index
j: site index
Low-energy subspace
2fold degenerate excited state
2fold degenerate ground state
Kondo!
Low-energy Fermi
liquid fixed point:
→ S well defined;
i tr reh = 4
The case Q = ±4
Hint,1
S
Hint,2
Interpolation between Q = 4 and Q = 4:
Hint(q) = Hint,1 sinq + Hint,2 cosq
1-4
Low-energy subspace
q≈0
e
transitions:
tunneling
2fold degenerate ground state
The case Q = ±4
Hint,1
S
Hint,2
Interpolation between Q = 4 and Q = 4:
Hint(q) = Hint,1 sinq + Hint,2 cosq
9-12
Low-energy subspace
q ≈ /2
e
transitions:
tunneling
2fold degenerate ground state
The case Q = ±4
Hint,1
S
Hint,2
Interpolation between Q = 4 and Q = 4:
Hint(q) = Hint,1 sinq + Hint,2 cosq
1-4
5-8
9-12
Low-energy subspace
generic q
e
transitions:
tunneling
to/from leads 1-4, 5-8, or 9-12
2fold degenerate ground state
3-channel Kondo!
Low-energy Fermi liquid
fixed point for generic q,
separated by Non-Fermi
liquid point.
i tr reh = 4
0
i tr reh = 4
/2
q
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=x/(N+1).
• An interacting multichannel Majorana wire can be mapped to an
effective Kondo problem if coupled to a normal-metal lead.
0
disorder strength
```