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
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. - PowerPoint PPT Presentation
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Multichannel Majorana Wires
Piet Brouwer
Dahlem Center for Complex Quantum SystemsPhysics DepartmentFreie Universität Berlin
Inanc AdagideliMathias DuckheimDganit MeidanGraham KellsFelix von OppenMaria-Theresa RiederAlessandro Romito
Capri, 2014
Excitations in superconductors
e
vu
vu
HH
e** eF = 0
u: “electron” v: “hole”
particle-hole symmetry: eigenvalue spectrum is +/- symmetric
Excitation spectrumEigenvalue equation:
Bogoliubov-de Gennes equation
superconducting order parameter
particle-hole conjugationu ↔ v*
one fermionic excitation → two solutions of BdG equation
Topological superconductors
e
vu
vu
HH
e** eF = 0
particle-hole symmetry: eigenvalue spectrum is +/- symmetric
Tewari, Sau (2012)Rieder, Kells, Duckheim, Meidan, Brouwer (2012)
As long as ’py remains a small perturbation, it is possible inprinciple that there are multiple Majorana states at each end, even in the presence of disorder.
Multichannel spinless p-wave wire? ?
L
W
bulk gap:coherence length
induced superconductivity is weak: and
Without ’py : chiral symmetry,
p+ip
H anticommutes with t2
Fulga, Hassler, Akhmerov, Beenakker (2011)
: integer number
Multichannel wire with disorder? ? W
bulk gap:coherence length
p+ip
Rieder, Brouwer, Adagideli (2013)
xx=0
Multichannel wire with disorder? ? Wp+ip
disorder strength0
Series of N topological phase transitions at
n=1,2,…,N
xx=0
Multichannel wire with disorder? ? Wp+ip
Without y’ and without disorder: N Majorana end states
xx=0
Multichannel wire with disorder? ? Wp+ip
Without y’ and without disorder: N Majorana end states
xx=0
Disordered normal metal with N channels
xx=0
For N channels, wavefunctions n increase exponentially at N different rates
Multichannel wire with disorder? ? Wp+ip
Without y’ but with disorder:
xx=0
Disordered normal metal with N channels
xx=0
For N channels, wavefunctions n increase exponentially at N different rates
Multichannel wire with disorder? ? Wp+ip
xx=0
disorder strength0
Without y’ but with disorder:
N N-1 N-2 N-3 number of Majorana end states
n = N, N1, N2, …,1
Series of topological phase transitions? ? Wp+ip
# Majorana end states
x/(N+1)l
disorder strength
xx=0
Scattering theory? p+ip
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
N S
L
Fulga, Hassler, Akhmerov, Beenakker (2011)
Rieder, Brouwer, Adagideli (2013)
? p+ipN S
L
Basis transformation:
Scattering theory
? p+ipN S
L
if and only if
Basis transformation: imaginary gauge field
Scattering theory
? p+ipN S
L
Basis transformation:
if and only if
imaginary gauge field
Scattering theory
? p+ipN S
L
if and only if
“gauge transformation”
Basis transformation: imaginary gauge field
Scattering theory
? p+ipN S
L
if and only if
“gauge transformation”
Basis transformation: imaginary gauge field
Scattering theory
? p+ipN S
L
“gauge transformation”
Basis transformation:
N, with disorder
L
Scattering theory
? p+ipN S
LBasis transformation:
“gauge transformation”
N, with disorder
L
Scattering theory
? p+ipN S
L
N, with disorder
L
: eigenvalues of
Scattering theory
? p+ipN S
L
N, with disorder
L
: eigenvalues of
Distribution of transmission eigenvalues is known:
with , self-averaging in limit L →∞
Scattering theory
Series of topological phase transitions? ? Wp+ip
Topological phase transitions at
xx=0
With y’ and with disorder:
disorder strength0
y’/
x’
(N+1)l /xdisorder strength
=
=
n = N, N1, N2, …,1
Series of topological phase transitions? ? Wp+ip
Topological phase transitions at
xx=0
With y’ and with disorder:
disorder strength0
y’/
x’
(N+1)l /xdisorder strength
=
=
n = N, N1, N2, …,1
Interacting multichannel Majorana wires? ? Wp+ip
Without ’py : effective “time-reversal symmetry”, t3Ht3 = H*
Interacting multichannel Majorana wires
HS is real: effective “time-reversal symmetry”,
Lattice model:
a: channel indexj: site index
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
a: channel indexj: site index
With interactions:Topological number Qint 8
Fidkowski and Kitaev (2010)
Qchiral = 0
Topological number Qchiral .
Qchiral is number of Majorana states at each end of the wire, counted with sign.
Qchiral = 1
Qchiral = 3Qchiral = 2
Qchiral = 4
Qchiral = -1Qchiral = -2Qchiral = -3Qchiral = -4
Interacting multichannel Majorana wires
a: channel indexj: site index
With interactions:Topological number Qint 8
Fidkowski and Kitaev (2010)
Qchiral = 0
Topological number Qchiral .
Qchiral is number of Majorana states at each end of the wire, counted with sign.
Qchiral = 1
Qchiral = 3Qchiral = 2
Qchiral = 4
Qchiral = -1Qchiral = -2Qchiral = -3Qchiral = -4
~
Interacting multichannel Majorana wires
a: channel indexj: site index
With interactions:Topological number Qint 8
Fidkowski and Kitaev (2010)
Topological number Qchiral .
Qchiral is number of Majorana states at each end of the wire, counted with sign.
ideal normal lead
S
With interactions?
Interacting multichannel Majorana wires
a: channel indexj: site index
ideal normal lead
S
With interactions?
Qchiral = 0Qchiral = 1
Qchiral = 3Qchiral = 2
Qchiral = 4
Qchiral = -1Qchiral = -2Qchiral = -3Qchiral = -4
Qint = -i tr reh
Qchiral = -i tr reh
S well defined;Qint = 0 , ±1, ±2, ±3
Meidan, Romito, Brouwer (2014)
The case Q = 4
a: channel indexj: site index
S
Low-energy subspace
2fold degenerate ground state
2fold degenerate excited state
tunneling to/from normal lead
Kondo!Low-energy Fermi liquid fixed point:
→ S well defined;
i tr reh = 4
The case Q = 4
a: channel indexj: site index
S
Low-energy subspace
2fold degenerate ground state
2fold degenerate excited state
tunneling to/from normal lead
Kondo!Low-energy Fermi liquid fixed point:
→ S well defined;
i tr reh = 4
The case Q = ±4
Hint,1
Hint,2
Hint(q) = Hint,1 sinq + Hint,2 cosqInterpolation between Q = 4 and Q = 4:
S
Low-energy subspace
2fold degenerate ground state
1-4
e
transitions:tunneling to/from leads 1-4
q ≈ 0
The case Q = ±4
Hint,1
Hint,2
Hint(q) = Hint,1 sinq + Hint,2 cosqInterpolation between Q = 4 and Q = 4:
S
Low-energy subspace
2fold degenerate ground state
9-12
e
transitions:tunneling to/from leads 9-12
q ≈ /2
The case Q = ±4
Hint,1
Hint,2
Hint(q) = Hint,1 sinq + Hint,2 cosqInterpolation between Q = 4 and Q = 4:
S
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.
0 /2qi tr reh = 4 i tr reh = 4
generic q
Summary
• 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).
disorder strength0
• 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.
• An interacting multichannel Majorana wire can be mapped to an effective Kondo problem if coupled to a normal-metal lead.