multichannel majorana wires
DESCRIPTION
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 PresentationTRANSCRIPT
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
v
u
v
u
H
H
** 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
v
u
v
u
H
H
** eF = 0
particle-hole symmetry: eigenvalue spectrum is +/- symmetric
Excitation spectrumEigenvalue equation: particle-hole
conjugationu ↔ v*
one fermionic excitation → two solutions of BdG equation
e
Spectra with and without single level at e = 0 are topologically distinct.
Topological superconductors
e
v
u
v
u
H
H
**
Excitation at e = 0 is particle-hole symmetric: “Majorana state”
Excitation spectrumEigenvalue equation: particle-hole
conjugationu ↔ v*
one fermionic excitation → two solutions of BdG equation
e
Spectra with and without single level at e = 0 are topologically distinct.
Topological superconductors
e
v
u
v
u
H
H
**
Excitation at e = 0 is particle-hole symmetric: “Majorana state”
Excitation spectrumEigenvalue equation: particle-hole
conjugationu ↔ v*
Excitation at e = 0 corresponds to ½ fermion: non-abelian statistics
e
Spectra with and without single level at e = 0 are topologically distinct.
Topological superconductors
e particle-hole conjugationu ↔ v*
Excitation at e = 0 corresponds to ½ fermion: non-abelian statistics
eIn 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”
Overview• Spinless superconductors as a habitat for Majorana fermions
-e
e
• Multichannel spinless superconducting wires
• Disordered multichannel superconducting wires
• Interacting multichannel spinless superconducting wires
=
Superconductor Superconductor
Particle-hole symmetric excitationCan one have a particle-hole symmetric excitation in a spinfull superconductor?
Particle-hole symmetric excitation
=
Superconductor Superconductor
Can one have a particle-hole symmetric excitation in a spinfull superconductor?
Particle-hole symmetric excitationsExistence of a single particle-hole symmetric excitation:
Superconductor• One needs a spinless (or
spin-polarized) superconductor.
Superconductor
Particle-hole symmetric excitations
• One needs a spinless (or spin-polarized) superconductor.
v
u
v
u
H
H
**
• D is an antisymmetric operator.• Without spin: D must be
an odd function of momentum.
p-wave:
Existence of a single particle-hole symmetric excitation:
Spinless superconductors are topological
scattering matrix for Andreev reflection:
S is unitary 2x2 matrix
S
h
particle-hole symmetry:
combine with unitarity:
Andreev reflection is either perfect or absent
if e = 0
Law, Lee, Ng (2009)Béri, Kupferschmidt, Beenakker, Brouwer (2009)
e
e
scattering matrix for point contact to S
Spinless superconductors are topological
scattering matrix for Andreev reflection:
S is unitary 2x2 matrix
S
h
particle-hole symmetry:
combine with unitarity:
if e = 0
e
e
scattering matrix for point contact to S
|rhe| = 1: “topologically nontrivial”|rhe| = 0: “topologically trivial”
Spinless superconductors are topological
scattering matrix for Andreev reflection:
S is unitary 2x2 matrix
S
h
particle-hole symmetry:
combine with unitarity:
if e = 0
e
e
scattering matrix for point contact to S
Q = det S = -1: “topologically nontrivial”Q = det S = 1: “topologically trivial”
Fulga, Hassler, Akhmerov, Beenakker (2011)
Spinless p-wave superconductors
one-dimensional spinless p-wave superconductor
Majorana fermion end statesbulk excitation gap: D = D’ pF
Kitaev (2001)
spinless p-wave superconductor
superconducting order parameter has the form
SN D(p)eif(p)rhe
p
Andreev reflection at NS interface
Andreev (1964)
reh-p
*p-wave:
Spinless p-wave superconductors
one-dimensional spinless p-wave superconductor
Majorana fermion end states Kitaev (2001)
spinless p-wave superconductor
superconducting order parameter has the form
SN D(p)eif(p)rhe
reh
p
-p
eih
e-ih
Bohr-Sommerfeld: Majorana state if
*
Always satisfied if |rhe|=1.
bulk excitation gap: D = D’ pF
Spinless p-wave superconductors
one-dimensional spinless p-wave superconductor
Majorana fermion end states Kitaev (2001)
spinless p-wave superconductor
superconducting order parameter has the form
Seh
Argument does not depend on length of normal-metal stub
bulk excitation gap: D = D’ pF
= x hvF/D
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)
Multichannel spinless p-wave wire
? ?
L
W
bulk gap:
coherence length
p+ip
Kells, Meidan, Brouwer (2012)
induced superconductivity is weak: and
Multichannel spinless p-wave wire
? ?
L
W
bulk gap:
coherence length
p+ip
Kells, Meidan, Brouwer (2012)
induced superconductivity is weak: and
Without superconductivity: transverse modes
xikW
yn xe )sin( n = 1,2,3,… n=1 n=2 n=3
2F
22
kkW
nx
Multichannel spinless p-wave wire
? ?
L
W
bulk gap:
coherence length
Majorana end-states→ …With D’px, but without D’py : transverse modes decouple
p+ip
D
0
induced superconductivity is weak: and
N
Multichannel spinless p-wave wire
? ?
L
W
bulk gap:
coherence length
Majorana end-states→ …With D’px, but without D’py : transverse modes decouple
p+ip
With D’py: effective Hamiltonian Hmn for end-states
Hmn is antisymmetric: Zero eigenvalue (= Majorana state) if and only if N is odd.
D
0
induced superconductivity is weak: and
Multichannel spinless p-wave wire
? ?
L
W
bulk gap:
coherence length
D
p+ip
Majorana if N odd
Black: bulk spectrum Red: end states
induced superconductivity is weak: and
Multichannel spinless p-wave wire
? ?
L
W
bulk gap:
coherence length
induced superconductivity is weak: and
p+ip
Tewari, Sau (2012)
Combine with particle-hole symmetry: chiral symmetry,
H anticommutes with t2
Without D’py : effective “time-reversal symmetry”, t3Ht3 = H*
Combine with particle-hole symmetry: chiral symmetry,
H anticommutes with t2
Without D’py : effective “time-reversal symmetry”, t3Ht3 = H*
Multichannel spinless p-wave wire
? ?
L
W
bulk gap:
coherence length
induced superconductivity is weak: and
p+ip
Tewari, Sau (2012)
“Periodic table of topological insulators”
IQHE
Schnyder, Ryu, Furusaki, Ludwig (2008)Kitaev (2009)
Q: Time-reversal symmetryX: Particle-hole symmetryP = QX: Chiral symmetry
3DTI
QSHE
Tewari, Sau (2012)
Combine with particle-hole symmetry: chiral symmetry,
H anticommutes with t2
Without D’py : effective “time-reversal symmetry”, t3Ht3 = H*
Multichannel spinless p-wave wire
? ?
L
W
bulk gap:
coherence length
induced superconductivity is weak: and
p+ip
“Periodic table of topological insulators”
IQHE
Schnyder, Ryu, Furusaki, Ludwig (2008)Kitaev (2009)
Q: Time-reversal symmetryX: Particle-hole symmetryP = QX: Chiral symmetry
3DTI
QSHE
Multichannel spinless p-wave wire
? ?
L
W
bulk gap:
coherence length
induced superconductivity is weak: and
p+ip
Tewari, Sau (2012)Rieder, Kells, Duckheim, Meidan, Brouwer (2012)
As long as D’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 D’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 Dy’ and without disorder: N Majorana end states
xx=0
Multichannel wire with disorder
? ? Wp+ip
Without Dy’ and without disorder: N Majorana end states
xx=0
Disordered normal metal with N channels
xx=0
For N channels, wavefunctions yn increase exponentially at N different rates
Multichannel wire with disorder
? ? Wp+ip
Without Dy’ but with disorder:
xx=0
Disordered normal metal with N channels
xx=0
For N channels, wavefunctions yn increase exponentially at N different rates
Multichannel wire with disorder
? ? Wp+ip
xx=0
disorder strength0
Without Dy’ but with disorder:
N N-1 N-2 N-3 number of Majorana end states
n = N, N-1, N-2, …,1
Series of topological phase transitions
? ? Wp+ip
# Majorana end states
x/(N+1)l
disorder strength
xx=0
Scattering theory
? p+ip
Without Dy’: 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 Dy’:
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
L
Basis 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 Dy’ and with disorder:
disorder strength0
Dy’
/Dx’
(N+1)l /xdisorder strength
=
=
n = N, N-1, N-2, …,1
Series of topological phase transitions
? ? Wp+ip
Topological phase transitions at
xx=0
With Dy’ and with disorder:
disorder strength0
Dy’
/Dx’
(N+1)l /xdisorder strength
=
=
n = N, N-1, N-2, …,1
Interacting multichannel Majorana wires
? ? Wp+ip
Without D’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 ≈ p/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 , qseparated by Non-Fermi liquid point.
0 p/2q-i 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.