multichannel majorana wires

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.