Multichannel Majorana Wires Piet Brouwer Dahlem Center for

Multichannel Majorana Wires Piet Brouwer Dahlem Center for Complex Quantum Systems Physics Department Inanc Adagideli 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 particle-hole conjugation u ↔ v* e. F = 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 Bd. G equation

Topological superconductors Excitation spectrum Eigenvalue equation: e e particle-hole conjugation u ↔ v* e. F = 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 Bd. G equation

Topological superconductors Excitation spectrum Eigenvalue equation: 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 Bd. G equation

Topological superconductors Excitation spectrum Eigenvalue equation: 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 =

Particle-hole symmetric excitation Can one have a particle-hole symmetric excitation in a spinfull superconductor? Superconductor =

Particle-hole symmetric excitations Existence of a single particle-hole symmetric excitation: • One needs a spinless (or spinpolarized) superconductor. Superconductor

Particle-hole symmetric excitations Existence of a single particle-hole symmetric excitation: • One needs a spinless (or spinpolarized) superconductor. • D is an antisymmetric operator. • Without spin: D must be an odd function of momentum. p-wave:

Spinless superconductors are topological S e scattering matrix for Andreev reflection: e h scattering matrix for point contact to S S is unitary 2 x 2 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: e h scattering matrix for point contact to S S is unitary 2 x 2 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: e h scattering matrix for point contact to S S is unitary 2 x 2 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 bulk excitation gap: D = D’ p. F Majorana fermion end states N p -p rhe reh Kitaev (2001) D 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 bulk excitation gap: D = D’ p. F Majorana fermion end states e ih e-ih N p -p rhe reh Kitaev (2001) D 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 bulk excitation gap: D = D’ p. F Majorana fermion end states e Kitaev (2001) S h x = hv. F/D Argument does not depend on length of normal-metal stub

Proposed physical realizations • fractional quantum Hall effect at ν=5/2 Moore, Read (1991) • unconventional superconductor Sr 2 Ru. O 4 • Fermionic atoms near Feshbach resonance Das Sarma, Nayak, Tewari (2006) Gurarie, Radzihovsky, Andreev (2005) 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, 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 = 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 D’px, but without D’py : transverse modes decouple … Majorana end-states → D N 0

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 → 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

Multichannel spinless p-wave wire ? p+ip ? W L bulk gap: coherence length induced superconductivity is weak: Black: bulk spectrum Red: end states and D Majorana if N odd

Multichannel spinless p-wave wire ? p+ip ? W L bulk gap: coherence length induced superconductivity is weak: and Without D’py : effective “time-reversal symmetry”, t 3 Ht 3 = H* Combine with particle-hole symmetry: chiral symmetry, H anticommutes with t 2 Tewari, Sau (2012)

“Periodic Multichannel table ofspinless topological p-wave insulators” wire ? p+ip IQHE ? W L bulk gap: coherence length 3 DTI induced superconductivity is weak: and QSHE Without D’py : effective “time-reversal symmetry”, t 3 Ht 3 = H* Combine with particle-hole symmetry: chiral symmetry, H anticommutes with t 2 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 3 DTI induced superconductivity is weak: and QSHE Without D’py : effective “time-reversal symmetry”, t 3 Ht 3 = H* Combine with particle-hole symmetry: chiral symmetry, H anticommutes with t 2 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 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)

Multichannel spinless p-wave wire ? p+ip ? L bulk gap: coherence length induced superconductivity is weak: and Without D’py : chiral symmetry, H anticommutes with t 2 : integer number Fulga, Hassler, Akhmerov, Beenakker (2011) W

Rieder, Brouwer, Adagideli (2013) 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 Dy’ and without disorder: N Majorana end states W

Multichannel Disordered normalwire metalwithdisorder N channels ? x=0 p+ip ? xx For N channels, wavefunctions exponentially Without Dy’ and without disorder: yn. Nincrease Majorana end states at N different rates W

Multichannel Disordered normalwire metalwithdisorder N channels ? x=0 p+ip ? xx For N channels, wavefunctions Without Dy’ but with disorder: yn increase exponentially at N different rates W

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 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

Rieder, Brouwer, Adagideli (2013) Scattering theory ? p+ip N S L Fulga, Hassler, Akhmerov, Beenakker (2011) Without Dy’: chiral symmetry (H anticommutes with ty) Topological number Qchiral is number of Majorana states at each end of the wire. Without disorder Qchiral = N. With Dy’: 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 Dy’/Dx’ With Dy’ 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 Dy’/Dx’ With Dy’ and with disorder: = 0 disorder strength (N+1)l /x disorder strength

Interacting multichannel Majorana wires ? p+ip ? W Without D’py : effective “time-reversal symmetry”, t 3 Ht 3 = H*

Interacting multichannel Majorana wires Lattice model: a: channel index j: site index HS is real: effective “time-reversal symmetry”, Topological number 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 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 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 ideal normal lead a: channel index j: site index With interactions? Topological number 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 ideal normal lead 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 2 fold degenerate excited state tunneling to/from normal lead 2 fold 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 2 fold degenerate excited state tunneling to/from normal lead 2 fold 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 to/from leads 1 -4 2 fold 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 ≈ p/2 e transitions: tunneling to/from leads 9 -12 2 fold 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 2 fold 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 p/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