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

Excitations in superconductors Excitation spectrum Eigenvalue equation: particle-hole conjugation u ↔ v* e eF = 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 BdG equation

Topological superconductors Excitation spectrum Eigenvalue equation: particle-hole conjugation u ↔ v* e e 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: particle-hole conjugation u ↔ v* e e 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: particle-hole conjugation u ↔ v* e e 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 particle-hole conjugation u ↔ v* e e In 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” 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: Superconductor One needs a spinless (or spin-polarized) superconductor. Superconductor

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

Spinless superconductors are topological scattering matrix for Andreev reflection: h e S is unitary 2x2 matrix scattering matrix for point contact to S 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 scattering matrix for Andreev reflection: h e S is unitary 2x2 matrix scattering matrix for point contact to S particle-hole symmetry: if e = 0 combine with unitarity: |rhe| = 1: “topologically nontrivial” |rhe| = 0: “topologically trivial”

Spinless superconductors are topological scattering matrix for Andreev reflection: h e S is unitary 2x2 matrix scattering matrix for point contact to S 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: D = D’ pF Majorana fermion end states Kitaev (2001) p rhe S N D(p)eif(p) -p reh Andreev reflection at NS interface * p-wave: Andreev (1964)

Spinless p-wave superconductors superconducting order parameter has the form one-dimensional spinless p-wave superconductor spinless p-wave superconductor bulk excitation gap: D = D’ pF Majorana fermion end states Kitaev (2001) eih p rhe S N e-ih D(p)eif(p) -p reh 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: D = D’ pF Majorana fermion end states Kitaev (2001) e S h x = hvF/D 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 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 Kells, Meidan, Brouwer (2012) Multichannel spinless p-wave wire ? p+ip ? W L bulk gap: coherence length induced superconductivity is weak: and

Multichannel spinless p-wave wire Kells, Meidan, Brouwer (2012) Multichannel spinless p-wave wire ? p+ip ? W L bulk gap: coherence length induced superconductivity is weak: and Without superconductivity: transverse modes n = 1,2,3,… n=1 n=2 n=3

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

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 With D’py: effective Hamiltonian Hmn for end-states Hmn is antisymmetric: Zero eigenvalue (= Majorana state) if and only if N is odd.

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

Multichannel spinless p-wave wire ? p+ip ? W L bulk gap: coherence length induced superconductivity is weak: and Combine with particle-hole symmetry: chiral symmetry, H anticommutes with t2 Without D’py : effective “time-reversal symmetry”, t3Ht3 = H* Tewari, Sau (2012)

Multichannel spinless p-wave wire “Periodic table of topological insulators” Multichannel spinless p-wave wire IQHE Schnyder, Ryu, Furusaki, Ludwig (2008) Kitaev (2009) Q: Time-reversal symmetry X: Particle-hole symmetry P = QX: Chiral symmetry 3DTI QSHE ? p+ip ? W L bulk gap: coherence length induced superconductivity is weak: and Combine with particle-hole symmetry: chiral symmetry, H anticommutes with t2 Without D’py : effective “time-reversal symmetry”, t3Ht3 = H* Tewari, Sau (2012)

Multichannel spinless p-wave wire “Periodic table of topological insulators” Multichannel spinless p-wave wire IQHE Schnyder, Ryu, Furusaki, Ludwig (2008) Kitaev (2009) Q: Time-reversal symmetry X: Particle-hole symmetry P = QX: Chiral symmetry 3DTI QSHE ? p+ip ? W L bulk gap: coherence length induced superconductivity is weak: and Combine with particle-hole symmetry: chiral symmetry, H anticommutes with t2 Without D’py : effective “time-reversal symmetry”, t3Ht3 = H* Tewari, 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 ? W L bulk gap: coherence length induced superconductivity is weak: and Without D’py : chiral symmetry, H anticommutes with t2 : integer number Fulga, Hassler, Akhmerov, Beenakker (2011)

Multichannel wire with disorder Rieder, Brouwer, Adagideli (2013) Multichannel wire with disorder ? p+ip ? W x x=0 bulk gap: coherence length

Multichannel wire with disorder ? p+ip ? W x x=0 Series of N topological phase transitions at n=1,2,…,N disorder strength

Multichannel wire with disorder ? p+ip ? W x x=0 Without Dy’ and without disorder: N Majorana end states

Multichannel wire with disorder Disordered normal metal with N channels x x=0 For N channels, wavefunctions yn increase exponentially at N different rates ? p+ip ? W x x=0 Without Dy’ and without disorder: N Majorana end states

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

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 number of Majorana end states disorder strength

Series of topological phase transitions ? p+ip ? W x x=0 # Majorana end states x/(N+1)l disorder strength

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

Scattering theory ? N p+ip S L Basis transformation:

Scattering theory ? N p+ip S L Basis transformation: imaginary gauge field if and only if

Scattering theory ? N p+ip S L Basis transformation: if and only if imaginary gauge field

Scattering theory ? N p+ip S L Basis transformation: imaginary gauge field if and only if “gauge transformation”

Scattering theory ? N p+ip S L Basis transformation: imaginary gauge field if and only if “gauge transformation”

Scattering theory ? N p+ip S L N, with disorder L Basis transformation: “gauge transformation” N, with disorder L

Scattering theory ? N p+ip S L N, with disorder L Basis transformation: “gauge transformation” N, with disorder L

Scattering theory ? N p+ip S L : eigenvalues of N, with disorder L

Scattering theory ? N p+ip S L N, with disorder L : eigenvalues of Distribution of transmission eigenvalues is known: with , self-averaging in limit L →∞

Series of topological phase transitions ? p+ip ? W x x=0 Dy’/Dx’ (N+1)l /x disorder strength = With Dy’ and with disorder: Topological phase transitions at n = N, N-1, N-2, …,1 disorder strength

Series of topological phase transitions ? p+ip ? W x x=0 Dy’/Dx’ (N+1)l /x disorder strength = With Dy’ and with disorder: Topological phase transitions at n = N, N-1, N-2, …,1 disorder strength

Interacting multichannel Majorana wires ? p+ip ? W Without D’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 a: channel index j: site index Qchiral = 2 Qchiral = 3 Qchiral = 4 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 a: channel index j: site index Qchiral = 2 Qchiral = 3 Qchiral = 4 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 ideal normal lead 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)

Interacting multichannel Majorana wires Meidan, Romito, Brouwer (2014) Interacting multichannel Majorana wires S ideal normal lead Qchiral = -i tr reh a: channel index j: site index With interactions? Qchiral = -4 Qchiral = -3 Qchiral = -2 Qint = 0 , ±1 , ±2 , ±3 Qchiral = -1 S well defined; Qchiral = 0 Qchiral = 1 Qint = -i tr reh Qchiral = 2 Qchiral = 3 Qchiral = 4

The case Q = 4 S Low-energy subspace Kondo! -i tr reh = 4 a: channel index j: site index Low-energy subspace Kondo! Low-energy Fermi liquid fixed point: 2fold degenerate excited state tunneling to/from normal lead → S well defined; 2fold degenerate ground state -i tr reh = 4

The case Q = -4 S Low-energy subspace Kondo! -i tr reh = -4 a: channel index j: site index Low-energy subspace Kondo! Low-energy Fermi liquid fixed point: 2fold degenerate excited state tunneling to/from normal lead → S well defined; 2fold degenerate ground state -i tr reh = -4

The case Q = ±4 S Hint(q) = Hint,1 sinq + Hint,2 cosq Interpolation between Q = 4 and Q = -4: Hint(q) = Hint,1 sinq + Hint,2 cosq Low-energy subspace 2fold degenerate ground state 1-4 e transitions: tunneling to/from leads 1-4 q ≈ 0

The case Q = ±4 S Hint(q) = Hint,1 sinq + Hint,2 cosq Interpolation between Q = 4 and Q = -4: Hint(q) = Hint,1 sinq + Hint,2 cosq Low-energy subspace 2fold degenerate ground state 9-12 e transitions: tunneling to/from leads 9-12 q ≈ p/2

The case Q = ±4 S Hint(q) = Hint,1 sinq + Hint,2 cosq Interpolation between Q = 4 and Q = -4: Hint(q) = Hint,1 sinq + Hint,2 cosq 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. p/2 q -i tr reh = 4 -i tr reh = -4 generic 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 time-reversal 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. disorder strength