Subgap States in 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 Aachen, 2013

excitations in superconductors D: antisymmetric operator ue, ve: solution of Bogoliubov-de Gennes equation: Eigenvalues of HBdG come in pairs ±e, with (particle-hole symmetry) e one fermionic excitation → two solutions of BdG equation Eigenstate of HBdG at e = 0: Majorana fermion, -e

Overview Spinless superconductors as a habitat for Majorana fermions Semiconductor nanowires as a spinless superconductor Disordered spinless supeconducting wires Multichannel spinless superconducting wires Disordered multichannel superconducting wires e -e

superconductor proximity effect ideal interface: S N rhe(e) = a e-if Deif reh(e) = a eif h e a = e-i arccos(e/D) n/nN e (mV) x1 x3 x2 S x1 x2 x3 S N I(nA) S N Guéron et al. (1996) Mur et al. (1996)

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 Béri, Kupferschmidt, Beenakker, Brouwer (2009)

spinless p-wave superconductors superconducting order parameter has the form one-dimensional spinless p-wave superconductor spinless p-wave superconductor bulk excitation gap 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 Majorana fermion end states Kitaev (2001) eih p rhe S N e-ih D(p)eif(p) -p reh Bohr-Sommerfeld: Majorana bound state if * Always satisfied if |rhe|=1.

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)

Semiconductor proposal spin-orbit coupling N B and SOI Zeeman field proximity coupling to superconductor Semiconducting wire with spin-orbit coupling, magnetic field Sau, Lutchyn, Tewari, Das Sarma (2009) Alicea (2010) Lutchyn, Sau, Das Sarma (2010) Oreg, von Oppen, Refael (2010)

Semiconductor proposal spin-orbit coupling N B and SOI Zeeman field proximity coupling to superconductor e p

Semiconductor proposal B and SOI e e e B -pF pF p p

Semiconductor proposal B and SOI e e e B B -pF pF -pF pF p p p

Semiconductor proposal B and SOI e e e e B B -pF pF -pF pF p p p p

Semiconductor proposal B and SOI e e e e B B -pF pF -pF pF D p p p p spinless p-wave superconductor

Semiconductor proposal B p -pF Mourik et al. (2012) S N B and SOI e e e e B B -pF pF -pF pF D p p p p spinless p-wave superconductor

Spinless superconductor B and SOI e e e e B B -pF pF -pF pF D p p p p spinless p-wave superconductor Effective description as a spinless superconductor

Spinless superconductor spinless p-wave superconductor e

Disorder-induced subgap states spinless p-wave superconductor with disorder: topological phase persists for e density of subgap states: at critical disorder strength: Motrunich, Damle, Huse (2001)

Disorder-induced subgap states Power-law tail for density of states: if 2lel > x ~ n n n e e e 2lel » x 2lel > x ~ 2lel < x ~ weak disorder almost critical beyond critical Disorder-induced subgap states are localized in the bulk of the wire. Localization length in topological regime: -1

Disorder-induced subgap states spinless p-wave superconductor L Finite L: discrete energy eigenvalues fermionic subgap states algebraically small energy for large L Majorana state exponentially small energy for large L e0,max: log-normal distribution e1,min: distribution has algebraic tail near zero energy

beyond one dimension semiconductor model: 1 2 3 3 2 1 If B D, ap: semiconductor model can be mapped to p+ip model projection onto “spinless” transverse channels Tewari, Stanescu, Sau, Das Sarma (2012) Rieder, Kells, Duckheim, Meidan, Brouwer (2012)

Multichannel spinless p-wave superconducting wire ? p+ip ? W L bulk gap: coherence length induced superconductivity is weak: and D

Multichannel spinless p-wave superconducting wire ? p+ip ? W L bulk gap: coherence length induced superconductivity is weak: and Without D’py : chiral symmetry … Majorana end-states → D inclusion of py: effective Hamiltonian Hmn for end-states Hmn is antisymmetric: One zero eigenvalue if N is odd, no zero eigenvalue if N is even.

Multichannel wire with disorder ? p+ip ? W L bulk gap: coherence length

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

Multichannel wire with disorder ? p+ip ? W L 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 Without Dy’: chiral symmetry 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.

Chiral limit ? N p+ip S L Basis transformation:

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

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

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

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

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

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

Chiral limit ? N p+ip S L : eigenvalues of N, with disorder L

Chiral limit ? 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 L Qchiral x/(N+1)l disorder strength Without Dy’: chiral limit topological phase transitions at n=1,2,…,N

Series of topological phase transitions ? p+ip ? W L Without Dy’: chiral limit With Dy’: topological phase transitions at n=1,2,…,N

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

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

Conclusions Majorana fermions may persist in the presence of disorder and with multiple channels Disorder leads to fermionic subgap states in the bulk; Density of states has power-law singularity near zero energy. Multiple channels may lead to fermionic subgap states at the wire ends. 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 strength