Delay times in chiral ensembles— signatures of chaotic scattering from Majorana zero modes Henning Schomerus Lancaster University Bielefeld, 12 December 2015

C Schellewald, Platonic solids: symmetry Earth Water Air Fire Quintessence (Heaven) solid liquid gas plasma New states of matter?

Order = broken symmetry → order parameter 2 nd order 1 st order phase transition

quasicrystals JP Sethna

liquid crystals nematic smectic chiral Wikipedia

Ψ → Superfluidity VUERQEX Helium (macroscopic) wave function Ψ is a possible order parameter

Ψ → Superconductivity (Cooper pairs: electrons+holes) metallurgyfordummies,.com (macroscopic) wave function Ψ is a possible order parameter

Ψ → Bose-Einstein condensates ultracold monatomic gas NIST (macroscopic) wave function Ψ is a possible order parameter

Edge dislocation in a crystal Defect in a nematic liquid Robust excitations from winding of the order parameter JP Sethna But none for a magnet!

Midgap state Transfer to electronic band structures: e.g. conjugated polymers (Su, Schrieffer, Heeger 1979) Winding of pseudospin

H = H † : unitary(complex) H = T H T = H *, T 2 = +1: orthogonal (real) H = T H T = H d, T 2 = ‒1: symplectic(quaternion) particle-hole symmetry C in superconductors: H = ‒ C H C  4 additional classes, including D chiral (anti)symmetry X H X = ‒ H :  3 additional classes, including BDI RMT classification: Hamiltonian Verbaschoot et al 1993, Altland & Zirnbauer 1996

Common features Symmetric spectrum Winding numbers/Berry phase Effect on quantization — from superconductivity — depend on class — zero modes Majoranas

Common features Symmetric spectrum Winding numbers/Berry phase Effect on quantization — from superconductivity — depend on class — zero modes Majoranas Half a fermion Z 2 parity of SC ground state Kitaev 2001

Ground-state fermion parity changes sign with each crossing of states at Fermi energy Fu & Kane 2009 Beenakker, HS, et al 2013

Mourik et al 2012 N S T midgap differential conductance peak [Law, Lee, and Ng (2009),...]  conductance peak as a signature Or weak antilocalization? Usually lost in magnetic field, but restored by particle-hole symmetry [Brouwer and Beenakker (1995), Altland and Zirnbauer (1996)] indium antimonide nanowires contacted with one normal (gold) and one superconducting (niobium titanium nitride) electrode

Majorana peak vs weak antilocalization… Pikulin, Dahlhaus, Wimmer, HS & Beenakker, New J Physics. 14, (2012)

N S T Conductance of nanowire Scattering formalism: Andreev reflection Wave matching  conductance Diffusive scattering with fixed T = T: RMT for Q: topological invariant

RMT of in symmetry class D: Dyson’s Brownian motion approach

RMT of in symmetry class BDI: Dyson’s Brownian motion approach

RMT of

 Average conductance Zero-bias anomaly no proof of Majorana fermions Q-independent!  Re-insert into large-N limit: Pikulin, Dahlhaus, Wimmer, HS & Beenakker, New J Physics. 14, (2012)

Deeper understanding: density of states independent of absence or presence of Majorana bound state Scattering matrix Density of states Scattering rate has distribution

RMT classification: Hamiltonian H = H † : unitary(complex) H = T H T = H *, T 2 = +1: orthogonal (real) H = T H T = H d, T 2 = ‒1: symplectic(quaternion) particle-hole symmetry C in superconductors: H = ‒ C H C  4 additional classes, including D chiral (anti)symmetry X H X = ‒ H :  3 additional classes, including BDI Z 2 quantum number Z quantum number

Midgap state conjugated polymers (Su, Schrieffer, Heeger 1979) pseudospin

chiral Boguliubov-De Gennes Hamiltonian: multiple Majorana modes Z quantum number

Scattering matrix Chiral Boguliubov-De Gennes Hamiltonian Top. quantum number Chiral symmetry

Meaning of the quantum number

Density of states Chiral symmetry which depends on ν! HS, M. Marciani, C. W. J. Beenakker, PRL 114, (2015)

Details Need nullspace of this, treat rest as perturbation

Test: RMT scattering rates versus direct sampling

Fermi-level density of states

partially transparent contacts Two sets of rates from Marginal distributions disentangle constraint

Summary In superconducting universality classes, signatures of Majorana zero modes compete with weak antilocalization effects chiral superconductors may show clearer signatures HS, M. Marciani, C. W. J. Beenakker, PRL 114, (2015) Pikulin, Dahlhaus, Wimmer, HS & Beenakker, New J Physics. 14, (2012)