Delay times in chiral ensembles—signatures of chaotic scattering from

Majorana zero modes

Henning SchomerusLancaster University

Bielefeld, 12 December 2015

Order = broken symmetry → order parameter

2nd order

1st 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

www.ndt-ed.org

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

Topological QuantumNumbers

Common features• Symmetric spectrum• Winding numbers/Berry phase• Effect on quantization

— from superconductivity— depend on class

— zero modes

Majoranas

Mourik et al 2012

N ST

midgap differential conductance peak [Law, Lee, and Ng (2009), ...] Þ 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, 125011 (2012)

N ST

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

Dyson’s Brownian motion approach

RMT of in symmetry class BDI:

Dyson’s Brownian motion approach

Dyson’s Brownian motion approach

Average conductance

Zero-bias anomaly no proof of Majorana fermionsQ-independent!

Re-insert into

large-N limit:

Pikulin, Dahlhaus, Wimmer, HS & Beenakker, New J Physics. 14, 125011 (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: HamiltonianH = 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

Z2 quantum number

Z quantum number

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, 166803 (2015)

Details

Need nullspace of this,treat rest as perturbation

Test: RMT scattering rates versus direct sampling

Fermi-level density of states

partially transparent contactsTwo 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, 166803 (2015)

Pikulin, Dahlhaus, Wimmer, HS & Beenakker, New J Physics. 14, 125011 (2012)