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