delay times in chiral ensembles signatures of chaotic scattering from majorana zero modes henning...
DESCRIPTION
Order = broken symmetry → order parameter 2 nd order 1 st order phase transitionTRANSCRIPT
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Delay times in chiral ensembles—signatures of chaotic scattering from
Majorana zero modes
Henning SchomerusLancaster University
Bielefeld, 12 December 2015
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Order = broken symmetry → order parameter
2nd order
1st order
phase transition
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quasicrystals
JP Sethna
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liquid crystals
nematic
smectic
chiral
Wikipedia
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Ψ
→ Superfluidity
VUERQEX
Helium
(macroscopic) wave function Ψ is a possible order parameter
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Ψ
→ Superconductivity
(Cooper pairs: electrons+holes)
metallurgyfordummies,.com
(macroscopic) wave function Ψ is a possible order parameter
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Ψ
→ Bose-Einstein condensates
ultracold monatomic gas
NIST
(macroscopic) wave function Ψ is a possible order parameter
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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!
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Midgap state
Transfer to electronic band structures:e.g. conjugated polymers (Su, Schrieffer, Heeger 1979)
Winding of pseudospin
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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
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Common features• Symmetric spectrum• Winding numbers/Berry phase• Effect on quantization
— from superconductivity— depend on class
— zero modes
Majoranas
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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
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Majorana peak vs weak antilocalization…
Pikulin, Dahlhaus, Wimmer, HS & Beenakker, New J Physics. 14, 125011 (2012)
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N ST
Conductance of nanowire
Scattering formalism: Andreev reflection
Wave matching conductance
Diffusive scattering with fixed T = T:
RMT for
Q: topological invariant
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RMT of in symmetry class D:
Dyson’s Brownian motion approach
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Dyson’s Brownian motion approach
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RMT of in symmetry class BDI:
Dyson’s Brownian motion approach
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Dyson’s Brownian motion approach
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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)
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Deeper understanding: density of states
independent of absence or presence of Majorana bound state
Scattering matrix
Density of states
Scattering rate has distribution
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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
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chiral Boguliubov-De Gennes Hamiltonian:multiple Majorana modes
Z quantum number
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Scattering matrix
Chiral Boguliubov-De Gennes Hamiltonian
Top. quantum number
Chiral symmetry
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Meaning of the quantum number
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Density of states
Chiral symmetry
which depends on ν!HS, M. Marciani, C. W. J. Beenakker, PRL 114, 166803 (2015)
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Details
Need nullspace of this,treat rest as perturbation
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Test: RMT scattering rates versus direct sampling
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Fermi-level density of states
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partially transparent contactsTwo sets of rates from
Marginal distributions
disentangle
constraint
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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)