[PPT] - Staggered level repulsion in lead-symmetric transport reflection PowerPoint Presentation

SLIDE 1

Staggered level repulsion in lead-symmetric transport

Henning Schomerus with: M Kopp, S Rotter Banff, 28 February 2008

reflection inversion

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Staggered level repulsion in lead-symmetric transport

Henning Schomerus with: M Kopp, S Rotter Banff, 28 February 2008

N

T

SLIDE 3

Overview

Motivation: Transport in mesoscopic systems
Symmetric systems
RMT: staggered level repulsion
Large number of channels
Appendix: details of the calculation

SLIDE 4

Transport in mesoscopic systems

Marcus group

SLIDE 5

Transport in mesoscopic systems

S matrix

                                = = = = ' ' r t t r S

†

evals Tn of

conductance
shot noise

∑ ∑ ∑ ∑

= = = =

N n n

T h e G

1 2

∑ ∑ ∑ ∑

= = = =

− − − − = = = =

N n n n

T T V h e P

1 2

) 1 ( 2

t t †

SLIDE 6

Scattering matrix from circular ensemble (COE: β=1; CUE: β=2; CSE: β=4) Joint pdf of transmission eigenvalues

RMT

Joint pdf of transmission eigenvalues level repulsion 1-point density (UCF) (WL)

{ { { { } } } } ( ( ( ( ) ) ) )

( ( ( ( ) ) ) )

∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏

− − − − < < < <

× × × × − − − − ∝ ∝ ∝ ∝

k k m n m n n

T T T T P

2

2 1 β

β β β β β β β

(Baranger & Mello 1994; Jalabert, Pichard & Beenakker 1994)

SLIDE 7

desymmetrization

lead-preserving symmetries

(Baranger & Mello 1996) Dirichlet

Superposition of transmission eigenvalues

(reduced repulsion) Neumann

SLIDE 8

desymmetrization

lead-transposing symmetries

reflection inversion

desymmetrization
transmission matrix
mixes parities ( [current,symmetry] ≠ 0 )

2 sin , ), 2 ( 4 1

2 † † † n n

T S S U U U t t Θ Θ Θ Θ = = = = = = = = − − − − − − − − = = = =

− − − − + + + +

) (

2 1 − − − − + + + + −

− − − = = = = S S t

SLIDE 9

(Baranger & Mello)

RMT: U from COE

var G increases by factor 2
no WL corrections

(S Rotter & co: Numerics)

previous observations

no WL corrections
one-point function

(S Rotter & co: Numerics) (Gopar, Rotter & HS)

( ( ( ( ) ) ) )

) 1 ( 1 uniform , 2 sin 2 T T T T

n n n

− − − − = = = = ⇒ ⇒ ⇒ ⇒ Θ Θ Θ Θ Θ Θ Θ Θ = = = = π π π π ρ ρ ρ ρ

SLIDE 10

from COE
here: complete statistics (β=1)

∏ ∏ ∏ ∏

< < < <

                        Θ Θ Θ Θ − − − − Θ Θ Θ Θ ∝ ∝ ∝ ∝ Θ Θ Θ Θ

m n m n n

P 2 sin ) ( ) 2 / ( sin2 T Θ Θ Θ Θ = = = =

† − − − − + + + +

= = = = S S U

can be realized by

n n

T arcsin 2 ± ± ± ± = = = = Θ Θ Θ Θ ) 2 / ( sin

n n

T Θ Θ Θ Θ = = = =

1

n

T

SLIDE 11

from COE
here: complete statistics (β=1)

∏ ∏ ∏ ∏

< < < <

                        Θ Θ Θ Θ − − − − Θ Θ Θ Θ ∝ ∝ ∝ ∝ Θ Θ Θ Θ

m n m n n

P 2 sin ) ( ) 2 / ( sin2 T Θ Θ Θ Θ = = = =

† − − − − + + + +

= = = = S S U

combinatorics over :

– order – pdf as a Vandermonde det. – sum over

determinant factorizes (odd indices vs even indices) For details see appendix

) 2 / ( sin

n n

T Θ Θ Θ Θ = = = =

N

T T T ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤

2

1 n

Θ Θ Θ Θ sgn

n

Θ Θ Θ Θ sgn

SLIDE 12

Final result

{ { { { } } } } ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) )

                                ⋅ ⋅ ⋅ ⋅                                 ⋅ ⋅ ⋅ ⋅                                         − − − − ⋅ ⋅ ⋅ ⋅                                         − − − − ∝ ∝ ∝ ∝

∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏

+ + + + − − − − > > > > > > > > even 1 1

dd

1 even both

dd

both N l T l T n m n m n m n m n

l l

T T T T T P

Reduced level repulsion enhanced fluctuations Symmetric weight no 1/N corrections (WL)

( ( ( ( ) ) ) )

) 1 ( 1 T T T − − − − = = = = π π π π ρ ρ ρ ρ

Staggered level sequence

N

T T T ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤

2

1

(magnitude, not: parity/ill defined)

SLIDE 13

staggered level sequences (lead-transposing) uncorrelated level sequences (lead-preserving)

SLIDE 14

Nearest-neighbour spacing s = Tn+1−Tn

(Wigner)

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Test: Model systems

SLIDE 16

n th-nearest neighbour statistics

large N:

statistics of staggered & independent superpositions converge

SLIDE 17

Observation: ignore weights uncorrelated superposition (2+2 levels)

large-N asymptotics

( ( ( ( ) ) ) )( ( ( ( ) ) ) ) ( ( ( ( ) ) ) )( ( ( ( ) ) ) ) ( ( ( ( ) ) ) )( ( ( ( ) ) ) ) ( ( ( ( ) ) ) )( ( ( ( ) ) ) )

2 3 1 4 2 4 1 3 3 4 1 2

T T T T T T T T T T T T − − − − − − − − + + + + − − − − − − − − + + + + − − − − − − − −

                                ⋅ ⋅ ⋅ ⋅                                

∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏

+ + + + − − − − even 1 1

dd

1 N l T l T

l l

staggered superposition

holds for all N
large N: continuum approx: weights constant

( ( ( ( ) ) ) )( ( ( ( ) ) ) )

2 4 1 3

2 T T T T − − − − − − − − = = = =

(low-order) correlation functions all converge to superposition of two uncorrelated level sequences (w/o WL)

SLIDE 18

Summary

Transport in systems with lead-transposing symmetry:

Mixes parities
Mixes parities
Joint pdf: staggered levels, no direct repulsion
For many channels: like uncorrelated level sequences

(as if system could be desymmetrized)

Dynamical mechanism? (semiclassics?)

preprint: arxiv:0708.0690

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Appendix: Details of the calculation

Order eigenphases
Vandermonde determinant

SLIDE 20

sum over

n

Θ Θ Θ Θ sgn

SLIDE 21

sum over

n

Θ Θ Θ Θ sgn

SLIDE 22

sum over

n

Θ Θ Θ Θ sgn

determinant factorises

SLIDE 23

SLIDE 24

SLIDE 25

2 sin2

n n

T Θ Θ Θ Θ = = = =

transformation to

SLIDE 26

Final result

{ { { { } } } } ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) )

                                ⋅ ⋅ ⋅ ⋅                                 ⋅ ⋅ ⋅ ⋅                                         − − − − ⋅ ⋅ ⋅ ⋅                                         − − − − ∝ ∝ ∝ ∝

∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏

− − − − > > > > > > > > even 1 1

dd

1 even both

dd

both l T l T n m n m n m n m n

l l

T T T T T P

N even:

{ { { { } } } } ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) )

                                ⋅ ⋅ ⋅ ⋅                                         − − − − ⋅ ⋅ ⋅ ⋅                                         − − − − ∝ ∝ ∝ ∝

∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏

− − − − > > > > > > > >

dd

) 1 ( 1 even both

dd

both l T T n m n m n m n m n

l l

T T T T T P

N odd:

r:

{ { { { } } } } ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) )

                                ⋅ ⋅ ⋅ ⋅                                 ⋅ ⋅ ⋅ ⋅                                         − − − − ⋅ ⋅ ⋅ ⋅                                         − − − − ∝ ∝ ∝ ∝

∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏

+ + + + − − − − > > > > > > > > even 1 1

dd

1 even both

dd

both N l T l T n m n m n m n m n

l l

T T T T T P