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

Staggered level repulsion in lead-symmetric transport reflection inversion Henning Schomerus with: M Kopp, S Rotter Banff, 28 February 2008

Staggered level repulsion in lead-symmetric transport T N Henning Schomerus with: M Kopp, S Rotter Banff, 28 February 2008

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

Transport in mesoscopic systems Marcus group

Transport in mesoscopic systems         r t '         S matrix = = = = S                 t r ' t † † t evals T n of 2 N e ∑ ∑ ∑ ∑ • conductance = = = = G T n h = = = = n 1 2 N 2 e ∑ ∑ ∑ ∑ • shot noise = = = = − − − − P V T T ( 1 ) n n h = = n = = 1

RMT Scattering matrix from circular ensemble (COE: β =1; CUE: β =2; CSE: β =4) Joint pdf of transmission eigenvalues Joint pdf of transmission eigenvalues 1 β ( ( ( ( ) ) ) ) β β β β ( ( ( ( { } { { { } } } ) ) ) ) β − − β β − − 2 ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∝ ∝ ∝ ∝ − − − − × × × × P T T T T 2 n n m k < < < < n m k level repulsion 1-point density (UCF) (WL) (Baranger & Mello 1994; Jalabert, Pichard & Beenakker 1994)

lead-preserving symmetries • desymmetrization (Baranger & Mello 1996) Dirichlet Neumann Superposition of transmission eigenvalues (reduced repulsion)

lead-transposing symmetries reflection inversion • desymmetrization • desymmetrization = = = = 1 + − − − − t ( S S ) • transmission matrix + − − + + − − 2 Θ Θ Θ Θ 1 † † † 2 n = = = = − − − − − − − − = = = = = = = = t t ( 2 U U ), U S S , T sin + + + + − − − − n 4 2 • mixes parities ( [current,symmetry] ≠ 0 )

previous observations RMT: U from COE • var G increases by factor 2 (Baranger & Mello) • no WL corrections • no WL corrections (S Rotter & co: Numerics) (S Rotter & co: Numerics) • one-point function (Gopar, Rotter & HS) Θ Θ Θ Θ 1 ⇒ ⇒ ⇒ ⇒ ( ( ) ( ( ) ) ) sin 2 n = = = = Θ Θ Θ Θ ρ ρ ρ ρ = = = = T , uniform T n n 2 π π π π − − − − T T ( 1 )

here: complete statistics (β= 1) † • from COE = = = = U S S + + − − + + − −         Θ Θ Θ Θ − − − − Θ Θ Θ Θ         ∏ ∏ ∏ ∏ n m Θ Θ Θ Θ ∝ ∝ ∝ ∝ • P ( ) sin n         2 < < < < n m sin 2 • • = = = = = = = = Θ Θ Θ Θ Θ Θ Θ Θ T T sin ( ( / / 2 2 ) ) n n • can be realized by Θ Θ Θ Θ = = = = ± ± ± ± 2 arcsin T n n T 1 0 n

here: complete statistics (β= 1) † • from COE = = = = U S S + + − − + + − −         Θ Θ Θ Θ − − − − Θ Θ Θ Θ         ∏ ∏ ∏ ∏ n m Θ Θ Θ Θ ∝ ∝ ∝ ∝ • P ( ) sin n         2 < < < < n m sin 2 • • = = = = = = = = Θ Θ Θ Θ Θ Θ Θ Θ T T sin ( ( / / 2 2 ) ) n n Θ Θ Θ Θ • combinatorics over : sgn n � ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ T T T – order 1 2 N – pdf as a Vandermonde det. Θ Θ Θ Θ sgn – sum over n � determinant factorizes (odd indices vs even indices) For details see appendix

Final result                                                 ( ( ( ( { { { { } } } } ) ) ) ) ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) )                                 ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∝ ∝ ∝ ∝ − − − − ⋅ ⋅ ⋅ ⋅ − − − − ⋅ ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ ⋅ 1 P T T T T T                 n m n m n                           T     1 − − − − T   l l > > > > > > > > m n m n + + l odd l + + N even                 both odd both even Symmetric weight Reduced level repulsion 1 enhanced fluctuations ( ( ) ( ( ) ) ) ρ ρ ρ ρ = = = = T π π π π − − − − T ( 1 T ) no 1/N corrections (WL) Staggered level sequence � ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ T T T N 1 2 (magnitude, not: parity/ill defined)

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

Nearest-neighbour spacing s = T n+1 − T n (Wigner)

Test: Model systems

n th-nearest neighbour statistics • large N : statistics of staggered & independent superpositions converge

large- N asymptotics                                 ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ 1 1 ⋅ ⋅ ⋅ ⋅ Observation: ignore weights                             − − − −     T 1 T l l + + + + l odd l N even uncorrelated superposition (2+2 levels) ( ( ( ( ) ( ) ) ) ( ( ( ) ) ( ) ) ( ( ( ) ) ) ) ( ( ( ( ) ( ) ) ) ( ( ( ) ) ( ) ) ( ( ( ) ) ) ) − − − − − − − − + + + + − − − − − − − − + + + + − − − − − − − − T T T T T T T T T T T T 2 1 4 3 3 1 4 2 4 1 3 2 ( ( ( ( ) ) ) ( ) ( ( ( ( ( ) ) ) ) ( ( ( ( ( ( ) ) ) ) ) ) ) ) = = = = − − − − − − − − T T T T 2 3 1 4 2 staggered superposition • holds for all N • large N : continuum approx: weights constant (low-order) correlation functions all converge to superposition of two uncorrelated level sequences (w/o WL)

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

Appendix: Details of the calculation • Order eigenphases • Vandermonde determinant

Θ Θ Θ Θ sgn • sum over n

Θ Θ Θ Θ sgn • sum over n •

Θ Θ Θ Θ sgn • sum over n • • determinant factorises

Θ Θ Θ Θ sin 2 n • transformation to = = = = T n 2