Uni-directional quantum graphs Boris Gutkin Georgia Tech & - PowerPoint PPT Presentation

Uni-directional quantum graphs Boris Gutkin Georgia Tech & Duisburg-Essen University Joint work with M. Akila QMath13: Atlanta, October 2016 – p. 1

Spectral Universality 0.5 ... 0.45 GUE λ 0.4 GOE n+2 s 0.35 λ n+2 0.3 n+1 s 0.25 λ n+1 0.2 n s λ 0.15 n 0.1 n−1 0.05 ... 0 0 1 2 3 4 5 6 7 ϕ n ∈ L 2 (Ω) ϕ n | ∂ Ω = 0 , (∆ + λ n ) ϕ n = 0 , – Hallmark of quantum chaos: Level repulsion Nearest neighbor distr. p β ( s ) ∼ s β – Chaotic systems fall into 3 symmetry classes : β = 1 , GOE: time reversal invariant (TRI) β = 2 , GUE: broken TRI β = 4 , GSE: TRI + half integer spin – p. 2

Uni-directional Systems ... λ λ n+1 n+2 λ n λ n−1 λ n−2 λ n−3 Classical: unidirectional (non-ergodic), but chaotic Quantum: both directions “weakly” coupled ( by dynamical tunneling, diffraction orbits) ⇒ – Quasi-degeneracies – Anomalous statistics B.G., J. Phys. A 40 , F761 (2007) B. Dietz, B.G et al., Phys. Rev. E 90 , 022903 (2014) – p. 3

Spectral properties � � � � � � � � Collaboration with � � � � � � � � the experimental group of A. Richter (Darmstadt) � � � � Smooth boundaries: δλ n ≪ mean level spacing In spite TRI, statistics are of GUE type Non-smooth boundaries: Strong tunneling due to diffraction ⇒ δλ n ∼ mean level spacing Anomalous spectral statistics – p. 4

Uni-directional Quantum Graphs � � S 0 det (1 − S L ( k )) = 0 , S = S T 0 � e ikl 1 , . . . , e ikl 2 B � L ( k ) = diag , l i = l i + B , B = # edges Spectrum of S L ( k ) is doubly degenerate Spectral statistics are of GUE type – p. 5

Adding back-scatterer � � i sin α cos α σ = e iα ˜ cos α i sin α α controls strength of back-scattering = ⇒ Lifting degeneracies Q. What is the nearest-neighbor distribution p ( s ) between eigenvalues? – p. 6

Adding back-scatterer Half of the spectrum doesn’t change: { ǫ i } Secular equation for other half: { λ i } B � λ − ǫ m � | A m | 2 cot � 1 /ν ≡ cot α = 2 m =1 | A m | 2 = Amplitude of original eigenstates at ˜ σ � � Nearest neighbor distribution: p ( s ) = 1 p in ( s ) + p ex ( s ) 2 p in ( s ) : distribution of ǫ i − λ i p ex ( s ) : distribution of λ i − ǫ i +1 – p. 7

Random Matrix model RMT assumptions: { ǫ n } ∼ CUE distributed p ( | A m | 2 ) = N exp − N | A m | 2 � � Joint probability:   N � N � 4 sin ǫ i − ǫ j sin λ i − λ j − N � �   P ( { ǫ i } , { λ j } ) ∝  exp ( λ i − ǫ i ) .   2 2 2 ν  i,j =1 i =1 i>j I. L. Aleiner et al., Phys. Rev. Lett. 80 , 814 (1998) – p. 8

Gap-Probability E = det F ( ǫ min , ǫ max ; λ min , λ max ) det F (0 , 0; 0 , 0) with the N × N matrix kernel � + π � + π 2 ν ( λ − ǫ ) e i ( k − 1) ǫ − i N − 1 ǫ e i ( l − 1) λ − i N − 1 dλ e − N λ F kl = dǫ 2 2 − π ǫ � �� � × 1 − θ ( ǫ − ǫ min ) θ ( ǫ max − ǫ ) 1 − θ ( λ − λ min ) θ ( λ max − λ ) Allows to write splitting distribution as derivative ∂ 2 � p in ( s ) ∝ E ( ǫ min , ǫ max ; λ min , λ max ) � ǫ min = λ min = − sπ/N ∂ǫ min ∂λ max ǫ max = λ max =+ sπ/N – p. 9

Nearest neighbor distribution Simple Surmise: Shifted Wigner-Distribution (for β = 2 ) gives a good approximation to p in ( s ) , p ex ( s ) : � 2 c N ( c ) = 4 � πc e − 4 c 2 π + erfc √ π p s ( s, c ) = p β =2 ( s − c ) / N ( c ) , . For p in ( s ) shift c determined by demanding: p s (0 , c ) = R 2 (0) Analytical Results: p in ( s ) , p ex ( s ) versus 2-point correlator R 2 ( s ) – p. 10

Generic position of scatterer Comparison with Quantum Graphs: If ˜ σ sits on “generic” edge = ⇒ RMT result holds – p. 11

Short loop scatterer ⇒ No RMT result σ on short cycle (i.e., self-loop) = ˜ Strong scarring of wave-functions on cycle affects | A m | 2 distribution. – p. 12

Scarring of wave-functions Deviations from Gaussian statistics Generic edge Short loop P ( | ψ n | 2 ) � = N exp( −| ψ n | 2 N ) – p. 13

Transition GUE → GOE Higher Rank perturbations: a) 2 scatterers b) 4 scatterers Solid line: p β =1 ( s ) Dashed line: 1-rank perturbation, Only for rank-one perturbation p (0) � = 0 , otherwise level repulsion ⇒ Fast transition to GOE Breaking unidirectionality = – p. 14

Summary M. Akila, B.G. J. Phys. A 48, 345101 � Analytic formula for p ( s ) . No level repulsion. Good agreement for generic position of ˜ σ ⇒ � No agreement for ˜ σ positioned on short loops = Strong scarring � Fast approach to GOE as # of scatterers increases � “Semiclassical” derivation of R 2 ( s ) through periodic orbit correlations ⇐ ⇒ Scarring – p. 15

Another interpretation Chain of unidirectional graphs Γ Γ Γ Γ Γ Γ Band structure λ λ Gaps No Gaps k k Nearly unidirectional Unidirectional – p. 16