The effect of spin in the spectral statistics of Pauli operator - PowerPoint PPT Presentation

RMT Pauli operator Spin contribution to form factor Effect of spin in spectral statistics Jon Harrison RMT The effect of spin in the spectral statistics of Pauli operator Spin contribution quantum graphs to form factor J. M. Harrison † and J. Bolte ∗ † Baylor University, ∗ Royal Holloway Banff – 28th February 08 Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Outline Effect of spin in spectral statistics Jon Harrison RMT Pauli operator Spin contribution to form factor 1 Random matrix theory 2 Pauli op. on graphs 3 Spin contribution to form factor Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Motivation Effect of spin in spectral statistics Jon Harrison Bohigas-Gianonni-Schmit conjecture RMT Spectral statistics of quantized chaotic systems with Pauli operator time-reversal symmetry depend on the spin quantum no. s . Spin contribution to form factor Bosons integer s COE statistics Fermions half-integer s CSE statistics Circular orthogonal ensemble (COE) U symmetric unitary matrix. µ ( U ) = µ ( O − 1 UO ) for O orthogonal. Circular symplectic ensemble (CSE) � 0 � I U symplectic unitary matrix U T JU = J , J = . − I 0 µ ( U ) = µ ( S − 1 US ) for S symplectic. Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Random matrix spectral statistics Effect of spin in spectral statistics Jon Harrison U unitary matrix, eigenvalues e i φ 1 , . . . , e i φ N . RMT ∞ N N Pauli operator ρ ( φ ) = 1 1 � � � e i n ( φ − φ i ) δ ( φ − φ i ) = Spin contribution 2 π N N to form factor i =1 i =1 n = −∞ ∞ 1 � (tr U n ) e i n φ = 2 π N n = −∞ Two-point correlation function R 2 (∆ φ ) = � ρ ( φ ) ρ ( φ + ∆ φ ) � . Definition (form factor) fourier transform of R 2 K ( τ ) = 1 τ = t | tr U t | 2 � � N N Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Effect of spin in spectral statistics 2 CUE COE CSE Jon Harrison 1.5 RMT K ( τ ) Pauli operator 1 Spin contribution to form factor 0.5 0 0 0.5 1 1.5 2 τ Power series expansion 2 + τ 2 4 + τ 3 8 + τ 4 K CSE ( τ ) = τ 12 + . . . 2 − τ 2 4 + τ 3 8 − τ 4 1 � τ = τ � 2 K COE 12 + . . . 2 Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Metric graph G Effect of spin in spectral statistics Jon Harrison RMT v 1 v 2 Pauli operator v 5 Spin contribution to form factor V set of vertices E set of edges, E = |E| . e = ( u , v ) ∈ E if u ∼ v . v 4 v 3 Metric graph Each edge e associated with interval [0 , L e ]. Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Free Pauli op. on G Effect of spin in spectral statistics Jon Harrison RMT Laplace op. − d 2 e on L 2 ( G ) ⊗ C n . Pauli operator d x 2 ( n = 2 s + 1 where s is the spin quantum no. ) Spin contribution to form factor Matching conditions (Kostrykin & Schrader) At vertex v valency k matching conditions defined by nk × nk matrices A v , B v A v ψ v + B v ψ ′ v = 0 . The operator is self-adjoint iff ( A v , B v ) maximal rank and A v B † v = B v A † v . Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Effect of spin in Vertex scattering matrices spectral statistics Matching conditions chosen s.t. S v = U − 1 � � Σ v ⊗ I n U v . Jon Harrison v RMT  R n ( u 1 )  Pauli operator ... U v =   Spin contribution   to form factor R n ( u k ) where u j ∈ Γ ⊆ SU (2) and R n (Γ) is an irrep. dim n . Σ v vertex scattering matrix of Laplace op. on L 2 ( G ). Σ v = − ( A ′ v + i k B ′ v ) − 1 ( A ′ v − i k B ′ v ) Then A v = ( A ′ v ⊗ I n ) U v and B v = ( B ′ v ⊗ I n ) U v . Time-reversal op. T n , T 2 n = ( − I ) n +1 . S v is time-reversal symmetric if Σ T v = Σ v . Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Effect of spin in S-matrix ensemble spectral statistics S ( ij )( lm ) Jon Harrison := δ jl σ ( ij )( jm ) R n ( u ( ij )( jm ) ) e i φ { ij } φ RMT u ( ij )( jm ) = ( u ( j ) i ) − 1 u ( j ) m spin transformation ( ij ) → ( jm ), Pauli operator u ( mj )( ji ) = ( u ( ij )( jm ) ) − 1 . Spin contribution to form factor Trace formula t A p e i πµ p χ R ( d p ) e i φ p � tr S t φ = r p p ∈ P t � � p = ( e 1 , e 2 , . . . , e t ) periodic orbit of G , χ R ( d ) = tr R n ( d ) . A p e i πµ p := σ e 1 e 2 σ e 2 e 3 . . . σ e t − 1 e t d p := u e 1 e 2 u e 2 e 3 . . . u e t − 1 e t φ p := φ e 1 + · · · + φ e t Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Graph form factor Effect of spin in spectral statistics Jon Harrison RMT Pauli operator Spin contribution 1 τ = 2 t | tr S t φ | 2 � � to form factor K orth / sym ( τ ) := φ , 2 × 2 En 2 En t 2 A p A q e i π ( µ p − µ q ) χ R ( d p ) χ ∗ � = R ( d q ) δ φ p ,φ q 2 × 2 En r p r q p , q ∈ P t Kramers’ degeneracy If T 2 = − I , half-integer spin ( n even), eigenvalues of S are doubly degenerate. Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Spin contribution to form factor Effect of spin in spectral statistics Jon Harrison Diagram D RMT A set of orbit pairs related by the same pattern of Pauli operator permutation and or time reversal of arcs between self Spin contribution to form factor intersections. t 2 A p A q e i π ( µ p − µ q ) χ R ( d p ) χ ∗ � K D orth / sym := R ( d q ) 2 × 2 En ( p , q ) ∈ D t Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Spin contribution to form factor Effect of spin in spectral statistics Jon Harrison Diagram D RMT A set of orbit pairs related by the same pattern of Pauli operator permutation and or time reversal of arcs between self Spin contribution to form factor intersections. t 2 A p A q e i π ( µ p − µ q ) χ R ( d p ) χ ∗ � K D orth / sym := R ( d q ) 2 × 2 En ( p , q ) ∈ D t Assume d p chosen randomly from Γ ⊆ SU (2).      t 2 orth / sym = 1 1 � � K D χ R ( d p ) χ ∗ A p A q e i π ( µ p − µ q ) R ( d q )    2 n | D t | 2 E ( p , q ) ∈ D t ( p , q ) ∈ D t Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Effect of spin in In semiclassical limit spectral statistics 1 1 Jon Harrison � χ R ( d p ) χ ∗ � χ R ( d p ) χ ∗ R ( d q ) → R ( d q ) | D t | | Γ | t RMT u 1 ,..., u t ∈ Γ ( p , q ) ∈ D t Pauli operator Spin contribution to form factor 1 4 6 3 2 5 d p = u 1 u 2 u 3 u 4 u 5 u 6 d q = u 1 ( u 2 u 3 u 4 ) − 1 u 5 u 6 Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Effect of spin in In semiclassical limit spectral statistics 1 1 Jon Harrison � χ R ( d p ) χ ∗ � χ R ( d p ) χ ∗ R ( d q ) → R ( d q ) | D t | | Γ | t RMT u 1 ,..., u t ∈ Γ ( p , q ) ∈ D t Pauli operator Spin contribution to form factor 1 4 6 3 2 5 d p = u 1 u 2 u 3 u 4 u 5 u 6 d q = u 1 ( u 2 u 3 u 4 ) − 1 u 5 u 6 Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Effect of spin in spectral statistics Theorem Jon Harrison 1 � c R � m D � χ R ( d p ) χ ∗ RMT R ( d q ) = | Γ | t n Pauli operator u 1 ,..., u t ∈ Γ Spin contribution where m D is the no. of self-intersections at which p was to form factor rearranged to produce q, c R = 1 for real irreps and c R = − 1 for R quaternionic. Idea of proof Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Effect of spin in spectral statistics Theorem Jon Harrison 1 � c R � m D � χ R ( d p ) χ ∗ RMT R ( d q ) = | Γ | t n Pauli operator u 1 ,..., u t ∈ Γ Spin contribution where m D is the no. of self-intersections at which p was to form factor rearranged to produce q, c R = 1 for real irreps and c R = − 1 for R quaternionic. Idea of proof R ( xy − 1 ) = c R � � χ R ( xy ) χ ∗ χ R ( xy ) χ ∗ R ( xy ) 1 n xy ∈ Γ xy ∈ Γ Jon Harrison Effect of spin in spectral statistics

RMT Pauli operator Spin contribution to form factor Effect of spin in spectral statistics Theorem Jon Harrison 1 � c R � m D � χ R ( d p ) χ ∗ RMT R ( d q ) = | Γ | t n Pauli operator u 1 ,..., u t ∈ Γ Spin contribution where m D is the no. of self-intersections at which p was to form factor rearranged to produce q, c R = 1 for real irreps and c R = − 1 for R quaternionic. Idea of proof R ( xy − 1 ) = c R � � χ R ( xy ) χ ∗ χ R ( xy ) χ ∗ R ( xy ) 1 n xy ∈ Γ xy ∈ Γ � 2 � � 1 � χ R ( xyz ) χ ∗ χ R ( xyz ) χ ∗ R ( xzy ) = R ( xyz ) 2 n xyz ∈ Γ xyz ∈ Γ Jon Harrison Effect of spin in spectral statistics