The effect of spin in the spectral statistics of quantum graphs

Transcription

1 The effect of spin in the of quantum graphs J. M. Harrison and J. Bolte Baylor University, Royal Holloway Banff 28th February 08

2 Outline Random matrix theory 2 Pauli op. on graphs 3

3 Motivation Bohigas-Gianonni-Schmit conjecture Spectral statistics of quantized chaotic systems with time-reversal symmetry depend on the spin quantum no. s. Bosons integer s COE statistics Fermions half-integer s CSE statistics Circular orthogonal ensemble (COE) U symmetric unitary matrix. µ(u) = µ(o UO) for O orthogonal. Circular symplectic ensemble (CSE) U symplectic unitary matrix U T JU = J, J = µ(u) = µ(s US) for S symplectic. ( ) 0 I. I 0

4 Random matrix U unitary matrix, eigenvalues e iφ,..., e iφ N. ρ(φ) = N N i= = 2πN δ(φ φ i ) = 2πN (tr U n )e inφ n= N i= n= e in(φ φ i ) Two-point correlation function R 2 ( φ) = ρ(φ)ρ(φ + φ). Definition (form factor) fourier transform of R 2 K(τ) = tr U t 2 τ = t N N

5 2.5 K(τ) CUE COE CSE Power series expansion τ.5 2 K CSE (τ) = τ 2 + τ τ τ ( τ ) 2 K COE = τ 2 2 τ τ 3 8 τ

6 Metric graph G v v 2 v 5 v 4 v 3 V set of vertices E set of edges, E = E. e = (u, v) E if u v. Metric graph Each edge e associated with interval [0, L e ].

7 Free Pauli op. on G Laplace op. d2 on L 2 (G) C n. dxe 2 (n = 2s + where s is the spin quantum no.) 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.

8 Vertex scattering matrices Matching conditions chosen s.t. S v = Uv ( ) Σv I n Uv. R n (u ) U v =... Rn(uk) 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 + ikb v ) (A v ikb 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+. S v is time-reversal symmetric if Σ T v = Σ v.

9 S-matrix ensemble S (ij)(lm) φ := δ jl σ (ij)(jm) R n (u (ij)(jm) ) e iφ {ij} u (ij)(jm) = (u (j) i ) u m (j) spin transformation (ij) (jm), u (mj)(ji) = (u (ij)(jm) ). Trace formula tr S t φ = p P t t r p A p e iπµp χ R (d p ) e iφp p = (e, e 2,..., e t ) periodic orbit of G, χ R (d) = tr ( R n (d) ). A p e iπµp := σ e e 2 σ e2 e 3... σ et e t d p := u e e 2 u e 2e 3... u e t e t φ p := φ e + + φ et

10 Graph form factor K orth/sym (τ) := 2 2En = t2 2 2En A p A q r p r q p,q P t tr S t φ 2 φ, τ = 2t 2En e iπ(µp µq) χ R (d p )χ R (d q) δ φp,φ q Kramers degeneracy If T 2 = I, half-integer spin (n even), eigenvalues of S are doubly degenerate.

11 Diagram D A set of orbit pairs related by the same pattern of permutation and or time reversal of arcs between self intersections. K D orth/sym := t2 2 2En (p,q) D t A p A q e iπ(µp µq) χ R (d p )χ R (d q)

12 Diagram D A set of orbit pairs related by the same pattern of permutation and or time reversal of arcs between self intersections. K D orth/sym := t2 2 2En (p,q) D t A p A q e iπ(µp µq) χ R (d p )χ R (d q) Assume d p chosen randomly from Γ SU(2). Korth/sym D = χ R (d p)χ R 2n D (dq) t2 A pa q e iπ(µp µq) t 2E (p,q) D t (p,q) D t

13 In semiclassical limit D t χ R (d p )χ R (d q) Γ t (p,q) D t u,...,u t Γ χ R (d p )χ R (d q) d p = u u 2 u 3 u 4 u 5 u 6 d q = u (u 2 u 3 u 4 ) u 5 u 6

14 In semiclassical limit D t χ R (d p )χ R (d q) Γ t (p,q) D t u,...,u t Γ χ R (d p )χ R (d q) d p = u u 2 u 3 u 4 u 5 u 6 d q = u (u 2 u 3 u 4 ) u 5 u 6

15 Theorem Γ t u,...,u t Γ ( χ R (d p )χ R (d cr q) = n ) md where m D is the no. of self-intersections at which p was rearranged to produce q, c R = for real irreps and c R = for R quaternionic. Idea of proof

16 Theorem Γ t u,...,u t Γ ( χ R (d p )χ R (d cr q) = n ) md where m D is the no. of self-intersections at which p was rearranged to produce q, c R = for real irreps and c R = for R quaternionic. Idea of proof χ R (xy)χ R (xy ) = c R n xy Γ χ R (xy)χ R (xy) xy Γ

17 Theorem Γ t u,...,u t Γ ( χ R (d p )χ R (d cr q) = n ) md where m D is the no. of self-intersections at which p was rearranged to produce q, c R = for real irreps and c R = for R quaternionic. Idea of proof χ R (xy)χ R (xy ) = c R χ R (xy)χ R n (xy) xy Γ xy Γ ( ) 2 2 χ R (xyz)χ R (xzy) = χ R (xyz)χ R n (xyz) xyz Γ xyz Γ

18 First step Lemma Γ g Γ χ R (ug 2 ) = c R n χ R(u) Let w = xy xy Γ χ R (xy)χ R (xy ) = wy Γ χ R (w)χ R (wy 2 ) = c R n Γ χ R (w)χ R (w) w Γ = c R χ R (xy)χ R n (xy) xy Γ

19 Result Korth/sym D = ( cr ) md t 2 2n n 2E (p,q) D t A p A q e iπ(µp µq) Diagram D contributes at order τ md+. Orthogonal case n odd, c R = and τ = t/2en. Korth D = K zero D

20 Result Korth/sym D = ( cr ) md t 2 2n n 2E Diagram D contributes at order τ m D+. Symplectic case n even and τ = 2t/2En. ( Ksym D cr ) md +2 = K D 2 zero (p,q) D t A p A q e iπ(µp µq) Compare with random matrix theory ( K CSE (τ) = ) m+ KCOE m τ m, 0 < τ 2. 2 m=

21 Conclusion Evaluated spin contribution of quantum graphs with random spin transformations. CSE or COE statistics depend on the repn of the group of spin transformations. Consistent with B-G-S conjecture if R(Γ) quaternionic irrep for half-integer spin, e.g. Γ = {±I, ±σ x, ±σ y, ±σ z }. J. Bolte and J. M. Harrison, J. Phys. A 36, L433 L440 (2003). arxiv nlin.cd/ J. Bolte and J. M. Harrison, In: Berkolaiko, et al. (Eds) Quantum Graphs and Their Applications, Contemporary Mathematics, 45 (AMS 2006) arxiv nlin.cd/050