Universality. Why? (Bohigas, Giannoni, Schmit 84; see also Casati, Vals-Gris, Guarneri; Berry, Tabor)

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2 Universality Many quantum properties of chaotic systems are universal and agree with predictions from random matrix theory, in particular the statistics of energy levels. (Bohigas, Giannoni, Schmit 84; see also Casati, Vals-Gris, Guarneri; Berry, Tabor) Why?

3 Outline Classical chaos Quantum chaos Semiclassical methods (Gutzwiller trace formula) Semiclassical approach to spectral statistics Resummation

4 1 Classical chaos

5 Billiards (classical) particle moving in domain B R 2 motion on straight line with constant velocity reflection at boundary (angle of incidence = angle of reflection) Billiards with irregular boundary show chaotic behaviour.

6 Chaos (a) sensitive dependence on initial conditions

8 Chaos (b) ergodicity long trajectories fill the interior almost uniformly + all angles are equally likely general definition: energy shell Ω = {(q, p), H(q, p) = E} a system is ergodic if for Ω 0 Ω and almost all trajectories time up to T spent in Ω 0 T vol(ω 0) vol(ω) (T )

9 Ergodicity billiards that are not ergodic:

10 2 Quantum chaos quantum properties of systems that are classically chaotic

11 Billiards (quantum) Schrödinger equation in domain B R 2, no potential 2m 2 ψ(q, t) = i ψ(q, t) t Dirichlet boundary conditions: 2 ψ(q, t) = 0 for q B time independent Schrödinger equation 2 2m 2 ψ n (q) = E n ψ n (q) ψ n (q) energy eigenfunctions E n energy levels

12 Quantum chaos (a) sensitive dependence on initial conditions? NO, because Schrödinger equation is linear

13 Quantum chaos (b) energy eigenfunctions plot ψ n (q) 2 for eigenfunctions with increasing energy: eigenfunctions become more and more equidistributed (quantum ergodicity)

14 Quantum chaos probability that a particle in state n is found in a part B 0 of position space B: µ n (B 0 ) = d 2 q ψ n (q) 2 B 0 in a quantum ergodic system: µ n (B 0 ) area(b 0) area(b) for increasing energies after taking out exceptional wavefunctions ( scars ) such as this:

15 Quantum chaos

16 Quantum chaos (c) statistics of energy levels (large) energy levels of a chaotic system repel each other!

18 4 Semiclassical approach to spectral statistics

19 Two-point correlation function measure likelihood of energy differences: ( de d E + y ) ( d E y ) = jk δ(y (E j E k )) make everything dimensionless and average over energy and energy differences: R(ɛ) = 1 d ( d E + ɛ ) ( d E ɛ ) 2 2 d 2 d

20 Random matrix predictions small energy differences are less likely! systems without time rev. invariance: Gaussian Unitary Ensemble ( ) sin(πɛ) 2 ( R GUE (ɛ) = 1 = Re 1 1 πɛ 2(πɛ) ) 2(πɛ) 2 e2πiɛ

21 Random matrix predictions systems with time rev. inv.: Gaussian Orthogonal Ensemble ( ) sin πɛ 2 ( Si πɛ R GOE (ɛ) = 1 + πɛ π sign ɛ )( ) cos πɛ sin πɛ 2 πɛ (πɛ) ( 2 ( ) 1 n = Re c n + ( ) ) 1 n d n e 2πiɛ ɛ ɛ n where Si(x) = x sin y 0 y dy in both cases non-oscillatory terms and oscillatory terms n

22 Spectral form factor Fourier transform K (τ) = Re dɛ (R(ɛ) 1)e 2πiɛτ systems without time rev. invariance: Gaussian Unitary Ensemble K (τ) = { τ (τ < 1) 1 (τ > 1) systems with time rev. inv.: Gaussian Orthogonal Ensemble 2τ τ ln(1 + 2τ) = 2τ 2τ 2 +2τ 3 8 K (τ)= 3 τ (τ < 1) 2 ln 2τ+1 2τ 1 (τ > 1)

23 Semiclassical approximation evaluate using we get the double sum R(ɛ) = 1 d 2 A p ( E ± S p ( E ± ( d E + ɛ ) ( d E 2 d d(e) d + 1 π Re p ɛ ) A p(e) 2 d ɛ ) 2 d S p(e) ± Tp(E)ɛ 2 d ɛ ) 2 d A pe isp(e)/ T H = 2π d (Heisenberg time) R(ɛ) Re A pa )/ TH 2 p ei(sp Sp e iπɛ(tp+tp )/TH p,p + similar term with e i(sp+s p )/

24 Semiclassical approximation R(ɛ) Re A pa )/ TH 2 p ei(sp Sp e iπɛ(tp+tp )/TH p,p + similar term with e i(sp+s p )/ energy average systematic contribution only from first term for small action differences (at most ) constructive interference! Fourier transform gives: K (τ) 1 ( Re A pa )/ p T ei(sp Sp δ τt H Tp + T ) p H 2 p,p relevant orbits have periods of the order Ω T H = 2π d = 2π ( 0) (2π ) n

25 Diagonal approximation (Berry; Hannay & Ozorio de Almeida) for systems without time reversal invariance: take p = p K diag (τ) = 1 A p 2 δ(τt H T p ) τ T H p sum over periodic orbits evaluated using ergodicity: A p 2 δ(t T p ) T p time reversal invariant systems: p = p or time reversed of p K diag (τ) = 2τ

26 Summary Gutzwiller trace formula: d(e) d + 1 π Re p A pe isp(e)/ Spectral form factor: K (τ) 1 T H Re p,p A p A p ei(sp S p )/ δ ( τt H T ) p + T p 2 RMT prediction: systems without time rev. invariance: Gaussian Unitary Ensemble K (τ) = { τ (τ < 1) 1 (τ > 1) systems with time rev. inv.: Gaussian Orthogonal Ensemble 2τ τ ln(1 + 2τ) = 2τ 2τ 2 +2τ 3 8 K (τ)= 3 τ (τ < 1) 2 ln 2τ+1 2τ 1 (τ > 1)

27 Orbit correlations Periodic orbits in chaotic systems come in bunches. Example (Sieber & Richter 2001): Realistic picture:

28 Orbit bunches Periodic orbits in chaotic systems come in bunches. Example (Sieber & Richter 2001): encounters = regions where parts of an orbit come close to each other (up to time reversal) can switch connections to get different (but very similar) orbits present example requires time reversal invariance

29 Underlying mechanism Phase space directions in hyperbolic systems: stable direction: deviations shrink asymptotically like e λt (λ=lyapunov exponent) unstable direction: deviations grow for t and shrink for t like e λt sensitive dependence on initial conditions Construction of partner orbit:

30 Underlying mechanism Phase space directions in hyperbolic systems: stable direction: deviations shrink asymptotically like e λt (λ=lyapunov exponent) unstable direction: deviations grow for t and shrink for t like e λt sensitive dependence on initial conditions Construction of partner orbit:

Underlying mechanism Phase space directions in hyperbolic systems: stable direction: deviations shrink asymptotically like e λt (λ=lyapunov

31 Underlying mechanism Phase space directions in hyperbolic systems: stable direction: deviations shrink asymptotically like e λt (λ=lyapunov exponent) unstable direction: deviations grow for t and shrink for t like e λt sensitive dependence on initial conditions Construction of partner orbit:

32 Underlying mechanism

34 Generalisation orbits can differ in arbitrarily many encounters where arbitrarily many stretches come close for time reversal invariant systems: stretches may be almost mutually time reversed

35 Generalisation contribution of each diagram where κ ( 1)V l lv l L V +1 τ L(L V 1)! κ = 2 with time-rev. invariance, 1 without v l = # of encounters with l stretches V = # of all encounters L = # of all stretches

36 Example: τ 3 orbit pairs in systems without time reversal invariance contributions cancel, agreement with GUE additional pairs requiring time reversal invariance 2τ 3, agreement with GOE (Heusler, S.M., Braun, Haake 2003)

37 Summary Spectral form factor: K (τ) 1 Re A p A p T ei(sp S p )/ δ H p,p ( τt H T ) p + T p 2 diagonal approximation: 2τ (τ without time reversal invariance) Sieber-Richter pairs: -2τ 2 2τ 3 from higher orders: e.g.

38 All orders Method I relate diagrams to each other by shrinking away links example: here the order of τ remains the same but the sign ( 1) V changes this leads to cancellation for systems without time-reversal invariance with time-reversal invariance: steps where shrinking changes the order are relevant get recursion between coefficients of K (τ) to implement this, describe diagrams in terms of permutations

39 All orders Result: systems without time-reversal invariance: agreement with GUE K (τ) = τ (τ < 1) time-reversal invariant systems: agreement with GOE K (τ) = 2τ τ ln(1 + 2τ) = 2τ 2τ 2 +2τ τ (τ < 1) (S.M., Heusler, Braun, Haake, Altland, PRL PRE 2005)

40 All orders in τ Method II Compare to nonlinear sigma model (field theoretical implemetation of RMT) encounters vertices links propagator lines diagrams and their contributions coincide Spectral statistics agrees with RMT

41 5 Resummation

42 Correlation function Idea: to get τ > 1 improve semiclassical approximation build in that E n R change setting to make this easier go back to correlation function ( R(ɛ) = d E + ɛ ) ( d E ɛ ) (take 2 2 d = 1) ( R GUE (ɛ) = Re 1 1 2(πɛ) ) 2(πɛ) 2 e2πiɛ ( ( ) 1 n R GOE (ɛ) = Re c n + ( ) ) 1 n d n e 2πiɛ ɛ ɛ n n non-oscillatory terms τ < 1 oscillatory terms τ > 1

43 Spectral determinant access level density from spectral determinant (E) = det(e H) = j (E E j ) = exp tr ln (E H) Motivation: used in random-matrix theory can now build in that (E) R for E R Relation to level density: Therefore 1 E (E) 1 = tr E H (E) 1 d(e) = 1 π Im tr 1 E H = 1 π Im (E) E (E ) E =E (where E is taken with a small positive imaginary part)

44 Generating function correlation function can thus be accessed through generating function: (E + γ) (E δ) Z (α, β, γ, δ) = (E + α) (E β) R(ɛ) Re 2 Z α β α=β=γ=δ=ɛ/2

45 Conventional semiclassics tr (E H) 1 iπ d }{{} Weyl A + sum over classical periodic orbits }{{} Gutzwiller A (E) exp ( E de tr ( E H ) 1) a e iπ de }{{} Weyl A F A( 1) n Ae is A(E)/ }{{} sum over sets of classical periodic orbits A

46 Resummed semiclassics But (E) should be real for real E Resummation (Berry & Keating) ( over orbit sets > TH /2) ( ) = over orbit sets < TH /2 Riemann-Siegel lookalike (E) = e iπ de F A ( 1) n A e isa(e)/ + c.c. A (T A <T H /2)

47 Generating function Z = (E + γ) (E δ) (E + α) (E β) e iπ(α+β γ δ) }{{} Weyl factor FA F B F CF D ( 1)n C+n D e i[(s A(E+α)+S C (E+γ) S B (E β) S D (E δ)]/ }{{} sum over orbit sets A, B, C, D (T C, T D < T H /2) + {γ δ, δ γ} Weyl factors 1 or e 2πiɛ as α, β, γ, δ ɛ/2 need small action differences S = S A (E + α) + S C (E + γ) S B (E β) S D (E δ) orbit correlations

48 Contributions Diagonal approximation: (A, C) contain the same orbits as (B, D) Encounters:

49 Contributions Diagonal approximation: (A, C) contain the same or orbits as (B, D) Encounters: Full agreement with RMT predictions. Heusler, SM, Altland, Braun, Haake (2007); Keating, SM (2007); SM, Heusler, Altland, Braun, Haake (2009)

50 Result without time reversal invariance Z = e iπ(α+γ β δ) (α+δ)(γ+β) (α+β)(γ+δ) + {γ δ, δ γ} with time reversal invariance Z = e iπ(α+γ β δ) +{γ δ, δ γ} ( (α+δ)(γ+β) (α+β)(γ+δ)) 2 + further terms R(ɛ) Re 2 Z α=β=γ=δ=ɛ/2 agrees with GUE and GOE predictions. α β

51 Conclusions Classically chaotic systems: sensitive dependence on initial conditions Gutzwiller formula: level density written as sum over periodic orbits Chaotic systems have universal spectral statistics semiclassical explanation involves orbit correlations due to encounters in phase space use generating functions and Riemann-Siegel lookalike to understand oscillatory terms in R(ɛ)

52 Applications symmetries higher order correlation functions transport through chaotic cavities need open trajectories

53 Mathematical status Orbit pairs: identical, time-reversed, differing in encounters universal result in agreement with RMT other systematic correlations: should be related to further symmetries universality requires absence of symmetries need a general definition of symmetries random orbit pairs with small action differences: contributions should cancel Conditions for universality: existence of orbit pairs requires hyperbolicity universal contribution obtained using ergodicity, mixing semiclassical limit no other symmetries

54 References F. Haake, Quantum Signatures of Chaos, 3rd ed., Springer, Berlin (2010) [in particular chapters 9 and 10] H.-J. Stöckmann, Quantum Chaos: An Introduction, Springer (2007) [in particular chapters 7 and 8.2] S. Müller and M. Sieber, Quantum Chaos and Quantum Graphs, The Oxford Handbook of Random Matrix Theory, eds. G. Akemann, J. Baik, and P. Di Francesco (2011) [ maxsm/chapter33.pdf] S. Müller, Quantum Chaos, Undergraduate lecture notes, University of Bristol (2013) [ maxsm/qcnotes.pdf]

Quantum Billiards. Martin Sieber (Bristol) Postgraduate Research Conference: Mathematical Billiard and their Applications

Quantum Billiards. Martin Sieber (Bristol) Postgraduate Research Conference: Mathematical Billiard and their Applications Quantum Billiards Martin Sieber (Bristol) Postgraduate Research Conference: Mathematical Billiard and their Applications University of Bristol, June 21-24 2010 Most pictures are courtesy of Arnd Bäcker