Quantum chaos in many-particle systems Boris Gutkin Georgia - PowerPoint PPT Presentation

Quantum chaos in many-particle systems Boris Gutkin Georgia Institute of Technology & Duisburg-Essen University QMath13: Atlanta, October 2016 – p. 1

Outline of the talk • “Single”-particle quantum chaos. Single (semiclassical) limit: � → 0 • Many-particle quantum chaos. Double limit: N → ∞ , � → 0 B.G. & V. Al. Osipov, Nonlinearity 29 (2016) arXiv:1503.02676 – p. 2

Chaos & Spectral universality Classical chaos: δ ( t ) ∼ δ (0) e λt Motivation ϕ n ∈ L 2 ( M ) Quantum: − ∆ ϕ n = λ n ϕ n , BGS conjecture G.Casati, et al. 1980; O. Bohigas, et al. 1984: Correlations of { λ n } ∞ n =1 are universal, described by Random Matrix Ensembles from the same symmetry class – p. 3

Semiclassical approach Gutzwiller’s trace formula: � i � � � ρ ( E ) = δ ( E − E n ) ∼ ¯ ρ ( E ) + ℜ A γ exp � S γ ( E ) ���� n γ ∈ PO Smooth � �� � Oscillating A γ stability factor, S γ ( E ) action of a periodic orbit γ γ Number of periodic orbits grows exponentially with length – No prediction on E n from an individual γ – All { γ } together ⇐ ⇒ spectrum – p. 4

Two-point correlation function R ( ε ) = 1 ρ 2 � ρ ( E + ε/ ¯ ρ ) ρ ( E ) � E − 1 ¯ � + ∞ R ( ε ) e − 2 πiτε dε ≈ (Semiclassically) K ( τ ) = −∞ �� �� � ≈ 1 τ − ( T γ + T γ ′ ) i � ( S γ − S γ ′ ) δ A γ A ∗ γ ′ e , T 2 2 T H H γ,γ ′ E T γ , T γ ′ are periods of γ, γ ′ , T H = 2 π � ¯ ρ (Heisenberg time) ⇐ ⇒ Spectral correlations Correlations between actions of periodic orbits – p. 5

Classical origins of universality K ( τ ) = c 1 τ + c 2 τ 2 . . . c 1 – diagonal approximation γ = γ ′ M. Berry 1985 Diagonal approximation Sieber−Richter pairs c 2 – non-trivial correlations (Sieber-Richter pairs) M. Sieber K. Richter 2001 ⇒ Duration of encounter ∼ τ E = λ − 1 | log � | S γ − S γ ′ ∼ � = � �� � Ehrenfest time All orders in τ = RMT result S. Müller, et. al., 2004 – p. 6

Symbolic Dynamics Continues flow = ⇒ Map T (Poincare section) p Phase space partition: 0 1 ... ... V = V 0 ∪ V 1 ∪ · · · ∪ V l − 1 l−2 l−1 q Point in the phase space: x = . . . x − 1 x 0 . x 1 x 2 . . . ; x i ∈ { 0 , 1 , . . . l − 1 } � �� � � �� � � �� � future past alphabet Tx = . . . x − 1 x 0 x 1 . x 2 x 3 . . . Periodic orbits ⇐ ⇒ [ x 1 x 2 . . . x n ] – p. 7

Partner orbits B. G, V. Osipov 2013 E A C D B F [ γ 1 ] = [ AECFBEDF ] , [ γ 2 ] = [ AEDFBECF ] E = e 1 e 2 . . . e p , F = f 1 f 2 . . . f p Each p-subsequence of symbols from γ 1 appears in γ 2 Locally similar but not identical = ⇒ Two orbits pass approximately the same points of the phase space: � γ 1 − γ 2 � ∼ Λ − p – p. 8

Many-particle systems N p 2 � n H = 2 m + V ( x n ) + V int ( x n − x n +1 ) n =1 Chaos, Local interactions, Invariance under n → n + 1 Two views on dynamics: Many−particle Periodic Orbit Single−particle Periodic Orbit d−dimensions Nd−dimensions 1 2 N Q: Is the single-particle theory of Quantum Chaos applicable? – p. 9

Semiclassical “Field Theory” Continuous limit: n → η ∈ [0 , ℓ ] , x n,t → φ ( η, t ) N q 2 ˙ 2 m + κ ( x n,t − x n +1 ,t ) 2 − V ( x n,t ) n,t � L = = ⇒ n =1 � ℓ dη ( ∂ t φ ( η, t )) 2 + ( ∂ η φ ( η, t )) 2 − V ( φ ( η, t )) L = 0 1) PO -are 2D toric surfaces in d -dim space (Rather than 1D lines in N · d -dim) 2) Encounters are “rings” (Rather than 1D stretches) of “width” ∼ λ − 1 | log � eff | – p. 10

2D Symbolic Dynamics T T 4 4 2 1 2 1 2 3 4 3 1 3 4 2 3 2 4 2 1 2 1 3 1 3 3 4 2 1 2 1 4 2 3 3 3 4 3 2 1 1 2 1 4 3 4 1 1 1 3 1 1 1 2 3 4 2 2 3 1 2 4 1 2 2 1 4 4 4 3 1 4 4 4 1 4 3 4 2 2 1 3 2 4 1 4 1 2 3 1 3 4 3 2 3 2 4 2 4 1 3 4 1 4 2 4 3 1 3 4 4 4 1 3 1 4 3 3 2 1 2 2 3 2 4 1 4 4 3 4 2 4 4 2 3 4 4 2 3 1 2 3 3 1 4 1 4 1 4 3 1 3 2 3 2 2 3 1 1 1 2 1 3 4 4 2 1 4 2 3 1 4 2 3 2 2 3 1 3 2 1 3 4 1 4 2 1 2 3 3 4 1 3 3 1 1 N 1) Small alphabet (does not grow with N ) 2) Uniqueness: Each PO Γ is uniquely encoded by M Γ 3) Locality: r × r square of symbols around ( n, t ) defines position of the n ’th particle at the time t up to error ∼ Λ − r Encounter - repeating region of symbols – p. 11

Different types of Partner Orbits A. Single particle partners: T T A A E E E D C F A F B C D B B E E C D F F A A F N N Dominant iff T � W � � N - Single particle theory W � ∼ Λ − 1 | log � eff | ≈ Width of encounter B. Dual partners: T T A D B C A A C B D A F E F E F E E F N N Dominant iff T � W � � N - Thermodynamic, short time regime – p. 12

Different types of Partner Orbits C. If T � W � , N � W � i.e. T and N are larger then “Ehrenfest scale” : T T A B E E C C B A E E ¯ Γ Γ N N Note: One encounter is enough, even if time reversal symmetry is broken B, C - Genuine many-particle Quantum Chaos! – p. 13

A Lone Cat Map: T 2 → T 2 Phase space: q t , p t ∈ [0 , 1) , windings m t = ( m q t , m p t ) ∈ Z Configuration Space q a � � � � � � � � m q q t +1 a 1 q t t = − , m p p t +1 ab − 1 b p t t a, b ∈ Z . Chaos if | a + b | > 2 Newton form: ∆ q t ≡ q t +1 − 2 q t + q t − 1 = ( a + b − 2) q t − m t – p. 14

Coupled-Cat Maps: T 2 N → T 2 N q i+2 q i+1 q i S ( q t , q t +1 ) = S 0 ( q t , q t +1 ) + S int ( q t ) , q t = ( q 1 ,t , q 2 ,t . . . q N ,t ) N Interacting cat maps, q n,t , p n,t ∈ [0 , 1) : N N � � S 0 = S cat ( q n,t , q n,t +1 ) + V ( q n,t ); S int = − q n,t q 1+ n,t n =1 n =1 � �� � interactions Equations of motion: p n,t = − ∂S ∂S p n,t +1 = ∂q n,t ∂q n,t +1 – p. 15

Classical Particle-time Duality Newtonian form: ∆ q n,t = ( a + b − 4) q n,t + V ′ ( q n,t ) − m n,t Discrete Laplacian: ∆ f n,t ≡ f n +1 ,t + f n − 1 ,t + f n,t +1 + f n,t − 1 − 4 f n +1 ,t Particle-time symmetry: t ← → n = ⇒ ⇒ T -particle POs { Γ ′ } N -particle POs { Γ } of period T ⇐ of period N S (Γ) = S (Γ ′ ) , A Γ = A Γ ′ { m n,t } - provide symbolic encoding of POs – p. 16

2D Symbolic Dynamics T 4 4 2 1 2 1 2 3 4 3   1 3 4 2 3 2 m 1 , 1 m 2 , 1 . . . m N, 1 2 1 2 1 3 3 3 4 2 1 2 1 4 2 3 3 4 3 2 1 1 2 1 4 3 4 1 1 1 3 1 1 3   4 2 2 3 1 2 4 1 2 2 1 4 4 4 3 m 1 , 2 m 2 , 2 . . . m N, 2   4 4 4 1 4 3 4 2 1 3 2 4 1 4 1 2 M Γ = 1   3 4 3 2 3 2 4 2 4 . . . 1 3 4 1 4 2 ... . . .   1 3 4 4 4 1 3 1 4 3 3 2 1 2 2 . . . 3 1 4 4 3 4 2 4   4 2 3 4 4 2 3 1 2 1 4 1 4 1 4 3 1 3 2 3 2 2 3 1 1 m 1 ,T m 2 ,T . . . m N,T 1 3 4 4 2 1 4 2 3 1 4 2 3 2 2 3 3 2 1 4 1 4 2 1 2 3 3 4 1 3 3 1 N √ Small alphabet (does not grow with N ) √ Uniqueness + Γ can be easily restored from M Γ √ Locality ( r × r square of symbols around ( n, t ) defines approx. position of the n ’th particle at the time t ) B.G. V. Osipov (2015), B.G., L Han, R. Jafari, A. K. Saremi, P Cvitanovi´ c (2016) – p. 17

Example of Partner Orbits T = 50 , N = 70 , a = 3 , b = 2 – p. 18

Example of Partner Orbits 1.0 0.34 0.8 0.32 0.6 p 0.30 p 0.4 0.28 0.2 0.26 0.0 0.06 0.08 0.10 0.12 0.14 0.0 0.2 0.4 0.6 0.8 1.0 q q All the points of Γ = { ( q n,t , p n,t ) } and ¯ Γ = { (¯ q n,t , ¯ p n,t ) } are paired – p. 19

Distances between paired points 50 40 1 30 10 � 3 t 20 10 � 6 10 10 � 9 1 1 10 20 30 40 50 60 70 n � 10 � 12 � q n ′ ,t ′ ) 2 + ( p n,t − ¯ p n ′ ,t ′ ) 2 , d n,t = ( q n,t − ¯ Largest distances ∼ 2 · 10 − 3 are between points in encounters – p. 20

Quantisation Hannay, Berry (1980); Keating (1991) U N is L N × L N unitary matrix, L = � − 1 eff Translational symmetries: = ⇒ N subspectra approximately of the same size = L N /N Gutzwiller trace formula Rivas, Saraceno, A. de Almeida (2000) 2 � Tr ( U N ) T = � − 1 � � � det( B T N − 1) exp( − i 2 πLS Γ ) . Γ ∈ PO All entries are symmetric under exchange N ↔ T – p. 21

Quantum Duality Tr ( U N ) T = Tr ( U T ) N �� 2 � � Tr ( U N ) T � 1 � � Form Factor: K N ( T ) = � 2 L N For short times T < n E = λ − 1 log L , N ∼ L T Regime dual to universal: K N ( T ) = L T − N K β ( TN/L T ) In particular for very short times L T /T < N , K β ≈ 1 K N ( T ) ≈ L T /L N Short time exponential growth instead of linear TN/L N – p. 22