Horizon in Random Matrix Theory, Hawking Radiation and Flow of Cold Atoms. Fabio Franchini

Transcription

1 Horizon in Random Matrix Theory, Hawking Radiation and Flow of Cold Atoms Coauthor: V. E. Kravtsov Thanks: Cargèse Summer School on Disordered Systems by Fabio Franchini Phys. Rev. Lett. R. Balbinot, S. Fagnocchi & I. Carusotto (also for some figures in this talk) 103,

2 The star of the talk: Two-Point (Density-Density) correlation function: (Anomalous: non-translational invariant) (Normal: translational invariant) Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 2 Fabio Franchini

3 Same correlator for different systems Invariant RME with Weak Confinement? Acoustic Black Hole in a BEC Luttinger Liquid in curved metric Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 3 Fabio Franchini

4 Outline Acoustic Black Hole in a BEC Hawking radiation RME with Weak Confinement Luttinger Liquid in curved metric & RME Conclusions Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 4 Fabio Franchini

5 Acoustic Black-Hole Fluid pushed to move faster than it s speed of sound: Sound waves cannot propagate up-stream Phonons feel effective Black-Hole metric Hawking radiation Hard to detect (Low T effect) Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 5 Fabio Franchini

6 Hawking radiation Prediction: a Black Hole radiates particles with an exact thermal (Black-Body) spectrum Solid result due only to horizon (kinematical) Different ways to understand it: Pair production close to horizon Red-shifting of last escaping modes Casimir effect Bogoliuobov overlap of positive frequency modes close to the horizon and at infinity Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 6 Fabio Franchini

7 QFT in Curved Space-Time Field quantization is basis-dependent: vacuum depends on the observer: For a different coordinate system: Plane waves Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 7 Fabio Franchini

8 Hawking Radiation If (uniform acceleration) black body radiation with temperature (pure quantum effect!) Free-falling observer not inertial w.r.t. far observer Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 8 Fabio Franchini

9 Search for Hawking radiation Not yet observed (too small) Solid theoretical prediction: general phenomenon Analogue Gravitational Models BEC:T H T BEC : still small for detection Search for different signature Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 9 Fabio Franchini

10 Acoustic BH in BEC Cool system Bose-Einstein Condensate Keep stream velocity constant & change speed of sound Effective dynamics is 1-D Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 10 Fabio Franchini

11 BEC Phonons Bose field: ρ = ρ 0 + δρ : number density φ = φ 0 + δφ : phase ρ 0 & φ 0 by mean-field Gross-Pitaaevskii δφ : Bogoliubov modes sound waves over background fluid: Effective D Alembert equation in curved metric Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 11 Fabio Franchini

12 Effective Gravity in fluids Bogoliubov modes: phonons near a black hole Hawking radiation Surface gravity 2-point correlator: κ hδρ(x, 0)δρ(x 0, 0)i cosh 2 2 (Balbinot et al. 08) µ x c r v + x0 v c l for xx 0 < 0 Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 12 Fabio Franchini

13 Numerical check Field theory prediction checked against ab-initio numerical simulation (Carusotto et al. 08) Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 13 Fabio Franchini

14 Acoustic BH Conclusions Low-energy modes are sound-waves Luttinger Liquid Effective metric simulates Black Hole effect Hawking radiation due to reference frame change Non-local correlation due to entangled pairs Let s apply what we learned to Random Matrix Theory Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 14 Fabio Franchini

15 Random Matrices Describe quantum chaotic systems Interactions between every degree of freedom Large matrices as Hamiltonian, Scattering Matrix... Take matrix entries randomly from a distribution Universality determined by symmetry: Orthogonal, Unitary and Simplectic Ensemble Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 15 Fabio Franchini

16 Invariant Ensembles Invariant Probability Distribution Function: Describe extended states (no localization) Wigner statistics Gaussian Ensemble: Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 16 Fabio Franchini

17 Weakly Confined Invariant Ensemble Critical Statistics (Spontaneous Breaking of U(N) Invariance?) Non-universal κ κ κ κ Exactly solvable (qdeformed Polynomial) (Muttalib et al. 93) Hermite Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 17 Fabio Franchini

18 Weakly Confined Invariant Ensemble Non-Trivial density eigenvalue distribution Unfolding to make density constant: Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 18 Fabio Franchini

19 Weakly Confined Invariant Ensemble For e -2π2 /κ << 1 semiclassical analysis (Canali et al 95): Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 19 Fabio Franchini

20 Weakly Confined Invariant Ensemble Numerical check (Canali et al 95): 1-Y 2 (x,x') x' Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 20 Fabio Franchini

21 Out-of-time Conclusions Weakly confined RME has critical eigenvalue structure Also non-local signature Exact results from orthogonal polynomial Lack of physical interpretation Effective model as Luttinger Liquid with Hawking radiation Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 21 Fabio Franchini

22 Invariant RME For an invariant ensemble: Plasma Model in 1-D Energy eigenvalues as 1-D interacting particles Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 22 Fabio Franchini

23 Effective Theory for RME Energy eigenvalues coordinates of interacting particles (fermions level repulsion) Parametric evolution of RME time coordinate Eigenvalue distribution ground state configuration of 1D quantum model Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 23 Fabio Franchini

24 Effective Theory for RME Low-Energy effective theory for 1-D system: Luttinger Liquid Low-Energy effective theory for 1-D system: Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 24 Fabio Franchini

25 Luttinger theory for RME Two-Point function (Kravtsov et al. 00): Unfolding: ρ 0 = 1 In flat space: 2-Point Function for Gaussian RME (K=1: Unitary) Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 25 Fabio Franchini

26 Luttinger theory in curved metric BEC system taught us that metric with horizons gives non-local correlation function In 1+1 D any horizon metric can be approximated by Rindler line element Let s choose: Horizon at y=0 Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 26 Fabio Franchini

27 Luttinger theory in Rindler space Periodic in imaginary time finite temperature t =const x =const Far from the origin: Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 27 Fabio Franchini

28 Luttinger Liquid in Rindler Space Remind two-point function: With the new coordinates: Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 28 Fabio Franchini

29 Luttinger Liquid in Rindler Space We recover exactly the RME correlation ( K=1): (Anomalous: non-translational invariant) (Normal: translational invariant) Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 29 Fabio Franchini

30 Summing up (part 1) Luttinger Liquid predicts oscillatory term in correlator Possible to detect them in a BEC in Tonks-Girardeau regime Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 30 Fabio Franchini

31 Summing up (part 2) RME is time-reversal invariant LL in thermal equilibrium with bath due to horizon (Hartle-Hawking effect) BEC is time-reversal broken actual Hawking radiation Kravtsov & Tsvelik (2001) already proposed a finite T LL for critical non-invariant ensemble relationship between the two models? Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 31 Fabio Franchini

32 Conclusions Luttinger Liquid in metric reproduces the 2-point function Horizons Hawking radiation (thermal bath + correlations) Equivalence with BEC system (+ oscillatory term) Same eigenvalue statistics as critical non-invariant RME Outlook Microscopical derivation: nature of thermal bath? U(N) SSB as Anderson Transition? Connection with Topological String theory Thank you! Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 32 Fabio Franchini

33 Topological String Theory Random Matrix Model (0+1 dim theory) U(N) Chern-Simons on S 3 (3 dim theory) Luttinger Liquid of Levels (1+1 dim theory)? Wess-Zumino-Witten (1+1 dim theory) Thank you! Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 33 Fabio Franchini

34 Non-Invariant Ensembles Non-Invariant PDF: Localized states (Poisson statistics) Multi-Fractal states (Critical Statistics) Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 34 Fabio Franchini

35 Invariant vs. non-invariant Ensembles Invariant: basis independent Wigner-Dyson eigenvalue statistics de-haar measure for eigenvector delocalized systems analytical techniques Non-Invariant: basis dependent Poisson/critical eigenvalue statistics eigenvector connected with eigenvalue localized/critical systems mostly numerical approaches Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 35 Fabio Franchini

36 Non-invariant Critical Ensemble Critical Random Banded Matrix (Multifractal spectrum) Thermal effective Luttinger Theory (Kravtsov & Tsvelik-2001) g*: critical conductance Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 36 Fabio Franchini

37 Thermal Field Theory Diagonal part of 2-point function common to weakly confined invariant ensemble Lorentzian banded matrix ensembles Standard thermal field theory How to generate the non-translational invariant part? Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 37 Fabio Franchini

38 2-Point Correlator In flat space: and for the density: Light-Cone coordinates Around the black hole: Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 38 Fabio Franchini

39 Conclusions We reproduced the asymptotic 2-point function of Random Matrix with a Luttinger Liquid in curved space-time description Curved metric with horizons Hawking radiation Equivalence with BEC system (oscillatory term) Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 39 Fabio Franchini

40 Outlook Underlying integrable model as interesting probe for emerging Quantum Gravity (transplanckian problem) Many unresolved questions (microscopical derivation?): nature of thermal bath Connection with Topological String theory Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 40 Fabio Franchini