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

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1 V BRUNEL Workshop on Random Matrix Theory (Brunel University West London) Horizon in Random Matrix Theory, Hawking Radiation and Flow of Cold Atoms by Fabio Franchini Phys. Rev. Lett. 103, Coauthor: V. E. Kravtsov (2009) Thanks: R. Balbinot, S. Fagnocchi & I. Carusotto (also for the figures)

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 space-time Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 3 Fabio Franchini

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

5 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. 5 Fabio Franchini

6 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. 6 Fabio Franchini

7 Weakly confined Invariant Ensemble Critical Statistics (Spontaneous Breaking of Invariance?) Exactly solvable (q-deformed Hermite Polynomial): (Muttalib et al. 93) Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 7 Fabio Franchini

8 Weakly Confined Invariant Ensemble Non-Trivial density eigenvalue distribution Unfolding to make density constant: Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 8 Fabio Franchini

9 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. 9 Fabio Franchini

10 Weakly Confined Invariant Ensemble Numerical check (Canali et al 95): Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 10 Fabio Franchini

11 First Interlude Interesting RME with interesting correlator Mathematically exact physical interpretation? Let s abandon RMT for a moment and look at something completely different Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 11 Fabio Franchini

12 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. 12 Fabio Franchini

13 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. 13 Fabio Franchini

14 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. 14 Fabio Franchini

15 Hawking Radiation If black body radiation with temperature (pure quantum effect!) Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 15 Fabio Franchini

16 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. 16 Fabio Franchini

17 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. 17 Fabio Franchini

18 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 : sound wave over background fluid δρ : algebraically related to δφ Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 18 Fabio Franchini

19 Effective Gravity in fluids Low-Energy excitations (phonons) propagate on top of bulk stream Effective dynamics: D Alembert equation in curved metric (c 2 =dp/dρ): Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 19 Fabio Franchini

20 2-Point Correlator In flat space: Light-Cone coordinates and for the density: Around the black hole: Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 20 Fabio Franchini

21 2-Point Correlator in curved metric (Balbinot et al. 08) Same non-local correlation as RME for ( except for the oscillatory term ) Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 21 Fabio Franchini

22 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. 22 Fabio Franchini

23 Second Interlude BEC system has non-local signature Low-Energy description in terms of free field in curved metric with horizon Let s go back to RMT and apply what we have learned Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 23 Fabio Franchini

24 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. 24 Fabio Franchini

25 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. 25 Fabio Franchini

26 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. 26 Fabio Franchini

27 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. 27 Fabio Franchini

28 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. 28 Fabio Franchini

29 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. 29 Fabio Franchini

30 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. 30 Fabio Franchini

31 Luttinger Liquid in Rindler Space Remind two-point function: With the new coordinates: Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 31 Fabio Franchini

32 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. 32 Fabio Franchini

33 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. 33 Fabio Franchini

34 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. 34 Fabio Franchini

35 Conclusions & Outlook We reproduced the asymptotic 2-point function in a Luttinger Liquid in curved space-time description Curved metric with horizons Hawking radiation Equivalence with BEC system (oscillatory term) Underlying integrable model as interesting probe for emerging Quantum Gravity (transplanckian problem) Many unresolved questions (microscopical derivation?): nature of thermal bath Thank you! Horizon in RME, Hawking Radiation & Flow of Cold Atoms n. 35 Fabio Franchini