q fermions or bosons drawn from static random distribution

Transkrypt

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2 q fermions or bosons drawn from static random distribution

3 modeling chaos in complex nuclei

4 discovery of conformal symmetries

5 Sachdev-Ye-Kitaev Model (15) A model of N randomly interacting Majorana fermions SYK model where the interaction constants are static and random, high energy scale Three perspectives: random matrix theory strong correlation physics holography

6 quantum chaos

7 random matrix correlations: Verbaarschot, Garcia-Garcia, 16 Note: depending on the value of N mod 8 the model realizes different symmetries Cotler et al. 17

8 strong correlations

9 Strong interactions: infinite range, strong, chaotic: amenable to large N mean field methods diagrammatic expansion of Majorana propagator i disorder average j k structureless

10 Solutions of mean field equation For numerical factor Non Fermi liquid mean field Green functions

11 Symmetries Hamiltonian action (neglecting time derivatives) invariant under reparameterization of time Elements of the diffeomorphism manifold describe reparameterizations of time. Infinitesimally: generated by Virasoro algebra. Weakly broken by time derivatives problem has NCFT1 symmetry (Maldacena and Stanford, 15).

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G f (, 0 )=f 0 ( ) 1/4 f 0 ( 0 ) 1/4 G(f( ),f( 0 ) emergence of infinite dimensional Goldstone mode manifold G f

13 Quantum fluctuations & ergodicity in the SYK model Würzburg, Aug. 2nd 2018 Alexander Altland, Dmitry Bagrets (Cologne), Alex Kamenev (Minnesota) conformal symmetry breaking, chaos and short time dynamics quantum ergodicity and long time dynamics Nucl. Phys. B 911, 191 (2016) Nucl. Phys. B 921, 727 (2017) Nucl. Phys. B 930, 45 (2018)

14 stages of SYK dynamics

15 SYK timeline? time II: large collective quantum fluctuations IV: ergodic (RMT) regime I: semiclassical short time regime III: quantum chaotic relaxation dynamics

16 conformal symmetry & Liouville quantum mechanics

17 Conformal Goldstone mode action flat measure action of Liouville QM aa, Bagrets, Kamenev, 16 effect of low energy Goldstone mode fluctuations encapsulated in Liouville QM. Universal feature (Shelton, Tsvelik 98): all operator correlation functions decay as

18 chaos and OTO correlation functions

19 OTO correlation function Out of time order (OTO) correlation function: a tool for diagnosing early stages of quantum chaotic dynamics (Larkin, Ovchinikov 69): X,Y one-body operators in many body context. Interpretation: quantum butterfly effect

20 Short time OTO: stationary phase At short times stationary phase analysis leads to (Maldacena 16) Result can be trusted up to effective Ehrenfest time (chaos bound maxed out!) At intermediate times

21 Long time OTO: conformal Goldstone modes At long times large Goldstone mode fluctuations generate power law tail

22 chaos and spectral correlation functions

23 Ergodic long time regime? Ergodic long time dynamics diagnosed in universal correlations of many body spectra Level spacing distribution Verbaarschot, Garcia-Garcia, 16 Verbaarschot, Garcia-Garcia, 16 Level number fluctuations Cotler et al. 17 Spectral form factor (Fourier trafo of energy level correlation function)

24 Pre ergodic regime For times shorter than an ergodic time observed.? universal deviations from RMT behavior are not RMT (still universal) Verbaarschot, Garcia-Garcia, 16 Cotler et al. 17 RMT

25 Compare to the physics of dirty metals Interested in fluctuations of Momentum q difference conserved in scattering. Idea: (i) basis change (ii) interpret this as a basis change in the space of Hilbert space operators.

26 dirty metals cont d Modes characterized by (i) discrete quantum numbers (ii) decay constants (iii) physical interpretation as irreversible relaxation modes

27 Dirty metals cont d Altshuler & Shklovskii, 86 Kravtsov & Mirlin, 94

28 relaxation dynamics in Fock space

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Dq 2 i! # of Majoranas in state (k) 2 N/2 k JN1/2 2 N/2 many body level spacing

31 SYK spectral correlation function <latexit sha1_base64="+w8ohqmtctt3y8ol/xvap6r7h6o=">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</latexit> <latexit sha1_base64="+w8ohqmtctt3y8ol/xvap6r7h6o=">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</latexit> <latexit sha1_base64="+w8ohqmtctt3y8ol/xvap6r7h6o=">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</latexit> <latexit sha1_base64="+w8ohqmtctt3y8ol/xvap6r7h6o=">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</latexit> R 2 (!) =R 2,RMT (!) Re X k6=0,even N k 1 (i! (k)) 2 comparison to numerical data I: number variance Σ RMT N=22 N= Verbaarschot, Garcia-Garcia, ϵ Δ

32 SYK spectral correlation function comparison to numerical data II: spectral form factor K(τ) K(τ)Δ N=22 N=26 N=30 N=34 RMT N=22 N=26 N=30 N= τ t/8j Cotler et al. 17 all in all: very good parameter free agreement with numerical data.

33 summary conformal Goldstone modes cause strong quantum fluctuations at intermediate time scales identified a high density set of complementary modes dominate fluctuations beyond a non-universal Thouless energy a case study of chaotic relaxation in a strongly nonlinear many body environment connection between these fluctutations? Ramifications in holography?

34 Holographic interpretation (Maldacena & Stanford, 16; Almheiri & Polchinksi, 16) Consider 2d Einstein-Hilbert action also constant positive cosmological constant gravitational constant action invariant under conformal deformations of 2d space (because it is topological) boundary (where SYK lives) deformation mode conformal compactified space AdS metric AdS metric (spontaneously) breaks symmetry to SL(2,R). Reparameterization Goldstone modes without action.

35 Holographic interpretation (continued) Improve situation by upgrading pure gravity action to dilaton action now a field Jackiw Teitelboim gravity This action (i) is non-topological, (ii) fluctuations of the dilaton field weakly break conformal symmetry and (iii) afford physical interpretation if AdS2 action is seen as boundary theory of higher dimensional extremal black hole. Combination (i-iii) motivates boundary with conformal invariance breaking and signatures of quantum chaos.