ASPECTS OF CLASSICAL DYNAMICS IN HOLOGRAPHIC MATRIX MODELS David - PowerPoint PPT Presentation

ASPECTS OF CLASSICAL DYNAMICS IN HOLOGRAPHIC MATRIX MODELS David Berenstein (UCSB/ DAMTP Cambridge)

GAUGE/GRAVITY DUALITY s a l a u ! q s e m e e r t a s y e s s e m h u T t n a u q

STANDARD DICTIONARY Correlators in QFT are computed by (Quantum) gravity solutions with modified boundary conditions. Kubo formulae (transport) can be related to CLASSICAL PHYSICS in ADS: retarded propagators with in-falling boundary conditions. Basically one computes Green functions on a classical geometry (PDE).

Viscosity of quark-gluon plasma AdS/CMT: people study full 3+1 and 4+1 codes to introduce disorder, superconductivity,…. Applications to hydrodinamics (with vortices in superfluids…) Thermalization: black hole formation dynamics in AdS (global or Poincare depending on choices)

FROM QFT TO GRAVITY An “In principle” solution to the BH information paradox: still no clue what goes on with geometry. Many SUSY counting formulae Localization (SUSY Wilson loops to all orders in coupling constant) Integrability program (Spectrum of strings on very special setups)

WISH LIST Evolve a quantum state in a strongly coupled field theory and compare with time dependent quantum gravity problems. Extract Geometry of bulk from first principles (for which states does it make sense?) How does geometry break down in detail (conversely: when do certain observables become geometric and how accurate is the geometric description?) Can we measure if there is a Firewall or not?

SIGN PROBLEM In real time QM there is quantum Z D X ( t ) exp( iS ) interference: wildly oscillating path integral. Not suitable for Monte Carlo Too many degrees of freedom for other methods (Solving the Schrodinger equation numerically by brute force)

WHAT CAN BE DONE Consider a single saddle point: classical trajectories (perhaps with some quantum corrected dynamics). These can be evolved numerically Hope that there is enough information left over so that comparisons with gravity can be made.

MODELS TO BE STUDIED BFSS matrix model: Dimensional reduction of 10 D SYM to 0+1 Related to M-theory in DLCQ quantization. BMN matrix model: Dimensional reduction of N=4 SYM on three sphere in an SU(2) invariant sector (Kim and Plefka) Related to M theory on plane wave

TOPICS TO BE COVERED Chaos in matrix dynamics Thermalization: how to think about time scales at equilibrium (microcanonical approach). Extracting D-branes and other geometric information from matrices. Special solutions: comparing classical geometric object and fuzzy counterparts.

CHAOS

EXPONENTIAL SENSITIVITY δ x ( t ) ' exp( κ t ) δ x (0) For asymptotically large times. Butterfly effect

Yang Mills (on a box, with translationally invariant configurations) is chaotic. Basenyan, Matinyan, Savvidy, Shepelyanskii, Chirikov (early 80’) Chaos in BFSS matrix theory: Aref’eva, Medvedev, Rytchkov, Volovich (1998) Chaos in BFSS/BMN: Asplund, D.B., Trancanelli, Dzienkowski (2011,2012), Asano, Kawai, Yoshida

Poincare section on a 2 dimensional ansatz ( Asano, Kawai, Yoshida)

Poincare sections are useless in higher dimensions: one needs to study time series of trajectories instead. One detects chaos by showing that the system does not have a quasi periodic spectrum in the time series like one has in systems with action-angle variables: instead one looks for continuous power spectrum (Fourier analysis of time series).

L = 2 P H w L L = 3 600 500 L = 4 400 300 200 100 500 w 100 200 300 400 The power spectrum of S ( a ) = h tr( X 1 + iX 2 ) L ( t )tr( X 1 � iX 2 ) L ( t + a ) i for various FIG. 6. L in random units. Results shown are for 13 ⇥ 13 matrices in the BFSS matrix model after thermalization. The results are averaged over 15 runs of the same length, taken from splitting a single time series in 15 equal parts. The jiggling of the data should be interpreted as an estimate of the statistical error bars for each frequency. Aspund, D.B., Dzienkowski CHAOS: No choice but numerics.

THERMALIZATION

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CLASSICAL MECHANICS IS IN WRONG REGIME: IS THAT A PROBLEM?

Mote Carlo Lattice Agnastopoulos, Hanada,Nishimura, Takeuchi, 2007 Caterall, Wiseman 2008 High temperature and low temperature in same phase: but no real time dynamics in MCL.

DIFFERENT INITIAL CONDITIONS AND PATHS TO THERMAL Fuzzy sphere + D0 brane+small noise in BMN, then quench to BFSS (what we did) Collapsing fuzzy sphere + noise in BFSS (Riggins +Sahakian, 2012) Black hole from brane collisions (Aoki, Hanada, Iizuka, 2015)

IS IT THERMAL? H � P 2 2 + V ( X ) Thermal implies time averaged distribution of some quantities (momenta) should match the Gibbs ensemble. P ( P ) ⇥ exp( � β P 2 2 ) This is the standard gaussian matrix model ensemble.

Need to study it with traceless matrices. The ridges include the finite N exact soln. Effective temperature is given by second moment.

N h Tr( P 2 0 ) i 0 h Tr( P 2 1 ) i 0 h Tr( P 2 2 ) i 0 h Tr( Q 2 1 ) i 0 h Tr( Q 2 2 ) i 0 h Tr( Q 2 3 ) i 0 h Tr( Q 2 4 ) i 0 h Tr( Q 2 5 ) i 0 h Tr( Q 2 6 ) i 0 4 23 . 2 ± 0 . 6 23 . 3 ± 0 . 4 23 . 2 ± 0 . 5 21 . 3 ± 0 . 5 21 . 3 ± 0 . 5 21 . 2 ± 0 . 6 21 . 2 ± 0 . 4 21 . 3 ± 0 . 4 21 . 0 ± 0 . 4 11 26 . 9 ± 0 . 3 27 . 2 ± 0 . 2 27 . 0 ± 0 . 3 26 . 6 ± 0 . 2 26 . 5 ± 0 . 3 26 . 6 ± 0 . 2 26 . 6 ± 0 . 3 26 . 6 ± 0 . 2 26 . 5 ± 0 . 2 23 32 . 2 ± 0 . 3 32 . 2 ± 0 . 2 32 . 1 ± 0 . 2 31 . 9 ± 0 . 2 31 . 9 ± 0 . 2 31 . 9 ± 0 . 2 31 . 9 ± 0 . 2 31 . 9 ± 0 . 2 32 . 0 ± 0 . 2 Need to be careful: there is a constraint. Symmetry breaking of X,Y eom’s symmetry breaking between X, Y, symmetry breaking between P , Q Need to consider a GGE to define T

TEST INDEPENDENCE OF N P H w L 1000 10 N = 47 0.1 N = 27 N = 13 N = 18 0.001 N = 10 N = 7 N = 4 500 w 100 200 300 400 Can compare different N: neither frequency nor amplitude normalized. Fit to max, and rescale power.

27 100 10 1 0.1 0.01 0.5 1.0 1.5 2.0 2.5 3.0 FIG. 8. The power spectrum of tr( X 1 + iX 2 ) 2 ( t ) for various sizes of N × N matrices. The axis of frequency has been rescaled for each N , to the frequency ω N , and we have also rescaled the power spectrum. The reference frequency for each N is located at 1 in the graph. Results shown for N = 7 , 10 , 47. We also have drawn additional suggestive straight lines superposed on the graph that serve as distinctive features of the power spectrum. Notice Log spectrum seems to have an absolute value singularity at 0: this would imply power law decays of correlation functions at long times.

IS THIS A FORM OF HYDRO ON THE HORIZON? P H Ω L 10 12 10 9 10 6 1000 1 800 Ω 200 400 600 FIG. 10. Power spectrum in arbitrary units for O L , with L = 2 , . . . 10, with values of L increasing from bottom to top in the graph. For each L we show two such sets. This data is from N = 27. Some dynamics is N independent. Can also do 3-pt functions and see 1/N behavior

WE CAN ADD ANGULAR MOMENTUM IN INITIAL CONDITION (CAN’T DO MC-SIGN PROBLEM: BEYOND LATTICE)

Add ( large) angular momentum in initial condition 15 10 5 After thermalization, one 500 1000 1500 2000 eigenvalue is expelled - 5 - 10 - 15 In progress.

System rotates “rigidly” at constant speed tr ( X 2 1 ) 250 200 150 100 50 tr ( X 2 100 150 200 250 0 )

Clock Log Power spectrum vs L 20 15 10 5 200 400 600 800 1000 1200 1400 - 5 - 10 L=2,4,6,8,9,10 in random units Critical angular momentum where eigenvalue peels off, and another one where the distribution deforms.

Positive versus negative frequency Clock 25 for high L. 20 15 10 5 200 400 600 800 1000 1200 1400 - 5

GEOMETRY?

Raw output of computations: Factory that spits out lists of matrices ordered in time.

Once you have a typical matrix configuration, how do you probe it geometrically?

EXTRACTING GEOMETRY:ADD A (OFF- SHELL) PROBE AT SOME POSITION

Typical idea of matrix models: add eigenvalue � X ⇥ Geometry: ∗ Want to study the x dependence of * ∗ † x String interpretation: * represent open strings starting on matrix configuration and ending at a point D0 brane located at x.

Mass of * represents roughly a distance M s ' T string ` Problem: there can be a zero point energy correction and M can become negative (instability). Not for fermionic states, so finding M near zero does measure something like zero length.

Add a D0 brane probe (extra eigenvalue) The probe lives in R 9 We can ask questions about the probe and locate information relative to this flat geometry.

Need to diagonalize ‘instantaneous’ effective Hamiltonian for od fermions. ( X i − x i ⊗ 1) ⊗ γ i X H eff = Define (spectral) distance as minimum eigenvalue (in absolute value) w. E. Dzienkowski See also Ishiki’s talk.

Scan over a 1 parameter set at fixed time Fermions are gapless in a region Size of X matrix

Effective field theory of probe independent of matrix breaks down in gapless region: can’t integrate out off-diagonal fermion modes.