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

T h e s e a r e e q u a l a s q u a n t u m s y s t e m s !

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

Z DX(t) exp(iS) In real time QM there is quantum 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

δx(t) ' exp(κt)δx(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:

• ne 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).

PHwL L=4 L=3 L=2

• FIG. 6.

The power spectrum of S(a) = htr(X1 + iX2)L(t)tr(X1 iX2)L(t + a)i for various 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

• f the statistical error bars for each frequency.

Aspund, D.B., Dzienkowski

CHAOS: No choice but numerics.

THERMALIZATION

฀฀

• ฀

Black holes have temperature: a collapse problem in gravity ends up with a temperature, so this is a thermalization trajectory in dual.

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

Mote Carlo Lattice

Caterall, Wiseman 2008

Agnastopoulos, Hanada,Nishimura, Takeuchi, 2007

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 hTr(P 2

0 )i0 hTr(P 2 1 )i0 hTr(P 2 2 )i0 hTr(Q2 1)i0 hTr(Q2 2)i0 hTr(Q2 3)i0 hTr(Q2 4)i0 hTr(Q2 5)i0 hTr(Q2 6)i0

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

N = 4 N = 7 N = 10 N = 13 N = 18 N = 27 N = 47

PHwL

Can compare different N: neither frequency nor amplitude normalized. Fit to max, and rescale power.

27

• FIG. 8. The power spectrum of tr(X1 + iX2)2(t) for various sizes of N × N matrices. The axis
• f frequency has been rescaled for each N, to the frequency ωN, 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?

PHΩL

• FIG. 10. Power spectrum in arbitrary units for OL, 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 eigenvalue is expelled In progress.

System rotates “rigidly” at constant speed

100 150 200 250 50 100 150 200 250

tr(X2

0)

tr(X2

1)

200 400 600 800 1000 1200 1400

• 10
• 5

5 10 15 20

Clock Log Power spectrum vs L L=2,4,6,8,9,10 in random units

Critical angular momentum where eigenvalue peels off, and another one where the distribution deforms.

200 400 600 800 1000 1200 1400

• 5

5 10 15 20 25

Positive versus negative frequency for high L. Clock

GEOMETRY?

Factory that spits out lists of matrices ordered in time.

Raw output of computations:

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 ∗ ∗† x ⇥

Geometry: Want to study the x dependence of * 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

Ms ' Tstring`

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 R9 We can ask questions about the probe and locate information relative to this flat geometry.

See also Ishiki’s talk.

Need to diagonalize ‘instantaneous’ effective Hamiltonian for od fermions.

Heff = X (Xi − xi ⊗ 1) ⊗ γi

Define (spectral) distance as minimum eigenvalue (in absolute value)

• w. E. Dzienkowski

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

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

Fix position of probe inside gapless region

Define spectral dimension using density of states near zero

dn dE |E'0 ' Eγ1

Same density of degrees of freedom as field theory in

γ + 1 dimensions spectral dim = γ

Spectral dimension = 1 Effective 1+1 field theory

A gas of D0 branes would have spectral dimension 9 Very non commuting configurations behave very different than ordinary D-branes: the IR is much richer. A fuzzy sphere would have spectral dimension 2

Physics can not be local in that region.

1 6= 9

Can not be both space filling and one-dimensional

Interpretation (speculation) Gapless region is ‘inside the black hole’ EFT breaks down as we get near black hole: gap becomes smaller than naive distance (could be interpreted as redshift)

Inside region can not be characterized as ‘no drama’: an observer falling in there could not help notice that there is a very hot IR full of degrees of

• freedom. At fixed T, the number of such degrees of

freedom grows with N.

Number of IR states with E < T ' N 3/4

C

• m

p l e t e l y s h a m e l e s s ! W e d

• n

’ t u n d e r s t a n d g e

• m

e t r y s u f fi c i e n t l y w e l l t

• fi

x i t

• r

t

• m

a k e t h i s c l a i m w i t h a s t r a i g h t f a c e .

Claim: this is evidence for the existence of firewalls. Geometry goes bunk and the inside of the black hole is not geometric.

Yet not saying it is ignoring the elephant in the room.

TO DO LIST: Understand the same with typical matrix configurations from Lattice Monte Carlo simulations. Then one is talking about setups where gauge theory has been matched to Black hole physics rather well.

SPECIAL CLASSICAL SOLUTIONS (EXTREMAL STATES IN BMN)

H = 1 2Tr(P 2

1 + P 2 2 + P 2 3 ) + 1

2Tr @

3

X

j=1

(Xj + i✏jmnXmXn)2 1 A

Look at SO(3) part of BMN matrix model Has SO(3) symmetry with generator J = LZ = Tr(XPY − Y PX)

H = Tr ✓1 2P 2

3 + 1

2(P1 ± (X2 + i✏231[X3, X1]))2 +1 2(P2 ⌥ (X1 + i✏123[X2, X3]))2 + 1 2(X3 + i✏312[X1, X2])2 ◆ ± J

Has a BPS inequality H ≥ |J| Can try to look for solutions that saturate equality: usually these rotate “rigidly”.

X+ = 1 2(X + iY ), X− = 1 2(X iY ) (

easier using complex matrices

Consider the ansatz

X+(t) =           a1 exp(iω1t) . . . a2 exp(iω2t) . . . ... ... ... . . . . . . aN−1 exp(iωN−1t) aN exp(iωNt) . . .          

− + †

Z time independent This rotates rigidly because there is still gauge invariance that can let us shift the w at the expense of introducing a non-trivial A_0.

Symmetry

U(θ)ZU −1(θ) = Z U(θ)XU −1(θ) = X cos(θ) Y sin(θ) ( U(θ)Y U −1(θ) = Y cos(θ) + X sin(θ)

Set of unitaries (gauge transformations) that rotate the configuration into itself. Ansatz we have is the only one that preserves a ZN D.B, Dzienkowski, Regas (2015)

Can’t have a round pancake, or a round torus.

Gauss law states that |ai|2ωi = |aj|2ωj J = 4Nωi|ai|2 An direct computation shows that

One can show that solutions are equivalent to having critical points

• f the energy function

E(J, |ai|, zi) = Ekin(J, |ai|) + V (|ai|, zi) (

= 1 8N 2

N

X

i=1

J2 |ai|2

) =

N

X

i=1

1 2[zi + 2|ai−1|2 − 2|ai|2]2 + 2(1 + zi+1 − zi)2|ai|2

Solve a nasty algebra problem

The fact that they are critical points of a function that depends on J implies that one can use Morse theory to analyze the problem: solutions come in continuous families parametrized by J and they can disappear only at multi-critical values (special values of J) The z appear as a quadratci form, so they can be eliminated as functions of the a.

0.305 0.305 0.305 0.31 0.31 0.31 0.315 0.32 0.325 0.33 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6

2x2 matrices solvable analytically. |a1| |a2|

0.5 1.0 1.5 2.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

“Phase diagram” for 3x3 matrices

|ai|

J

Some a values for 4x4 Some z values for 4x4 There is always a critical value after which only one solution remains

Can also take large N (potential is nearest neighbor finite differences), then

P R Vpot = V (|ai|, zi) ! N 3 Z d✓ 1 2

• z 2@θ|a|22 + 2 (1 + @θz)2 |a|2
• Ekin = N 3

Z d✓ ˜ |2 8|a(✓)|2

Z dθ = 1 Constrained to

Can do a variational principle and interpret in terms of a non-trivial dynamics on a curved space with a background magnetic field and a non-trivial effective potential Need to shoot for periodic orbits of the right period

z |a|2

• 1.5
• 1.0
• 0.5

0.5 1.0 1.5 0.5 1.0 1.5

For BPS states, it is simpler:

@✓z = 1 + ˜ | 4|a|2 @✓|a|2 = 1 2z

L = ˙ q2 q + ˜ | 4 log q

Equivalent to a dynamics with the Following Lagrangian q = |a2| Where

W = ˙ q2 + q ˜ | 4 log(q) = z2 + q ˜ | 4 log(q) (

One gets a constant of motion Shape of the 2-dimensional D-brane in 3-dimensions. It’s rotationally invariant 2-torus (possibly multiply wound).

For some 4x4 matrices, finding zeroes of effective mass matrix for fermions we get Very deformed geometry due to finiteness of N.

We go from sphere to torus at finite J. Torus is similar to giant torus of Nishioka+Takayanagi (2008). It arises from condensation of strings that stretch between north and south pole.

Understanding how when geometries for other shapes at large N make sense Recent work by Hudoba, Karczmarek, Sabella-Garnier. Yeh (2015), also Ishiki. Older work by Hoppe et al (2004-2014).

General BPS equations without the discrete rotation symmetry Bak (2002) Need to solve in general the following equations:

[X0, [X0, Z]] + [Y0, [Y0, Z]] − 2

µ

3

2

Z = 0 , [X0, Y0] = iµ 3 Z .

END COMMENTS

Classical dynamics of holographic matrix models is interesting. Can address questions related to black holes, geometry, scrambling, shape of membranes. Chaos forces numerical approach. Interesting observables related to probe D0-branes: not just traces, but spectra of effective Hamiltonians contain a lot of

• ther information.