Thermodynamics of the BMN matrix model at strong coupling

Transcription

1 hermodynamics of the BMN matrix model at strong coupling Miguel S. Costa Faculdade de Ciências da Universidade do Porto Work with L. Greenspan, J. Penedones and J. Santos Correlations, criticality, and coherence in quantum systems Évora - October 04

2 Motivation Gauge/gravity duality as definition of quantum gravity in AdS Dual CF is renormalizable and unitary. Problem: how to decode the hologram? Unfortunately field theory is strongly coupled in region of interest for quantum gravity (classical gravity N!, /N expansion loop expansion). Would like examples where computations in both sides are within reach est and understand the gauge/gravity duality with observables that are not protected by SUSY and can not be computed using integrability. How does gravitation phenomena, like black holes, emerge from gauge theory side? Idea: Study thermodynamics of black holes dual to Matrix Quantum Mechanics that can be simulated on a computer using Monte-Carlo methods.

3 he case of D0-branes Closed strings interact with D0-branes in flat space D0-brane Closed string Open string, Closed strings interact with geometry produced by D0-branes Closed string D0-brane geometry [Itzhaki, Maldacena, Sonnenschein, Yankielowicz 98]

4 D0-branes: field theory description (matrix quantum mechanics) [Itzhaki et al 98] Z apple S D0 = N dt r (D t X i ) + D t + X i,x j + i j [,Xj ] X i SU(N) bosonicmatrices(i =,...,9) SU(N) fermionicmatrices(6realcomponents) SO(9) global symmetry t Hooft coupling is dimensionfull (relevant) = g YMN = g sn ( ) l 3 s mass 3 eff = E 3 E!(UV) weak coupling E! 0(IR) strong coupling Dual 0D gravitational coupling 6 G N l 8 s =( ) ( l3 s) N

5 heory on Euclidean time circle with periodicity =/ Z apple S D0 = N 0 dt r Dimensionless temperature = /3 Low temperatures is strong coupling Can put theory on a computer using Monte Carlo simulations, accessing in particular strongly coupled region. Dimensionless mean energy N = E N /3 [Catterall, Wiseman 07, 08, 09; Anagnostopoulos et al 07; Hanada et al 08, 3]

6 D0-branes: gravitational description D SUGRA solution (near horizon geometry of non-extremal D0-brane) ds = dr f(r) + r d 8 + R r 7 dz + f(r)dt dz r0 R 7 dt f(r) = r0 r 7, R`s 7 =60 3 g s N, r0 `s 5 = 0 49 ( g s N) 5 3 Classical gravity domain (at horizon) D SUGRA IIA Horizon at l s scale lsr(r 0 ) ) g s e (r 0) ) N 0 N 0 = / /3

7 Standard gravitation thermodynamics S = A H 4G N = d N 9 5 d = N = c 4 5 c = 9 4 d (because de = ds) 0 corrections give next term in expansion, at large N 6 G N Z d 0 x p ge R R ) S N = d 9 5 +d 9 5 [Hanada et al 08] fixes next power in expansion l P corrections to D SUGRA give /N correction (solution purely gravitational) 6 G Z d x p g R + lp 6 R 4 ) S N = d N d fixes both coefficient and power of /N correction [Hanada et al 3]

8 Low temperature expansion predicted from gravity h i N = c c N h c i +... /N E/N N=7,!=6 N=7,!= SUGRA SUGRA + 0 corrections Monte-Carlo simulation of MQM negligible checked predicted [Hanada, Hyakutake, Nishimura, akeuchi 08]

9 Low temperature expansion predicted from quantum gravity h i N = c c N h c i +... Monte-Carlo simulation of MQM /N negligible checked [Hyakutake 3] [Hanada, Hyakutake, Ishiki, Nishimura 3]

10 oday s talk is not about D0-brane matrix model Caveat: canonical ensemble ill defined - IR divergences from flat directions in D0-brane moduli space. his is suppressed at large N (metastable state), but it is a source of tension in Monte Carlo simulations [Catterall, Wiseman 09] F (,r) N F finite ( )+ 9 N ln r Instability corresponds to Hawking radiation of D0-branes. At large N this is suppressed and black hole is stable (positive specific heat). oday s talk is about BMN matrix model [Berenstein, Maldacena, Nastase 0] Mass deformation resolves IR divergence - canonical ensemble well defined. Much richer thermodynamics with a st order phase transition (at large N there are two dimensionless parameters).

11 BMN matrix model S = S D0 N Z dt r apple µ 3 (Xi ) + µ 6 (Xa ) + µ i µ 3 ijkx i X j X k Massive deformation of D0-brane MQM. Preserves SUSY but breaks SO(9)! SO(6) SO(3) a =4,...,9 i =,, 3 In large N t Hooft limit dimensionless coupling constant = g YM N µ 3 Many vacua X a =0 X i = µ 3 J i [J i,j j ]=i ijk J k Focus on trivial vacuum (single M5-brane) that is SO(9) invariant X i = X a =0 Canonical ensemble is well defined and may still be simulated on a computer.

12 hermodynamics ( N!) Start here Dimensionless coupling = g YM N µ 3 µ H µ W E A K C O U P L I N G Deconfined phase F = O(N ) Confined phase F = O(N 0 )? S R O N G C O U P L I N G Dimensionless temperature µ oday: strongly coupled limit µ! 0, µ Dual geometry is SO(9) invariant non-extremal D0-brane with deformation turned on fixed and large Exponential growth of spectrum with energy! = g YM N µ 3 Hagedorn transition First-order phase transition at c µ = log 3 apple c + O( 3 ) O( ) [Hadizadeh, Ramadanovic, Semenoff, Young 04]

13 µ H µ W E A K C O U P L I N G Deconfined phase F = O(N ) Confined phase F = O(N 0 )? S R O N G C O U P L I N G Solutions known as the Lin-Maldacena vacuum geometries = g YM N µ 3 At strong coupling, for large temperature, dual geometry is SO(9) invariant and is approximately the non-extremal D0-brane solution ds = dr f(r) + r d 8 + R7 dc = µdt^ dx ^ dx ^ dx 3 r 7 dz + f(r)dt Non-normalizable mode responsible for massive deformation dz r0 7 R 7 dt Need back-reaction to decrease temperature and study phase transition at strong coupling. In particular, SO(9)! SO(6) SO(3)

14 Ansatz for D SUGRA x = S pole ds = A ( y7 ) y 7 d + 4 y 7 + y " B apple d + ( y7 )d y 7 # (dy + Fdx) 4dx ( y 7 )y + x + x ( x )d + 3 ( x ) d 5 {z } S 5 x =0 equator d 8 if = = 3 = y = y =0 C =(Md + Ld ) ^ d horizon S 8 S boundary S 8 S S M-theory circle + x x is a angular coordinate on compact 8-dimensional space with S 8 topology y is a radial coordinate from boundary ( y =0) to horizon ( y =) S 5 collapses A, B, F,,, 3, 4,,M,L x are functions of and y Horizon ailored to numerical implementation (domain of unknown is the unit square; everything dimensionless) 0 S collapses y

15 Ansatz for D SUGRA x = S pole ds = A ( y7 ) y 7 d + 4 y 7 + y " B apple d + ( y7 )d y 7 # (dy + Fdx) 4dx ( y 7 )y + x + x ( x )d + 3 ( x ) d 5 {z } S 5 x =0 equator C =(Md + Ld ) ^ d Non-extremal D0-brane solution corresponds to d 8 if = = 3 = y = horizon S 8 S y =0 boundary S 8 S S A = B = = = 3 = 4 = =, F = M = L =0, = 4 7 (Euclidean time circle) and need to use scaling symmetry of D SUGRA action g µ! s g µ, C µ! s 3 C µ ) I! s 9 I +! + s 0 ) I! s 0 I with s 0 = s = r 0 7 R g s`s r 0 r 0 his scaling symmetry will be important later...

16 x Boundary conditions S 5 collapses At infinity ( y =0): A, B,,, 3, 4,!, F! 0 Recall that C =(Md + Ld ) ^ d M! ˆµ x3 ( x ) 3 y 3, L! 3 ˆµy4 x 3 ( x ) 3 SO(6) SO(3) invariant tensor hamonic 0 S collapses Horizon y S S 5 Regularity at the axis of symmetry: horizon ( y =), pole ( x =) and equator ( x =0). Perform above scalings, then geometry has asymptotics of non-extremal D0-brane with temperature and mass deformation turned on. here is a single parameter ˆµ = 7 µ Important! Just learned that I = s9 s 0 6 G N Î µ = N µ Î S = s9 s 0 4G N Ŝ µ = N µ Ŝ

17 Smarr formulae (good to check numerics) Let v µ be a killing vector. From field equations it follows that (K v ) µ = r µ v + 3 F µ v C + 6 v[µ F ] C is a conserved antisymmetric tensor, i.e.? d K v =0 H y! 0 Integrate? d K v =0 over surface of constant time with y <y<y 0= Z? d K v = v = Z H?K v Z?K v y!0 For example take (time translations Smarr formula relates horizon area to boundary data v 7 Ŝ = Z?K v y!0

18 he solution Einstein-deurck equations Rµ r(µ ) = Deurck term that makes Einstein equations elliptic µ Fµ =g [Headrick, Kitchen, Wiseman 09] Fµ = µ gµ F µ = + - Descretize PDEs with N N Chebyshev grid Derivatives are estimated using polynomial approximation that involves all points in the grid spectral methods - exponential convergence

19 Horizon area and shape Horizon area Ratio of maximal radius of S to S 5 Ŝ R / R 5 ˆµ ˆµ After scaling symmetry to obtain physical metric: S = N µ Ŝ R i = a i 3 5 ˆR i µ Reproduces scalings predicted from strongly coupled low energy moduli estimate [Wiseman 3]

20 Black hole thermodynamics - critical temperature F (,µ)=f (,0)f(ˆµ) = c 4 5 f(ˆµ) both using st law or holographic renormalization Phase transition occurs when free energy f changes sign, since for < c geometry without horizon is favoured F O N 0 [Lin, Maldacena 05] - ˆµ ˆµ c.75 c µ = 7 ˆµ c 0.06 BH is thermodynamically stable for ˆµ <ˆµ c µ ) c S = 9 ˆµ log s(ˆµ) > 0

21 Phase diagram at large N µ W E A K C O U P L I N G Deconfined phase F = O(N ) Confined phase F = O(N 0 )? S R O N G C O U P L I N G Very similar to SYM on a 3-sphere (µ /R) [Aharony, Marsano, Minwalla, Papadodimas, van Raamsdonk 03] = g YM N µ 3

22 Boundary data he 0 functions Q i (x, y) admit expansion E 3 X i near the boundary (y =0) Q i (x, y) = X j y j Qj i (x) o preserve SO(6) SO(3) depends on ratio of radii sin = R 5 X a X a = R X i X i / X a E 6 Q j i (x) Boundary metric has symmetry, so are harmonic functions on. SO(9) S 8 hus we can classify the SO(6) SO(3) invariant perturbations according to SO(9) spin. his helps to establish bulk field / operator correspondence.

23 - form modes in the asymptotic expansion C =(Md + Ld ) ^ d v(x, y) = X lodd l f l (y)+ fl l (y) H l (x) + back reaction SO(6) SO(3) invariant harmonic -form f l (y) y +l f l (y) y l n(l) v v normalizable non-normalizable 0 - =ˆµ v v O ijk r X i X j X k X A...X Al, l odd

24 scalar modes in the asymptotic expansion s(x, y) = X l even l g l (y)+ l g l (y) S l (x) + back reaction SO(6) SO(3) invariant harmonic scalar n(l) , s s s 3 s normalizable non-normalizable zero modes -6 s s O r (X A...X Al ), l even

25 Vevs read from normalizable modes appear first at order y -form (l =) (l =) - - Scalar ˆµ ˆµ ijk hr (X i X j X k )i hr X i X i X a X a i - - Smarr formulae involve coefficients in asymptotic expansion up to order y Numerics pass this highly non-trival check with 0.05% accuracy -? d K v =0 - - H y! 0

26 Future work Confirm phase diagram with Monte-Carlo simulations of PWMM; confirm predictions for expectation values of operators dual to normalizable modes that are turned on Study dynamical stability of our BH Construct BH duals of other vacua (different horizon topology) (caveat: we really only determined upper limit on critical temperature) Deeper question: What makes the PWMM special? What are the minimal ingredients of a quantum mechanical system such that it gives rise to classical gravity in the limit of many degrees of freedom?

27 HANK YOU