Probing Universality in AdS/CFT

Transcription

1 1 / 24 Probing Universality in AdS/CFT Adam Ritz University of Victoria with P. Kovtun [ , ] and J. Ward [ ] Shifmania workshop Minneapolis May 2009

2 2 / 24 Happy Birthday Misha! Oops...some fine print: this talk contains no strong gauge dynamics...just the weakly-coupled dual this talk crosses no (phase) boundaries...just sticks to the critical endpoint!

3 3 / 24 Happy Birthday Misha! Oops...some fine print: this talk contains no strong gauge dynamics...just the weakly-coupled dual this talk crosses no (phase) boundaries...just sticks to the critical endpoint!

4 Are all fixed points alike? 4 / 24 Which d-dimensional fixed points behave like classical GR in AdS d+1? For example, when is AdS 4 /CFT 3 most applicable to phenomena in condensed matter physics [Herzog et al. 07, Hartnoll et al. 07,...] There are strongly-interacting near-critical regimes for which AdS/CFT may provide a novel (and unique) toolkit Reverse engineering may allow CM physics to unlock some generic aspects of gravity, and the AdS/CFT correspondence

5 5 / 24 Static Universality Static universality class criteria spatial dimension symmetry of the order paramater Both critical points are described by the same Ising CFT in d=3.

6 6 / 24 Dynamic Universality Dynamic universality class criteria [Hohenberg & Halperin 77] spatial dimension symmetry of the order parameter conserved currents,... The dynamic universality classes of these critical points are different!

7 Quantum Critical Universality 7 / 24 When are these notions of universality the same? CFTs with classical AdS gravity duals comprise a special subset of this class with correlated vacuum, thermodynamic and (hydro)dynamic properties NB: this implies taking a single dominant scale (temperature)

8 Quantum Critical Universality 8 / 24 When are these notions of universality the same? CFTs with classical AdS gravity duals comprise a special subset of this class with correlated vacuum, thermodynamic and (hydro)dynamic properties NB: this implies taking a single dominant scale (temperature)

9 Outline 9 / 24 1 Thermodynamics from central charges 2 Hydrodynamics from central charges 3 Conclusions

10 Outline 10 / 24 1 Thermodynamics from central charges 2 Hydrodynamics from central charges 3 Conclusions

11 Vacuum CFTs and central charges 11 / 24 For a CFT vacuum state, symmetry determines the correlators of conserved currents: T µν (x)t αβ (0) = c x 2d Π µναβ c "measures" the total degrees of freedom J µ (x)j ν (0) = k x 2(d 1) Π µν k "measures" the charged degrees of freedom

12 Thermal CFTs 12 / 24 For a CFT, with temperature the only scale: p(t) = T V ln Tr [e β(h µq)] µ 0 = c T d k µ 2 T d 2 + c measures entropy density s = c dt d 1 k measures charge susceptibility χ = k T d 2

13 13 / 24 Claim In d = 2, conformal symmetry relates the vacuum and thermal states [Bloete et al. 86, Affleck et al. 86]: d = 2 c = π 6 c k = 1 2π k so thermodynamics is uniquely fixed by the central charges. In d > 2, conformal symmetry does not imply this constraint BUT: it is true in general for CFTs with gravity duals d = 3 c = π3 162 c k = π 24 k d = 4 c = π2 80 c k = 1 12 k

14 AdS gravity dual 14 / 24 The dual bulk geometry describing the CFT state, and the fluctuations which determine correlators of the conserved currents, follow at leading order in EFT from the Einstein-Maxwell action: 1 S = d d+1 x g 16πG d+1 [ R + d(d 1) L 2 where G d+1 and g 2 d+1 encode details of the CFT. ] 1 4g 2 d d+1 x g F 2 + d+1 Vacuum state AdS d+1 : ds 2 = z 2 (dz 2 + dx 2 ) Large volume thermal state RNAdS d+1 BH: T = T BH, µ = A t z 0

15 Central charges and thermodynamics Perturbing the vacuum state, we obtain the correlators: T xy (x 1 )T xy (x 2 ) 2 S on shell h xy (x 1 )h xy (x 2 ) J x (x 1 )J x (x 2 ) 2 S on shell A x (x 1 )A x (x 2 ) = c = const(d) Ld 1 G d+1 = k = const(d) Ld 3 g 2 d+1 From the thermal state dual to the black hole (with µ 0): s = A D 2 4G d+1 V = c dt d 1 = c = const(d) Ld 1 G d+1 χ µ=0 = ρ µ = k T d 2 = k = const(d) Ld 3 g 2 d+1 15 / 24

16 Thermodynamics from central charges 16 / 24 Central charges (c, k) determine (s, χ) in CFTs with AdS gravity duals c c k k = = 1 4π d/2 1 2π d/2 ( ) 4π d Γ(d/2) 3 d Γ(d) ( ) 4π d 2 Γ(d/2) 3 d Γ(d) (d 1) d(d+1) In the most relevant cases of d = 3, 4: d = 3 c = π3 162 c k = π 24 k d = 4 c = π2 80 c k = 1 12 k

17 Outline 17 / 24 1 Thermodynamics from central charges 2 Hydrodynamics from central charges 3 Conclusions

18 Hydrodynamics 18 / 24 Assuming hydrodynamics controls the IR dynamics near the critical point: µ T µν = µ J µ = 0, where (for a conformal fluid) T µν = ɛ 3 (4uµ u ν + η µν ) 2 η µ u ν + J µ = ρu µ D Π µ ν ν ρ +, D = σ/χ is the diffusion const η is the shear viscosity [NB: Strictly we require J µ ej µ so σ and χ are implicitly proportional to e 2 ]

19 Transport from thermodynamics 19 / 24 Using Kubo relations, and the AdS/CFT prescription [Policastro et al 02]: η = lim ω 0 Im 1 ω T xyt xy (ω, 0) R = 1 16πG d+1 σ = lim ω 0 Im 1 ω J xj x (ω, 0) R = 1 g 2 d+1 ( ) 4πL d 1 T d 1 d ( ) 4πL d 3 T d 3. It follows that CFTs with gravity duals have transport coefficients determined by thermodynamics! d η s = 1 4π, σ χ = D = 1 d 4πT d 2.

20 Transport from central charges 20 / 24 Putting the pieces together [Kovtun & AR 08] : [ σ = χ ( ) ] d 4πT d 2 = 1 d 4π d 2 Γ(d/2) 3 8π d/2+1 kt d 3. d 2 d Γ(d) [ η = s ( ) ] 4π = 1 d 1 4π d Γ(d/2) 3 16π d/2+1 ct d 1. d + 1 d Γ(d) General relations, independent of field content, symmetries, etc! The two limits ω/t 0 and ω/t are generally determined by different physics [Sachdev 99]

21 Central charges and symmetry breaking 21 / 24 A corrollary (cf. weak gavity conjecture [Arkani-Hamed et al 06]) η σ = [ 8π 2 (d 1)(d 2) d 3 (d+1) ] c k T2 g2 d+1 G d+1 The ratio c/k also determines when the U(1) symmetry is unstable to condensation forming a superfluid (or superconducting) phase. In d = 3, the instability occurs when [Denef & Hartnoll 09]: q 2 (3 + 2 ( 3)) k c with q ( ) the charge (dimension) of the operator breaking U(1).

22 Comparisons 22 / 24 d=4: The AdS value is possible if 2n s + n f = 8n v, and in general: 3 8 c/c free c/c AdS 9 4. d=3: In this case (c/c ) free 1/ζ(3) so the AdS value is not possible at weak coupling. For the O(N) model at large N [Sachdev 93, Chubukov et al 94] c /c O(N) c /c AdS 1.07, k /k O(N) k /k AdS 1.31, BUT σ/χ O(N) O(N). [Cf. The proposed higher-spin AdS dual [Klebanov & Polyakov 02]]

23 Outline 23 / 24 1 Thermodynamics from central charges 2 Hydrodynamics from central charges 3 Conclusions

24 24 / 24 Conclusions CFTs with AdS (gravity) duals are a precise subset with linked vacuum, thermodynamic, and hydrodynamic properties Are there physical quantum critical regimes with these features? Bulk higher dimension operators generically correct these universal relations: R n>1 terms correct η/s [Buchel et al 04; Brigante et al 07, Kats and Petrov 07] RF 2 corrects σ/χ [AR & Ward 08] Can symmetries protect these relations beyond the classical gravity (large N, λ) limit? Are some regimes in the swampland? Are there other universal relations of this type, associated with perturbing by other control" paramaters, or for NR quantum critical pts? finite chemical potential or magnetic field (and zero T) - extremal BHs the confinement or χsb scales?