Investigations on the SYK Model and its Dual Space-Time

Transcription

1 2nd Mandelstam Theoretical Physics Workshop Investigations on the SYK Model and its Dual Space-Time Kenta Suzuki A. Jevicki, KS, & J. Yoon; [hep-th] A. Jevicki, & KS; [hep-th] S. Das, A. Ghosh, A. Jevicki, & KS; [hep-th]

2 MOTIVATION & INTRODUCTION Simple and Solvable examples of AdS/CFT are needed for better understanding of holography itself and quantum gravity problems. Based on earlier Sachdev-Ye model [Sachdev & Ye 93], Sachdev-Ye-Kitaev (SYK) model was proposed as a simpler version of AdS/CFT, which is a quantum mechanical many-body system based on N( 1) fermionic sites. [Kitaev 15] From maximally chaotic behavior of the model, dual gravity theory was conjectured to be AdS 2 black hole theory. [Kitaev 15] As a first step of understanding of this SYK duality, in this talk I will explore some aspects of the SYK model, in the large N limit. Kenta Suzuki Brown University 2/28

3 Outline 1 SYK MODEL 2 QUADRATIC FLUCTUATION 3 SCHWARZIAN ACTION 4 RADON TRANSFORM and LEG FACTORS 5 FURTHER STUDIES Kenta Suzuki Brown University 3/28

4 1 SYK MODEL SYK model [Kitaev 15] consists of Majorana fermions on N sites (N 1): with H = 1 4! N i,j,k,l=1 J ijkl χ i χ j χ k χ l, {χ i, χ j} = δ ij. J ijkl are all-to-all & random; distributions are Gaussian: ( P (J ijkl ) exp N ) 3 Jijkl 2. 12J 2 Generalization to q-point interaction model is straightforward. Kenta Suzuki Brown University 4/28

5 Replica Trick Replica Trick is one way to deal with the random average: ( ) Z n J 1 ln Z J = lim, n 0 n where the disordered average is O J N i,j,k,l dj ijkl P (J ijkl ) O. At least in the leading order of 1/N, the model is replica symmetric [Sachdev & Ye 93], [Anninos, Anous & Denef 16]. Kenta Suzuki Brown University 5/28

6 O(N) Gauge Symmetry After evaluating the disorder average [Sachdev 15]: Z n J = Dχ i T exp [ 1 2 dt N i=1 a=1 + J 2 8N 3 n χ a i tχ a i dt 1dt 2 ( n N ) q ] χ a i (t 1)χ b i(t 2). a,b=1 i=1 Now O(N) symmetry χ i O ijχ j dynamically appeared. Therefore, SYK Model = O(N) Vector Model! Kenta Suzuki Brown University 6/28

7 Collective Theory The Large N theory is represented through a (replica diagonal) bi-local collective field: Ψ(t 1, t 2) 1 N N χ i(t 1)χ i(t 2). i=1 The corresponding path-integral is Z = DΨ(t 1, t 2) e Scol[Ψ], t 1,t 2 where S col is the (Euclidean time) collective action [Jevicki, K.S. & Yoon 16]: S col [Ψ] = N [ ] dt tψ(t, t ) + N 2 t =t 2 Tr log Ψ J 2 N dt 1dt 2 Ψ q (t 1, t 2). 2q Kenta Suzuki Brown University 7/28

8 Saddle-Point In strongly coupling limit J t 1 (critical IR fixed point), one can drop the Kinetic term: S c = N 2 Tr(log Ψ) J 2 N dt 1dt 2 [Ψ(t 1, t 2)] q. 2q Saddle-Point Equation: 0 = δs c δψ(t 1, t 2) = N 2 Critical Saddle-Point Solution [Kitaev 15]: [ ] Ψ 1 (t 2, t 1) J 2 Ψ q 1 (t 1, t 2). Ψ 0(t 1, t 2) = b J 2 q sgn(t 12), t 12 q 2 where b is a dimensionless number depending on q and t 12 t 1 t 2. Kenta Suzuki Brown University 8/28

9 Finite-Temperature Ψ 0 The critical action is invariant under S c = N 2 Tr(log Ψ) J 2 N 2q dt 1dt 2 [Ψ(t 1, t 2)] q, with an arbitrary function f(t). Ψ(t 1, t 2) f (t 1)f (t 2) 1 q Ψ(f(t 1), f(t 2)), Finite-temperature solution can be obtained by f(t) = β π Ψ 0,β (t 1, t 2) = b tan( πt β ) as [ ] 2 π q Jβ sin( πt 12 ) sgn(t 12). β Kenta Suzuki Brown University 9/28

10 2 QUADRATIC FLUCTUATION Expansion around the critical IR background Ψ 0 as 2 Ψ(t 1, t 2) = Ψ 0(t 1, t 2) + η(t1, t2), N the collective action S col leads to the systematic 1/N expansion: S col = N S (0) + S (2) + 1 N S (3) +. Bi-local Quadratic Action: S (2) = dt i η(t 1, t 2) K(t 1, t 2; t 3, t 4) η(t 3, t 4). i=1 Kenta Suzuki Brown University 10/28

11 SL(2, R) Casimir Quadratic kernel K is a function of the bi-local SL(2, R) Casimir C 1+2 = ( ˆD1 + ˆD ) 2 1( 2 ˆP1 + 2 ˆP )( 2 ˆK1 + ˆK ) 1( 2 ˆK1 + 2 ˆK )( 2 ˆP1 + ˆP ) 2 = (t 1 t 2) The eigenfunctions of the bi-local SL(2, R) Casimir C 1+2 are, because of the properties of conformal blocks, given by the three-point function of the form O h (t 0) O (t 1) O (t 2) = sgn(t 12) t 10 h t 20 h t 12 2 h. Kenta Suzuki Brown University 11/28

12 Diagonalizing the Kernel It is more useful to Fourier transform from t 0 to ω by Õh (ω) O (t 1) O (t 2) dt 0 e iωt 0 O h (t 0) O (t 1) O (t 2) sgn(t12) t e iω( t 1 +t 2 2 ) Z ν( ωt 12 ), 2 where h = ν + 1/2 and Z ν(x) = J ν(x) + ξ ν J ν(x), ξ ν = tan(πν/2) + 1 tan(πν/2) 1. The complete set of ν can be fixed from the representation theory of the conformal group [Kitaev 17]. The discrete modes ν = 2n + 3/2 (n = 0, 1, 2, ) and the continuous modes ν = ir (0 < r < ). Kenta Suzuki Brown University 12/28

13 Bi-local Propagator Therefore, the bi-local propagator (D = K 1 ) is [ D(t 1, t 2; t 1, t 2) J 1 dr 0 + n=0 dw r u ir,w(t 1, t 2) u ir,w(t 1, t 2) sinh(πr) g(ir) 1 ], dw ν u νw(t 1, t 2) u νw(t 1, t 2) g(ν) 1 ν= n where for q = 4 g(ν) 2ν ( ) πν 3 cot. 2 The zero mode ν = 3, (n = 0) gives a divergence since 2 g ( 3 2) = 1, (for any q) Kenta Suzuki Brown University 13/28

14 3 SCHWARZIAN ACTION The effective action associated with the zero mode was conjectured to be given by the Schwarzian derivative [Kitaev 15]. This was confirmed at the quadratic level together with symmetry discussions [Maldacena & Stanford 16]. We demonstrate how to derive Schwarzian action for all order, up to a numerical constant, (which needs to be fixed by a numerical evaluation). [Jevicki, K.S. & Yoon 16], [Jevicki & K.S. 16] Kenta Suzuki Brown University 14/28

15 Conformal Symmetry The critical action S c = N 2 Tr(log Ψ) J 2 N 2q dt 1dt 2 [Ψ(t 1, t 2)] q. exhibits the conformal reparametrization symmetry (t 1, t 2) (f(t 1), f(t 2)) [Kitaev 15] together with Ψ(t 1, t 2) Ψ f (t 1, t 2) f (t 1)f (t 2) 1 q Ψ(f(t 1), f(t 2)). f(t) is dynamical collective time coordinate [Jevicki, K.S. & Yoon 16], and in dual gravity theory, it corresponds to a dynamical boundary time (or boundary graviton) [Jensen 16] [Maldacena, Stanford & Yang 16] [Engelsöy, Mertens & Verlinde 16] Kenta Suzuki Brown University 15/28

16 IR Breaking The kinetic term breaks the conformal symmetry: S col [Ψ] = N ] dt 1 [ 1Ψ(t 1, t 2) + S c. 2 t 2 =t 1 Hence, the (leading order in 1/J) effective action is given by this form: S[f] = N ] dt 1 [ 1Ψ 0,f (t 1, t 2). 2 t 2 =t 1 Now the question is how to evaluate this action. Kenta Suzuki Brown University 16/28

17 q = 2 Model q = 2 case is simple, [Jevicki, K.S. & Yoon 16]: ( ) Ψ 0,f (t 1, t 2) = 1 f (t 1)f (t 2) πj f(t 1) f(t 2) = 1 πj ( 1 t 12 + t12 12 Sch(f(t2), t2) + ), where we expanded in the t 1 t 2 limit. Our regularization scheme is to eliminate the first term, which leads to S[f] = N dt Sch(f(t), t). 24πJ For q > 2 models, more complicated regularization is needed. Nevertheless, still one can derive the Schwarzian action for all order, up to a numerical coefficient. Kenta Suzuki Brown University 17/28

18 1/J Saddle-Point Correction We expand the exact saddle-point solution by 1/J as: Ψ(t 1, t 2) = Ψ 0(t 1, t 2) + Ψ 1(t 1, t 2) +. The saddle-point equation of Ψ 1 is given by dt 3dt 4 K(t 1, t 2; t 3, t 4)Ψ 1(t 3, t 4) = 1δ(t 12). Using an ansatz for Ψ 1 (s > 0): Ψ (s) 1 (t1, t2) = B1 sgn(t 12) (J t 12 ) 2 q +2s, is turned out that s = 1/2 is the correct answer. Kenta Suzuki Brown University 18/28

19 s-regularization Using the Ψ 1 equation, we rewrite S[f] = N 2 lim s 1/2 dt 1dt 2dt 3dt 4 Ψ 0,f (t 1, t 2)K(t 1, t 2; t 3, t 4)Ψ (s) 1 (t3, t4). Here, we define our regularization by exchanging the order of the integrations and s 1/2 limit. After evaluating the integrals and limit, we obtain [Jevicki & K.S. 16] S[f] = γb1n dt Sch(f(t), t), 2πJ with a q-dependent function γ. Kenta Suzuki Brown University 19/28

20 Another Way [Kitaev & Suh 17] The delta function source leads to δ (t 12) h=2 α h J 2 sgn(t 12)v(t 12) (J t 12 ) h+1 2 q. We define a regularization by S[f] = N 2 ] dt 1dt 2 [δ (t 12)Ψ 0(t 1, t 2). t f(t) Evaluating the leading (h = 2) contribution, we get S[f] = ˆα2N dt Sch(f(t), t). 2J Kenta Suzuki Brown University 20/28

21 4 RADON TRANSFORM and LEG FACTORS After change of coordinates by t t1 + t2 2 the bi-local SL(2, R) Casimir becomes, η t1 t2 2, C 1+2 = η 2 ( 2 η + 2 t ), which is the Laplacian of Lorentzian ds 2 space (or AdS 2) In fact, the eigenfunctions ds 2 = dη2 + dt 2 η 2. u ν,ω(η, t) = sgn(η) η 1 2 e iωt Z ν( ωη ), are particular choice of α-vacuum wave functions in Lorentzian ds 2 space. Kenta Suzuki Brown University 21/28

22 α-vacuum There is a unique vacuum for Lorentzian de-sitter space, which satisfies the Hadamard condition: Euclidean (Bunch-Davis) vacuum [Bunch & Davis 78] φ E ω(η) = η 1 2 H (2) ν ( ω η). The α-vacuum wave function is defined by Bogoliubov transformation from this Euclidean wave function [Mottola 84] [Allen 85] as [ ] φ α ω(η) N α φ E ω (η) + e α φ E ω (η) [ ] = N α η 1 2 H ν (2) ( ω η) + e α H ν (1) ( ω η). The choice with α = iπ(ν + 1/2) recovers the SYK eigenfunctions. Kenta Suzuki Brown University 22/28

23 i -Problem However, this Lorentzian ds 2 space cannot be the correct dual spacetime. If the dual gravity theory is on Lorentzian ds 2, it must has [ ( Z = DΦ m exp i S grav[φ] + S matter[φ]) ], while the SYK partition function is Z = DΨ e S col[ψ]. The mismatch of the factor i proceeds for all n-point functions [Das, Ghosh, Jevicki & K.S. 17]. Kenta Suzuki Brown University 23/28

24 Radon Transform There is a natural transform from Euclidean AdS to Lorentzian ds, given by Radon transform: [ ] t+η ( ) dz Rf (η, t) = 2η dτ z δ η 2 (τ t) 2 z 2 f (τ, z). t η 0 The inverse transformation is R 1 ψ (ds 2) ω,ν (η, t) = L 1 (ν) φ (EAdS 2) ω,ν (τ, z). where L(ν) (Leg Factor) = 2i π Γ( ν 2 ) Γ( ν 2 ). Kenta Suzuki Brown University 24/28

25 Bi-local Propagator and Leg factors The bi-local propagator with η = (t 1 t 2)/2 and t = (t 1 + t 2)/2: [ G(η, t; η, t 4 sin πν n ) dω g(ν n) 1 ψ ω,ν n (η, t) ψ ω,νn (η, t ) n=0 + 0 ν n=2n+ 3 2 dr ψ ω,ν (η, t) ψ ω,ν (η, t ) g(ν) 1 ν=ir Using the inverse Radon transform to bring the ds wave functions into the EAdS wave functions: [ G(τ, z; τ, z 4 sin πν n ) dω g(ν n) 1 L 1 (ν n) 2 φ ω,ν n (τ, z) φ ω,νn (τ, z ) n=0 + 0 dr L 1 (ν) 2 φ ω,ν(τ, z) φ ω,ν (τ, z ) g(ν) 1 ] ν=ir ].. Kenta Suzuki Brown University 25/28

26 Implications of the Leg Factors The extra Leg factors L 1 (ν) 2 in the integrand leads to additional poles in the propagator. This is analog to the old c = 1 matrix model/2d string duality, where the extra leg pole factors are characterizing the so-called discrete modes in the 2D string theory. We speculate that our leg factors in the SYK model also play a central role to identify the gauge degree of freedom in the dual gravity theory. [Das, Ghosh, Jevicki & K.S. 17] Kenta Suzuki Brown University 26/28

27 5 FURTHER STUDIES We are currently investigating what type of gravity theory can reproduce such leg-factor contributions in its propagator. We also need to explicitly evaluate higher-point function in this Radon transform and leg-factors scheme. It is also interesting to see whether this type of transformation and leg-factors are needed for other dualities. (e.g. N = 4 SYM/ IIB string or higher spin/vector model dualities) The next order correction for the Schwarzian action which recently discussed by [Kitaev & Suh 17] is also important to understand the complete dual gravity theory. Kenta Suzuki Brown University 27/28

28 Thank you!