Universal features of Lifshitz Green's functions from holography

Transcription

1 Universal features of Lifshitz Green's functions from holography Gino Knodel University of Michigan October 19, 2015 C. Keeler, G.K., J.T. Liu, and K. Sun; arxiv: C. Keeler, G.K., and J.T. Liu; arxiv:

2 Motivation Challenges of AdS/CMT: Holographic dictionary less clear than in relativistic case. Holographic models often describe an idealized theory. Universal predictions of nonrelativistic gauge/gravity duality (e.g. analog of η/s)? Main focus: Holographic Green's functions. 1. What do we know? 2. Which features are robust w.r.t. deformations of the theory? 3. Comparison with eld theory

3 Constraining holographic observables via symmetries Type IIB on AdS 5 S 5 N = 4 SYM. Consider retarded Green's function of scalar operator O : G R (ω, k) = i d d+2 x e ik x θ(t) [O (x), O (0)] Superconformal symmetry = G R (q 2 ) = A( q 2 ) 2, q 2 = ω 2 k 2, A = const. What about non-relativistic symmetries? Lifshitz scaling

4 Lifshitz Green's functions Lifshitz scaling symmetry: x Λ 1 z x, t Λt z > 1: dynamical critical exponent Assuming rotational invariance: G R (ω, k) = k 2νz G (ˆω), ˆω = ω k z Properties of Green's functions in Lifshitz eld theories without additional symmetries?

5 Lifshitz Green's functions - general properties G R (ω, k) = k 2νz G (ˆω) Generic features: 1. G is analytic in the upper half plane (causality) 2. Scaling behavior: 2.1 G(ˆω 0) const. (so that G R k 2νz as ω 0) 2.2 G(ˆω ) ˆω 2ν (so that G R ω 2ν as k 0) No (obvious) further constraints. Lifshitz eld theory: Family of theories with dierent dynamics, and thus G(ˆω).

6 Holographic Lifshitz Green's function However: On the gravity side, there is a canonical result. Lifshitz spacetime: ds 2 d+2 = dt2 + dρ 2 ρ 2 + d x 2 ρ 2/z isometry: x Λ 1 z x, t Λt, ρ Λρ. For z = 2: G(ˆω) = K 0 ˆω 2ν Γ ( ν + ) i 4 ˆω Γ ( 1 2 ν + ) i 4 ˆω [Kachru, Liu, Mulligan; ] Where is the freedom on the gravity side?

7 Review: The two-derivative holographic Green's function Consider a probe scalar in Lifshitz spacetime: ( m 2 )φ = 0 Ansatz: φ = e i( k x ωt) ρ d/2z ψ(ρ) = Schrödinger-like equation: ψ (ρ) + U 0ψ(ρ) = 0 U 0 = ν2 1 4 ρ 2 + k ρ 2 2/z ω2, ν 2 = m 2 + ( ) 2 d + z 2z

8 Using scale-invariant coordinates ˆρ = ρ k z : ψ (ˆρ) + Û0(ˆρ)ψ(ˆρ) = 0, Û 0 (ˆρ) = ν ˆρ 2 ˆω2 ˆρ 2 2/z normalizable mode: non-normalizable mode: Near the horizon: Near the boundary: ψ(ˆρ ) ae i ˆω ˆρ i ˆω ˆρ + be ψ(ˆρ 0) Aˆρ 1 2 ν + B ˆρ 1 2 +ν

9 Holographic Green's functions ψ(ˆρ ) ae i ˆω ˆρ i ˆω ˆρ + be ψ(ˆρ 0) Aˆρ 1 2 ν + B ˆρ 1 2 +ν Choose infalling boundary conditions for retarded Green's function: G(ˆω) = ˆω 2ν B A Find relation between {A, B} and {a, b} by solving a quantum mechanical scattering problem. b=0

10 Introducing higher derivatives Where does the freedom of nding dierent G arise in the gravity theory? 1. Choice of background metric? 2. Dynamics of probe scalar, e.g. higher derivative corrections: ( ) i = ω t i, j = k j : modied Schrödinger potential Û = Û0(ˆρ) + 1ˆρ 2 λ i,j ˆω i ˆρ i+j/z, ˆρ = ρ k z, ˆω = ω i+j>2 k z = Û0(ˆρ) + 1ˆρ 2 λ i,j (ωρ) i ( k ρ 1/z ) j i+j>2 ( ) where λ i,j l i+j 2, L l L (curvature scale). ) n: ( ρ e.g. : ψ + Ûψ = λψ (4) ψ + Uψ λ(ûψ) = O(λ 2 ) ψ + Ũ ψ 0 Higher derivative corrections can be captured by modifying the eective potential.

11 Keeping higher derivatives under control Consider only a single higher derivative term (i temporal, j spatial derivatives): Û(ˆρ) = Û0(ˆρ) + λ i,j ˆω i ˆρ i+j/z 2 Want to compute Green's function perturbatively, i.e. G = G 0 + δg. Problem: higher derivatives generically blow up near horizon (ˆρ ). But: Green's function is given by B/A (near-boundary behavior of ψ) and thus only contains information about the eective potential in the tunneling regime.

12 WKB approximation Recall: G(ˆω) = ˆω 2ν B A ψ(ˆρ 0) Aˆρ 2 1 ν + B ˆρ 1 2 +ν b=0 Infalling boundary conditions can be imposed order by order in λ i,j. WKB approximation : ψ WKB (ˆρ) = { νû 1 4 S(ˆρ, ˆρ 0) = ( ) S( ˆρ, ˆ C 1(a)e ρ0) S( ˆρ, ˆ + C 2(a)e ρ 0) 1 a( Û) 4 e iφ( ˆρ, ρˆ 0 ) ˆρ0 ˆρ d ˆρ Û, Φ(ˆρ, ˆρ0) = d ˆρ Û ˆρ 0 ˆρ, ˆρ < ˆρ 0, ˆρ > ˆρ 0 ˆρ 0 : classical turning point.

13 ψ WKB (ˆρ) = { νû 1 4 ( ) S( ˆρ, ˆ C 1(a)e ρ0) S( ˆρ, ˆ + C 2(a)e ρ 0) 1 a( Û) 4 e iφ( ˆρ, ρˆ 0 ), ˆρ < ˆρ 0, ˆρ > ˆρ 0 Coecients of non-normalizable/normalizable mode: A = lim ɛ 0 C 1ɛ ν e S(ɛ,ρ 0), B = lim ɛ 0 C 2ɛ ν e S(ɛ,ρ 0) S(ɛ, ˆρ 0) = = ˆρ0 ɛ ˆρ0 ɛ d ˆρ Û d ˆρ ν 2 ˆρ ˆρ 2 2/z ˆω2 + λ i,j^ω i+j/z 2 i^ρ Coecients A, B contain information about Û up to the classical turning point ˆρ 0. Can calculate Green's function with higher derivatives, as long as h.d. are subdominant compared to two-derivative terms.

14 ˆρ0 ν S = d ˆρ 2 ɛ ˆρ ˆρ 2 2/z ˆω2 + λ i,j^ω i+j/z 2 i^ρ! = S 0 + δs + O(λ 2 i,j) Will have to impose upper bound on λ i,j and lower cuto for ˆω.

15 Calculating the spectral function Calculate ImG (spectral function) via ( ) ImG(ˆω) = ˆω 2ν B Im A (can nd ReG via Kramers-Kronig relation) = K 0 ˆω 2ν lim ɛ 0 ɛ 2ν e 2S(ɛ, ˆρ 0) Can expand by expanding integral ImG = ImG 0 + δimg + O(λ 2 ) S(ˆρ, ˆρ 0) = ˆρ0 ˆρ d ˆρ Û 0 + λ i,j ˆω i ˆρ i+j/z 2 linearly in λ i,j.

16 Results Recall: Two interesting limits: Û(ˆρ) = ν2 1 4 ˆρ ˆρ 2 2/z ˆω2 + λ i,j ˆω i ˆρ i+j/z 2

17 High frequency limit - two derivative result 1. ˆω ν 1 z : Classical turning point lies at ˆρ 0 νˆω. To leading order: S 0 = ˆρ0 ɛ d ˆρ ν 2 ˆρ ˆρ 2 2/z ˆω2 ˆρ0 ɛ d ˆρ ν 2 ˆρ 2 ˆω2 νlog(ɛ) + const. Gives rise to power law scaling near the boundary: ( ) ψ(ɛ 1) ɛ 1 2 C 1e S 0(ɛ, ρˆ 0 ) + C 2e S 0(ɛ, ρˆ 0 ) C 1ɛ 1 2 ν + C 2ɛ 1 2 +ν 2-derivative Green's function: ImG(ˆω) = ˆω 2ν Im ( ) ( ) B C 2 = Im ˆω 2ν A C 1

18 High frequency limit with higher derivatives Computing S to linear order in λ yields: ImG(ˆω) C ˆω 2ν, [ ] C = (2ν) 2ν exp 2ν(1 δ j,0 c i λ i,0 ν i ) Expected large frequency behavior G R = k 2νz G(ˆω) ω 2ν (featureless; similar to AdS) Higher derivatives normalize prefactor. Spatial derivatives derivatives (j 0) are subleading in the ˆω limit. H.d. corrections become of order one if λ i,0 ν i 2 1 δu hd (ˆρ 0 ) = λ i,0 ˆω i ˆρ i 2 0 ˆω 2 Size of corrections to spectral function is controlled by relative size of h.d. corrections to the potential up to the classical turning point. Translates into condition on mass: λ i,0 ν i 2 ( l L ν) i 2 (ml) i 2 1

19 Low frequency limit Eective potential Û(ˆρ) = ν2 1 4 ˆρ ˆρ 2 2/z ˆω2 + λ i,j ˆω i ˆρ i+j/z 2

20 Low frequency limit - two derivative result 2. ˆω ν 1 z : Classical turning point lies at ˆρ 0 ˆω To leading order: z 1 z. S 0 = ˆρ0 ɛ d ˆρ ν 2 ˆρ ˆρ 2 2/z ˆω2 νlog(ɛ) + νlog(ɛ) + ˆω 1 z 1 + const. z ˆω z 1 1 d ˆρ + const. ˆρ 1 1/z Wavefunction near the boundary: ( ) ψ(ɛ 1) ɛ 1 2 C 1e S 0(ɛ, ρˆ 0 ) + C 2e S 0(ɛ, ρˆ 0 ) ( ) ( ) C 1ɛ 1 2 ν E0 1 exp 2 ˆω z 1 + C 2ɛ 1 2 +ν exp E0 1 2 ˆω z 1

21 e.g. normalizable mode: ψ(ɛ 1) C 1ɛ 1 2 ν exp ( ) ( ) E0 1 2 ˆω z 1 + C 2ɛ 1 2 +ν exp E0 1 2 ˆω z 1 exponential growth/decay due to tunneling under ˆρ 2/z 2 barrier. 2-derivative Green's function: ImG(ˆω) = D ˆω 2ν exp[ E 0 ˆω 1 z 1 ].

22 Low frequency limit with higher derivatives Computing S to linear order in λ yields: E 0 = ImG(ˆω) = D ˆω 2ν exp[ ˆω 1 z 1 (E 0 + δe(ˆω))] [ ] D = (2ν) 2zν z 2ν(1 z) exp 2zν(1 + δ i,0 d j λ 0,j ν j ), d j = πγ ( ( zγ 1 2(z 1) z 2(z 1) ) ), δe(ˆω) = e i,j λ i,j ˆω z 1 1 (i+j 2) +..., e i,j = x=0 dx x j x 1 0 dx x i 1+ j 1 z 1 x 2 2 z Prefactor is renormalized by terms λ 0,j ν j 2. Exponent is modied by frequency-dependent terms. Recall: Classical turning point lies at Û(ˆρ) = ν2 1 4 ˆρ ˆρ 2 2/z ˆω2 + λ i,j ˆω i ˆρ i+j/z 2 ˆρ 0 ˆω H.d. corrections become of order one if z 1 z λ i,j ˆω 1 z 1 (i+j 2) 1 λ i,j ˆω i ˆρ i+j/z 2 0 ˆω 2

23

24 Exponential suppression At low frequencies: ImG(ˆω ν 1 z ) D ˆω 2ν exp[ ˆω 1 z 1 (E 0 + δe(ˆω))] δe(ˆω) = e i,j λ i,j ˆω 1 z 1 (i+j 2) +... Strict low-frequency limit ˆω 0 is non-universal (h.d. dominate), but corrections are under control if λ i,j ˆω z 1 1 (i+j 2) 1. Using λ i,j ( l L ) i+j 2: ν 1 z ˆω ( ) z 1 l L Recall: ν L m 1 exponential suppression is a robust feature in a wide l l window of frequencies. Corrections can be computed order-by-order for dierent models.

25 Interpretation of exponential behavior Exponential behavior of ImG is a robust result at low frequencies. Can we nd the same behavior of ImG on the eld theory side?

26 Example: Quadratic band crossing model Scale-invariant theory with z = 2: { S = d 2 xdt [ Ψ iγ γ 1( 2 x 2 ] } y ) + 2γ 2 x y Ψ gψ 1 ψ 2 ψ2ψ1 where Ψ = (ψ 1, ψ 2) T. Dispersion relation: ɛ ±( k) = ± k 2 Can consider bosonic particle-hole operators b i = Ψγ i Ψ, i = 0, 1, 2, 3

27 Calculating self-energy corrections Want to calculate spectral weight ImG (b). Optical theorem: ImG (b) (ω, k) 0 Γ bi...(ω, k) 0 1-loop: 5-loop: etc. At n-loop order, O(g 2n ), ImG (b) is related to decay rate for b{ω, k} n particles{ω pi, k pi } + n holes{ω hi, k hi }.

28 Energy and momentum conservation: n+1 n+1 k = kpi khi, i=1 i=1 i=1 n+1 n+1 n+1 ω = ω pi ω hi = ( k pi 2 + k hi 2 k 2 ) 2(n + 1), i=1 i=1 At each xed order, spectral weight is zero below a certain threshold. for xed ω k 2, spectral weight can only arise via process of O(g 2n ), where n k2 2ω ImG (b) (ω k) g 2n g k2 /ω For g 1 this implies ImG (b) (ω k 2 ) exp( const. ω ), ˆω = ˆω k. 2

29 Result can be generalized to z 2: ImG (b) (ω k 2 ) g ˆω 1 z 1 exp( const. ˆω 1 z 1 ), ˆω = ω k z. Agrees with the holographic prediction ImG(ˆω) exp[ E 0 ˆω 1 z 1 ]. Exp. behavior is generic for Lifshitz theories with bosonic decay channels. Aside: In Dirac theory (z = 1), dispersion relation is ω = k, so energy conservation implies n+1 n+1 k = kpi khi, i=1 i=1 n+1 n+1 n+1 ω = ω pi ω hi = ( k pi + k hi ) k, i=1 i=1 i=1 = ImG (b) (ω < k ) = 0

30 Summary Field theory: Family of theories with Lifshitz symmetry. Gravity: Family of theories with Lifshitz symmetry; dynamical information encoded in higher derivative couplings λ i,j. Higher derivative corrections can be computed systematically: ImG(ˆω ν 1 z ) C ˆω 2ν ; C has perturbative expansion in λν i 2 (ml) i 2 1. ImG(ˆω ν 1 z ) D ˆω 2ν exp[ E^ω 1 z 1 ] D has perturbative expansion in λν j 2. E has perturbative expansion in λ i,j ˆω 1 z 1 (i+j 2).. Naive limit ˆω 0 is not under perturbative control, but results are robust after imposing the cuto ˆω ( ) l z 1. L Can conrm exponential behavior of spectral function in a broad class of eld theories.

31 Open questions 1. Constraints on sign of λ i,j from bulk causality/unitarity? 2. Compute corrections to ImG(ˆω) in string embeddings? 3. Measure spectral function experimentally?

32 Thank you!