Pole skipping away from maximal chaos
CCERN-TH-2020-171
Prepared for submission to JHEP
Pole skipping away from maximal chaos
Changha Choi, a,b,c
M´ark Mezei, a G´abor S´arosi d a Simons Center for Geometry and Physics, SUNY, Stony Brook, NY 11794, USA b C.N. Yang Institute for Theoretical Physics, SUNY, Stony Brook, NY 11794, USA c Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA d CERN, Theoretical Physics Department, 1211 Geneva 23, Switzerland
E-mail: [email protected] , [email protected] , [email protected] Abstract:
Pole skipping is a recently discovered subtle effect in the thermal energy densityretarded two point function at a special point in the complex ( ω, p ) planes. We propose thatpole skipping is determined by the stress tensor contribution to many-body chaos, and thespecial point is at ( ω, p ) p.s. = iλ ( T ) (1 , /u ( T ) B ), where λ ( T ) = 2 π/β and u ( T ) B are the stresstensor contributions to the Lyapunov exponent and the butterfly velocity respectively. Whilethis proposal is consistent with previous studies conducted for maximally chaotic theories,where the stress tensor dominates chaos, it clarifies that one cannot use pole skipping toextract the Lyapunov exponent of a theory, which obeys λ ≤ λ ( T ) . On the other hand, in alarge class of strongly coupled but non-maximally chaotic theories u ( T ) B is the true butterflyvelocity and we conjecture that u B ≤ u ( T ) B is a universal bound. While it remains a challengeto explain pole skipping in a general framework, we provide a stringent test of our proposalin the large- q limit of the SYK chain, where we determine λ, u B , and the energy density twopoint function in closed form for all values of the coupling, interpolating between the free andmaximally chaotic limits. Since such an explicit expression for a thermal correlator is one ofa kind, we take the opportunity to analyze many of its properties: the coupling dependenceof the diffusion constant, the dispersion relations of poles, and the convergence properties ofall order hydrodynamics. a r X i v : . [ h e p - t h ] O c t ontents q SYK chain 71.4 Outline 8 q SYK chain 9
A Four point function for p = 0 – 1 – Introduction and summary of results
There is a rich variety of phenomena and signatures associated to quantum chaos. Sharpeningour understanding of one of these, the growth of operator complexity and its probe, the out-of-time order correlation function (OTOC) has lead to great advances in the understandingof many-body chaos and its relation to gravity through AdS/CFT in recent years.Recently, a novel signature of chaos was proposed, the pole skipping phenomenon [1, 2].While this effect was very convincingly demonstrated in [1–6] for theories saturating the chaosbound of [7], it was not understood what the fate of this effect was away from maximal chaos.In this paper, we propose a generalization of pole skipping away from maximal chaos. Weprovide a stringent test of the proposal by exact computations in a solvable chaotic theory,the large- q SYK chain [8, 9].We spend the rest of the introduction reviewing the behavior of OTOCs in large- N theories, introducing the pole skipping phenomenon, formulating it away from maximal chaos,and providing evidence for the proposal. Since our investigation led us to a precious simpleclosed form thermal correlator in an interacting chaotic theory, we close the introduction withthis formula and a brief discussion of its properties. In maximally chaotic theories in the regime β (cid:28) t, | x | (cid:28) t scr = O (log N ) the OTOC behavesas OTOC( t, x ) = 1 − N exp (cid:20) πβ ( t − | x | /u B ) (cid:21) , maximal chaos (1.1)where the λ L = 2 π/β is the maximal Lyapunov exponent and u B is the butterfly velocitythat determines the boundary of the butterfly cone, the region in which the OTOC grows.Recently the pole skipping phenomenon in the thermal energy density retarded two pointfunction G Rεε ( ω, p ) was discovered [1, 2] and tested in maximally chaotic theories [1–6]. Theenergy density retarded two point function has a family of hydrodynamic poles on the complex ω plane as we vary p : ω pole ( p ) = (cid:40) ± c s p + . . . (sound) − iDp + . . . (energy diffusion) (1.2)where the dispersion relation depends on whether the system conserves momentum or not.The discovery of [1] was that this family of poles goes through a special point determined by λ L and u B : ( ω, p ) p.s. = iλ L (cid:18) , u B (cid:19) , maximal chaos (1.3)– 2 –nd at this precise location its residue vanishes. The proposed explanation of this poleskipping is that the growth of the OTOC comes from the same hydrodynamic mode that isresponsible for energy transport [2]. In the OTOC, this pole gives rise to the exponentialgrowth, while in the energy-energy correlator, the growth must be absent, and therefore thepole must be skipped. This prediction of pole skipping in the thermal energy density twopoint function for maximally chaotic system was verified in a wide class of holographic systemsdual to Einstein gravity with a general matter content [3].Hence it is natural to expect pole skipping to take place at the point (1.3) also away frommaximal chaos. This would make pole skipping extremely exciting, as it would suggest thatone can extract some data about chaos from a two point function, which is a much simplerobservable than the OTO four point function. However, the naive conjecture (1.3) cannotbe true. The most elementary way to see this, is to realize that while not all 2d CFTs aremaximally chaotic, their retarded energy density two point function G Rεε is universal [4], G Rεε ( ω, p ) = − πc ω (cid:0) ω + (2 π/β ) (cid:1) (cid:20) ω − p + i(cid:15) + 1 ω + p + i(cid:15) (cid:21) , (1.4)where (cid:15) > ω, p ) p.s. = i (2 π/β )(1 , λ L in general. Note that u B = 1 in any 2d CFT [11]. Weexplain below how to modify (1.3) so that it holds away from maximal chaos. Away from maximal chaos (1.1) is replaced byOTOC( t, x ) = 1 − N exp (cid:20) λ (cid:18) | x | t (cid:19) t (cid:21) , (1.5)where we introduced the velocity dependent Lyapunov exponent (VDLE) λ ( u ) [11–13]. Theordinary Lyapunov exponent λ L is obtained by setting u = 0: λ L = λ (0), and u B is definedby the edge of the growing region, λ ( u B ) = 0. Note that while in the maximally chaotic casechaos is characterized by two numbers, λ L and u B , in the non-maximal case there is a wholefunction λ ( u ) worth of freedom. In terms of this function, the chaos bound translates to thepointwise bound λ ( u ) ≤ πβ − (1 − | u | /u B ) [11].Importantly, in all examples we know λ ( u ) is a result of a competition between two terms,one coming from the “leading Regge trajectory” and another describing the contribution ofthe stress tensor. For small | u | the Regge contribution dominates. For larger | u | ≤ u B thereare two possible scenarios depending on the details of λ ( u ) [11]: In the first case the Reggecontribution dominates for all 0 ≤ | u | ≤ u B . In the second case, above some critical velocity However, pole skipping for a 2d CFT on a compact spatial manifold is not universal, instead it dependson the stress tensor one point function [10]. The stress tensor itself is part of the leading Regge trajectory, but it plays a distinguished role. – 3 – ∗ ≤ | u | ≤ u B the stress tensor dominates, and we have λ ( u ) = λ ( T ) L (cid:32) − | u | u ( T ) B (cid:33) (cid:16) u ∗ ≤ | u | ≤ u ( T ) B (cid:17) . (1.6)The stress tensor contribution to chaos is determined by two numbers λ ( T ) L = 2 π/β and u ( T ) B .In the second scenario described in (1.6), we have u B = u ( T ) B , while in the first case we have u B < u ( T ) B . In maximally chaotic theories one has u ∗ = 0, i.e. the stress tensor dominatesfor all u , and we recover (1.1).We are now ready to formulate our proposal. We conjecture that the pole skipping pointin the retarded energy density two point function is at ( ω, p ) p.s. = iλ ( T ) L (cid:32) , u ( T ) B (cid:33) , (1.7)i.e. it encodes the stress tensor contribution to chaos. While the stress tensor determineschaotic dynamics at maximal chaos, its contribution can be decreased in non-maximallychaotic theories or completely cancelled in integrable systems; an example of the latter isprovided in [14]. The proposal is consistent with the literature on pole skipping in maximallychaotic theories, since in that case λ L = λ ( T ) L and u B = u ( T ) B . It is also consistent with the 2dCFT result discussed around (1.4).Let us discuss the existing evidence for (1.7) in the literature. The main evidence comesfrom conformal field theories on Rindler space. Rindler space is a patch of flat spacetime butit is conformal to S × H d − and therefore it is an example of thermal physics that can bestudied using vacuum correlators in a conformal field theory. For example, the thermal energydensity two point function can be obtained from the vacuum one via a Weyl transformationand therefore is universal to all CFTs. Pole skipping in this context was recently studiedin [6] and it was shown that it happens at the values corresponding to maximal chaos. Note that most CFTs on Rindler space do not display maximal chaos, instead the Lyapunovexponent is determined by the resummation of operators on the leading conformal Reggetrajectory and is related to the analytic continuation of the spin of this trajectory to non-physical operator dimensions [7, 11]. However, for a large class of theories the stress tensor This inequality follows from the concavity of the contribution of the leading Regge trajectory to theVDLE. In all the examples we know, this is true, and we suspect it holds generally. This formula is valid in the presence of (possibly discrete) translational symmetry. Since the spatial manifold H d − is curved, (1.7) must be interpreted so that it applies to the appropriatenotion of Fourier transform. In flat space the Fourier mode at the poles skipping point takes the formexp [ ip p.s. x ] = exp (cid:20) − λ ( T ) L u ( T ) B x (cid:21) . In analogy with this result, the right interpretation in hyperbolic space is thatfor large geodesic separations, the Fourier mode at the pole skipping point must behave as exp (cid:20) − λ ( T ) L u ( T ) B ρ (cid:21) , where ρ is the spatial geodesic distance. – 4 –ontribution dominates near the butterfly front and u ( T ) B = ( d − − gives the true butterflyspeed; for example this happens in planar N = 4 SYM theory whenever the ’t Hooft couplingis greater than 37.7384 [11], while the theory is only maximally chaotic at infinite coupling.Another piece of non-trivial evidence comes from holographic theories with higher deriva-tive corrections performed using shockwaves [7, 15–17]. Such corrections do not affect theLyapunov exponent but they change the butterfly speed. It was shown in [5] that both forGauss-Bonnet coupling and the leading α (cid:48) R correction the change in the butterfly speedand the energy-energy correlator are such that pole skipping remains valid. This is consis-tent with (1.7) since the Lyapunov exponent stays maximal. In fact, when we consider finitecoupling corrections to the thermal OTOC in N = 4 SYM we need to include worldsheeteffects that correct the Lyapunov exponent [19] in addition to the leading higher derivativeterm α (cid:48) R that corrects the butterfly speed. The fact that in this case pole skipping stillhappens at maximal Lyapunov exponent is additional evidence for (1.7) (and contradicts thenaive propsal (1.3)).The main technical result of this paper is the confirmation of (1.7) in the large- q limitof the SYK chain introduced in [8], where we obtain both λ ( u ) and G Rεε in closed form, andcan prove (1.7) analytically. The model has two dimensionless coupling constants 0 < v < < γ ≤
1. The latter controls the strength of inter-site coupling, while the formercontrols the interaction strength; as a function of v the model interpolates between a free anda maximally chaotic theory. There exists a critical line in coupling space v ∗ ( γ ), above which(in the regime v ∗ ( γ ) ≤ v <
1) chaos is non-maximal, but the butterfly speed is maximal u B = u ( T ) B , while for 0 < v < v ∗ ( γ ) we get that both λ L < λ ( T ) L and u B < u ( T ) B , see Fig. 1.Contrary to the 2d CFT case, here both u B and u ( T ) B are nontrivial functions of the couplings. Shockwave computations only capture t -channel graviton exchange, but correctly compute the VDLE λ ( v ) in Einstein gravity. It was found in [18] that there are contributions to the four graviton S-matrix inhigher derivative gravity theories that are not captured by the shockwave computations and that change thevalue of the Lyapunov exponent. It is an open problem to determine how these terms change the VDLE and v B . We thank Shiraz Minwalla and Douglas Stanford for a discussion on this point. Note that here we put the theory on S × R d − as opposed to the Rindler discussion of the previousparagraph. The first correction to the Lyapunov exponent comes at order 1 / √ λ while to the butterfly speed at order1 /λ / , where λ is the ’t Hooft coupling. – 5 – .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 Figure 1 : The two dimensional coupling space of the model. There is a critical line v ∗ ( γ )that separates the blue region, where above a critical velocity u ∗ the VDLE is dominated bythe pole and is maximal, from the white region, where the saddle always dominates, and theVDLE is nowhere maximal.We end this section with some comments about (1.7). • The proposal is somewhat anticlimactic, as it shows that just from determining the poleskipping point, we cannot read off the true λ L and u B characterizing the OTOC. Onthe other hand, while λ ( T ) L ≡ π/β does not carry any non-trivial information, u ( T ) B stilldepends on the theory. Moreover, for a large set of strongly coupled but not maximallychaotic theories, one has u B = u ( T ) B . In these theories, the OTOC butterfly speed canbe read from the location of the pole skipping point, however, we caution that thereis no way to tell whether u ( T ) B is the true butterfly speed just by having access to theenergy density two point function. • The proposal (1.7) refers to the stress tensor contribution to the growth of the OTOC.This can be defined in models where we can analytically compute the OTOC, but itis not entirely clear how to define it in complete generality. However, when there is acritical velocity u ∗ < u B , we can read this data directly from the behavior of the VDLEnear u = u B . • There is strong evidence that u B ≤ u ( T ) B is true generally [11], in which case we can reada bound on the butterfly effect from the pole skipping point using the VDLE bound of[11] (given in the text after (1.5)). – 6 – It would be very interesting to prove (1.7) in full generality. This would likely require thedevelopment of an effective field theory framework for the four point functions of non-maximally chaotic theories generalizing the works [2, 20] that apply at maximal chaos.The theory would have to apply for large Lorentzian times for OTOC configurations,and would also have to govern energy transport at finite frequency and momentum. q SYK chain
A byproduct of our investigation is a closed form expression for the energy density two pointfunction in the large- q SYK chain, see Sec. 2 for the definition of the model. It is given bythe simple formula: G Rεε ( ω ) = − N v q (cid:18) ∂ θ log ψ n ( θ v ) + tan πv h ( h − (cid:19) (cid:12)(cid:12)(cid:12) n →− iω + (cid:15) , (1.8)where v is the SYK coupling constant, we chose the inverse temperature β = 2 π (withoutloss of generality), and the function ψ n ( θ ) is a sum of two hypergeometric functions givenexplicitly by: ψ n ( θ ) = c o ψ on ( θ ) + c e ψ en ( θ ) ,c o = Γ (cid:0) − h − n v (cid:1) sin (cid:0) πh + πn v (cid:1) sin (cid:0) nπ (cid:1) Γ (cid:0) − h + n v (cid:1) ,c e = Γ (cid:0) − h − n v (cid:1) cos (cid:0) πh + πn v (cid:1) cos (cid:0) nπ (cid:1) (cid:0) − h + n v (cid:1) ,ψ en = sin( θ ) h F (cid:18) h − n/v , h + n/v , , cos θ (cid:19) ,ψ on = cos( θ ) sin( θ ) h F (cid:18) h − n/v , h + n/v , , cos θ (cid:19) ,h = 12 (cid:16) (cid:112) γ (cos( p ) − (cid:17) ,θ v = π − v ) , (1.9)where γ is the inter site coupling constant and p is the momentum. We believe this result isprecious: this is the only closed form thermal correlator in a chaotic system, which changesnontrivially as we go from weak to strong coupling. The only analogous known expressionsis (1.4) in 2d CFT (and similar results for higher dimensional CFTs in hyperbolic space [6]),which however is universal for any CFT integrable or chaotic, and is completely determinedby symmetry. Beyond this, there exist results either for small coupling [21–25], or in theholographic setting corresponding to large coupling [21, 26–29].We extract (1.8) from a complicated formula for the four point function by taking an OPElimit. The main technical result of the paper is that we find the unique analytic continuation– 7 –f this Euclidean Green function defined at discrete Matsubara frequencies to the entirecomplex plane, which allows us to prove (1.7) and to study its analytic structure.In more detail, is easy to verify that (1.8) exhibits pole skipping at( ω, h ( p )) p.s. = (cid:18) i, v (cid:19) . (1.10)With the explicit simple formulas for λ ( u ) , u B , λ ( T ) L , u ( T ) B determined in Sec. 3, we can verifythat this confirms the proposal (1.7) for all values of the couplings v and γ .The only singularities of the correlator (1.8) are poles. For small p , the closest pole to thereal ω axis is a diffusion pole with dispersion relation ω ( p ) = − iDp + . . . . As we increase p thispole can collide with a non-hydrodynamic pole on the imaginary ω axis, and subsequentlythe poles can depart the imaginary ω axis to the complex plane. While this scenario isreminiscent of what happens in theories with weakly broken momentum conservation [30],in our model these poles never become propagating quasiparticles (sound modes), as theyalways have comparable real and imaginary frequencies. We also show that besides a Drudepeak (present for small momentum) the spectral function is featureless: it does not show thepresence of quasiparticles.The collision of poles delineates the range of momenta for which all order hydrodynamicsconverges [31, 32]. We determine the radius of convergence by finding pole collisions and thereconnection of pole trajectories for complex momenta. Unlike in more familiar examples,here the radius of convergence of hydrodynamics decreases with increasing coupling v . The outline of the paper is as follows. In Sec. 2 we review the SYK chain and its large- q limit,where the quantum many-body problem can be reduced to solving differential equations. InSec. 3 we use the retarded kernel approach for computing λ ( u ). This is considerably simplerthan computing the full four point function that we undertake in Sec. 4. In Sec. 5 we extracta simple formula for the Euclidean energy density two point function from the complicatedformula for the four point function by taking an OPE limit. In Sec. 6 we find all the poleskipping points in the energy density correlator, and verify the proposal (1.7). In Sec. 7, weperform a detailed study of the analytic structure of this correlator, and extract informationabout the (all order) hydrodynamics of the SYK chain. In Appendix A, we give a newderivation of the coordinate space four point function of the large- q SYK dot first derived in[33, 34] using two different methods. – 8 –
Large- q SYK chain
We will be studying the SYK chain introduced in [8], which is a 1+1 dimensional generaliza-tion of the SYK dot [9, 35–37]. It is a disordered chain of Majorana fermions with a large N number of on-site degrees of freedom that is chaotic, yet solvable using methods resemblingDynamical Mean Field Theory. The fermions interact in groups of q , and by taking thisparameter to be large, we can analytically solve the model for all values of the two remainingdimensionless coupling constants of the theory: v that controls the strength of interactions(and can be thought of as a proxy for temperature measured in units of the dimensionfulcoupling constant J ) and γ that controls the relative strength between on-site and inter-siteinteractions and that enters in all results in a simple kinematical way.The Hamiltonian of the system is given by H = i q/ M − (cid:88) x =0 (cid:16) (cid:88) ≤ i <... u ∗ , (3.4)where the stress tensor butterfly speed u ( T ) B and critical velocity u ∗ are defined via solving κ (cid:32) iu ( T ) B (cid:33) = 1 , u ∗ = i dκ ( p ) dp (cid:12)(cid:12)(cid:12)(cid:12) p = i/u ( T ) B . (3.5) This identification is inspired by the analogy to what one gets in conformal Regge theory [11]. It wouldbe interesting to establish this connection firmly. We will also encounter further poles for (pure imaginary) p , which never dominate the integral. It would be interesting to identify whether they are also stress tensorcontributions or correspond to other operators. – 13 –xplicit expressions for these quantities read as u ( T ) B = (cid:18) arccosh 1 + v + ( γ − v γv (cid:19) − , u ∗ = (cid:112) (1 + v − v )(1 + v + 2( γ − v )2 + v . (3.6)Let us, for completeness, also write down the explicit formula for the Legendre transform partof the VDLE (first line of (3.4)) λ saddle ( u ) = 12 v (cid:118)(cid:117)(cid:117)(cid:116) − γ + 4 (cid:16) u + (cid:112) u + (9 − γ ) u v + γ v (cid:17) v + 9 − − u arccosh (cid:32) u + (cid:112) u + (9 − γ ) u v + γ v γv (cid:33) . (3.7)It is important that (3.4) is valid only for u ∗ < u ( T ) B , in which case the true butterflyspeed (defined via λ ( u B ) = 0) is the stress tensor one u B = u ( T ) B , and the velocity dependentexponent is maximal for u > u ∗ [11]. On the other hand, when u ∗ > u ( T ) B , one only has thefirst branch of (3.4) and the exponent is nowhere maximal. In this case, the true butterflyspeed is determined by solving λ saddle ( u B ) = 0. Note that due to concavity of λ saddle ( u )we have u B ≤ u ( T ) B for all couplings. There is a critical line in coupling space marking theboundary between these two types of behaviors, given by equating the two velocities in (3.6).This is shown on Fig. 1.It is perhaps educational to give explicit formulas in the strong coupling limit [11]: λ ( u ) = − u ∗ u ( T ) B (cid:20) (cid:16) uu ∗ (cid:17) (cid:21) when u < u ∗ , − uu ( T ) B when u > u ∗ ,u ∗ = (cid:112) γ δv , u ( T ) B = (cid:114) γ δv . (3.8)We see that u ∗ goes to zero, while u ( T ) B diverges. If take u comparable to u ( T ) B , we see thatthe theory is maximally chaotic, while if we take very small u ’s, we can detect the deviationfrom maximal chaos (that itself goes to zero as δv → t and x dependence ofthe OTOC, where t = i ( τ + τ − τ − τ ) / x is the spatial distance between the two operator pairs (see (2.9)). However,the OTOC also depends on the relative times ξ = iv ( τ − τ + π ), ξ (cid:48) = iv ( τ − τ + π ). Thisdependence is encoded in the wave function factors in the retarded kernel eigenvalue equation– 14 –3.1) [9, 20]. The full position dependence of the growing piece of the OTOC is therefore (cid:90) dp ξ cosh ξ (cid:48) ] κ ( p ) /v (cid:16) πκ ( p )2 (cid:17) e κ ( p ) t + ipx . (3.9)In the chaos limit, the relative positions ξ , ξ (cid:48) remain bounded, so this extra dependence onthem does not affect the steepest descent contour and the discussion in this section. In this section we invert the kinetic kernel of (2.10). Instead of starting from its spectralrepresentation, we solve Green’s equation with the appropriate delta sources on the righthand side. We switch to Matsubara representation using δ ( Y − Y (cid:48) ) = π (cid:80) n e in ( Y − Y (cid:48) ) , andexpand G p ( θ, Y, θ (cid:48) , Y (cid:48) ) = (cid:88) n π e in ( Y − Y (cid:48) ) G p,n ( θ, θ (cid:48) ) , (4.1)so that Green’s equation for the kerenel (2.14) reads as Nq (cid:20) v (cid:18) ∂ θ − h ( h − ( θ ) (cid:19) + n (cid:21) G p,n ( θ, θ (cid:48) ) = 2 v (cid:2) δ ( θ − θ (cid:48) ) + ( − n δ ( θ + θ (cid:48) − π ) (cid:3) . (4.2)This equation is supplemented with the boundary condition G p,n ( θ v , θ (cid:48) ) = G p,n ( π − θ v , θ (cid:48) ) = 0,where θ v ≡ (1 − v ) π .We can find G n using the homogeneous solutions Ψ ± n that satisfy Ψ − n ( θ v , v ) = Ψ + n ( π − θ v , v ) = 0. We explain how to do this on Fig. 2. The resulting formula is Nq G p,n ( θ , θ ) = 8 v W (cid:2) Ψ − n ( θ , v )Ψ + n ( θ , v )Θ( − θ + θ ) + Ψ + n ( θ , v )Ψ − n ( θ , v )Θ( θ − θ ) (cid:3) + ( − n v W (cid:2) Ψ − n ( θ , v )Ψ + n ( π − θ , v )Θ( π − θ − θ ) + Ψ + n ( θ , v )Ψ − n ( π − θ , v )Θ( − π + θ + θ ) (cid:3) . (4.3)Here the Wronskian is given by W = Ψ − n ∂ θ Ψ + n − Ψ + n ∂ θ Ψ − n and is independent of θ andis inserted to correctly normalize the delta functions on the RHS of (4.2). We can easilyconstruct homogeneous solutions with the required properties using the zero modes (2.17):Ψ − n ( θ, v ) = ψ en ( θ ) ψ on ( θ v ) − ψ on ( θ ) ψ en ( θ v ) , Ψ + n ( θ, v ) = ψ en ( θ ) ψ on ( θ v ) + ψ on ( θ ) ψ en ( θ v ) . (4.4)Note that Ψ − n ( θ, v ) = Ψ + n ( π − θ, v ) due to ψ e/on ( θ ) = ± ψ e/on ( π − θ ). Since the Wronskian isindependent of θ , we may evaluate it at θ = θ v that results in W = − Ψ + n ( θ v , v )Ψ − n (cid:48) ( θ v , v )= − ψ en ( θ v ) ψ on ( θ v )[ ψ en (cid:48) ( θ v ) ψ on ( θ v ) − ψ on (cid:48) ( θ v ) ψ em ( θ v )] , (4.5)– 15 – igure 2 : We piece together the solution to (4.2) using the homogeneous solution Ψ − n forregions with red boundaries and Ψ + n for regions with blue boundaries. Along green dashedlines, there are jumps in the first derivatives giving rise to the delta functions in the RHS of(4.2).where prime denotes θ derivative. We recognize in the brackets the Wronskian between ψ en and ψ on which must be independent of θ v since they solve the same second order homogeneousODE without first order terms. We may therefore replace θ v → π/ W = − ψ en ( θ v ) ψ on ( θ v ) . (4.6)It would be interesting to obtain a position space expression by summing (4.3). We didnot manage to do this in general, however, note that for p = 0 ( h = 2) the result must agreewith the four point function in the case of the SYK dot [33, 34]. We confirm that this is thecase in Appendix A. Here we would like to compute the Matsubara amplitudes of the connected energy densitytwo point function (recall that we take the lattice spacing to be 1 and set β = 2 π ) G εε ( τ, x ) = (cid:104)T ε x + y ( τ ) ε y (0) (cid:105) conn = 12 π (cid:88) n ∈ Z e inτ (cid:90) π − π dp π e ipx G Mεε ( n, p ) , (5.1)– 16 –here we define the energy density as a local term in the Hamiltonian (2.1) H = M (cid:88) x =1 ε x (0) ,ε x (0) = i q/ (cid:110) (cid:88) i <...
2. The other combination can be evaluated as ∂ θ Ψ − n ( π − θ v , v ) − ( − n ∂ θ Ψ − n ( θ v , v ) = − (1 + ( − n ) ∂ θ ψ en ( θ v ) ψ on ( θ v ) − (1 − ( − n ) ∂ θ ψ on ( θ v ) ψ en ( θ v )= (cid:40) − ∂ θ ψ en ( θ v ) ψ on ( θ v ) n ∈ Z , − ∂ θ ψ on ( θ v ) ψ en ( θ v ) n ∈ Z + 1 . (5.9)Therefore we find the Matsubara Green’s function for the energy density q N G
Mεε ( n, p ) = − v∂ θ ψ en ( θ v )2 ψ en ( θ v ) + contact term n ∈ Z , − v∂ θ ψ on ( θ v )2 ψ on ( θ v ) + contact term n ∈ Z + 1 . (5.10)Next, we proceed to obtain the analytic continuation in n after which we determine thecontact term. In order to obtain retarded correlators we need to determine the analytic continuation of(5.10) from integer n to the complex n plane. This is unique due to Carlson’s theorem oncewe eliminate the exponential growth in the n → ± i ∞ directions. We can do this by defininga master function that unifies ψ e/on of (2.17) for even/odd n and does not grow exponentially– 18 –n the limit n → ± i ∞ at θ = θ v . It turns out that the right master function is ψ n ( θ ) = c o ψ on ( θ ) + c e ψ en ( θ ) , (5.11)where c o = Γ (cid:0) − h − n v (cid:1) sin (cid:0) πh + πn v (cid:1) sin (cid:0) nπ (cid:1) Γ (cid:0) − h + n v (cid:1) , c e = Γ (cid:0) − h − n v (cid:1) cos (cid:0) πh + πn v (cid:1) cos (cid:0) nπ (cid:1) (cid:0) − h + n v (cid:1) . (5.12)For integer n , ψ n ( θ ) ∝ ψ e/on ( θ ) depending on the parity of n , up to a θ independent prefactorthat cancels in the ratio (5.10). We choose to introduce this prefactor so that c e/o can beentire functions in n ; this will soon be useful. With the use of this master function, we maywrite the retarded correlator as q N G
Rεε ( ω ) = − v ∂ θ ψ n ( θ v ) ψ n ( θ v ) n →− iω + (cid:15) + contact term= − v c o ∂ θ ψ on ( θ v ) + c e ∂ θ ψ en ( θ v ) c o ψ on ( θ v ) + c e ψ en ( θ v ) n →− iω + (cid:15) + contact term . (5.13)One may confirm that this is analytic in ω in the upper half plane for real momentum(1 ≤ h ≤ Now we determine the contact term in the energy density two point function. One neces-sary condition on the contact term comes from the Ward identity corresponding to energyconservation which gives [44]: lim p → G Rεε ( ω (cid:54) = 0 , p ) = 0 . (5.14)This condition however is not sufficient to fix the contact term, since it leaves the freedomof adding a p dependent contact term which vanishes at p = 0. We can fix this freedom bymatching to the expected short time behavior of the energy correlator. Since we are analyzinga quantum mechanical model of Majorana fermions, which in the UV become asymptoticallyfree (at leading order in N ), and the operator ε x ( τ ) in (5.2) is a polynomial of fermions, butnot their time derivatives, we expect that G Rεε ( t = 0 + , p ) = finite . (5.15) One may arrive at this by taking the v → ψ e/o in (2.17) are simple functions of θ ,but involves certain gamma function prefactors. We obtain the answer for general v by replacing n → n/v inside the resulting gamma functions. This leaves us with an undetermined phase: in the sin (cid:0) πh + πn v (cid:1) termin c o in (5.12), the phase shift πh is not fixed by this argument (but it must agree with the phase shift in thecos (cid:0) πh + πn v (cid:1) term in c e ). We guessed the right value based on matching with (rather high order) perturbationtheory in δv ≡ − v , where the analytic continuation is straightforward. – 19 –his is only possible, if the correlator decays fast enough in frequency space. For conveniencewe take the UV limit in the Euclidean theory, which corresponds to taking ω → i ∞ with0 < v < h ) fixed in (5.13). We get: q N G
Rεε ( ω ) | ω → i ∞ = (cid:20) v (cid:16) πv (cid:17) h ( h −
1) + O (cid:18) ω (cid:19)(cid:21) + contact term . (5.16)The Fourier integral producing (5.15) will only converge, if the contact term cancels theconstant in the above equation. This can indeed be done by a contact term, since h ( h −
1) isa (first order) polynomial in cos( p ), see (2.15). Fourier transforming to real lattice space, thisexpression produces a combination of δ ( τ − τ ) δ x,x (cid:48) and δ ( τ − τ ) δ x,x (cid:48) ± , which are contactterms in both time and space. Getting rid of these terms, we get the complete expression ofthe retarded energy density two point function as announced in (1.8): G Rεε ( ω ) = − N v q (cid:18) ∂ θ log ψ n ( θ v ) + tan (cid:16) πv (cid:17) h ( h − (cid:19) (cid:12)(cid:12)(cid:12) n →− iω + (cid:15) . (5.17) We are interested in points where zero and pole lines of (5.17) meet, which are called poleskipping points. The necessary condition of pole skipping is that the denominator and thenumerator of the first term in the parenthesis of (5.17) goes to zero simultaneously ψ on ( θ v ) ψ en ( θ v ) ∂ θ ψ on ( θ v ) ∂ θ ψ en ( θ v ) c o c e = . (6.1)– 20 –et us denote this equation as A (cid:126)c = (cid:126)
0. Since det A = W ( ψ on , ψ en )( θ ) = 1, pole skipping canonly happen when c e = c o = 0. Inspecting (5.12) we find that there are six classes of solutions(i) cos (cid:16) nπ (cid:17) = 0 & 1 / Γ (cid:18) − h n v (cid:19) = 0= ⇒ n = 2 Z + 1 & h = 1 + nv + 2 N (ii) cos (cid:16) nπ (cid:17) = 0 & Γ (cid:18) − h − n v (cid:19) sin (cid:18) πh πn v (cid:19) = 0= ⇒ n = 2 Z + 1 & h = − nv − N (iii) sin (cid:16) nπ (cid:17) = 0 & 1 / Γ (cid:18) − h n v (cid:19) = 0= ⇒ n = 2 Z & h = 2 + nv + 2 N (iv) sin (cid:16) nπ (cid:17) = 0 & Γ (cid:18) − h − n v (cid:19) cos (cid:18) πh πn v (cid:19) = 0= ⇒ n = 2 Z & h = − − nv − N (v) 1 / Γ (cid:18) − h n v (cid:19) = 0 & Γ (cid:18) − h − n v (cid:19) cos (cid:18) πh πn v (cid:19) = 0= ⇒ n = − v ( a + b + 1) & h = − a + b, a, b ∈ N (vi) 1 / Γ (cid:18) − h n v (cid:19) = 0 & Γ (cid:18) − h − n v (cid:19) sin (cid:18) πh πn v (cid:19) = 0= ⇒ n = − v ( a + b + 1) & h = − a + b + 1 , a, b ∈ N (6.2)THus we see that pole skipping points are only possible for real-valued h and so it is naturalto divide into the following three cases { p ∈ i R , p ∈ R , p ∈ (2 Z + 1) π + i R } . We note thatonly the first case has pole skipping points on the upper half ω plane, while all three caseshave additional pole skipping points on the lower half ω plane. • Purely imaginary momentum p ∈ i R : h ≥ n > h ≥ (cid:26) ( h, n ) = (cid:18) − n nv + 2 k, n (cid:19) | n, k ∈ N (cid:27) . (6.3)The pole skipping point connected to the diffusion pole is the one at n = 1 and h = 1 + v , weshow this on Fig. 3. Our modified pole skipping conjecture (1.7) is that pole skipping on thispole line happens at n = 1 and p = i/u ( T ) B , which indeed translates to h = 1 + 1 /v based onour discussion of the OTOC in Sec. 3. Therefore, the conjecture (1.7) indeed holds exactly– 21 –n the SYK chain at any coupling.Furthermore, we note that there are additional pole skipping points on the lower halfplane n < h ≥
2. The complete collection of pole skipping points for imaginarymomentum can be observed in Fig. 4, where we see that pole skipping points on the lowerhalf plane exist for both integer and non-integer values of n , whereas on the upper half plane,all pole skipping points have integer n . • Real momentum p ∈ R : √ − γ ≤ h ≤ p the retarded Green’s function can only have poles (and hence pole skippingpoints) in the lower half plane. In this case, the list of pole skipping points coming from (i),(ii), (iii), (iv) of (6.2) simplifies into the following expression which corresponds to integer n (cid:110) h = 2 + nv − (cid:100) nv (cid:101) , − nv + (cid:100) nv (cid:101) (cid:111) ∩ (cid:110) n ∈ − N | n − (cid:100) nv (cid:101) = 0 mod 2 (cid:111) . (6.4)Furthermore (v) and (vi) of (6.2) generate pole skipping points at h = 2 (equivalently p = 0)and generically non-integer n given by { ( h, n ) = (2 , − v (1 + r )) | r ∈ N + } . (6.5)We plot the pole skipping point for the real momentum case on Fig. 5. • p ∈ (2 Z + 1) π + i R : ≤ h ≤ √ − γ .Finally, there is one more class of pole skipping points which happens on the line ofcomplex momentum whose real part lies on the edge of the Brillouin zone Re( p ) = (2 Z + 1) π .Similarly to the real momentum case, pole skipping only happens on the lower half of thecomplex frequency plane and (i), (ii), (iii), (iv) of (6.2) correspond to the following unifiedexpression (cid:26)(cid:26) h = 1 + nv − (cid:100) nv (cid:101) , − nv + (cid:100) nv (cid:101) | ≤ h ≤ (cid:27) ∩ (cid:110) n ∈ − N | n − (cid:100) nv (cid:101) = 1 mod 2 (cid:111)(cid:27) ∪ (cid:26)(cid:26) h = 2 + nv − (cid:100) nv (cid:101) , − nv + (cid:100) nv (cid:101) | ≤ h ≤ √ − γ (cid:27) ∩ (cid:110) n ∈ − N | n − (cid:100) nv (cid:101) = 0 mod 2 (cid:111)(cid:27) . (6.6)And finally, (v), (vi) of (6.2) generate pole skipping points at h = 1 with non-integer n { ( h, n ) = (1 , − vr ) | r ∈ N + } . (6.7) One may wonder how the sequence of pole skipping points (6.3) on the upper half plane isrelated to Lyapunov growth. Examining (3.3) the poles that can be picked up in the OTOCare at ( h − v = 2 m + 1, m ∈ Z , or equivalently, h = 1 + m +1 v . This is only a subset of the– 22 – igure 3 : Density plot of the retarded energy-energy two point function for imaginarymomentum and frequency for v = 0 . , γ = 1. Hot lines (white) are pole lines, cold lines(blue) are zero lines. Black dots are the pole skipping points corresponding to chaos, whoselocation agrees with the proposal (1.7). Left:
We plot the complete correlator.
Right:
Wedrop the momentum dependent contact term and plot only ∂ θ log ψ n ( θ v ), as we do in the restof the density plots in this paper. The pole lines and the pole skipping points are unaffectedby this, while the shape of the zero lines change. These shapes are therefore not physical. Figure 4 : Density plot of the numerator and denominator of (5.13) overlaid, with poleskipping points marked with black dot. There are pole/zero lines starting from negative n that make it to positive n that are not visible on the overall density plot of Fig. 3. Each ofthe additional pole zero line pairs cross an odd number of times. The diffusion pair crossesonce, the next one three times, the next one five times and so on. The pole skipping pointsthat are not on the diffusion pole line do not contribute to chaos.– 23 – igure 5 : Two point function for imaginary frequency and real momentum. The pole lines(hot) are confined to negative n as they should be. There are various pole skipping points inthis case too, marked by black dots. The left plot is for v = 0 .
6, the right one is for v = 0 . γ = 1.pole skipping points in (6.3). We may ask if these poles ever dominate the OTOC. It turnsout that their contribution is already negative at their respective critical velocities where theyare activated (where the steepest descent contour in (3.3) crosses them), or in other words,their critical velocities are higher than u ( T ) B . Therefore they do not contribute to the growthof the OTOC, except for m = 0. We show this in Fig. 6. In the discussion above, we have found that the large q SYK chain has pole skipping points onthe lower half plane. These points can be divided into two classes: one class involves negativeinteger Matsubara frequencies, while the others are at non-integer values of n frequencies.Note that both classes occur both at real and complex momenta p . While pole skippingpoints on the lower half plane cannot contribute to an exponential growth of the OTOC, weremark that the existence of pole skipping at non-integer multiples of the unit Matsubarafrequency is interesting in its own right.Pole skipping in the thermal energy two point function at negative integer multiplesof the unit Matsubara frequency was discovered in [45] in the context of holography. The– 24 – .5 1.0 1.5 2.0 2.5 3.0 - - - - Figure 6 : The solid colored lines are contribution of the higher poles h = 1 + m +1 v tothe VDLE, while the dashed gridlines are their respective critical velocities where they getactivated. The only pole that gets activated where it gives a positive exponent, and hencecan contribute to the growth of the OTOC is the one with m = 0. (Even when they getactivated, they do not dominate over the m = 0 contribution.) The dotted black line is thesaddle contribution to the VDLE that touches the pole contributions at critical velocitiesmarked with dashed gridlines. The plot is for v = 0 . γ = 0 . ω, p ), the quasinormalmodes of linearized Einstein gravity are not unique, implying that at these frequencies, theholographically dual retarded Green’s function [26] is indefinite, which is another indicationof pole skipping. In the discussion of [45], the universal structure of the black hole metricand the near horizon expansion always seems to yield such locations at negative Matsubarafrequencies ω = − i πnβ , n ∈ N (this is also robust under higher derivative corrections [46]). Itwould be interesting to see whether non-integral valued pole skipping points like (6.5), (6.7)van be understood in the context of holography by finding singular quasinormal modes atnon-integral frequencies. The energy density retarded two point function determines the linear response behavior ofthe system. If we take | ω | , | p | (cid:28) T , we are in the hydrodynamic regime, and from the poleclosest to the origin, we can determine the hydrodynamic transport coefficients. We findthat for all values of the coupling the transport of energy is diffusive, and is controlled by adiffusion pole.We find that the two point function is meromorphic: it only has poles, but no branch cuts.We investigate the motion of the hydrodynamic and non-hydrodynamic poles on the complex ω plane as we change p . We already analyzed this problem partially: it is the continuation ofthe diffusion pole line to p ∼ T that participates in pole skipping (1.7). Here we ask, whetherpoles collide as we change p ; the collision between the diffusion pole and a non-hydrodynamicpole delineates the applicability of the (all order) hydrodynamic expansion. These phenomenahave been thoroughly analyzed in the planar four-dimensional N = 4 super Yang-Mills (SYM)– 25 –heory both at weak and at strong coupling [21, 26–28, 31, 32, 47]. Our system allows usto perform the analysis at all values of the coupling, and we comment on similarities anddifferences between the SYK model and SYM theory. We can extract the diffusion constant by examining the ω = in (cid:28) p (cid:28) → h ≈ q N G
Rεε ≈ sec (cid:0) πv (cid:1) (cid:0) sin( πv )(( h − v − n ) + π ( h − v (cid:1) (2 − h )[ πv tan (cid:0) πv (cid:1) + 2 v ] + 2 n , (7.1)which by using the relation between h and p in (2.15) leads to the diffusion constant: D = 112 γv (cid:16) πv tan (cid:16) πv (cid:17) + 2 (cid:17) . (7.2)The strong coupling limit of this result is D = γ δv with δv ≡ − v , whose analog for q = 4 was derived already in [8]. (See also the discussion around (3.2).) Note that D is anincreasing function of the coupling v , unlike in the more familiar cases of field theories, wherethe diffusion constant diverges at weak coupling.Note that one has D ≤ ( u ( T ) B ) (or D ≤ β π ( u ( T ) B ) when we reinstate the temperature)for all values of the couplings v and γ , with saturation at strong coupling v →
1. This isconsistent with (in fact stronger than) the diffusivity bound of [48]. One may also examinethe bound in terms if the true butterfly speed when it is not given by u ( T ) B . In these cases, u B can be determined numerically by equating (3.7) to zero. It turns out that D ≤ u B canbe violated at weak coupling, but we get a correct bound by dividing with the Lyapunovexponent, that is, D ≤ u B /v is always true, again consistently with [48]. One may confirmthis analytically in the weak coupling limit, where we can solve for the true butterfly speed u B = 12 (cid:114) γ ev + O ( v ) , = ⇒ u B v = 18 eγv + O ( v ) , (7.3)where e is the natural number, while D = γv + O ( v ) in this limit.Note that the pole line giving rise to the diffusion pole is the same as the pole lineparticipating in pole skipping at (1.7). Let us concentrate on the movement of poles for real p first. To explore the full range of h ∈ [1 ,
2] as we move around in the Brillouin zone p ∈ [ − π, π ], we set γ = 1. All the plots in thissection can be easily converted to any value of γ using the relation p γ = arccos (cid:104) cos( p here ) − γγ (cid:105) that follows from (5.13). Relatedly, for γ < h range.From Fig. 5, we can already start building intuition about the movement of pole trajecto-ries (hot lines). In Fig. 7 we plot the dispersion relations of the first few poles for representative– 26 –alues of v as we go from weak to strong coupling. We see that at weak coupling the first fewpoles stay at pure imaginary ω . At stronger coupling, we encounter collisions of poles, whichis followed by a gap in momentum, after which another pair of poles appears. Next we seekto understand the details of pole collisions. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Figure 7 : The dispersion relation of the first few poles from weak to strong coupling. Orangedots mark the collision of two pole lines and determine the radius of convergence of all orderhydrodynamics. For v = 0 .
65 and v = 0 . p that we analyze further in Figs. 12 and 8, 11 respectively.It turns out that after the collision the poles move off to the complex plane. We illustratethis on Fig. 8, by examining the location of poles on the complex ω plane for the two valuesof momenta marked by gridlines on the v = 0 . p the pair of poles return to the imaginary axis and continue to livethere until we reach the boundary of the Brillouin zone.The fact that poles never wander too far from the imaginary axis implies that the spectralfunction ρ ( ω, p ) = 2Im G Rεε ( ω, p ) is rather featureless: there are no quasiparticle peaks emerg-ing even at weak coupling, see Fig. 9. Besides the usual sharp Drude peak for small fixed– 27 –omentum (at small frequency), which is the signature of the hydrodynamic diffusion poleclose to the real ω axis, a noteworthy feature is that for larger momentum and sufficientlystrong coupling, the peak can shift away from zero frequency. This can be observed on theright panel of Fig. 9. This happens when the retarded correlator has a zero on the negativeimaginary ω axis that is closer to the real axis than the first pole. Note that this is verydifferent behavior than what one finds in SYM theory slightly away from infinite coupling[49]. Figure 8 : Movement of poles in the complex plane for v = 0 .
8. The hot locations are poles(also marked by red dots), and the cold ones are zeros of the two point function.
Left:
For p = 1 .
297 the poles are on the negative imaginary axis, as we have learned from Fig. 7.
Right:
As we increase the momentum, the poles collide and depart the imaginary axis. Theyfollow the trajectory marked by the red dashed line, and eventually return to the imaginaryaxis, as can also be seen clearly from Fig. 7. Here we show the density plot for p = 1 . Figure 9 : The spectral function for various momenta h and couplings v .– 28 –ext, following [31, 32], we ask about the convergence radius of all order hydrodynamics.As explained in [31], we can use the analytic implicit function theorem to find the radius ofconvergence. This amounts to finding a point on the diffusion pole line n ( p ) such that ∂ n ψ n ( θ v ) | n ( p c ) ,p c = 0 . (7.4)We plot the solution p c to this equation on Fig. 10. We find that the radius of convergence | p c | increases as we decrease the coupling. , For strong coupling v c < v <
1, where v c =0 . ± . p c is real, while for v < v c it develops an imaginarypart. This corresponds to the fact that we have a dispersion relation for the diffusion modethat has finite p support for v > v c , while it reaches the edge of the Brillouin zone for v < v c ,see Figs. 7 and 5. There is another distinguished value of the coupling which we denote by v (cid:48) c = 0 . ± . v < v (cid:48) c we have Re p c = π , therefore the region of convergence of allorder hydrodynamics contains the entire Brillouin zone.Note that the convergence radius shrinks to zero size as we go to maximal coupling v → D diverging as v → iω = O (1) that has mildmomentum dependence. This clearly leads to a decreasing p c as we increase the coupling. Figure 10 : The numerical solutions to (7.4) for various values of v . The absolute value | p c | is the radius of convergence in momentum space of the hydrodynamic expansion. For v > v c ≈ .
69, the resulting p c is real, while for v < v c it develops an imaginary part. The realpart of p c reaches π shown in dashed (i.e. the edge of the Brillouin zone) at v (cid:48) c = 0 . ± . p ’s. We fix the modulus and vary the phase of p = p m e iφ . We obtain the reconnection of poletrajectories as shown in Fig. 11 for the two selected values of p m we used in Fig. 8. At strong The case of the limit v → An analogous result for real fluids recently appeared in [50]. – 29 –oupling, the reconnection always happens at real p , and we can read off the convergenceradius of all order hydrodynamics from Fig. 7. Figure 11 : Motion of the hydrodynamic and a non-hydrodynamic pole on the complex ω plane for v = 0 . p m and varying phase of the momentum φ . Left:
For p m = 1 .
297 thepoles sit on the imaginary axis of the complex ω plane (horizontal plane) close to each otherfor φ = 0 , π . As we increase φ (vertical axis) from 0 to 2 π the projection of their trajectoryonto the ω plane goes around a circular curve twice, since the function only depends on p .Note that the vertical axis represents a periodic direction, so the upper and lower end planesare identified. Right:
For p = 1 .
35 the two curves reconnect, and we wind around theprojection of the curve to the ω plane once as we change the phase from 0 to 2 π . There isalso another curve whose plot coincides with this one, but has phase shifted by π . Insets:
The insets zooms onto the point where the reconnection takes place, and are marked withred cuboids on the large figures.At weaker coupling, we find a pole collision of a different flavor. Instead of the recon-nection of pole trajectories, they are unlinked at small p , and become linked at a complex– 30 – c plotted on Fig. 10, without the reorganization of each individual curve. We show this onFig. 12. Figure 12 : The analog of Fig. 11 for v = 0 .
65. Here we show only the region near p c ,the whole curve is too complicated to comprehend. Left:
For p m = 1 . < | p c | the poletrajectories are unlinked. Right:
They become linked for p m = 1 . > | p c | . Both the blueand the orange curve only change significantly near the crossing point, and remain virtuallyunchanged far away from it. For fixed h, n we can take the strong coupling limit corresponding to small δv ≡ − v , andobtain the following simple formula valid for 2 > Re( h ) > / q N G
Rεε ( ω ) = (2 − h )( h − πδv − (2 − h )( h − π + ( πδv ) (cid:18) h ( h − h −
1) + 6 ω h − / (cid:19) + (cid:18) πδv (cid:19) h − cot (cid:0) πh (cid:1) Γ (cid:0) − h (cid:1) Γ ( h − iω )2 Γ (cid:0) − + h (cid:1) Γ (1 − h − iω ) + O (cid:16) δv , δv h − (cid:17) . (7.5)Note that the terms in the first line are analytic in ω ; they play the important role of cancellingthe singular contribution of the second line at h = 3 /
2, where the powers δv and δv h − coincide. We can simply verify pole skipping on this function, and we plot the motion ofthe poles for real momentum on Fig. 13. This plots is helpful in that it explains where somefeatures we saw on Fig. 7 originate from.Note however that this formula misses the important diffusion pole, as that happens for2 − h = O ( δv ). If we take another limit, we can recover the results of [8] including thediffusion pole, but we miss the higher poles discussed above. To analyze the hydrodynamiclimit ( p, ω →
0) of the retarded correlator at strong coupling, we rescale the momentum as p = ˜ p δv and expand (5.13) in terms of δv (see also the discussion around (3.2)). Together– 31 – .5 1.0 1.5 2.0 2.5 3.0 - - - - Figure 13 : The dispersion relation of the poles of (7.5) is drawn with blue lines for realmomenta. We also include a sketch of the diffusion pole with an orange line, which is notcaptured by the formula (since the pole degenerates to the vertical axis). Compare with thelast plot of Fig. 7.with a first order approximation h = 2 − γ p and the diffusion constant D = γ δv we get q N G
Rεε ( ω ) = δv πDp (1 + ω )4( − iω + Dp ) + (contact terms) + O ( δv ) , (contact terms) = Dp π − δv Dp + ( Dp ) π + O ( δv ) . (7.6)This matches the result obtained in [8] up to the contact terms that they did not keeptrack of. We note that we do not see the poles that were plotted in Fig. 13, as they havesubleading residues compared to the diffusion pole. This leads to the curious conclusion thatthe convergence radius of hydrodynamics in this scaling limit is infinite, as opposed to theresult of Fig. 10 that took into account the collision with the aforementioned subleading poles.Another consequence is that the dispersion relation of the pole, ω ( p ) = − iDp is an entirefunction of p , and according to the result of [51] this implies the relation D = ( u ( T ) B ) that wefound in the strong coupling limit Sec. 7.1. Here we wish to analyze (1.8) in the weak coupling limit v →
0. In analogy with the strongcoupling limit, we will consider two different scalings of the parameters of the Green function.Let us first consider fixed real momentum, h ∈ [1 , v , and in thelimit v → As shown in [51] various bounds can be derived for D using the analyticity properties of the dispersionrelation ω ( p ) and pole skipping, and it would be interesting to explore them for our system. – 32 –ince the density of poles on the imaginary axis diverges in the weak coupling limit, it isreasonable to consider another limit, in which we define the rescaled imaginary frequency ˜ n = n/v and keep this fixed as v →
0. If we do this, the hypergeometric functions F ( a, b, c ; z ) in(1.9) can be approximated by 1, since their z argument goes to 0, while their a, b, c argumentsremains finite. We get the simple approximation: q N G
Rεε ( in ) ≈ πv (cid:32) n Γ (cid:0) ( − h + ˜ n + 2) (cid:1) Γ (cid:0) ( h + ˜ n + 1) (cid:1) Γ (cid:0) ( − h + ˜ n + 1) (cid:1) Γ (cid:0) ( h + ˜ n ) (cid:1) − ˜ n + h ( h − (cid:33) . (7.7)Similarly to (7.5), this expression is useful, as the movement of poles and pole skipping canbe easily analyzed. At the level of the fermion two point function, the SYK chain is ultralocal in space, i.e. (cid:104) χ i,x ( τ ) χ i,x (cid:48) (0) (cid:105) ∝ δ x,x (cid:48) . It is locally critical [8], since in the vacuum the correlator decaysas a power law, and at small temperature it takes the form dictated by finite temperatureconformal quantum mechanics (2.7). If we move on to the study of the energy density, wesaw that our state of matter exhibits diffusive energy transport. Next, we ask, if our system has a continuum limit. We have been working with di-mensionless positions and momenta, x and p . If we worked with dimensionful versions, y ≡ xa, k ≡ p/a , we would find that the diffusion constant is D phys = (cid:18) πa β (cid:19) γv (cid:16) πv tan (cid:16) πv (cid:17) + 2 (cid:17) = π J γ a (for β J → ∞ ) , J γ a (for β J → . (7.8)The only meaningful way to take the continuum limit of the system is to implement thedouble scaling limit βa → ∞ , β J → ∞ , D phys β = πγ β J ) (cid:18) aβ (cid:19) = fixed . (7.9)We find that for all other O (1) values of the dimensionless coupling constant β J , i.e. allvalues of v not infinitesimally close to 1, we get a lattice scale diffusion constant.Let us now examine the propagation of chaos in the model. The physical butterfly velocitycorresponding to the spatial coordinate y is u B, phys = π aβ u B , which in the β J → ∞ ( v → These properties make it analogous to extremal black holes in holography, whose dual is a semi-localquantum liquid [52]. A related phenomenon is the slow spreading of quantum information (measured by R´enyientropies) in the system [53]. The SYK chain built from charged fermions also transports charge diffusively [54]. – 33 –imit takes the value u B, phys = (cid:115) π γ β J ) (cid:18) aβ (cid:19) , (7.10)which is exactly the same combination as appearing in (7.9), and hence remains finite in thecontinuum limit. This is how it had to be since the remarkable relation D phys = β π u B, phys has to be satisfied in the strong coupling limit as discussed in Sec. 7.1.In the continuum limit the theory is maximally chaotic, with λ ( u ) = πβ (cid:16) − | u | u B (cid:17) , see(3.8). The energy density Green function given by (7.6) is extremely simple: it only has apole with dispersion relation ω = − iD phys k and an infinite radius of convergence in k .That the continuum limit is so simple, clearly demonstrates that momentum is not anapproximately conserved quantity at any scale in the SYK chain. While the collision of thediffusion and a non-hydrodynamic pole on the imaginary ω axis shown on Fig. 8 is reminiscentof the scenario articulated in [30], whereby at large k the collision of poles create two soundmodes with ω = ± c s k + . . . , here the poles only depart the imaginary ω axis for a short while,they always have comparable real and imaginary parts, and return to the imaginary ω axisfor larger values of the momentum. At fixed momentum, tuning the coupling from weak tostrong has a dramatic effect in SYM theory: the closely spaced branch cuts at zero ’t Hooftcoupling [21] break up into families of poles that form multiple branches of a “Christmas tree”at strong coupling [55], only for the top branch to remain at infinite coupling. The motionof poles in our model is less rich, but fully calculable and should complement the recentstudies of the analytic structure of thermal correlators at small but finite coupling [22–25]:changing the coupling at fixed p leads to occasional collisions of poles that move them outto the complex ω plane in some window of v , and sometimes pole skipping happens, when aline of zeros intersects with a line of poles. Acknowledgements
We thank Felix Haehl, Hong Liu, Shiraz Minwalla, Douglas Stanford, Alexander Zamolod-chikov, and especially Saˇso Grozdanov for useful discussions and comments on earlier versionsof the draft. CC is supported in part by the Simons Foundation grant 488657 (Simons Col-laboration on the Non-Perturbative Bootstrap) and also by the KITP Graduate Fellowship.MM is supported by the Simons Center for Geometry and Physics.– 34 –
Four point function for p = 0 As mentioned in the main text, the four point function (4.3) must agree with that of the SYKdot for p = 0 ( h = 2). Here we confirm this by doing the n sum. For h = 2 the zero modes(2.17) reduce to ψ en ( θ ) = vn n cos (cid:16) n ( π − θ )2 v (cid:17) v + cot( θ ) sin (cid:18) n ( π − θ )2 v (cid:19) ,ψ on ( θ ) = 1 n v − n sin (cid:16) n ( π − θ )2 v (cid:17) v − cot( θ ) cos (cid:18) n ( π − θ )2 v (cid:19) . (A.1)These agree with the eigenfunctions considered in [34], but with a different normalization.Let us introduce a rescaled version of the solutions (4.4)Φ − n ( θ ) = nv (cid:18) n v − (cid:19) Ψ − n ( θ ) , Φ + n ( θ ) = nv (cid:18) n v − (cid:19) Ψ − n ( θ ) . (A.2)The advantage of scaling out the n dependence from the denominator is that now these modesdepend polynomially on n apart from the Fourier modes in θ . That is, one can writeΦ ± n ( θ ) = p ± + ( n, θ ) e i n v θ + p ±− ( n, θ ) e − i n v θ , (A.3)where p ±± ( n, θ ) are degree two polynomials in n . Now (4.3) is invariant under rescaling themodes, so we have the expression Nq G ,n ( θ , θ ) = v cot πv n ( n − v ) (cid:104) ( − n Φ − n ( θ , v )Φ + n ( θ , v )Θ( − θ + θ )+ ( − n Φ + n ( θ , v )Φ − n ( θ , v )Θ( θ − θ ) + Φ − n ( θ , v )Φ − n ( θ , v )Θ( π − θ − θ )+ Φ + n ( θ , v )Φ + n ( θ , v )Θ( − π + θ + θ ) (cid:105) , (A.4)where we used the Wrosnkian W (Φ − n , Φ + n ) = ( − n n (cid:0) n − v (cid:1) tan (cid:0) πv (cid:1) v . (A.5)– 35 –sing the property (A.3), we may pull out the polynomial dependence on n as derivativesand write the position space expression as Nq (cid:88) n e inY G ,n ( θ , θ ) = 8 v cot πv (cid:104) Θ( π − θ − θ ) × (cid:88) a,b = ± p − a ( − i∂ Y , θ ) p − b ( − i∂ Y , θ ) F ( Y + a n v θ + b n v θ )+ Θ( − π + θ + θ ) (cid:88) a,b = ± p + a ( − i∂ Y , θ ) p + b ( − i∂ Y , θ ) F ( Y + a n v θ + b n v θ )+ Θ( − θ + θ ) (cid:88) a,b = ± p − a ( − i∂ Y , θ ) p + b ( − i∂ Y , θ ) F ( Y + π + a n v θ + b n v θ )+ Θ( θ − θ ) (cid:88) a,b = ± p + a ( − i∂ Y , θ ) p − b ( − i∂ Y , θ ) F ( Y + π + a n v θ + b n v θ ) (cid:105) , (A.6)where F ( Y ) = (cid:88) n (cid:54) =0 n ( n − v ) e inY = (cid:0) − Y + 6 π | Y | − π (cid:1) v − πv csc( πv ) cos(( | Y | − π ) v ) + 66 v , (A.7)and the sum for F ( Y ) was done by Sommerfeld-Watson resummation. Since all the p ± a in(A.6) are degree two polynomials in the derivatives, it is easy to now explicitly evaluate (A.6)and confirm that it agrees with the results in [33, 34]. References [1] S. Grozdanov, K. Schalm, and V. Scopelliti,
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