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Quantum hydrodynamics and the SYK model at next-to-leading order Reeves, Wyatt 2019

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Quantum Hydrodynamics and the SYK Model atNext-to-Leading OrderbyWyatt ReevesBSc with Honors in Mathematical Physics, University of Alberta, 2017a thesis submitted in partial fulfillmentof the requirements for the degree ofMaster of Scienceinthe faculty of graduate and postdoctoral studies(Physics)The University of British Columbia(Vancouver)April 2019©Wyatt Reeves, 2019The following individuals certify that they have read, and recommend to the Facultyof Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Quantum Hydrodynamics and the SYK Model at Next-to-Leading Or-dersubmitted byWyatt Reeves in partial fulfillment of the requirements for the degreeof Master of Science in Physics.Examining Committee:Moshe Rozali, PhysicsSupervisorMark van Raamsdonk, PhysicsExamining Committee MemberiiAbstractThere has been renewed interest in understanding the details and origins ofchaos in quantum systems with many degrees of freedom. Chaos plays a significantrole in holographic theories, hydrodynamic transport, and even the strange metalphase of condensed matter systems. With this importance, discovering a unifiedorigin that yields universal results for chaotic systems is clearly desirable.In this thesis, we investigate the conjectured hydrodynamic origin of quantummany-body chaos, first posited in [1], by testing it with the next-to-leading orderSachdev-Ye-Kitaev (syk) model. We provide a review of how hydrodynamic the-ories are constructed, and how hydrodynamic theories with a certain symmetrypossess all the standard features of chaos. We then review the leading order sykmodel, demonstrate its chaotic behaviour, and compare it with the predictions ofthe hydrodynamic theory. We finally perform an in-depth investigation of the next-to-leading order syk model, demonstrating that, while one sector of the theorysatisfies the conjecture, another sector does not admit a hydrodynamic description.This shows that the conjecture must be modified to account for near-maximallychaotic theories.iiiLay SummaryChaos, or the butterfly effect, refers to the phenomenon bywhich small perturba-tions in a physical system cause changes that grow exponentially with time. Whilebest known for its manifestations in weather and other classical systems, chaoticbehaviour can be observed on microscopic scales, where quantum mechanical ef-fects become relevant. Understanding such quantum chaotic systems is importantfor understanding the properties of fluids, superconductors, and even black holes.In this thesis, we make further steps towards understanding why certain quantummechanical systems are chaotic, by investigating a proposal for the origin of chaos.ivPrefaceThis dissertation is original, unpublished, independent work by the author, W.Reeves, with supervision by M. Rozali. Chapters 2 and 3 consist primarily ofreview of relevant material, while chapters 4 and 5 are primarily original work.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Lagrangian and Shift Symmetry . . . . . . . . . . . . . . . . . . 72.3 Shift Symmetry Results . . . . . . . . . . . . . . . . . . . . . . . 93 SYK Model-Leading Order . . . . . . . . . . . . . . . . . . . . . . . 143.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14vi3.2 Reparameterizations . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Four point function . . . . . . . . . . . . . . . . . . . . . . . . . 184 Energy Two-Point Function . . . . . . . . . . . . . . . . . . . . . . . 214.1 SYK Direct Calculation . . . . . . . . . . . . . . . . . . . . . . . 214.2 Hydrodynamics in 0+1 dimensions . . . . . . . . . . . . . . . . . 235 Next-to-Leading Order Corrections . . . . . . . . . . . . . . . . . . 255.1 NLO Corrections Summary . . . . . . . . . . . . . . . . . . . . . 265.2 Energy Two-point Function . . . . . . . . . . . . . . . . . . . . . 305.3 Orthogonal Modes . . . . . . . . . . . . . . . . . . . . . . . . . 336 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37A Schwinger-Keldysh Path Integral . . . . . . . . . . . . . . . . . . . . 40B Thermal Correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . 45C NLO Energy Two-Point Function Details . . . . . . . . . . . . . . . 47viiList of TablesTable 5.1 Various definitions used to obtain the non-local action for thesoft mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28viiiList of FiguresFigure 2.1 A diagram of an out-of-time-ordered correlator. The placementof the operators along the doubly-folded contour ensures theproper ordering of the operators. . . . . . . . . . . . . . . . 12Figure 3.1 A diagrammatic representation of the Schwinger-Dyson equa-tions. The first line is the standard representation of the two-point function in terms of the self-energy, while the second lineexpresses the self energy in terms of the two point function. . 16Figure 3.2 A Feynman diagram of the nth term in the four-point function.Each rung contains q − 2 lines in the general case. . . . . . . 19Figure A.1 An example of a closed time path. The dotted line demonstratesthat the two ends of the contour are identified with each other. 41ixList of Abbreviationsads Anti-de Sittercft Conformal Field Theoryctp Closed Time Patheft Effective Field theorynlo Next-to-Leading Orderotoc Out-of-Time-Ordered Correlatorsyk Sachdev-Ye-Kitaevtoc Time-Ordered CorrelatorxAcknowledgmentsI would like to thank Moshe Rozali for providing me with this project and hissupervision. I would like to thank Sean Cooper, Alex May, Dominik Neuenfeld,Chris Waddell, David Wakeham, and Jordan Wilson-Gerow for several helpfulconversations.xiDedicationTo my parents, for their love and support.xiiChapter 1IntroductionQuantum many-body chaos lacks a universal description despite the ubiquity ofchaos in thermal systems. The use of Out-of-Time-Ordered Correlators (otocs) ofoperators consisting of a few degrees of freedom,C(t) = 〈V(t)W(0)V(t)W(0)〉, (1.1)to diagnose chaos has been well documented since its proposal [2]. In theorieswith a large parameter N (e.g., the number of particles) the otoc is small at earlytimes, C(t) ∼ O(N−1), until the characteristic relaxation time tr when all two-pointfunctions have died off. The typical signifier of chaos is exponential growth in theotoc after the relaxation time:C(t) ∼ 1Neλt, tr  t  ts, (1.2)where λ is the Lyapunov exponent, and ts = 1λ log N is the scrambling time whenthe otoc becomes O(1). If the operators are also separated by large distances, thenthis exponential behaviour often becomes C(t, x) ∼ 1N eλ(t−|x |vB), where vB is thebutterfly velocity. Other forms of spatial dependence such as diffusive spreadingcan also arise, C(t, x) ∼ 1N e(λt− |x |2D0 t ). This exponential growth in time has beencoined “scrambling”, and indicates the growth in the number of degrees of freedomaffected by V after perturbing the system withW .1It was shown in [3] that such growth is bounded: λ ≤ 2pi/β. This bound issaturated for theories which have a holographic dual description of an Anti-de Sit-ter (ads) black hole, where the otoc can be viewed as a scattering amplitude nearthe black hole horizon [4, 5]. Thus, saturation of the chaos bound in a many-bodyquantum theory is a key indicator that the theory is holographic. Unfortunately, de-termining the behaviour for such otocs often requires complicated, model-specificcalculations [6, 7]. This is partially a result of the lack of a general description forthe origin of scrambling and chaos.Blake, Lee, and Liu [1] have proposed that the origin of chaos lies in quantumhydrodynamics. They conjecture that chaos arises due to a “shift symmetry” inthe hydrodynamic effective action, leading to a buildup of a “hydrodynamic cloud”that serves as the mechanism which scrambles operators among many degrees offreedom. This symmetry yields an exponentially growing fluid mode, and leads toa variety of results such as exponential growth in the otocs of generic operators,the lack of such growth in Time-Ordered Correlators (tocs), and an interesting“pole-skipping” phenomenon in the energy two-point function. This pole-skippingin particular is very useful, as it provides a straightforward way to calculate both theLyapunov exponent and butterfly velocity. This conjecture has been verified for boththe Sachdev-Ye-Kitaev (syk) [6] and syk chain models [8], chaotic Conformal FieldTheorys (cfts) [9], Einstein gravity [10], and higher-derivative gravity theories [11].In all of these models, the conjecture was checked at leading order in pertur-bation theory (e.g., at leading order in the inverse coupling for the syk model),where each model is maximally chaotic. However, at Next-to-Leading Order (nlo),negative corrections to the Lyapunov exponent have been shown to arise in many ofthese models, most notably from stringy effects in ads black holes [5] and from themassive modes in the syk model [6], making these theories only near-maximallychaotic.In this thesis, we will investigate the syk model at nlo, comparing syk model-specific calculations to general predictions from hydrodynamics. We first introducehydrodynamics and the conjectured origin of chaos in Chapter 2, followed by thesyk model at leading order in Chapter 3. We then determine how hydrodynamicalpole-skipping is modified in 0+ 1 dimensional theories, comparing the predictionsto the leading order syk energy two-point function in Chapter 4. We will then2discuss the NLO correction to the soft mode theory in Chapter 5, demonstratingthat the model remains maximally chaotic at nlo, retains the shift symmetry (inthis case a full SL(2,R) symmetry), and possesses the 0+ 1 dimensional version ofpole-skipping. Finally, we will discuss the massive modes, their role in reducingthe Lyapunov exponent, and their impact on the conjectured hydrodynamic originof chaos.3Chapter 2HydrodynamicsIn this chapter, we review the hydrodynamic effective theory and its relation tochaotic theories, closely following [1].2.1 PreliminariesQuantum hydrodynamics is a universal sector of all quantum many-body sys-tems. It is the low energy Effective Field theory (eft) for gapless modes associatedwith conserved quantities, such as energy and momentum. We follow the theory aslaid out in [1, 12], formulating it for systems where the only conserved quantity isenergy.We describe the theory using what is essentially the Lagrange description offluids; we define fluid spacetime coordinates σA = (σ0, σi), where the σi labelseach fluid element via its position in the fluid spacetime, and σ0 is interpreted asthe “internal clock” for each fluid element. The hydrodynamic degrees of freedomare given by mappings from the fluid spacetime to two copies of physical spacetimeon the legs of a Closed Time Path (ctp) contour, Xµs (σ), s = 1,2. The need for actp (also known as a Schwinger-Keldysh contour) is familiar from the study of non-equilibrium systems (see Appendix A for details regarding the Schwinger-Keldyshformalism). Additionally, a local inverse temperature for each fluid element isgiven, β(σA). The system begins in some initial state ρ0, often a thermal state, andits evolution is governed by the Schwinger-Keldysh path integral with an effective4hydrodynamic action Ihydro.To justify the above description and find the correct form of the action, wefollow the standard story of efts: identify the correct symmetries, write down themost general action that satisfies them, and find the correct physical interpretationof the degrees of freedom. To find the relevant symmetries, consider a theory witha single conserved current Jµ and its generating functional, with sources Asµ oneach leg of the ctp:eW [A1µ ,A2µ ] = Tr(ρ0TCei∫A1µJµ1 −A2µJµ2). (2.1)Because Jµ is conserved, the generating functional is invariant under gauge trans-formations Asµ → Asµ + ∂µλs for any function λs. We can then find a generatingfunctional written as a path integral over a local action S by promoting the symmetryparameters λs to physical degrees of freedom:eW [A1µ ,A2µ ] =∫Dϕ1Dϕ2eiS[B1µ ,B2µ ], Bsµ B Asµ + ∂µϕs . (2.2)The generating functional defined this way is manifestly invariant under gaugetransformations, and we have a local action that yields a generating functional forour conserved current.If our conserved current is the stress-energy tensor, we know that turning onsources corresponds to putting the system in a curved spacetime with metric gsµνon each leg of the ctp. Conservation of the stress-energy tensor then leads to thegauge symmetry being identified as diffeomorphism invariance of the metric,gsµν(xs) → gsρσ(ys(xs))∂yρs∂xµs∂yσs∂xνs.Promoting the symmetry parameters to degrees of freedom means our hydrody-namic action is given byeW [g1,g2] =∫DX1DX2Dβ eiIhydro[h1,h2,β],hsAB = gsµν(Xs) ∂Xµs∂σA∂Xνs∂σB,(2.3)5where we have also introduced a local temperature β(σ). Interpreting the σAas coordinates on the fluid spacetime, and Xs as the coordinates on the physicalspacetime, we’ve found the general form of our action and justified the descriptiongiven at the start of the section. This is an eft for a system whose only conservedcurrent is the stress-energy tensor, described with the associated long-lived gaplessdegrees of freedom.There are a few additional symmetries to impose. The action should not dependon how we initially choose to label the fluid elements in the fluid spacetime, nor onhow each fluid element tracks time. We thus demand the action Ihydro be invariantunder diffeomorphisms of the spatial or time components of σA, but not both:σ0 → σ0, σi → σ′i(σi),σ0 → σ′0(σ0, σi), σi → σi .(2.4)This is weaker than general diffeomorphism invariance by necessity: allowing, forexample, σi → σ′i(σ0, σi) would mean that the fluid element σi changes withtime, thus treating actual fluid motion as relabeling.We also require that the action satisfies the unitarity conditionsI∗hydro[h1, h2, β] = −Ihydro[h2, h1, β], (2.5)Ihydro[h1 = h2, β] = 0, (2.6)ImIhydro ≥ 0. (2.7)The first condition is from CPT invariance of the Schwinger-Keldysh path integral,the second from unitarity of time evolution (since setting h1 = h2 amounts toevolving forward and backward in time in the exact same way), and the third fromrequiring that the path integral be well defined (since a negative imaginary part ofthe action would lead to exponential growth in the path integral).Finally, we require the action to be invariant under what is referred to in [12] as“a Z2 dynamical KMS symmetry”. This symmetry takes a simple form by usingtime diffeomorphism to fix the local temperature,β(σA) = β = β0 12 (√−h100 +√−h200), (2.8)6where β0 is some reference scale (e.g., the background inverse-temperature if weare in a thermal state). With this, the metric transforms under this symmetry ash˜1(−σ,−xi) = h1(σ+iθ, xi), h˜2(−σ,−xi) = h2(σ−i(β0−θ), xi), θ ∈ [0, β0], (2.9)and the action is invariant under this symmetry, Ihydro[h1, h2, β] = Ihydro[h˜1, h˜2, β].This imposes local equilibrium and microscopic time reversal symmetry.2.2 Lagrangian and Shift SymmetryFor our purposes, we can consider the action without external sources for asystem with only energy conservation, as in [1]. We use spatial diffeomorphisminvariance to set σi = X ir =: xi, leaving X01,2(σ0, xi) as the remaining dynamicalvariables (recall we used time diffeomorphism invariance to fix β). We thenidentify Xr = 12 (X01 + X02 ) as physical motion and Xa = X01 − X02 as quantum-statistical noises1, and invert Xr (σ0, xi) ≡ t to express the theory in terms ofσ(t, xi) B σ0(t, xi) and Xa(t, xi) B Xa(σ, xi). This has the additional benefit thatwe now have β = β0/∂tσ.We can then write down the action to quadratic order in the noise field Xa usingstandard eft techniques:Lhydro = −H∂tXa − Gi∂iXa + i2∂tXaM1∂tXa +i2∂iXaM2∂iXa +O(a3), (2.10)where H and Gi are functions of β = β0/∂tσ and its derivatives, and M1,2 aredifferential operators constructed out of ∂t, ∂i, and β. The equilibrium configurationσ = t,Xa = 0 is always a solution to this Lagrangian.We now expand around equilibrium:σ = t + r (t, xi), Xa = −a(t, xi), β = β0 + δβ, δβ = β0(1 − ∂tr ), (2.11)1This is the standard “average-difference” basis found in Schwinger-Keldysh theory. This basisgenerally admits the interpretation of physical motion and noise [13]. We will use this basis regularlythroughout this thesis.7and expand Equation 2.10 to quadratic order in a:Lhydro = aKr − i2aMa, K = β0( f1∂t + h1∂2i )∂t, M =(M1∂t + M2∂2i)|β=β0,(2.12)where we have written H and Gi as H = f1δβ, Gi = h1∂iδβ, where f1(∂t, ∂i) andh1(∂t, ∂i) are differential operators. All β dependence in M1,2 has been set to β0.The equation of motion for this Lagrangian is( f1∂t + h1∂2i )∂tr = 0, a = 0. (2.13)Everything thus far has been for general theories where the only conservedquantity is energy, and the only long-lived gapless modes are those associated withenergy conservation. We now demonstrate how certain hydrodynamic efts canpredict chaotic behaviour: the key is demanding that Equation 2.12 be invariantunder the “shift symmetry”,u(t, xi) → u(t, xi) + f (t, xi), u = e−λσ, ∂t f = κ(∂i) f , (2.14)where f is some function satisfying the above differential equation, λ is a constant,and κ(∂i) is a differential operator with at least one derivative (or equal to zero), suchthat f (t, xi) = c is always a solution. Under this symmetry, we find an exponentiallygrowing solution to Equation 2.13: = − fλeλt, ∂t = λ˜(∂i), λ˜(∂i) = λ + κ(∂i). (2.15)This exponentially growing mode is responsible for the exponential growth found inotocs, while the symmetry protects tocs from this exponential growth (as expectedfor systems without instabilities). Both the syk and syk chain models satisfy thissymmetry; for the syk model, with no spatial dependence, the symmetry is justa constant shift in u(t), while for the syk chain the shift is an arbitrary time-independent shift u(t, xi) → u(t, xi) + a(xi), both corresponding to κ = 0.82.3 Shift Symmetry ResultsWe now review the results demonstrating the chaotic behaviour of a shiftsymmetric theory. We investigate the behaviour of the retarded two-point functionsfor both the hydrodynamic mode and energy density, along with the behaviour ofotocs and tocs of generic few-body operators. These results will establish that anyhydrodynamic theory with a shift symmetry is chaotic, and predict a phenomenonin the retarded energy two-point function called pole-skipping.First, note that this symmetry implies that the operators f1, h1 have the formf1 = (∂t − λ˜(∂2i ))a(∂t, ∂2i ), h1 = (∂t − λ˜(∂2i ))b(∂t, ∂2i ), (2.16)where a(∂t, ∂i), b(∂t, ∂i) are new differential operators. This ensures that Equa-tion 2.12 satisfies the shift symmetry.We now need the retarded two-point function for the hydrodynamic modes,GR(x) = i〈r (x)a(0)〉; (2.17)see Appendix A for the origin of this expression. Since Equation 2.12 is quadratic,with the term aKr , we can find the retarded two-point function by inverting theK = β0( f1∂t + h1∂2i )∂t operator, subject to retarded boundary conditions. This isaccomplished by taking the inverse Fourier transform of GR(ω, k) = −1/K(ω, k)with an open contour C that goes above all its poles:GR(x) = −∫Cddk(2pi)de−iωt+iki xiK(ω) . (2.18)Using Equation 2.12 and Equation 2.16 we can express GR(ω, k) asGR(ω, k) = − 1iβ0ω2a(ω, k)(ω − iλ˜(k))(ω + iD(ω, k)k2), D(ω, k) B −b(ω, k)a(ω, k),(2.19)where we have introduced the “diffusion operator” D(ω, k). The choice of contourmeans that when t < 0, we can close the contour in the upper half plane withoutpicking up any poles, yielding zero. We will assume that the only poles in thecomplex ω upper half plane come from the ω − iλ˜(k) term. There is a line of poles9in the lower half plane fromω + iD(ω, k)k2 = 0. (2.20)In the limit of small ω, k, these yield the standard energy diffusion poles ω =−iDE k2,DE B D(0,0) (hence the name diffusion operator).For a 0+1 dimensional system (such as the sykmodel), obtained by suppressingall spatial dependence, we haveGR(ω) = − 1iβ0ω2a(ω)(ω − iλ),GR(t) = −θ(t) 1β0a(iλ)λ2 eλt + · · · ,(2.21)where we have only kept the exponentially growing term. For systems with spa-tial dependence, we get a similar result using Equation 2.19: we perform the ωintegration, and look at the exponentially growing term from the pole atω − iλ˜(k) = 0 (2.22)We see that if there is a solution to the equationλ˜(k) + k2D(iλ˜, k) = 0 (2.23)for some k2 = −k2C < 0, then we obtainGR(x) = cθ(t)eλ(t−|x |vB), (2.24)λ B λ˜(−k2C), λ − k2CD(iλ,−k2c) = 0, vB BλkC. (2.25)Thus we see that assuming shift symmetry gives us the expected behaviour for theretarded two-point function in a chaotic system.Using these correlators with the expressions for the energy density found in [1],Appendix A, we can also find the retarded energy density two-point functionGEER (x) = i〈Er (x)Ea(0)〉 (2.26)10to beGEER = β0(ω − iλ˜(k))k2b(w, k)ω + iD(ω, k)k2 . (2.27)We see the same line of diffusion poles from Equation 2.20, ω + iD(ω, k2) = 0.Crucially, we see that this line of poles coincides with the zero from the ω − iλ˜(k)factor in the numerator precisely when ω and k obtain the same values that yieldedthe Lyapunov exponent and butterfly velocity in Equation 2.25. This phenomenonof “pole-skipping” implies that calculating the retarded energy two-point functionand determining its singular behaviour allows one to simply read off the Lyapunovexponent and butterfly velocity. Things must be modified slightly for theories withno spatial dependence, which will be discussed in Section 4.2.The final important result is regarding otocs and tocs. Without going into toomuch detail, we split any operator involving only a few degrees of freedom V(t)into a “bare operator” V̂ that doesn’t communicate with any other bare operator,along with the bare operator dressed by a “hydrodynamic cloud”:V(t) = V̂(t) + L(1)t [V̂ε](t) + · · · , 〈V̂Ŵ〉 = 0, (2.28)where L(1)t is a differential operator that couples the bare operator with the hydro-dynamic mode.Generically, otocs require a “doubly folded” time-path that goes forward andback twice, as in Figure 2.1, not once like with a ctp. Given this path, any four-pointfunction between any four operators A,B,C,D can be given by〈TCAi1(t1)Bi2(t2)Ci3(t3)Di4(t4)〉, (2.29)where ik = 1,2,3,4 tells us which leg of the path the operator is inserted on. Forexample, to get the otoc in Equation 1.1, we can use〈TCV1(t1)V2(t2)W1(t3)W3(t4)〉, (2.30)where t1,2 ≈ 0 and t3,4 ≈ t, tr  t  ts. Note that we can also obtain a more11Re tIm t−iβV (0)W (t)V (0)W (t)Figure 2.1: A diagram of an out-of-time-ordered correlator. The placementof the operators along the doubly-folded contour ensures the properordering of the operators.standard toc, e.g., (with t1 < t2 < t3 < t4),〈TCV1(t1)V2(t2)W3(t3)W4(t4)〉. (2.31)Note that there are many other orderings of operators one could use for a toc; forexample, [1] often uses the ordering 〈V(0)W(t)W(t)V(0)〉, corresponding to theexpectation value of the operatorW(t)W(t) in the state V(0)|0〉. What unites themis that the second fold in the contour can actually be made redundant; for example,one could put each of the operators in Equation 2.31 on the same leg and get thesame result.Even with the otoc, we can actually avoid using this doubly-folded time path,thanks to the fact that, at quadratic order in  , any four-point function of operatorsV,W reduces to various two-point functions of  . Two-point functions of operatorson any contour (with any number of folds) can always be reduced to two-pointfunctions on a single ctp, due to the unitarity of time evolution. Thus, we are freeto use the standard Schwinger-Keldysh formalism we have been using so far.By assuming that the coupling respects both the shift symmetry and a “time12reversed” shift symmetry,L(1)t1 [〈Vˆ(t1)Vˆ(t2)〉e±λt1] + L(1)t2 [〈Vˆ(t1)Vˆ(t2)〉e±λt2] = 0, (2.32)one can show that the toc between operators V and W does not contain an expo-nentially growing part, while the otoc does.In conclusion, we have the observation that a hydrodynamic theory with a shiftsymmetry is chaotic, and does not contain instabilities in tocs; the conjecture isthat this is the origin of chaos (at least for a certain class of theories). An importantcorollary to this conjecture is that pole-skipping in the energy two-point functionoccurs whenever a large N theory is chaotic, making pole-skipping a necessarycondition for chaos, and can be used to determine the Lyapunov exponent andbutterfly velocity.This conjecture appears to hold for a number of maximally chaotic theories,particularly the syk models at leading order mentioned above and Einstein gravity[10]. However, the status of non-maximally chaotic theories and their relation tothis conjecture is yet to be determined. The syk model at nlo is an example of a“near-maximally” chaotic theory; the Lyapunov exponent receives a small negativecorrection proportional to the perturbation parameter, λ = 2piβ − δλ. It is thistheory we will be investigating to gain insight on whether near-maximal chaos ishydrodynamic in origin, or whether there is another sector to chaotic theories thatmust be found.13Chapter 3SYK Model-Leading OrderIn this chapter, we provide a quick overview of the sykmodel and its descriptionat leading order. Much of this section utilizes results from Section 3.3 and Section4 in [6].3.1 PreliminariesThe syk model [14, 15] has received a great deal of focus since its proposal,thanks to the wealth of phenomena it exhibits, along with its computational sim-plicity. The model describes N Majorana fermions with all-to-all interactions,with couplings sampled from independent randomGaussian distributions with zeromean and the same variance:H =iq/2q!∑j1, j2 · · · jqjj1, j2 · · · jqψj1ψj2 · · ·ψjq ,〈 j2j1, j2 · · · jq 〉J =J2(q − 1)!Nq−1=2q−1J 2(q − 1)!qNq−1, {ψi,ψj} = 2δi j .(3.1)ψi are the Majorana operators, which are effectively N dimensional Dirac matrices.The variance of the random couplings j· · · defines the effective one dimensionalcoupling J, and its scaling with N is chosen to give an interesting large N limit,while the (q − 1)! term simplifies some expressions. The second formula definesa rescaled coupling J that is useful for some formula (e.g., in [6]). We study the14model in the regime N  βJ  1, allowing us to consider only leading ordereffects in N , while working up to nlo in the inverse dimensionless coupling 1/βJ.We will be working in Euclidean time τ = it at finite temperature. For anyparticular sampling of couplings j· · ·, the path integral isZ(β) =∫Dψ exp−∫dτ ©­«12∑iψi∂τψi +∑j1, j2 · · · jqjj1, j2 · · · jqψj1ψj2 · · ·ψjqª®¬ .(3.2)It turns out that taking the disorder average over the randomcouplings gives a simple,classical result. We introduce a bilocal field G˜(τ1, τ2) and Lagrange multiplier fieldΣ˜(τ1, τ2) that sets G˜ equal to the averaged two-point function 1N∑j ψj(τ1)ψj(τ2).We can then perform the integral over ψ:〈Z(β)〉J =∫DG˜DΣ˜ exp(−N I[G˜, Σ˜]),I[G,Σ] = −log Pf(∂t − Σ) + 12∫dτ1dτ2[ΣG − J2qGq].(3.3)The factor of N multiplying the whole action I means that taking the large N limitis a classical limit (similar to taking ~ → 0). The classical equations of motionare then found from finding the saddle point, and one can show that the equationsare the same as the Schwinger-Dyson equations for the two-point function G andself-energy Σ,G(τ) = (∂τ − Σ(τ))−1 , Σ(τ) = J2G(τ)q−1. (3.4)The first equation can be seen as the standard relationship between the full two-pointfunction G(τ), the free propagator ∂−1τ , and the self-energy Σ(τ). This is showndiagrammatically in Figure 3.1.In the limit of strong coupling (or equivalently, long time), we can neglectthe ∂τ term; the resulting action and SD equations are invariant under arbitraryreparameterizations of time f (τ),G(τ1, τ2) → G( f (τ1), f (τ2))| f ′(τ1)|∆ | f ′(τ2)|∆,Σ(τ1, τ2) → Σ( f (τ1), f (τ2))| f ′(τ1)|1−∆ | f ′(τ2)|1−∆.(3.5)15= + + + · · ·=G(τ) =Σ(τ) =Figure 3.1: A diagrammatic representation of the Schwinger-Dyson equa-tions. The first line is the standard representation of the two-pointfunction in terms of the self-energy, while the second line expresses theself energy in terms of the two point function.where ∆ = 1/q is the conformal weight of the Green’s function. The limit wherewe neglect the ∂τ term is thus named the conformal limit.For example, starting with the zero temperature solutionGβ=∞(τ1, τ2) = −b∆ |J(τ1 − τ2)|−2∆sgn(τ1 − τ2), (3.6)we can obtain the set of all solutions at a given inverse-temperature β by applyingEquation 3.5 with f (τ) = e 2pi iβ g(τ),where g is a diffeomorphism. This is similarto the way finite temperature two-dimensional cfts are obtained by exponentiatingthe zero temperature theory [16]. Every individual solution is unchanged by thetransformation f → af+bc f+d ,a, b, c, d ∈ R; the reparameterization symmetry has beenspontaneously broken to SL(2,R)3.2 ReparameterizationsUnfortunately, the conformal limit is inconsistent. While part of the four-pointfunction has a finite part in the limit, known as the “conformal four-point function”,another piece yields infinity, as will be seen in Section 3.3. This forces us to includethe leading non-conformal contribution to the action, leading to a parametricallylarge O(βJ) term in the four-point function. This means that we are includingthe leading effects from the ∂τ term, explicitly breaking the reparameterizationsymmetry in Equation 3.5.In this limit, the leading solution to the Schwinger-Dyson equations at a given16temperature is still found by reparameterizing the zero-temperature solution Gβ=∞,but only g(τ) = τ is a saddle point:Gc(τ1, τ2) = Gβ=∞(e2pi iβ τ1, e2pi iβ τ2)|(e 2pi iβ τ1)′ |∆ |(e 2pi iβ τ2)′ |∆. (3.7)We call this the conformal two-point function. Including the leading effects frombreaking reparameterization symmetry means that the reparameterization modesg(τ) become pseudo-Nambu-Goldstone modes, or soft modes. Thus we can writethe leading action for Equation 3.3 by considering fluctuations away from the confor-mal Green’s function generated by the reparameterization modes. For infinitesimalfluctuations g(τ) = τ + (τ), the fluctuation in the correlator isδGc(τ1, τ2) = (∆ ′(τ1) + ∆ ′(τ2) + (τ1)∂τ1 + (τ2)∂τ2)Gc(τ1, τ2), (3.8)and the corresponding action isS ≈ Sl = NαSJ∫dτ12( ′′2 − λ2 ′2), (3.9)where αS is a constant determined by q and numerical data (see [6],[17] for moredetails). The action for finite reparameterizations can be determined as well,yielding the Schwarzian action,Sl = −NαSJ∫ β0dτ Sch(e 2pi iβ g(τ), τ), Sch( f (t), t) = f′′′(t)f ′(t) −32(f ′′(t)f ′(t))2, (3.10)where we have defined the Schwarzian derivative. One can reproduce (3.9) bysetting g(τ) = τ + (τ) and expanding to quadratic order.One important thing to note is that Equation 3.9 is zero for the reparameteriza-tions (τ) = eimτ,m = 0,±1. These are not true zero modes, but instead the resultof an SL(2,R) gauge symmetry, coming from the fact that the conformal two-pointfunction is unchanged by SL(2,R) transformations acting on f = e 2pi iβ g(τ). Sincewe should only consider fluctuations leading to physically distinct configurations,such reparameterizations are gauge transformations.Another important thing to note about these reparameterization modes is thatthey are actually the hydrodynamic modes of the system, up to a constant, making17the Schwarzian theory a hydrodynamic theory. One way to see this is that theyrepresent a mapping from physical time, τ, into a “fluid time” g(τ). Anotherway is that the Schwarzian action, analytically continued to the Schwinger-Keldyshcontour, is an example of a model with a Lagrangian of the form in Equation 2.12.Appendix B of [1] explicitly shows how one can obtain the Schwarzian actionin Lorentzian time from the general hydrodynamic effective action; they find ahydrodynamic Lagrangian with a shift symmetryLhydro = L(σ1) − L(σ2), L(σ) = −a2Sch(e−2piβ σ, t). (3.11)Under analytic continuation to Euclidean time, −σ = ig, and the shift symmetry isjust a subset of the full SL(2,R) symmetry.We can write the Schwarzian action another way1, which will be more usefulwhen we consider the nlo corrections;Sl = −NαSε∫ 2pi0dθ Sch(eiϕ(θ), θ), (3.12)where ε = 2piβJ ,θ =2piβ τ, and ϕ(θ) = 2piβ g(βθ2pi ) is a diffeomorphism of the unit circle.In summary, the leading action for the syk model comes from the explicitbreaking of reparameterization invariance, and is characterized by a single gaplessreparameterization mode that is equivalent to the hydrodynamic mode of Chapter 2.3.3 Four point function(3.8) and (3.9) can be used to find the Lorentzian connected four point func-tion, both in time-ordered and out-of-time-ordered configurations, by finding theEuclidean time-ordered, averaged, connected four-point function,1NF(τ1, τ2, τ3, τ4) B 1N2∑i, j〈Tψi(τ1)ψi(τ2)ψj(τ3)ψj(τ4)〉−G(τ12)G(τ34), τi j B τi−τj,(3.13)where we have subtracted off the disconnected piece and noted the N scaling ofthe connected piece. This can be found using Feynman diagrams, and is a sum of1This is the convention found in [17]18· · ·Figure 3.2: A Feynman diagram of the nth term in the four-point function.Each rung contains q − 2 lines in the general case.ladder diagrams generated by a kernel K(τ1, · · · , τ4):F =∞∑n=0KnF0, F0 = −G(τ13)G(τ24) + G(τ14)G(τ23), (3.14)K(τ1, · · · , τ4) = −J2(q − 1)G(τ13)G(τ24)G(τ34)q−2, (3.15)represented diagrammatically in Figure 3.2. This is the starting point in [6], wherethey solve this equation directly by diagonalizing the kernel and taking into consid-eration the leading effects of the ∂τ term.An alternative method is to note that, using Equation 3.3, we can find F nearthe conformal limit usingF(τ1, τ2, τ3, τ4) = 〈(G˜(τ12) − Gc(τ12)) (G˜(τ34) − Gc(τ34))〉, 2 (3.16)i.e., by looking at the two-point function between fluctuations around the conformalcorrelator. Since we know that near the conformal limit, reparameterizations oughtto be the most dominant, we look at the part of the connected four-point functionthat comes from infinitesimal reparameterizations,F(τ1, τ2, τ3, τ4)(−1) = 〈δGc(τ1, τ2)δGc(τ3, τ4)〉, (3.17)where the (−1) superscript indicates the order in 1/βJ we will end up finding. Wecan now use Equation 3.9 to obtain the correlator for  and find F (−1); to leadingorder the result only depends on two point functions of  . The final expression2Recall that G˜ represent the integration variable found in the path integral of Equation 3.3.19differs depending on whether the time-ordering from Equation 3.13 results in ii j jorder or i ji j order; the former yieldsF(τ1, τ2, τ3, τ4)(−1)G(τ12)G(τ34) = cβJ(piτ12β tan piτ12β− 1) (piτ34β tan piτ34β− 1), (3.18)where c is some constant. We now see why the conformal limit is inconsistent;setting βJ = ∞ means Equation 3.18 diverges.This expression can be analytically continued to a real time toc by settingτi = δi − iti, δ1 > · · · > δ4, where the δi are simple regulators imposing thetime-ordering. Doing so shows there is no exponential growth in the toc; ifτ1 ≈ τ2 ≈ −it, τ3 ≈ τ4 ≈ i0 as in Equation 2.31, then all of the terms in Equation 3.18have finite limits as t →∞.The i ji j ordering takes a simple expression when setting τ3 = 0, τ3 = β2 :F(τ1, τ2,0, β2 )(−1)Gc(τ12)Gc(β2 )= −cβJ(piτ12β tan piτ12β− 1 − pisin piτ12β sinpiτ34β| sin piτ12β). (3.19)This formula is useful for obtaining a common variant of the otoc,C˜(t) = 〈ρ1/4ψi(t)ρ1/4ψj(0)ρ1/4ψi(t)ρ1/4ψj(0)〉, ρ = e−βH . (3.20)This correlator has the same late-time growth in the chaos limit, tr  t  ts,as the traditional otoc from Equation 1.1, but has each operator moved a quarteraround the thermal circle, making for easier calculations. We can obtain it fromEquation 3.19 by setting θ1 = −β4 − it, θ2 = β4 − it: the result isC˜(t)(−1) = cβJ(1 − cosh 2pitβ)Gc( β2 )Gc(β2), (3.21)demonstrating the exponential growth of the otocwith Lyapunov exponent λ = 2piβ .In summary, direct calculation of the four-point function in the syk model canbe calculated using the Schwarzian theory, establishing the theory as chaotic. Itis a hydrodynamic theory with a shift symmetry, and as we will show in the nextsection, satisfies the 0 + 1 dimensional version of pole-skipping.20Chapter 4Energy Two-Point FunctionWe are now ready to calculate the syk energy two-point function to leading orderin 1/βJ. We will start by obtaining an explicit expression for the energy densityE(τ) for the action in (3.10), along with the leading order in (τ) contribution.This, along with the correlator for Fourier modes of (τ), will give us the energyMatsubara correlator in frequency and time, which can be analytically continued toobtain various real time correlators. 1Wewill then calculate the energy two-point function for a generic hydrodynamicaction with no spatial dependence, specializing to the syk model afterwards andchecking for agreement with the direct result.4.1 SYK Direct CalculationThe action in (3.10) has time translation symmetry, yielding the Noether currentfor the energy,E(τ)(1) = −B(g′′′g′− 32(g′′g′)2+λ22g′2)= −B Sch(tan pig(τ)β, τ), (4.1)where B = Nαs/J , and the superscript indicates we are at O(1/βJ).Considering infinitesimal reparameterizations g = τ + (τ) and expanding toleading order in  , neglecting total derivative terms, we get the quadratic action1See Appendix B for details on thermal correlators.21(3.9) and energyE (1) = −B( ′′′ + λ2 ′). (4.2)To derive the two-point function for the energy density, we first obtain thetwo-point function for the Fourier modes of infinitesimal reparameterizations: =∑m∈Zme−iωmτ,Sl =12B∑mβω2m(ω2m − λ2)m−m,∴ 〈mn〉 = 1Bβδm+nω2m(ω2m − λ2),(4.3)where ωm = 2pin/β = λm are Matsubara frequencies. It is important to note thatthe SL(2,R) gauge symmetry is generated by the m = 0,±1 Fourier modes of  .Thus we can gauge-fix the action by simply dropping these modes, i.e., we set0 = ±1 = 0, yielding a finite correlator for all valid m.With this, the Matsubara correlator for the energy can be obtained:E(iωm)(1) = −iBωm(ω2m − λ2)n,GEEM (iωm)(1) =Bβ(ω2m − λ2),〈E(τ)E(0)〉(1) =∑m,0,±1GEEM (iωm)e−iωmτ =NαS(2pi)2J β3 [1 − β(δ′′(τ) + δ(τ))].(4.4)After dropping contact terms, this yields a constant energy two-point function, theexpected result for a conserved charge. Interestingly, it yields the same result as〈(δE)2〉 = ∂2βlog Z =cβ32, (4.5)where c/β = (2pi)2αSN/βJ is the specific heat. This result leads to the surprisingstatement that the leading order connected four point function in the time-ordered2log Z = −βE0 + S0 + c2β , where E0,S0 are the ground state energy and zero temperature entropyrespectively. See [6], Section 2.6 for details.22configuration comes entirely from energy fluctuations. Using the result from [6](Equation 3.129) and calculating the variation in the conformal correlator producedby variations in β, along with the saddle point relation E = c/2β2, one obtainsF (−1)N= ∂βG(τ1, τ2)∂βG(τ3, τ4) β6c2〈T(τ)T(0)〉(1), (4.6)where τ = 12 (τ1 + τ2 − τ3 − τ4)4.2 Hydrodynamics in 0+1 dimensionsWe now calculate the retarded energy two-point function for a hydrodynamiceft with no spatial dependence, GEER (t) = i〈Er (t)Ea(0)〉. The theory is formulatedon the the ctp contour in real time, and we have Er = 12 (E1 + E2), Ea = E1 − E2.We will then use this to find the energy two-point function for the syk model.We start by considering a general hydrodynamic Lagrangian with no spatialdependence to quadratic order; this can be obtained from (2.12) by setting allspatial derivatives equal to zero:Lhydro = βa f1∂2t r −i2aM1∂2t a . (4.7)(Note that we have dropped the subscript from β0, for consistency with the expres-sions in Chapter 3.Using the results from [1], Appendix A, where f ∗1 (∂t ) B f1(−∂t ) is the operatorobtained by doing integration by parts for f , we obtain the retarded energy two-point function for a general hydrodynamic action with no spatial dependence, aswell as the two-point function whenever the action has a shift symmetry:GEER (t1 − t2) = β2 f1(∂t1) f ∗1 (∂t2)∂t1∂t2GR(t1 − t2)= −β2(∂t − λ)2a(∂t )2∂2t GR(t),(4.8)where GR(t) = i〈r (t)a(0)〉 is the propagator of near-equilibrium hydrodynamicmodes, t B t1 − t2.Dropping all spatial dependence for 2.19, wefindGR(ω) = − 1β(iβω2a(ω)(ω − iλ))−1,23and so 3GEER (ω) = −i(ω − iλ)a(ω). (4.9)This is the 0+1 dimensional version of pole-skipping: a hydrodynamic theory withshift symmetry possesses an energy two-point function that vanishes at ω = iλ.Wenowconsider the sykmodel, whose hydrodynamicLagrangian (the Schwarzianaction of the reparameterization modes) contains a shift symmetry. Recall that theLagrangian that yields the Schwarzian action on two legs of a ctp contour is Equa-tion 3.11; one finds that f1 = Bβ (∂t − λ)(∂t + λ), yieldingGEER (ω) = −Bβ(ω2 + λ2),GEEM (iωm) =Bβ(ω2m − λ2).(4.10)where in the last line we have analytically continued the retarded, real frequencycorrelator to give theMatsubara frequency correlator,ω→ iωm (see Equation B.5).This is equal to the result from the direct syk calculation, Equation 4.4.In summary, we have found the leading order retarded energy two-point functionfor the sykmodel, and found that it satisfies the 0 + 1 dimensional version of pole-skipping, i.e., it possesses a zero at ω = iλ. Since it is also a hydrodynamic theory,as shown in Section 3.2, the syk model satisfies the conjecture. We now aim toinvestigate themodel at nlo, and see if themodel continues to satisfy the conjecture.3We have inserted a factor of β into the definition of the Fourier transform, compared to [1], forconsistency with our definition of the discrete Fourier series in Equation 4.3.24Chapter 5Next-to-Leading OrderCorrectionsIn this chapter, we investigate the nlo correction to the syk action for thereparameterization mode, derived in [17]. We will calculate the retarded energytwo-point function at nlo and look for its zeros, demonstrating that even at nlothe soft mode theory is maximally chaotic, thus satisfying the conjecture. We thenshow how to consistently incorporate the remaining degrees of freedom of the sykmodel, establish their role in reducing the Lyapunov exponent, and comment on thedifficulty in interpreting the theory as hydrodynamic.The correction to the soft mode action retains the non-locality present in Equa-tion 3.3; it readsSnl = −Nγε22[∬dθ12pidθ22piϕ′(θ1)2ϕ′(θ2)2ϕ412(ln(ϕ212ϕ′(θ1)ϕ′(θ2)ε2)+ c) ]fin,ϕ12 B 2 sinϕ(θ1) − ϕ(θ2)2.1(5.1)Here and in the following, we will use the field definition and angular timecoordinate as in Equation 3.12; ϕ is the reparameterization mode, θ ∈ [0,2pi) is the1Note that in this section, the subscript i j no longer means just the difference: θ12 = 2 sin θ1−θ22 .This is due to the prevalence of terms such as this when investigating the nlo soft mode theory.Differences such as θ1 − θ2 will be explicitly written out, or denoted by θ− when appropriate.25(angular) time coordinate, and ε = 2piβJ . The constant γ is related toαS , c is a constantdetermined by the method chosen to regularize the ∂τ perturbation in Equation 3.3,and the “fin” subscript indicates that we regularize the action by discarding anycutoff-dependent local terms that arise when one imposes a cutoff to regularizethe ϕ12 divergence.2 This action is SL(2,R) invariant, just as Equation 3.12 is,implying a shift symmetry; under the transformation eiϕ → eiϕ + δ, the variationof the Lagrangian vanishes.5.1 NLO Corrections SummaryLet us discuss the origin of the non-local action. The ∂τ term in Equation 3.3is an irrelevant perturbation that produces UV corrections to the Green’s function.The procedure in [17] is to replace this perturbation with a simpler UV perturba-tion, possessing a non-singular integral kernel σ(τ1, τ2), and match the leading IRresponse of this perturbation to the numerical result from solving the full Schwinger-Dyson equations. This provides an analytically tractable way to determine the nloaction for the soft mode, along with corrections to the soft mode propagator andthe four-point function, Equation 3.13.After replacing the ∂τ term with σ(τ1, τ2), redefining our variables as in Ta-ble 5.1, and expanding to quadratic order in fluctuations around the conformalsaddle point, (G˜, Σ˜) → (G˜c + δG˜, Σ˜c + δΣ˜), Equation 3.3 becomesI[δG˜, δΣ˜]N≈ 14Tr(G˜cδΣ˜)2 + 12∫dϕ1 dϕ2(δΣ˜(ϕ1, ϕ2)δG˜(ϕ1, ϕ2)− q − 12|G˜c(ϕ1, ϕ2)|q−2δG˜(ϕ1, ϕ2)2 − σ˜(ϕ1, ϕ2)(G˜c(ϕ1, ϕ2) + δG˜(ϕ1, ϕ2))).3(5.2)The action here has been expressed in a frame-invariant way, using ϕ(θ) as a timevariable as opposed to the physical time θ (recall that ϕ is a generic diffeomorphismon S1, which will later represent the reparameterization mode). We can now set δΣ˜equal to its saddle point to get an action for δG˜, and subsequently set δG˜ equal to2This procedure is actually unique, as argued in [17]3See Table 5.1 for a definition of the symbols used throughout this chapter. These symbols aredefined for consistency with [17].26its saddle point, resulting in (after another change of variables)I[δg]N= −12〈s |gc + δg〉 + 14 〈δg |K−1c − 1|δg〉, (5.3a)I∗N= −12〈s |gc〉 − 14 〈s |Kc(Kc − 1)−1 |s〉, (5.3b)where I∗ is the action evaluated at the saddle point for a given UV perturbationσ, and Kc is the “conformal kernel”, which is a symmetrized version of the ker-nel from Equation 3.15 evaluated in the conformal limit; see Table 5.1 for thedefinition. The inner products are defined as integrals over S1 × S1, 〈 f |g〉 B∫dθ1dθ2 f (θ1, θ2)g(θ1, θ2), and A| f 〉 B∫dθ3dθ4A(θ1, θ2, θ3, θ4) f (θ3, θ4). Notethat Kc has eigenfunctions with eigenvalue 1 generated by infinitesimal reparame-terizationmodes δϕ, δG˜ | | = |G | q−22 δδϕGc, where δδϕGc was given in Equation 3.8.Hence, Kc(Kc −1)−1 is only defined on the orthogonal complement of the reparam-eterization mode subspace, with elements labeled by δG˜⊥. The conformal kernelgenerates the conformal four-point function F˜⊥c , as calculated in [6].The first term in Equation 5.3a generates the Schwarzian action Equation 3.12,while the second will lead to the nlo correction to the action, Equation 5.1. Mean-while, the second term in Equation 5.3b is interpreted as giving a correction tothe conformal Green’s function for a given source, δgUV = 12KcKc−1 |s〉. These UVcorrections to the Green’s function have been found in the zero-temperature limitnumerically for the exact ∂τ source to beδGUV (τ1, τ2) ∝ |J(τ1 − τ2)|1−hGβ=∞(τ1, τ2), Re h ≥ 1. (5.4)The next step is to choose an appropriate UV perturbation source σ(θ1, θ2) (orequivalently s(θ1, θ2)) that reproduces the UV corrections to the Green’s function,in order to facilitate calculation of the second term in Equation 5.3a. The sourceshould only be supported for |θ1 − θ2 |  1, while still impacting the IR propertiesof the model. Any source can be expressed in the basis of eigenfunctions ofthe conformal kernel, Wh(θ1, θ2) (each with eigenvalue kc(h)), multiplied by asmooth window function u(ξ B ln |θ1−θ2 |ε ), which provides the proper support.Terms with kc(h) , 1 only contribute at short times, as Kc1−Kc |uWh〉 will only27Symbol Meaning Analytic Expressionεϕ(ϕ)Renormalizingfield represent-ing the softmode in a givenframeεdϕdθG˜ϕ(ϕ1, ϕ2)RenormalizedGreen’s func-tion in a givenframeG(τ1, τ2) = G˜(ϕ1, ϕ2)(εϕ(ϕ1))∆(εϕ(ϕ2))∆G˜c(θ1, θ2)RenormalizedconformalGreen’s func-tionG˜ϕ(θ)=θ(θ1, θ2)Σ˜ϕ(ϕ1, ϕ2)Renormalizedand redefinedself-energyΣ(τ1, τ2) = J2(Σ˜ϕ(ϕ1, ϕ2)−σ˜ϕ(ϕ1, ϕ2))(εϕ(ϕ1))1−∆(εϕ(ϕ2))1−∆σ˜ϕ(ϕ1, ϕ2)Renormalizedperturbationsourceσ(τ1, τ2) = J2σ˜ϕ(ϕ1, ϕ2)(εϕ(ϕ1))1−∆(εϕ(ϕ2))1−∆Rc(ϕ1, ϕ2) Common func-tion (q − 1)1/2 |G˜c(ϕ1, ϕ2)|q−22g(ϕ1, ϕ2)Normalizedtwo-pointfunctionRc(ϕ1, ϕ2)G˜(ϕ1, ϕ2)s(ϕ1, ϕ2) Normalizedsource Rc(ϕ1, ϕ2)−1σ˜(ϕ1, ϕ2)Kc(ϕ1, ϕ2; ϕ3, ϕ4)Kernel for 4-point function inconformal limitRc(ϕ1, ϕ2)G˜c(ϕ1, ϕ3)G˜c(ϕ4, ϕ2)Rc(ϕ3, ϕ4)F˜⊥c (ϕ1, ϕ2; ϕ3, ϕ4)Conformal four-point functionRc(ϕ1, ϕ2)−1[ Kc1−Kc (ϕ1, ϕ2; ϕ3, ϕ4)− Kc1−Kc (ϕ1, ϕ2; ϕ4, ϕ3)]Rc(ϕ3, ϕ4)−1Table 5.1: Various definitions used to obtain the non-local action for the softmode.28be supported at short times. However, when kc(hI ) = 1, we have a resonance;since Kc1−Kc |WhI 〉 =kc (hI )1−kc (hI ) |WhI 〉 = ∞, including the window function meansδgUV =12Kc1−Kc |uWhI 〉 can affect the model at larger times. Thus, we considersources corresponding to these eigenfunctions.The source for a given solution of kc(hI ) = 1 issI (θ1, θ2) = −aIεhI−1 |θ1 − θ2 |−hI sgn(θ1 − θ2)u(ξ), (5.5)where aI is a constant to be determine numerically, ε−hI−1 is to provide properunits, and |θ1 − θ2 |−hI sgn(θ1 − θ2) is an approximate eigenfunction of Kc. ThisyieldsδgUV ,I ≈ aI−k ′c(hI )εhI−1 |θ1 − θ2 |−hI sgn(θ1 − θ2). (5.6)This has the same form as Equation 5.4. The dominant source and response aregiven by h0 = 2; this is the source we will use to derive Equation 5.1.Finally, we can change our degrees of freedom to allow for easier calculations.Our primary degrees of freedom are represented by δG˜, which separates intoeigenfunctions of Kc with eigenvalue 1 and all other eigenvalues (the orthogonalcompliment), δG˜ = δG˜ | | + δG˜⊥. To make calculations easier, for each δG˜ we canchange variables into a frame ϕ(θ) where δG˜ | |ϕ = 0; this is called the “conformalframe”. The advantage of this frame is that we’ve eliminated fluctuations whereKc1−Kc |δG˜〉 = ∞. We can now use a variation of Equation 5.3a with calculationsdone in the conformal frame, allowing us to use the conformal kernel as defined onthe orthogonal compliment:S[δg⊥ϕ , ϕ]N= −12〈sϕ |gc + δg⊥ϕ 〉 +14〈δg⊥ϕ |K−1c − 1|δg⊥ϕ 〉, (5.7a)S∗[ϕ]N= −12〈sϕ |gc〉 − 14 〈sϕ |Kc(Kc − 1)−1 |sϕ〉, (5.7b)where all of the inner products are taken in the conformal frame, and in the secondline we again set δg⊥ϕ to its saddle point. Our remaining degree of freedom is the softmode ϕ. Equation 5.1 can now be obtained from the second term of Equation 5.7b.295.2 Energy Two-point FunctionWe now extend the results from Section 4.1 to nlo. Equation 5.1 is invariantunder simultaneous translation of both times, and has an associatedNoether current,Tnl(θ1, θ2) = −Nγε28pi2ϕ−412[ϕ′(θ1)2ϕ′(θ2)2(ln(ϕ212ϕ′(θ1)ϕ′(θ2)ε2)+ c − 1)],Tnl(θ1, θ2) ≈ −Nγε28pi2θ−412[2 ln θ12 − 2 ln ε + c − 1+ (4 ln θ12 − 4 ln ε + 2c − 3)(δϕ′1 + δϕ′2 −δϕ1 − δϕ2tan θ1−θ22) ],(5.8)where (∂1 + ∂2)Tnl = 0, and in the second line we expanded to linear order inδϕ = ϕ − θ.We also find that the Noether current for the local action Equation 3.12 isTl(θ+) = −Nαsε Sch(eiϕ(θ+), θ+),Tl(θ+) ' −Nαsε(12+ (∂3θ+ + ∂θ+)δϕ(θ+)).(5.9)where we use the fact that the Schwarzian is obtained with respect to the averageof the two times, θ+ = θ1+θ22 , as shown in [17], Section 3.2. Note that the energyof the system from Section 4.1 is related to the generator of θ+ translations byE(τ+) = λT( θ+λ ), so GEEM (iωm) = λ2〈TmT−m〉.Now, for the contribution from the non-local action, we must integrate thecurrent with respect to θ− B θ1 − θ2 to obtain the generator of average-timetranslations. We can then calculate theO(ε2) part of the energy two-point function:T(θ+) = T(l)(θ+) +∫dθ−2piT(nl)(θ1, θ2) +O(ε3) C T(l)(θ+) + T(nl)(θ+), (5.10)〈T(θ+)T(θ ′+)〉(2) = 〈T(l)T(l)〉(0) + 〈T(l)T(nl)〉(−1) + 〈T(nl)T(l)〉(−1). (5.11)where the superscripts on the right-hand side indicate the order in ε of the correlatorfor the soft mode used: (−1) indicates the soft mode correlator in Equation 4.3, and(0) the correlator from the non-local action Equation 5.1. The local-local piece was30found in [17], Equation 197:〈T(l)mT(l)−m〉(0) = −Nγε24pi2(∂huh,mh=2 + (2 ln ε + 2 − c)u2,m +16− 14δm,0).(5.12)We calculate the Fourier modes of the non-local-local piece, yielding〈T(l)mT(nl)−m〉(−1) = −Nγε28pi2(−6 ln ε + 3c − 92− 83∂h) (uh, m2 + uh,−m2) h=2,(5.13)whereuh,m =∫ 2pi0(2 sin θ2)−2heimθ dθ2pi= eipimΓ(1 − 2h)Γ(1 − h + m)Γ(1 − h − m) . (5.14)We thus obtain the O(ε2) part of the energy Matsubara correlator:GEEM (iωm) = −Nγε248pi2(ωmλ(ω2m − λ2)[2 ln ε + 2 − c − 2ψ(4) + 2ψ(ωmλ+ 2) ]− 3(ω2m − λ2) − 3λ2δm,0 + cos2ωmpi2λ(ωm2λ[(ωm2)2 − λ2][−6 ln ε + 3c − 92− 83(−2ψ(4) + 2ψ(ωm2λ+ 2))]+ 2ω2m −83λ2)),(5.15)where ψ is the digamma function. The expression is zero when ωm = λ, so onanalytic continuation GEER (ω = iλ) = 0. According to the conjecture from 4.2, thisshould occur for a maximally chaotic theory; indeed, [17] found that at nlo the softmode contribution to the exponentially growing part of the otoc at is31F (0)soft mode(θ1, θ2, θ3, θ4) ≈1(q − 1)b[θ12θ34 f ‖(θ1, · · · , θ4)− 3(−k ′c(2))12pi sin θ1−θ22 sinθ3−θ42((pi − 2∆θ+) cos∆θ++(2 − pi − (θ1 − θ2)tan θ1−θ22− pi − (θ3 − θ4)tan θ3−θ42)sin∆θ+)],(5.16)where ∆θ+ = θ1+θ2−(θ3+θ4)2 and f‖ is a (rather complicated) function that arisesin the analysis; see [17], Section 5.1.5 for its exact form. This expression onlyincludes the terms that grow exponentially upon analytic continuation to the timeconfiguration from Equation 3.20.To find any change in the Lyapunov exponent, we analytically continue Equa-tion 5.16 to real time as we did in Section 3.3 and look for terms that go like te2piβ t ;these correspond to adding a small negative correction to the Lyapunov exponentin Equation 3.21, ase(2piβ − 1βJ δλ)t ≈ e 2piβ t(1 − 1βJδλt). (5.17)While at first glance it appears there are terms proportional to te2piβ t in Equation 5.16(e.g., the ∆θ+ cos∆θ+ term), we have explicitly checked that f ‖ contains terms thatdirectly cancels them. Thus the exponential growth remains proportional to e2piβ t .This confirms the fact that the correction to the soft mode action does not changethe Lyapunov exponent. This makes sense when considering the symmetries of thetheory: the soft mode action must be SL(2,R) invariant to all orders in perturbationtheory, as it’s a gauge symmetry, causing the maximally chaotic behaviour. Sincethe soft mode still maintains its status as a hydrodynamic mode, the soft modesector of the syk model at nlo satisfies the conjecture.325.3 Orthogonal ModesThe conclusion of the previous section supports the shift symmetry conjecture,demonstrating another example of amaximally chaotic theorywith a shift symmetry.However, it doesn’t achieve our goal of checking the conjecture for a near-maximallychaotic theory. It is known that the sykmodel is only maximally chaotic at leadingorder; at nlo the Lyapunov exponent receives a negative correction. The soft modeaction in 5.1 yielded a maximal Lyapunov exponent. What’s missing?The answer lies in the step from Equation 5.7a to Equation 5.7b, where we setδg⊥ equal to its saddle point. This means that in Equation 5.7b, we are only lookingat fluctuations in the two-point function generated by the reparameterization mode,and nothing else. However, these are not all possible fluctuations, and it is theseother “orthogonal modes” that reduce the Lyapunov exponent. This was first notedin [6], whose authors calculated the conformal four-point function from Table 5.1by diagonalizing the kernel K (excluding the eigenfunctions corresponding to thereparameterization mode) and demonstrated that it contains a term proportional tote2piβ t . We will now show how to consistently incorporate the orthogonal modeswith the soft mode theory, reproducing the conformal four-point function.We start with the action Equation 5.7a, restated here for convenience:S[δg⊥, ϕ] = −12〈sϕ |gc + δg⊥〉 + 14 〈δg⊥ |K−1c − 1|δg⊥〉.Recall that the ϕ dependence is encoded in the fact that the integrals are taken withrespect to ϕ B ϕ(θ). We now replace δg⊥ with its saddle point plus fluctuations,δg⊥ = δg⊥∗ + η; one finds δg⊥∗ = −12 KcKc−1 |sϕ〉, andS[η, ϕ] = −12〈sϕ |gc〉 − 14 〈sϕ |Kc(Kc − 1−1)|sϕ〉 + 14 〈η |K−1c − 1|η〉= Sl[ϕ] + Snl[ϕ] + S⊥[η, ϕ].(5.18)The first term yields the Schwarzian, the second yields the non-local correction tothe soft mode action, and the final yields the action for the orthogonal modes, whichrepresent all fluctuations of the two point function away from the saddle point thatare not generated by reparameterizations.33Explicitly, the action for the orthogonal modes, including their coupling to thesoft mode, isS⊥[η, ϕ] =14∫dϕ1dϕ2dϕ3dϕ4η(ϕ1, ϕ2)[K−1c (ϕ1, ..., ϕ4) − 1(ϕ1, ..., ϕ4)]η(ϕ3, ϕ4)=14∫dθ1dθ2dθ3dθ4η(θ1, θ2)[K−1c (θ1, ..., θ4)(∏iϕ′(θi))− 1(θ1, ..., θ4)]η(θ3, θ4),(5.19)where in the last line we have switched to the physical frame, where the transforma-tion of Kc and η under reparameterizations was determined via their relationship tothe original two-point function. 1(θ1, θ2, θ3, θ4) is the kernel of the identity operatoron antisymmetric functions.We can easily find the two-point function of orthogonal modes, since Equa-tion 5.19 is Gaussian with respect to η(θ1, θ2). If S⊥ was a function of η(θ1, θ2)only, then the two-point function would just be the inverse-kernel, (K−1c −1)−1. Butsince Equation 5.19 does have ϕ dependence, we must take the expectation valuewith respect to ϕ of the inverse-kernel:〈η(θ1, θ2)η(θ3, θ4)〉 = 〈2[K−1c (θ1, ..., θ4)(∏iϕ′(θi))− 1(θ1, ..., θ4)]−1〉ϕ . (5.20)Using the antisymmetry of Equation 5.19, we can alsowrite it in an antisymmetrizedform:〈η(θ1, θ2)η(θ3, θ4)〉 =〈[K−1c (θ1, θ2; θ3, θ4)(∏iϕ′(θi))− 1(θ1, θ2; θ3, θ4)]−1−[K−1c (θ1, θ2; θ4, θ3)(∏iϕ′(θi))− 1(θ1, θ2; θ4, θ3)]−1〉ϕ .(5.21)However, note that if we set ϕ(θ) to the identity, we reproduce the conformal34four-point function:〈η(θ1, θ2)η(θ3, θ4)〉 = KcKc − 1 (θ1, θ2; θ3, θ4) −KcKc − 1 (θ1, θ2; θ4, θ3)= Rc(θ1, θ2)F˜⊥c (θ1, θ2, θ3, θ4)Rc(θ3, θ4).(5.22)As shown in [6], this is already leading order in 1/N . This implies that includingthe effects of the fluctuating soft mode will be lower order in 1/N , since the softmode propagator isO(N−1). This justifies setting ϕ(θ) = θ; the orthogonal and softmodes decouple.Finally, since δG˜⊥ = R−1c η are the fluctuations due to orthogonal modes,〈δG˜⊥(θ1, θ2)δG˜⊥(θ3, θ4)〉 = F˜⊥c (θ1, θ2, θ4, θ3). (5.23)Thus, by including fluctuations in the two-point function due to orthogonal modes,we have properly reproduced the conformal four-point function. This is added tothe contribution from the soft mode in Equation 5.16 to give the full nlo four-pointfunction. As [6] showed, it is this piece of the four-point function that is responsiblefor the reduction of the Lyapunov exponent; in the chaos limit tr  t  ts, itcontributes an O((βJ)0) term proportional to te 2piβ t .The orthogonal modes can be characterized by their eigenvalue h(h−1)with theconformal casimir, thanks to the conformal symmetry of Kc. There are infinitelymany such values of h; a discrete subset h = 2n,n ≥ 2, and a continuum pieceh = 12 + is. While the continuum subset is unimportant in the chaos limit, one mustsum over the entire discrete subset to get a convergent result. This is similar to howinfinitely many stringy modes must be summed over for ads black holes at nlo.These orthogonalmodes, unlike the soft mode, do not admit any sort of interpre-tation as hydrodynamic variables. For one, they parameterize all of the remainingdegrees of freedom of the microscopic model near the saddle point; hydrodynamicmodes, on the other hand, only parameterize the long-lived, gapless degrees of free-dom. Additionally, there is no clear way to interpret even a subset of the orthogonalmodes as mappings from physical spacetime to fluid spacetime, as is required forthe description to be hydrodynamical. This suggests that another ingredient needsto be added to the conjectured hydrodynamic origin of chaos.35Chapter 6ConclusionThe conjectured hydrodynamic origin of chaos, with shift symmetry as the causeand pole-skipping as a necessary condition, has been tested against another maxi-mally chaotic theory and survived, but against a near-maximally chaotic theory itcomes up short. After modifying the conjecture for theories with no spatial degreesof freedom, we predict that chaotic theories admit a hydrodynamic description,contain a shift symmetry, and possess a zero in the frequency space retarded energytwo-point function at ω = iλ. These all hold true for the syk model at leadingorder, and we have demonstrated that they hold even at nlo in the soft mode sector.However, by carefully incorporating the other degrees of freedom of the sykmodel,we see that the orthogonal modes reduce the Lyapunov exponent, decouple fromthe soft mode sector, and admit no clear hydrodynamic interpretation.Understanding how to change the conjecture to correctly account for near-maximally chaotic theories is of paramount importance if we want to determinea universal origin of chaos. Several proposals have been made to this end; in [1]it is suggested that, similar to Regge physics, the infinitely many stringy modesmentioned in Section 5.3 might be captured by a single mode. In [18], it issuggested that a type of inelastic scattering between “stringy states” dominates theotocs, accurately describing even nearly-maximally chaotic theories like the sykmodel. The possibility of treating the hydrodynamic theory as on open field theoryas in [19] may also play a role in the reduction of chaos. By investigating theseavenues, we hope to gain deeper understanding into the origin of chaos.36Bibliography[1] M. Blake, H. Lee, and H. Liu, “A quantum hydrodynamical description forscrambling and many-body chaos,” arXiv:1801.00010 [hep-th].[2] A. I. Larkin and Y. N. Ovchinnikov, “Quasiclassical method in the theory ofsuperconductivity,” Soviet Physics, JETP (1969) 1200.[3] J. Maldacena, S. H. Shenker, and D. Stanford, “A bound on chaos,” Journalof High Energy Physics 2016 no. 8, (Aug, 2016) 106, arXiv:1503.01409[hep-th].[4] A. Kitaev, “Hidden correlations in the Hawking radiation and thermalnoise.,” 2015. Talks at KITP.[5] S. H. Shenker and D. Stanford, “Stringy effects in scrambling,” Journal ofHigh Energy Physics 2015 no. 5, (May, 2015) 132, arXiv:1412.6087[hep-th].[6] J. Maldacena and D. Stanford, “Remarks on the Sachdev-Ye-Kitaev model,”Phys. Rev. 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Liu, “Effective field theory of dissipativefluids,” arXiv:1511.03646 [hep-th].[13] A. Kamenev, Field Theory of Non-Equilibrium Systems. CambridgeUniversity Press, 2011.[14] A. Kitaev, “A simple model of quantum holography,” 2017. and[15] S. Sachdev and J. Ye, “Gapless spin-fluid ground state in a random quantumHeisenberg magnet,” Phys. Rev. Lett. 70 (May, 1993) 3339–3342,arXiv:cond-mat/9212030.[16] D. A. Roberts and D. Stanford, “Diagnosing chaos using four-point functionsin two-dimensional conformal field theory,” Phys. Rev. Lett. 115 (Sep, 2015)131603, arXiv:1412.5123 [hep-th].[17] A. Kitaev and S. J. Suh, “The soft mode in the Sachdev-Ye-Kitaev model andits gravity dual,” Journal of High Energy Physics 2018 no. 5, (May, 2018)183, arXiv:1711.08467 [hep-th].[18] Y. Gu and A. Kitaev, “On the relation between the magnitude and exponentof OTOCs,” Journal of High Energy Physics 2019 no. 2, (Feb, 2019) 75,arXiv:1812.00120 [hep-th].[19] Avinash, C. Jana, R. Loganayagam, and A. Rudra, “Renormalization in OpenQuantum Field theory I: Scalar field theory,” arXiv:1704.08335[hep-th].38[20] L. M. Sieberer, M. Buchhold, and S. Diehl, “Keldysh field theory for drivenopen quantum systems,” Reports on Progress in Physics 79 no. 9, (Aug,2016) 096001, arXiv:1512.00637 [cond-mat.quant-gas].[21] H.-P. Breuer and F. Petruccione, The theory of open quantum systems.Oxford University Press, Great Clarendon Street, 2002.39Appendix ASchwinger-Keldysh Path IntegralThe Schwinger-Keldysh formalism is an incredibly powerful and widely usedtool. Not only does it allow the computation of all Lorentzian correlators withoutgoing to Euclidean time (compared to the standard path integral formalism whichonly allows computation of time-ordered correlators), but it also provides a unifieddescription of unitary and non-unitary theories, equilibrium and non-equilibriumsystems, and the evolution of both pure and mixed states.The key is performing the path integral along a ctp as in Figure A.1, or equiv-alently doubling the degrees of freedom. The ctp is a curve in complex-time thatstarts at some initial time ti, proceeds to some future time t f , does a small imaginaryshift to t f − i , and finally returns to ti − i . For thermal theories, an additional shiftby −iβ is added to the end, and the two ends of the path are identified. The degreesof freedom φ(tC) and operators O(tC) 1 are defined for any tC ∈ C along the path.We can then obtain a generating functional for a system beginning in state |Ω〉at time ti:ZSK [J] = 〈Ω|TC exp(∫CL[φ] + Jφ)|Ω〉, (A.1)allowing us to define correlation functions of operators similarly to the standardpath integral, except operators are ordered based off their position on the path viathe contour ordering operator TC . 21We suppress the spatial dependence of φ and O2For example, an operator on the reverse leg, at t1 − i , always comes after one on the forwardleg, at t2, regardless of the values of t1 and t2. Also note that operators solely on the reverse path are40Re tIm t−iβti tf−iǫFigure A.1: An example of a closed time path. The dotted line demonstratesthat the two ends of the contour are identified with each other.One can then split the degrees of freedom into one set that lives on the forwardleg of the contour, φ1(t), and a second that lives on the reverse leg of the contour,φ2(t), where t ∈ [ti, t f ] is the real, physical time; similarly, operators are defined tolive on one path or the other, with separate sources for the forward and reverse paths.These degrees of freedom are subject to the boundary condition φ1(t f ) = φ2(t f ),enforcing the fact that the degrees of freedom were originally identified with eachother.There are several ways to justify the need for the ctp. One of the most intuitiveways is to consider the standard expectation value of some operator O(t), startingin some state |Ω〉 at an initial time ti:〈Ω|O(t)|Ω〉 = 〈Ω|U(ti, t)O(ti)U(t, ti)|Ω〉, (A.2)U(t f , ti) = e−iH(t f −ti ), (A.3)where we have expressed the operator using the Heisenberg equation. This can beinterpreted as evolving the state |Ω〉 up to the insertion time t with the evolutionoperator U, inserting the operator O(ti), and then evolving back to the initial time.anti-time-ordered, since the path runs backwards in real time.41The justification for the introduction of the standard path integral is to assume ourinitial state is the ground state of the theory, |0〉, and that it remains unchanged (upto a phase rotation) under the evolution operator, U(t f , ti)|0〉 = eiδ |0〉; then we canwrite〈0|O(t)|0〉 = eiα〈0|U(∞, t)O(−∞)U(t,−∞)|0〉. (A.4)Thus we can interpret this as evolving the vacuum from −∞ up to t, inserting theoperator O(−∞), and then continuing the evolution up to +∞. This eliminatesthe need to “evolve backwards in time”, justifying the use of the standard, singlecontour path integral, with time running from −∞ to +∞.However, such a procedure isn’t justified for generic states. For example, anon-equilibrium state certainly changes upon time-evolution, preventing us fromreplacing the backwards evolution to the initial time (and hence back to our initialstate) with the forward evolution to future infinity. We are forced to keep both theforward and backward evolution operators. We can then say that the degrees offreedom "live" on this single time path that goes forward then backwards, or we cansplit the degrees of freedom into a set that lives on the forward path and a set onthe reverse path, with boundary conditions enforcing equality where the path turnsaround. The physical operators will always live on the forward path, but it turns outto be more convenient to allow operators to live on both paths, as we will see later.A more formal way to justify the Schwinger-Keldysh formalism is to considerthe evolution of the density matrix of the system from ti to t f , i.e.,ρ(ti) =∑jλj |Ψj(ti)〉〈Ψj(ti)|, (A.5)ρ(t f ) = U(t f , ti)ρ(ti)U(t f , ti)†, (A.6)Z = Trρ(t f ), (A.7)where we have expressed the density matrix as a generic mixed state, and Z isthe partition function corresponding to the evolution of ρ(ti). Since U(t f , ti)† =U(ti, t f ), we begin to see (heuristically) that the ket states |Ψj(ti)〉 are evolved forwardin time, while the bra states 〈Ψj(ti)| are evolved backwards in time, correspondingto the doubling of the degrees of freedom where one set evolves backwards intime. The trace operation in Z then forces us to identify the states as equal at t f ,42corresponding to the joining of the forward and reverse legs of the ctp. An excellentsource for the details of this interpretation can be found in [20, 21].The path integral one gets from following this procedure, written in terms ofthe doubled degrees of freedom (rather than the ctp), isZSK =∫ φ1(t f )=φ2(t f )ρ(ti )Dφ1 Dφ2 exp(i∫ t ftidt∫dd−1xL[φ1] − L[φ2]), (A.8)where φ1,2 are degrees of freedom defined on the forward and backward contoursrespectively. The negative in front of L[φ2] is because of the time reversal. Theboundary conditions on the functional integral state we are starting in the statedefined by ρ(ti), and that we identify φ1 and φ2 at the time where we reverse thecontour.One of the benefits of the Schwinger-Keldysh formalism is that it allows usto obtain all the standard Lorentzian correlators, rather than just the Feynmancorrelator. Since operators on the forward contour always come before those on thereverse contour (and since the reverse contour is reverse-time-ordered), we can getfour different correlators based on where we insert the operators:GF (t1, t2) = −i∫ φ1(t f )=φ2(t f )ρ(ti )Dφ1 Dφ2 exp(i∫ddxL[φ1] − L[φ2])O1(t1)O1(t2)= 〈T O(t1)O(t2)〉,GF˜ (t1, t2) = −i∫ φ1(t f )=φ2(t f )ρ(ti )Dφ1 Dφ2 exp(i∫ddxL[φ1] − L[φ2])O2(t1)O2(t2)= 〈T˜ O(t1)O(t2)〉,G+(t1, t2) = −i∫ φ1(t f )=φ2(t f )ρ(ti )Dφ1 Dφ2 exp(i∫ddxL[φ1] − L[φ2])O1(t1)O2(t2)= 〈O(t1)O(t2)〉,G−(t1, t2) = −i∫ φ1(t f )=φ2(t f )ρ(ti )Dφ1 Dφ2 exp(i∫ddxL[φ1] − L[φ2])O2(t1)O1(t2)= 〈O(t2)O(t1)〉,(A.9)43which are the Feynman, anti-Feynman (T˜ is the anti time-ordering operator), posi-tive Wightman and negativeWightman correlators. Note thatGF +GF˜ = G++G−.The retarded, advanced, and Keldysh correlators can be obtained from these 4correlators as well; in fact, upon a changing from the “forward-backward” basis ofdegrees of freedom to the “average-difference” basis φr = (φ1+φ2)/2, φa = φ1−φ2,one directly obtains these correlators; for example,GR(t1, t2) = i〈φr (t1)φa(t2)〉SK = iΘ(t1 − t2)〈[φ(t1), φ(t2)]〉, (A.10)where 〈·〉SK means inserting these operators into the SK action. This is the basisused in Chapter 2 to describe the hydrodynamic effective theory.44Appendix BThermal CorrelatorsWe provide a brief review of some of the fundamental results regarding thermalcorrelators in quantum field theory. We will be working in 0 + 1 dimensions i.e.,thermal quantum mechanics; the results can be easily extended to systems withspatial dimensions.InMinkowski space, there are several types of correlators that appear, includingFeynman, retarded, and advanced; determining these propagators in thermal fieldtheories can be a difficult task. Oftentimes, the easiest way is to analyticallycontinue the Lorentzian time to a pure imaginary Euclidean time, t → −iτ. In this"imaginary time formalism", there is only one well-defined correlator, which canbe analytically continued in different ways to obtain all the Lorentzian correlators.We startwith a quantummechanical systemat temperature βwith, for simplicity,a discrete energy spectrum. We start by considering the real-time Wightmancorrelators of position operators; the positive Wightman correlator isG+(t1, t2) = 〈q(t1)q(t2)〉 = 1Z(β)Tr[e−βHq(t1)q(t2)]=1Z(β)∑n,me−βEneiEn(t1−t2)e−iEm(t1−t2) |〈n|q(0)|m〉|2,(B.1)where in the second line we inserted a complete basis of energy eigenstates. Sinceour energy spectrum is bounded from below, we can expect this sum to converge dueto the e−βEn factor. (Note that the negative Wightman correlator is just G−(t1, t2) =45G+(t2, t1).)Now let us analytically continue t1, t2 to complex values of time. We imme-diately run into an issue: e−βEneiEn(t1−t2) = eEn(−β+i(t1−t2)) and e−iEm(t1−t2) candiverge unless− β ≤ Im(t1 − t2) ≤ 0. (B.2)This leads to the Matsubara correlator, the correlator of position operators at imag-inary times τ and 0,GM (τ) = G+(−iτ,0) = 〈q(−iτ)q(0)〉 = 1Z(β)Tr[e−βHq(−iτ)q(0)] , (B.3)where Equation B.2 demands β ≥ τ ≥ 0. This restriction is why some say theMatsubara correlator is “automatically time-ordered”, as it is only guaranteed toexist in time-ordered configurations. Then, the cyclicity of the trace allows one todemonstrate that the function is periodic, GM (τ) = GM (τ + β), thus extending therange of τ.We can then take the Fourier transform of the Matsubara correlator:GM (iωn) =∫ β0dτeiωnτGM (τ), (B.4)where ωn = 2piβ n are the Matsubara frequencies. Note thatGM (iωn) is defined onlyon a countable subset of the complex plane.Finally, demanding that the analytic continuation from the discrete, imaginaryMatsubara frequencies to complex z vanish at ∞ and be analytic outside the realaxis yields a unique analytic continuation of Equation B.4, GM (z). This functioncan be shown to exactly reproduce various real-time correlators. For example, theFourier transforms of the retarded and advanced correlators are given byGR(ω) = GM (ω + iη),GA(ω) = GM (ω − iη),(B.5)i.e., by performing the replacement iωn → ω± iη, we obtain the retarded/advancedcorrelators.46Appendix CNLO Energy Two-Point FunctionDetailsIn this appendix we provide some of the details for the calculation in Section 5.2.The Lagrangian isL = CΦ2[ln1Φε2+ c], Φ Bϕ′(θ1)2ϕ′(θ2)2ϕ412, ϕ12 B 2 sinϕ(θ1) − ϕ(θ2)2, (C.1)whereC B −Nγε28pi2 is the constant appearing in Equation 5.1. We are looking for theNoether current associated with simultaneous translation of both times; the fieldstransform as δϕi = ϕ′i, ϕi B ϕ(θi). The Noether current can then be obtained via(∂1 + ∂2)Tnl B ∂1(δϕ1∂L∂ϕ′1)+ ∂2(δϕ2∂L∂ϕ′2)− δL. We findδL = (∂1 + ∂2)C(Φ2[ln1Φε2+ c] ), (C.2)∂1(δϕ1∂L∂ϕ′1)+ ∂2(δϕ2∂L∂ϕ′2)= (∂1 + ∂2)C(Φ2[2 ln1Φε2+ 2c − 1] ), (C.3)∴ Tnl(θ1, θ2) = C(Φ2[ln1Φε2+ c − 1] ). (C.4)We now want the generator of θ+ B θ1+θ22 translations, as this is what givesus the energy of the soft mode theory, E(τ+) = λT( θ+λ ). Since we are ultimately47interested in the energy two-point function near equilibrium, this means we wantEquation C.4 at linear order in δϕi = ϕi − θi :Tnl(θ+) =∫dθ− θ−412[(δϕ1 + δϕ2 − δϕ1 − δϕ2tan θ−2)(−4 ln ε + 2c − 3 + 4 ln θ12)].(C.5)We can now calculate the full energy two-point function at O(ε2) as in Equa-tion 5.11. As previouslymentioned, the local-local piecewas calculated in [17]. Wecalculate the non-local-local piece by using the leading order soft mode correlator,derived from the Schwarzian action 3.12〈δϕmδϕn〉(−1) = 12piNαSεδm,−nm2(m2 − 1) . (C.6)We can now calculate the non-local-local piece〈Tl(θ+)Tnl(θ ′+)〉(−1) =∑m,nNαSεNγε24pi∫dθ ′−2pi(−im3 + im)eimθ+θ−434 (4 ln θ34 − 4 ln ε + 2c − 3)[(in − 1tan θ−/2 )einθ3 + (in + 1tan θ−/2 )einθ4]〈δϕmδϕn〉(−1).(C.7)By using θ3,4 = θ ′+ ± 12θ ′−, and Equation 5.14, we can solve this integral, giving usEquation 5.13 , restated here for convenience:〈T(l)mT(nl)−m〉(−1) = −Nγε28pi2(−6 ln ε + 3c − 92− 83∂h) (uh, m2 + uh,−m2) h=2.48Finally, since m is an integer,uh, m2 + uh,−m2 =Γ(h + m2)2Γ(2h)Γ (1 − h + m2 ) cos2 mpi2cos pih,∂h(uh, m2 + uh,−m2)=Γ(h + m2)2Γ(2h)Γ (1 − h + m2 ) cos2 mpi2cos pih[−2ψ(2h) + ψ(1 − h + m2)+ ψ(h +m2)+ pi tan hpi].(C.8)With these formulae, we obtain Equation 5.15.49


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