Non-local reparametrization action in coupled Sachdev-Ye-Kitaev models
PPrepared for submission to JHEP
Non-local reparametrization action in coupled Sachdev–Ye–Kitaev models
Alexey Milekhin
Department of Physics, University of California at Santa Barbara, Santa Barbara, CA93106, U.S.A.
E-mail: [email protected]
Abstract:
We continue the investigation of coupled Sachdev–Ye–Kitaev(SYK)models without Schwartzian action dominance. Like the original SYK, at large N and low energies these models have an approximate reparametrization symmetry.However, the dominant action for reparametrizations is non-local due to the presenceof irrelevant local operator with small conformal dimension. We semi-analyticallystudy different thermodynamic properties and the 4-point function and demonstratethat they significantly differ from the Schwartzian prediction. However, the residualentropy and maximal chaos exponent are the same as in Majorana SYK. We alsodiscuss chain models and finite N corrections. a r X i v : . [ h e p - t h ] F e b ontents ξ dependence 11 N correction 20 N Sachdev–Ye–Kitaev(SYK) model [1–3] and related Kondo models [4, 5] and tensormodels [6–9] are remarkable quantum mechanical models which exhibit emergentconformal symmetry, maximal chaos [10] and non-zero residual entropy. The mostsalient feature shared by all these models is the emergence of reparametrization– 1 –ymmetry at low energies which is explicitly(by kinetic term) and spontaneously(bythe form of 2-point function) broken. It was shown by Maldacena and Stanford[11] and Kitaev and Suh [12] that in the original SYK model the corresponding(Euclidean) action for reparametrizations is governed by the Schwartzian action: S Sch = − N α
SSch J (cid:90) du Sch ( τ [ u ] , u ) , Sch ( τ [ u ] , u ) = τ (cid:48)(cid:48)(cid:48) τ (cid:48) − (cid:18) τ (cid:48)(cid:48) τ (cid:48) (cid:19) (1.1)Because of this it has been conjectured that the original SYK model provides a UVcompletion for two-dimensional Jackiw–Teitelboim (JT) gravity.In a variety of examples a it has been explicitly demonstrated that the Schwartziandoes indeed dominate in various physical observables at low energies in both in andout of equilibrium. However, it had remained an open question whether there weremodels where the reparametrizations are governed by some other action. The purpose of this paper is to present such model and argue that the low energyphysics is dominated by a non-local action for reparametrizations: S nonloc = − N α S h J h − (cid:90) du du (cid:18) τ (cid:48) ( u ) τ (cid:48) ( u )( τ ( u ) − τ ( u )) (cid:19) h (1.2)This action was conjectured by Maldacena, Stanford and Yang(MSY) [25] when thespectrum of conformal dimensions contains an irrelevant local operator with thedimension h within the interval 1 < h < /
2. The original SYK does not have suchoperators. In this study we present a microscopic model where such operators arepresent. And then provide some analytic and extensive numerical evidence that thenon-local action indeed dominates. Our strategy is to study various large N exactequations numerically. This paper is a more extensive and detailed presentation ofour results reported in [26]. In addition, at the end of this paper we present someresults about finite N corrections.It is important to emphasize that the Schwartzian is still present in the modelwe explored. Numerical results clearly shows its presence. The main point is that atlarge N it gives a subleading(in 1 /βJ ) contribution.The microscopic model we consider is simply two coupled SYK models withtwisted kinetic terms. It has 2 N Majorana fermions ψ ai , i = 1 , . . . , N, a = 1 ,
2. TheLagrangian has the following form: L T = L + L int , (1.3)where L = (cid:88) i (1 − ξ )2 ψ i ∂ u ψ i + (1 + ξ )2 ψ i ∂ u ψ i + 14! (cid:88) ijkl (cid:32) J ijkl ψ i ψ j ψ k ψ l + J ijkl ψ i ψ j ψ k ψ l (cid:33) (1.4) a Refs. [13–24] and many others. – 2 – int = 32 α (cid:88) ijkl C ij ; kl ψ i ψ j ψ k ψ l (1.5)Disorder tensors J , J , C are all independent and drawn from Gaussian ensemble.We specify their variances and symmetry properties in the main text.Let us state some elementary properties of this model: • Without the two-side coupling, α = 0, it is just two decoupled Majorana SYKmodels and 1 ± ξ can be reabsorbed into J , J . Obviously, Schwartzian dom-inates in this case. • Without twisting, ξ = 0, it is just coupled SYK model with a marginal inter-action which was studied in [21, 27, 28]. This model has Z symmetry and isdominated by Schwartzian at any coupling α .For general α, ξ the discussion is very similar to standard SYK model. However, wedo not expect the Schwartzian to dominate. In the large N limit one can integrateout the disorders and write down exact Schwinger–Dyson(SD) equations. Recall thatthe low energy conformal solution in SYK is obtained by neglecting the kinetic term.The same happens here. In fact, at low energies parameter ξ drops out and the2-point function has the same form as in the original SYK.Parameter α controls the anomalous dimension h in the non-local action. Thedimension h can be anywhere between 1 and 2. Specifically the interesting range is | α | >
1, where 1 < h < /
2. The interaction strength α S h depends on both ξ and α . For small ξ we expect it to depend quadratically on ξ , however we have observedthat for large ξ there are deviations from this behavior. We expect the dependenceon α in α S h to be complicated.From a holographic point of view the action (1.2) has a simple interpretation: wehave a matter field in AdS dual to a boundary operator O h of dimension h . In thelarge N limit we expect the matter to be non-interacting. Adding O h to the boundaryaction and integrating out the matter produces a boundary-to-boundary propagator1 / ( u − u ) h integrated over the whole boundary. Dressing it with reparametriza-tions produces exactly the action (1.2). From this point of view, non-quadratic ξ dependence is quite puzzling. Perhaps a simple explanation is that operator O h en-ters in the action with non-linear ξ coefficient: holographic description of SYK(andour coupled model) works in the IR only and various operators undergo a finiterenormalization between UV and IR. For this reason one should not treat the ξ de-formation in the UV Lagrangian (1.3) as a simple addition of ξ O h in the IR. Wediscuss this issue more in the main text.Unfortunately, we were not able to demonstrate analytically that the non-localaction indeed dominates in this coupled SYK model. Therefore our strategy is toobtain various physical predictions of the non-local action analytically and checkthem against the numerics. – 3 –e performed an extensive numerical analysis of large N exact equations. Aswe mentioned above, at infinite N it is possible to write down SD equations for2-point functions. Our strategy was to first solve the Euclidean SD equations toobtain exact(valid at all times, not just in low energy) 2-point functions. We did thisusing a uniform discretization in the time/frequency domain and a standard iterationprocedure [11]. It is straightforward to extract the energy from the 2-point functions.Also we studied the connected 4-point function. It is 1 /N effect, but one can obtainan exact(but somewhat formal) expression in terms of a certain functional kernelbuild from 2-point functions. By numerically diagonalizing the kernel we arguedthat the non-local action dominates in the 4-point function too. In fact, one canlook at 4-point function computation as the derivation of the non-local action.Also we discuss the physics of the non-local action. In general it is applicable atlow temperatures: T (cid:28) J . If temperatures are not too low, J/N / (2 h − (cid:28) T it canbe treated classically. We mostly study this temperature range. We show that theresidual entropy and chaos exponent are the same as in SYK. We have studies theelementary thermodynamics quantities and also transport coefficients in the chainmodels. We demonstrate that the diffusion constant becomes temperature depen-dent(in the Schwartzian-dominating case it does not depend on the temperature).However, the thermal conductivity remains linear in the temperature. We have sum-marized our findings in Table b
1. It is worth noting that the leading non-conformalcorrection δG to 2-point function is always different from SYK answer as long as α (cid:54) = 0 , ξ (cid:54) = 0. We have found that at zero temperature δGG ∝ J u ) h − (1.6)whereas in SYK: δG SY K G SY K ∝ J u (1.7)This happens because the coupled model always has operator with dimension 1 2. It is only for | α | > / b Residual entropy depends on the form on the conformal solution only so the matching betweenthe two columns in trivial. We included it for completeness. – 4 –chwartzian Non-local actionResidual entropy(Sec. 3.1) 2 S ,SY K S ,SY K Energy vs temperature(Sec. 2.2), T T h − Late-time OTOC(Sec. 3.2) βJ e πt/β ( βJ ) h − e πt/β Diffusion constant(chain models, Sec. 3.4) const T − h Thermal conductance(chain models, Sec. 3.4) T T Table 1 . Summary of our results. We kept only the most relevant factors: temperature T , inverse temperature β , Lorentzian time t . the non-local action and study the thermodynamics numerically. Then we continuethis analysis and discuss ξ -dependence in Section 2.3.In Section 3 we investigate the physics of the non-local action. We start bydiscussing the residual entropy in Section 3.1. Section 3.2 contains the computationof the out-of-time ordered 4-point function and demonstrates the maximality of chaosexponent. In Section 3.3 we examine the time ordered 4-point function and itsrelation to energy-energy correlators. Section 3.5 computes 1-loop N correction tothe free energy. We conclude by studying the chain models in Section 3.4, where wederive the low-energy effective action and study transport.Section 4 is dedicated to a detailed discussion of the 4-point function and deriva-tion of the non-local action. We start by reviewing Maldacena–Stanford [11] deriva-tion of the Schwartzian in Section 4.1. After that Section 4.2 explores the subleadingcorrection to the conformal 2-point functions in the coupled model. In Section 4.3 wediscuss the properties of the kernel. Section 4.4 contains the results of the numericaldiagonalization of the kernel. Kernel spectrum is sensitive to the precise form of thenon-local action. We see a good agreement with the analytical prediction, which wetake as the most important evidence for the non-local action dominance. In Section4.5 we continue the exploration of the kernel eigenvalues and discuss the prefactorin the non-local action.Section 5 contains some exact diagonalization(ED) results at finite N . In Section5.1 we compare the ground state energy obtained two ways: by numerically solvinglarge N Schwinger–Dyson equations and performing ED. In Section 5.2 we probe thedensity of states near the ground state. This quantity is sensitive to 1 /N corrections.Section 5.3 contains the numerical evaluation of 2-point function at very late times, τ (cid:29) N/J . Section 5.4 is dedicated to the study of the energy levels statistics.In Conclusion we summarize our results and describe numerous open questions.In Appendix A we write Schwinger–Dyson equations in Lorentzian signature.– 5 – The model The model we consider has two c independent Majorana SYK with a marginal inter-action: H T = N (cid:88) ijkl =1 (cid:18) J ijkl ψ i ψ j ψ k ψ l + 14! J ijkl ψ i ψ j ψ k ψ l + 6 α (2!) C ij ; kl ψ i ψ j ψ k ψ l (cid:19) (2.1)However the anti-commutation relations are twisted because of the twisted kineticterm: { ψ ai , ψ bj } = 11 − ξ a δ ij δ ab , ξ = ξ, ξ = − ξ (2.2)In principle, we can make the kinetic term standard by rescaling the fermions. How-ever, we prefer not to do that. Tensors J , J , are usual SYK disorders: totallyantisymmetric and the components are independent and Gaussian. Tensor C ij ; kl has a Gaussian distribution too, but it has a separate skew-symmetry in ij and kl indices: C ij ; kl = − C ji ; kl = − C ij ; lk (2.3)However, it does not mix ij and kl . Because of that, integrating it out only produces G and G and would not introduce mixed correlators G , G . We adopt thefollowing normalizing for the variances: (cid:104) (cid:0) J aijkl (cid:1) (cid:105) = 3! J N , a = 1 , (cid:104) ( C ij ; kl ) (cid:105) = J N (2.4)As in SYK, up to 1 /N corrections there is no difference between quenched andannealed averages. Treating J , , C as annealed(i.e. normal quantum fields) andintegrating them out, we get the following Euclidean G Σ action: S G Σ = 12 (cid:88) a =11 , (cid:18) Tr log ((1 − ξ a ) ∂ u − Σ a ) − (cid:90) du du Σ a ( u , u ) G a ( u , u ) (cid:19) ++ 18 (cid:90) du du (cid:0) G + G + 6 α G G (cid:1) , ξ = ξ, ξ = − ξ (2.5)and Euclidean Schwinger–Dyson equations:(1 − ξ ) ∂ u G − J ( G + 3 α G G ) ∗ G = δ ( u )(1 + ξ ) ∂ u G − J ( G + 3 α G G ) ∗ G = δ ( u ) (2.6)where ∗ denotes convolution in imaginary time u . c Throughout the paper index a labels the two sides. It will be equal either 1 , , – 6 –t low energies(Euclidean times u (cid:29) /J ) and low temperatures( βJ (cid:29) 1) wecan neglect the kinetic term. Notice that ξ parameter drops out. Then SD equationsadmit symmetric G = G solution given by SYK conformal solution: G = G = G conf = b sgn( u )(1 + 3 α ) / πJ β sin (cid:16) π | u | β (cid:17) / , /J (cid:28) | u | , βJ (cid:29) b = 1 / (4 π ) / . By dropping the kinetic term, we acquired time-reparametrizationsymmetry. However, because of non-zero α , G and G are still coupled, so thereis only one reparametrization mode which acts on G a as G a ( u , u ) → ( τ ( u ) (cid:48) τ ( u ) (cid:48) ) / G a ( τ ( u ) , τ ( u )) , a = 11 , 22 (2.8)Above conformal solution (2.7) tells us that elementary fermions ψ , i has confor-mal dimension 1 / 4. Let us discuss the spectrum of conformal dimension of bilinearoperators. Using standard techniques, it can be shown [29] that the dimension h ofoperator O , = (cid:88) i (cid:0) ψ i ∂ u ψ i − ψ i ∂ u ψ i (cid:1) , (2.9)is determined by the smallest d h solution of1 − α α g A ( h ) = 1 , g A ( h ) = − 32 tan ( π ( h − / / h − / α . One can easily see that for | α | > 1, thedimension h is in the range we are looking for: 1 < h < / Z -odd.Before proceeding to the detailed investigation of this operator, let us discussthe possible symmetry breaking in this model. It is important because in the Z symmetry-broken phase the conformal solutions (2.7) do not represent the ther-modynamically dominating phase and the whole argument would not work. Aclosely related model, but with ξ = 0, was studied by Kim–Klebanov–Tarnopolsky–Zhao(KKTZ) [29]: H Z = 14! (cid:88) ijkl J ijkl (cid:0) ψ i ψ j ψ k ψ l + ψ i ψ j ψ k ψ l + 6 αψ i ψ j ψ k ψ l (cid:1) (2.11)In fact, the above models have the same spectrum of conformal dimension in the an-tisymmetric e ψ ψ , ψ ψ bilinear sector. However the problem is that in the original d The rest of the solutions determine the dimensions of O ,n = (cid:80) i ψ i ∂ n +1 u ψ i − ψ i ∂ n +1 u ψ i .Also, there is Z -even sector O ,n = (cid:80) i ψ i ∂ n +1 u ψ i + ψ i ∂ n +1 u ψ i with the same dimensions as inSYK, which are determined by h A ( h ) = 1. e “antisymmetric” refers to time dependence, not Z parity. Operators O ,n , O ,n are said tobe in antisymmetric sector because the 2-point function (cid:104) T ψ i ( u ) ψ i (0) (cid:105) is antisymmetric under u → − u . In contrast, under general assumptions the correlator (cid:104) T ψ i ( u ) ψ i (0) (cid:105) is symmetric in u . – 7 – h vs αα h Figure 1 . The dimension h of operator (2.9) as a function of α . h approaches 1 for α → ∞ . KKTZ there is Z symmetry breaking for | α | > 1. Actual ground state is separatedby a gap from the rest of the spectrum. At certain critical temperature T crit ( α ) ∼ N there is second-order phase transition. Below this temperature Z symmetry is bro-ken and the actual physical behavior is not described by the conformal solution.However above T crit the physics is described by the conformal solution. Hence weexpect that KKTZ model, once augmented with ξ -term, is also dominated by thenon-local action, but only in some window of temperatures T crit ( α ) < T (cid:28) J . Noticethat after integrating out J ijkl in KKTZ model, SD equations contain mixed Green’sfunctions G , G . The symmetry breaking is triggered by the operator O = (cid:88) i ψ i ψ i (2.12)in the symmetric sector which acquires complex scaling dimension for | α | > 1. Inour case mixed correlators G do not appear at all up to 1 /N order. Therefore weconjecture that the symmetry breaking does not occur in our model and the non-local action dominates all the way to temperatures as low as J/N / (2 h − . We verifythis statement with finite N exact diagonalization in Section 5. As we just found out, the coupled model does contain an operator with dimension1 < h < / 2. Obviously, this irrelevant operator does not affect the conformalsolution. How do we describe the influence of this operator on thermodynamics andother physical observables?In the standard SYK story(and in our coupled model) one obtains the conformalsolution by neglecting the kinetic term in the SD equations. One way to recover the– 8 –ow energy physics is to consider conformal perturbation theory [12](see [30] for arecent discussion). One starts from the artificial “exactly conformal” SYK withoutthe kinetic term: L conf = (cid:88) ijkl J ijkl ψ i ψ j ψ k ψ l (2.13)This theory taken literary is obviously pathological, as ψ i operators square to zero andlead to null states. However, the exact 2-functions are given by conformal solutionsproportional to the one in eq. (2.7). We proceed by perturbing this theory by a setof irrelevant operators which are meant to mimic the kinetic term: L SY K = L conf + (cid:88) h α h O h (2.14)The most important operator in this sum is h = 2 operator: O h =2 = ψ i ∂ u ψ i (2.15)However, there are other terms with higher conformal dimensions. Notice that allof them come with unknown f coefficients α h . Therefore one should be very carefulin translating the operators in the UV Lagrangian to IR expansion in eq. (2.14).Specifically, we expect that in our case some α h are non-linear in ξ .Operator with h = 2 gives rise to Schwartzian and has to be treated separately.We can try to treat other, h (cid:54) = 2 operators O h in our model in a perturbative fashion.Naively, the leading contribution to the free energy comes from dressing the two-pointfunction (cid:104)O h O h (cid:105) with reparametrizations: (cid:104)O h ( u ) O h ( u ) (cid:105) ∝ u − u ) h → (cid:18) τ (cid:48) ( u ) τ (cid:48) ( u )( τ ( u ) − τ ( u )) (cid:19) h (2.16)This leads to a non-local action for reparametrizations (1.2) with some unknowncoefficient α S h . Crucially, the above computation assumes that 1-pt function (cid:104)O h (cid:105) vanishes.Let us now describe elementary consequences of this. As long as temperatures arenot too low, T (cid:29) J/N / (2 h − , the action (1.2) can be treated classically because ofthe overall factor of N . It is easy to check that the thermal solution is the same as inthe Schwartzian case: τ ( u ) = tan( πu/β ). Plugging this solution into the Schwartzianaction trivially yields the following free energy:∆ F Sch /N = − π α SSch J T → ∆ E Sch /N = 2 π α SSch J T (2.17) f To the best of our knowledge, there are no recipes for computing them ab initio . One possibilityin the original SYK is to find them in 1 /q expansion. Unfortunately, for our coupled model large q limit is more complicated. – 9 –he non-local action requires a bit more work. Assuming a fixed energy cutoff at ∼ J , naive evaluation of the action yields a divergent term − β ∆ F nonloc /N = α S h J h − β (cid:90) d (cid:101) u (cid:18) πβ sin ( π (cid:101) u ) (cid:19) h = T + T h − α S h π h − / J h − Γ (1 / − h )Γ (1 − h )(2.18)Fortunately, this divergent term is proportional to 1 /T , hence it is simply a shift inthe ground state energy [25]. Throughout the paper we will be using the followingnotation for the free energy: F/N = E /N − T S − f h T h − − f Sch T + . . . (2.19)and energy: E/N = E /N + c h T h − + c Sch T + . . . (2.20)These coefficients are related by: c Sch = f Sch , c h = (2 h − f h (2.21)In this notation eq. (2.18) says that f h = α S h π h − / J h − Γ (1 / − h )Γ (1 − h ) (2.22) T − − − − E n er gy × − − T α = ξ = Figure 2 . Energy vs T for J = 2 π . Blue points are numerical data. For ξ = 0 we expectSchwartzian answer. We can easily check predictions from the Schwartzian and the non-local actionagainst the numerical solution of SD equations. Using by now standard methods– 10 –f solving SD equation in Euclidean time, we plotted energy versus temperaturesquared T . First consider the benchmark case with ξ = 0 and α = 1 . ξ = 0, we expect Schwartzian answer. We see that the energy is indeedproportional to T . We have performed this analysis for a wide range of β between50 and 500(not shown) and verified that the energy stays proportional to T . Nowwe switch to non-zero ξ = 0 . T law. Toquantify it, we have fitted the data with eq. (2.20) keeping g c Sch , c h and power h unknown(i.e. they are extracted from the data). We see that h best are very close totheoretical values. Analysis at other values of α (not shown) lead to similar results. T − − − − − − − E n er gy × − − T T + T h − , h best = α = ξ = T − − − − − − − E n er gy × − − T T + T h − , h best = α = ξ = Figure 3 . Energy vs T for J = 2 π . Blue points are numerical data. We see a cleardeviation from the Schwartzian prediction(dashed green is a straight line to guide the eye).For α = 1 . h theor = 1 . 24 and for α = 1 . h theor = 1 . 31. Changing the number ofdiscretization points, temperature range and removing c Sch T term from the fit produces h best = 1 . ± . 04 for α = 1 . h best = 1 . ± . 02 for α = 1 . ξ dependence As we have mentioned before, we do not really know ξ -dependence of coefficients α h in the expansion (2.14). We addressed this question by numerically extractingcoefficients c h and c Sch in the energy, eq. (2.20) for different values of ξ . It ischallenging to perform this computation because time discretization has to be smallerthan the inverse J eff = J/ (1 − ξ ) , which becomes very big for ξ → 1. This is whywe plotted c Sch , c h vs ξ for different number of discretization points to make sure weconverge. The results are presented in Figure 4. We see that both c Sch and c h start g It is computationally costly to go to very low temperatures, therefore we have included thesubleading Schwartzian c Sch T term. By performing a fit with and without it one can estimate theuncertainty in h . – 11 – .0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ξ c S c h α = J = π ξ f it f or ξ c h α = J = π ξ f it f or Figure 4 . Coefficients c Sch (Left) and c h (Right) versus ξ for different number of discretiza-tion points 2 , . quadratic but then deviate from ξ law. For large ξ the dependence becomes slowerthan quadratic.The Schwartzian coefficient c Sch decreases with ξ . From JT gravity perspective,adding ξ terms introduces extra light matter in the bulk. We can try to comparethis result to a similar problem: Schwartzian coupled to 2D CFT. This problem isexactly soluble [31] and CFT does lower Schwartzian coefficient. This Section is dedicated to various physical properties of the non-local action. Ev-erywhere, except Section 3.5, we assume that N is large and the temperatures arenot too low( T (cid:29) J/N / (2 h − ) so that the non-local action can be treated classically.In Section 3.5 we compute leading N correction to the free energy, which amountsto 1-loop computation. We do not perform any numerics here.In many places we will need the form of quadratic fluctuations around the ther-mal solution. Expanding the non-local action (1.2) near the zero-temperature solu-tion τ ( u ) = u + (cid:15) ( u ) would yield S nonloc,β = ∞ ∝ (cid:90) dp (cid:15) ( p ) | p | h +1 (cid:15) ( − p ) (3.1)However, we are interested in the finite temperature case τ ( u ) = tan ( π ( u + β(cid:15) ( u )) /β ).In this case the fluctuations can be expanded in Fourier modes (cid:15) = (cid:80) n (cid:15) n e πiun/β – 12 –iving rise to the following action h S nonloc,β = N α S h m h ( βJ ) h − (cid:88) n (cid:15) n g h ( n ) (cid:15) − n (3.2)with g h ( n ) = n (cid:18) Γ( n + h )Γ(1 + n − h ) − Γ( h − − h ) (cid:19) (3.3)and numerical coefficient m h is m h = − (2 π ) h +1 π ( h − cos( πh )Γ(2 h ) (3.4)For large n we expect zero-temperature answer g h ( n ) ∝ n h +1 . However even atsmall n this is a good approximation. For bookkeeping, (cid:15) ( u ) and (cid:15) n will always bedimensionless. For a warm-up, let us start from the zero-temperature entropy. Recall that the resid-ual entropy at T = 0 can be computed i from evaluating Tr log G on the conformalsolution [4]. In the model we are considering the conformal solution is exactly thesame as in SYK model. Therefore the residual entropy is just twice Majorana SYKresidual entropy: S = 2 S ,SY K (3.5) S ,SY K = (cid:90) / dx πx tan( πx ) = 0 . . . . (3.6)The actual residual entropy is N times this, eq. (2.19). Our numerical results areconsistent with this prediction. In this Section we show that the non-local action (1.2) leads to a maximal chaosexponent [10] in the out-of-time ordered correlation(OTOC) function.The OTOC can be computed as follows. Since the reparametrizations is theonly dominant physical mode at low energies, we need to dress the product of two2-point functions with reparametrizations and average over them. In SYK one hasto use the Schwartzian action (1.1), however in our case it is the non-local action h We are grateful to D. Stanford and Z. Yang for the discussion about this computation and thesubsequent chaos exponent computation. i Modulo some UV subtleties – 13 –3.2). Leading 1 /N contribution comes from using the linearized action (3.2): F G conf ( x ) G conf ( x (cid:48) ) = (cid:104) ψ i ( θ ) ψ i ( θ ) ψ j ( θ ) ψ j ( θ ) (cid:105) conn G conf ( x ) G conf ( x (cid:48) ) == ( βJ ) h − N π − h m h α S h (cid:88) | n |≥ e in ( y (cid:48) − y ) g h ( n ) (cid:20) sin nx tan x − n cos nx (cid:21) (cid:34) sin nx (cid:48) tan x (cid:48) − n cos nx (cid:48) (cid:35) (3.7)where θ i are angle variables on the thermal circle θ = 2 πu/β and y, y (cid:48) , x, x (cid:48) are certaincombinations of angles: x = θ − θ , x (cid:48) = θ − θ , y = θ + θ , y (cid:48) = θ + θ βJ ) h − enhancement.In general this expression is complicated for ordering θ < θ < θ < θ which isrelevant for OTOC. Fortunately, it simplifies a lot when the points are antipodal onthe circle. Specifically, we put θ = − π − θ, θ = 0 , θ = π − θ, θ = π (3.9)which corresponds to x = x (cid:48) = − π, y = − θ, y (cid:48) = π (cid:88) | n |≥ e in ( π/ θ ) n cos πn g h ( n ) (3.11)We see that the sum goes over even n only. We can convert the sum into the integralby introducing a factor 1 / ( e iπn − 1) and integrating over the contour C enclosing ± , ± , . . . : 12 (cid:73) C dn n e iπn − e in ( π/ θ ) g h ( n ) (3.12)Now we can move the contour to infinity, since the integrand decays along the imag-inary axis. It will pick up the pole at n = 0 where e iπn = 1 and at locations where g h ( n ) has zeroes. The zeroes are located at n = 1 , , − n .Poles at negative n are not relevant for us, because after analytically continuing toOTOC, namely θ → − πit/β , they will produce exponentially decaying contribu-tions(or a constant for n = 0). The pole at n = 1 yields j maximal chaos exponent F ( t ) G conf ( π ) G conf ( π ) = − p h ( βJ ) h − N α S h m h exp (cid:18) πtβ (cid:19) + [non-increasing] (3.13) j The coefficient p h here is π − h Γ(2 − h )4Γ(1+ h )( ψ (1+ h ) − ψ (2 − h )) where ψ is Digamma function, ψ ( x ) =Γ (cid:48) ( x ) / Γ( x ) – 14 –he fact that the chaos exponent is still maximal should not be too surprising: latetime asymptotic growth ∼ e πt/β at late times t (cid:29) β can be found directly from theconformal solution [2](and Section 3.6.1 of [11]) by computing the OTOC in the realtime domain. However, the prefactor is parametrically smaller: for Schwartzian itis ( βJ ) . It would be interesting to compute finite βJ corrections to the Lyapunovexponent and see that they satisfy the bound proposed in [32]. It is also interesting to consider time-ordered θ < θ < θ < θ (3.14)4-point function. This computation will highlight a certain difference with SYK: inSYK h = 2 mode(Schwartzian) is the energy operator, whereas in our case it is not.We take the general expression (3.7) for the 4-point function and convert it intoa contour integral:( βJ ) h − N α S h (cid:73) C dn e πin − e in ( y (cid:48) − y ) g h ( n ) (cid:20) sin nx tan x − n cos nx (cid:21) (cid:34) sin nx (cid:48) tan x (cid:48) − n cos nx (cid:48) (cid:35) (3.15)where the contour C encloses ± , ± , . . . . In the time-ordered case one can actuallyclose the contour at infinity and pick up poles of e πin − g h ( n ). In the usualSYK story, g h ( n ) ∝ n ( n − 1) and the only contributing pole is at n = 0. Because ofthis, the leading contribution to the 4-point function depends only on two variables θ − θ and θ − θ ( y, y (cid:48) drop out). Further taking the OPE limit θ → θ , θ → θ willproduce the expression which is independent of θ i at all. This is usually interpretedas follows: the OPE limit has produced the operator ψ∂ τ ψ which is just the stress-energy operator T . Obviously, the correlator (cid:104) T ( θ ) T ( θ ) (cid:105) (3.16)should not depend on θ , θ from the energy conservation. It turns out to be indeedthe case. This allows one to compute the energy-energy correlators in SYK-chainmodels at any frequency | ω | (cid:28) J .Let us return to our expression with complicated g h ( n ). Now we have to take intoaccount poles of g h ( n ) at negative n . Because of that, the OPE limit θ → θ , θ → θ will produce an expression which is θ − θ dependent . We are forced to concludethat ψ∂ τ ψ is no longer proportional to energy. So we cannot extract energy-energycorrelators easily.However, we can still do it, but only in the hydrodynamic regime | ω | (cid:28) /β (cid:28) J [33]. We consider a general time reparametrization (2.8) of the 2-point function. Inthe hydrodynamic regime we keep only the leading derivative of τ ( u ): τ ( u ) = u + (cid:15) (cid:48) βu + . . . (3.17)– 15 –y looking at the form of conformal 2-point function, eq. (2.7), we see that (cid:15) (cid:48) simplyrescales β : δβ = − β (cid:15) (cid:48) (3.18)Now we propagate this change into energy: δE = − β dEdβ (cid:15) (cid:48) (3.19)This way we identify (cid:15) (cid:48) and δE . Hence, knowing the correlators of (cid:15) (cid:48) we can obtainthe correlators of energy. Obviously, correlators of (cid:15) (cid:48) are governed by the non-localaction.One can cross-check this relation. First, from explicitly differentiating the par-tition function one has (cid:104) ( δE ) (cid:105) = − dEdβ (3.20)This implies (cid:104) (cid:15) (cid:48) (cid:15) (cid:48) (cid:105) = 4 π n β (cid:104) (cid:15) n (cid:15) − n (cid:105) = − β dEdβ (3.21)Now, using the explicit (cid:15) propagator (3.2) and the expression for the energy(2.18) one can indeed verify the above relation. The identification between (cid:15) (cid:48) and δE will be very useful when we discuss the chain model. We can also study a chain build from our model and study transport properties. Themodel we will consider is very similar to the ones discussed in the literature [27, 33].It was previously shown that in SYK chain models thermal conductivity and electri-cal resistivity are linear in the temperature, similar to strange metals. Also, whenthe Schwartzian dominates, the diffusion constant is temperature-independent. Inthis Section we will show that once the non-local action becomes dominant, the ther-mal conductivity is still linear in the temperature. However, the diffusion constantbecomes temperature dependent.The model we consider is simply 1D array of independent dots: L T,chain = (cid:88) x L T,x (3.22)where for each x Lagrangian L T,x is given by eq. (1.3). After integrating out thedisorder we get a bunch of non-interacting models (2.5). To make them interact, weadd a tight-binding(in x ) random interaction: L int,chain = 12! (cid:88) x,ijkl (cid:0) V ,xij ; kl ψ i,x ψ j,x ψ k,x +1 ψ l,x +1 + V ,xij ; kl ψ i,x ψ j,x ψ k,x +1 ψ l,x +1 (cid:1) (3.23)– 16 –here each V / ,xij ; kl is skew symmetric in ij and kl : V / ,xij ; kl = − V / ,xji ; kl = − V / ,xij ; lk (3.24)but do not mix the two pairs ij and kl . The full configuration is illustrated in Figure5. Assuming x − independent variance Figure 5 . Illustration of couplings in the chain model. (cid:104) (cid:16) V / ,xij ; kl (cid:17) (cid:105) = V N (3.25)we get the following extra terms in the G Σ-action:∆ S G Σ = V (cid:90) du du (cid:88) x (cid:0) G ,x ( u , u ) G ,x +1 ( u , u ) + G ,x ( u , u ) G ,x +1 ( u , u ) (cid:1) (3.26)We have chosen the interaction term such that it does not lead to mixed G corre-lators, which might cause instability. As usual, choosing x − independent ansatz forSD equations G x, = G , G x, = G (3.27)we arrive at single-site SD equations (2.6) with effective (cid:101) J and (cid:101) α : (cid:101) J = J + V , (cid:101) α = α J J + V (3.28)Now can discuss the kernel and the effective action for reparametrizations. Wewill assume that the reader is familiar with SYK 4-point function computation vialadder diagrams. This computation is pedagogically reviewed in Section 4.1 of thispaper. We have two types of interactions in the chain model: on-site and between-site next-neighbor interaction, eq. (3.23). Performing a Fourier transform in the x space makes the ladder diagrams(and the kernel) depend on momentum p . On-site– 17 –nteraction would produce p -independent part. Next-neighbor interaction will yieldcos( p ) dependence. So that the total kernel is K chain = K ren + K p (3.29) K ren is simply single-site kernel (4.30), but with renormalized J, α : K ren v = (cid:32) (cid:101) J G ∗ (( G + (cid:101) α G ) v + 2 (cid:101) α G G v ) ∗ G (cid:101) J G ∗ (( G + (cid:101) α G ) v + 2 (cid:101) α G G v ) ∗ G (cid:33) (3.30)The remaining part is proportional to cos( p ) − K p v = 2 V (cos( p ) − (cid:18) G ∗ ( G v ) ∗ G G ∗ ( G v ) ∗ G (cid:19) (3.31)Note that K p is not proportional to K ren . So in general it would be hard to findeigenvalues of K chain . Fortunately, K p has the form of SYK kernel and in the leadingconformal approximation G , G are proportional to the standard SYK conformalsolution. So reparametrizations of G again produce the kernel eigenvector with theeigenvalue close to 1. In the small p limit this is enough for us, since this term isproportional to 1 − cos( p ) ≈ p / K ren is analysed in Section 4.3 in detail. The upshot is that at | (cid:101) α | > (cid:101) α , not α . Putting this togetherwe learn that the leading(in 1 /β (cid:101) J and p ) eigenvalue shift for reparametrizations is1 − k (2 , n, p ) = α K h ( β (cid:101) J ) h − g h ( n ) | n | ( n − 1) + p V (cid:101) J (1 + 3 (cid:101) α ) (3.32)So that the action for infinitesimal reparametrizations is given by k S = π b N (cid:88) n,p (cid:15) n,p (cid:32) α K h ( β (cid:101) J ) h − g h ( n ) + p | n | ( n − V (cid:101) J (1 + 3 (cid:101) α ) (cid:33) (cid:15) − n, − p (3.33)One final step is to switch to Minkowski space and consider the limit of small timefrequencies. Recall the analytic expression eq. (3.3) for g h ( n ). We need to continue in → βω π and consider the limit ω → β . This way only the factor n in g h ( n ) gives a finite contribution. Putting everything together we get: S hydro = π b N β π (cid:90) dωdp (cid:15) ω,p (cid:32) α K h ( β (cid:101) J ) h − (2 h − h )Γ(2 − h ) ω β π + ip ωβ π V (cid:101) J (1 + 3 (cid:101) α ) (cid:33) (cid:15) − ω, − p (3.34) k The overall factor can be determined from requiring that p = 0 reparametrizations reproducethe answer for a single copy of our model, see Sections 4.1, 4.4 of this paper for a detailed discussion. – 18 –imilarly to a single-site case, we identify ∂ t (cid:15) x with energy at site x , eq. (3.19). Wesee a typical diffusion pole ω + iDp in the energy-energy correlator l . The diffusionconstant is given by D = T − h π Γ(2 − h )3(2 h − h ) V (cid:101) J h − α K h (cid:101) J (1 + 3 (cid:101) α ) (3.35)A few comments are in order.The important part is the temperature dependence T − h . Schwartzian yieldstemperature-independent diffusion constant [27, 33]. Here, the dimension h of theirrelevant operator controls the temperature power.Using the identification (3.19) between the energy and reparametrizations, onecan also compute the energy-energy correlator and extract the thermal conductivity.It is proportional to the specific heat, eq. (2.20): c v = N T h − (2 h − h − α S h π h − / (cid:101) J h − Γ (1 / − h )Γ (1 − h ) (3.36)times the diffusion constant, eq. (3.35). Thus it is linear in the temperature as inthe chains where Schwartzian dominate: κ = c v D ∝ N T (3.37)In the conventional SYK chain [27] the butterfly velocity is v B = 2 πDT (3.38)which agrees with the holographic expectations [34]. However, careful computationof the butterfly velocity requires the knowledge of the subleading correction to the4-point function [27]. In our case this seems complicated because the two kernels K ren and K p are not the same. Nonetheless, we conjecture that in our model therelation (3.38) still holds. Physically it is motivated by the fact that the samemode(reparametrizations) governs both OTOC chaos exponent and energy diffusion.This happens in the conventional SYK chains too. On the computational level,ignoring the subleading correction to 4-point function results in picking the pole at ω + iDp = 0, leading to eq. (3.38).Unfortunately, we cannot study electric conductivity in our model because we donot have U (1) symmetry. There are two ways to introduce U (1) symmetry. We cansimply promote the Majorana fermions to complex fermions. In this case one hasto study operators in the symmetric sector which is not related to the Schwartzian.For example, in complex SYK [23] fluctuations in U (1) phase are governed by simple U (1)-sigma model. However, there are also t-J models where resistivity is relatedto time reparametrization mode [35]. It would be interesting to see how the changefrom Schwartzian to non-local action affects the transport in t-J models. We leavethis question for future work. l Note that p is dimensionless in our conventions. – 19 – .5 N correction In this Section we will compute N correction to the free energy and try to infer thedensity of states ρ ( E ) near the ground state. We would like to emphasize that in thisSection we will be interested in the energies close to ground state: | E − E | ∼ N .Whereas the thermodynamic results in eq. (2.18) corresponded to | E − E | ∼ N .Also we will discuss below the validity of our N computation.We can easily compute N correction to the free energy. Again, we will needsome knowledge about the kernel eigenvalues so we refer to Sections 4.1 and 4.4 forthe detailed discussion. The N correction is given by the determinant of fluctuationsaround the thermal solution. Since reparametrizations are enhanced we expect thatthey will dominate in the determinant too. It can be argued diagramatically [36]and by path-integral techniques [11] that N correction to log Z is given the sum ofkernel eigenvalues − (cid:88) h,n log (1 − k ( h, n )) (3.39)Obviously, the leading contribution will come from k close to 1, which are exactlyreparametrizations.As usual, let us start from the standard SYK case, where the Schwartzian dom-inates. Then the eigenvalue shift is proportional to, eq. (4.16):(1 − k (2 , n )) Sch ∝ | n | βJ , | n | ≥ n has to be cut at n ∼ βJ . This produces the following answer:(log Z ) Sch, − loop ∝ β − 32 log ( βJ ) (3.41)The term proportional to β has unknown coefficient, but it simply gives the shift tothe ground state energy.Let us discuss the non-local action now. From the linearized action (3.2), weexpect the eigenvalue shift to be, eq. (4.42):(1 − k (2 , n )) nonloc ∝ βJ ) h − g h ( n ) | n | ( n − 1) (3.42)Recall that g h ( n ) is given by eq. (3.3). This sum is harder to evaluate. It can besimplified by noticing that one can separate the first term in the parenthesis: g h ( n ) = n Γ( n + h )Γ(1 + n − h ) η h ( n ) (3.43)with η h ( n ) = (cid:18) − Γ( h − n − h )Γ( − h )Γ( n + h ) (cid:19) (3.44)– 20 –e see that at large n , η h ∝ − /n h − hence the sum (cid:80) | n |≥ log( η h ) actually con-verges and gives something of order ( βJ ) . We are not interested in this contribution.Now it is possible to evaluate the sum (cid:80) ≤| n |≤ βJ log ( g h ( n ) / ( | n | ( n − η h ( n ))). Theanswer is (log Z ) nonloc, − loop ∝ β − 32 (2 h − 2) log ( βJ ) (3.45)We can try to convert this into the energy density by doing the inverse Laplacetransform: ρ ( E ) = (cid:90) dβ Z ( β ) e − βE (3.46)In this equation the energy E is measured from the ground state and it includesa factor of N . The most interesting regime, which actually can be probed withexact diagonalization is the density near the ground state, E ∼ N J . However, onehas to be extremely careful with the range of validity of (3.41) and (3.45). In thisregime the above integral is dominated by β ∼ N/J (in the Schwartzian case) andby β ∼ N / (2 h − /J (in the non-local case) and we cannot trust the above 1-loopcomputation anymore.Fortunately for the Schwartzian, it is 1-loop exact [37], so we can actually trusteq. (3.41) and obtain square-root edge: ρ ( E ) Sch, − loop = ρ ( E ) Sch,exact ∝ √ E (3.47)Unfortunately, we do not know if the non-local action has the same property.Naively using 1-loop result (3.45) we get ρ ( E ) nonloc, − loop ∝ E h − (3.48) In fact, our exact diagonalization results at finite N do not do not supportthis result. This suggests that the non-local action partition function is not 1-loopexact. We will discuss this more in Section 5. This Section is entirely devoted to numerical, but ab initio • Write down ladder diagrams and find the corresponding kernel.– 21 – Find the leading non-conformal correction to the 2-point function. • Compute the kernel eigenvalue shift coming from this correction. • Interpret the answer as an integral over reparametrizations with some action. Let us recall how Maldacena–Stanford(MS) [11] argued that in SYK the low-energyphysics is dominated by the Schwartzian action. Standard SYK has the followingHamiltonian: H SY K = 14! (cid:88) ijkl J ijkl ψ i ψ j ψ k ψ l , (cid:104) J ijkl (cid:105) = 3! J N (4.1)Summation of melonic diagrams lead to (Euclidean) Schwinger–Dyson equations forthe two-point function G ( u − u ) ≡ G (12) = (cid:104) T ψ i ( u ) ψ i ( u ) (cid:105) :( − iω n − Σ( ω n )) G ( ω n ) = 1 , ω n = 2 πβ (cid:18) n + 12 (cid:19) (4.2)Σ( u ) = J G ( u ) (4.3)In the strict large N these equations are exact. Compared to the rest of the paper,here the conformal solution differs by a factor of √ α : G conf = b sgn( u ) π (cid:114) J β sin (cid:16) π | u | β (cid:17) , /J (cid:28) | u | , βJ (cid:29) Figure 6 . First few ladder diagrams contributing to the (connected) 4-point function.Solid lines are fermionic propagators and dashed lines indicate disorder contractions. (cid:104) ψ i ( θ ) ψ i ( θ ) ψ j ( θ ) ψ j ( θ ) (cid:105) = G (12) G (34) + 1 N F (4.5) (cid:104) ψ i ( θ ) ψ i ( θ ) ψ j ( θ ) ψ j ( θ ) (cid:105) conn = 1 N F ( θ , . . . , θ ) = 1 N − K F (4.6)where F is the leading (connected) 4-point function: F (12; 34) = − G (13) G (24) + G (14) G (23) (4.7)– 22 –xplicitly the kernel K = K (12; 34) is given by K (12; 34) = − J G (13) G (24) G (34) (4.8)The kernel acts by convolution with the last two (34) variables. For example, thenext-to-leading answer F is F (12; 34) = K F = (cid:90) d (cid:48) d (cid:48) K (12; 3 (cid:48) (cid:48) ) [ − G (3 (cid:48) G (4 (cid:48) 4) + G (3 (cid:48) G (4 (cid:48) F in the basis of eigenfunctions of K . Schemat-ically: F = (cid:88) k − k (cid:104) Ψ k |F (cid:105)(cid:104) Ψ k | Ψ k (cid:105) Ψ k (4.10)It turned out, in the conformal limit K has a set of eigenvalues equal to one: k (2 , n ) =1. The corresponding eigenfunctions are proportional to reparametrizations of G .The convenient basis of reparametrizations is u → u + β (cid:88) n (cid:15) n e − πinu/β (4.11)They give enhanced contribution to the 4-point function(4-pt) ⊃ N π b (cid:88) n | n | ( n − k (2 , n )1 − k (2 , n ) δ (cid:15) n G δ (cid:15) − n G (4.12)with δ (cid:15) n G ( u − u ) = πG ( u − u ) (cid:34) sin nπuβ tan πuβ − n cos nπuβ (cid:35) e iπn ( u + u ) /β (4.13)The factor 2 k (2 , n ) / ( b ) came from the overlap between the F and the kernel eigen-vector and π | n | ( n − 1) came from the normalization of δ (cid:15) n G . The leading orderanswer can be obtained by taking the conformal limit answer everywhere expect in1 − k (2 , n ). The difference 1 − k (2 , n ) is determined by the leading non-conformalcorrection δG to G . By analysing the large q limit MS argued that the leadingcorrection goes as 1 /βJ : δGG conf = − α GSch βJ f (4.14)Function f is given by f = 2 + π − | θ | tan | θ | , θ = 2 πu/β (4.15)In the next sub-Section we will describe how to find these corrections in a systematicway. – 23 –rom now on it is straightforward(but tedious) to find 1 − k (2 , n ). MS did itanalytically and found: k (2 , n ) = 1 − α KSch | n | βJ (4.16)Hence (4-pt) ⊃ N βJα KSch π b (cid:88) n n ( n − δ (cid:15) n G δ (cid:15) − n G (4.17)This answer can be understood as follows. We start from the leading (disconnected)contribution to the 4-point function, eq. (4.5): G ( u − u ) G ( u − u ) (4.18)where both Green functions are taken at finite (inverse) temperature β . Then wedress it with an infinitesimal reparametrization u → u + β(cid:15) ( u ) and average over (cid:15) with the action S Sch = NβJ π α SSch (cid:88) n (cid:15) n n ( n − (cid:15) − n , α SSch = α KSch b π 32 (4.19)Now, if we take the Schwartzian action S = − N α SSch J (cid:90) du Sch( τ [ u ] , u ) (4.20)and expand it near the thermal solution: τ ( u ) = tan (cid:18) πβ ( u + β(cid:15) ( u )) (cid:19) (4.21)we get exactly the action (4.19). Hence the Schwartzian reproduces the correctanswer for the 4-point function. Notice that it has the correct n dependence andcorrect βJ dependence. In the previous Section we promised to present a general approach for computingcorrections to the conformal 2-point function. This approach is nothing than a simpleconformal perturbation theory associated with the deformation (2.14). For example[30], the leading correction from operator O h is given by the 3-point function:( δG ) h ( u ) = α h (cid:90) du (cid:48) (cid:104)O h ( u (cid:48) ) ψ i ( u ) ψ i (0) (cid:105) ∝ J u ) h − / (4.22)Answer f h = u / − h is valid either at zero temperature or in the regime 1 /J (cid:28) | u | (cid:28) β . In general this correction is given by a hypergeometric function. In principle, onecan compute even the second-order correction [30]:( δ G ) h h ( u ) = α h α h (cid:90) du du (cid:104)O h ( u ) O h ( u ) ψ i ( u ) ψ i (0) (cid:105) ∝ J u ) h + h − / (4.23)– 24 –gain, the final answer f h + h ∝ /u h + h − / is valid for 1 /J (cid:28) | u | (cid:28) β .It can be shown that the leading correction in SYK(eq. (4.14)) comes from h = 2operator. However, we see right away that any operator with h < h < / 2) will dominate over this h = 2. We would like to verify this statement inour coupled model.Given the simplicity of β = ∞ answers, we will find the exact 2-point functionnumerically at zero temperature in Lorentzian time . The procedure is described inAppendix A. In particular, we will examine the spectral density: ρ / = Im G R, / ( ω ) (4.24)where G R is retarded 2-point function and ω is real frequency. From eqns. (4.22)and (4.23) ρ has the following expansion: ρ / × √ ω = const + (cid:88) h A / h ω h − + (cid:88) h ,h B / h h ω h + h − + . . . (4.25)We have multiplied ρ by √ ω because the leading conformal answer goes as 1 / √ ω .We will concentrate on the leading ω h − correction. We expect it to be ξ -odd. Soit should have a different sign for G and G . Our strategy is the following: find ρ / √ ω at a given J and α and perform the fit with b + b ω h − (4.26)with unknown b , b in two ways: put h = h theor ( α ) obtained from eq. (2.10) or allow h to be inferred from the data. In other words, perform the fit with unknown h andobtain h best . The uncertainty in h best arises from changing the fitting interval in ω .One important thing to notice is that h best tends to overestimate h by about 0 . • Schwartzian benchmark: α = 0Here for α = 0 the two systems decouple and for any ξ we have two independentSYK. The result for α = ξ = 0 is presented in Figure 7. We indeed see thatfor small ω there is a linear term coming from the Schwartzian: ρ / √ ω = b + b ω (4.27) • | α | > 1: Here operator O , , eq. (2.9), has the dimension in the interval1 < h < / 2, as can be easily seen from eq. (2.10). The results for G arepresented in Figure 8. To make the graphs more expressive we took ratherlarge ξ = 0 . 9. Note that the leading correction has to be ξ -odd. Therefore weexpect ρ to curve in different directions for 11. This is indeed the case as canbe seen from Figure 9. – 25 – .0 0.5 1.0 1.5 ω s pe c t r a l den s i t y ρ × √ ω J = α = ξ = Large N SD h best = ± h theor = Figure 7 . Spectral density for original SYK. For comparison we have fitted using thetheoretical value h theor = 2 and arbitrary h . The fit was performed with b + b ω h − . ω s pe c t r a l den s i t y ρ × √ ω J = α = ξ = Large N SD h best = ± h theor = ω s pe c t r a l den s i t y ρ × √ ω J = α = ξ = Large N SD h best = ± h theor = Figure 8 . Results for G , α > 1. The fit was performed with b + b ω h − . • | α | < 1: Now O , has the dimension 3 / < h < 2. In this interval we do notexpect the non-local action to dominate in the 4-point function or free energy.However, it still should dominate in the non-conformal correction. The resultsare presented in Figure 10. Again, to make the graphs more expressive we tookrather large ξ = 0 . 9. – 26 – .0 0.2 0.4 0.6 0.8 ω s pe c t r a l den s i t y ρ × √ ω J = α = ξ = Large N SD h best = ± h theor = ω s pe c t r a l den s i t y ρ × √ ω J = α = ξ = Large N SD h best = ± h theor = Figure 9 . Results for G , | α | > 1. The fit was performed with b + b ω h − . ω s pe c t r a l den s i t y ρ × √ ω J = α = ξ = Large N SD h best = ± h theor = ω s pe c t r a l den s i t y ρ × √ ω J = α = ξ = Large N SD h best = ± h theor = Figure 10 . Results for G , | α | < 1. The fit was performed with b + b ω h − . Let us now discuss the kernel for our coupled model. In this Section we put β = 2 π .It is straightforward to draw ladder diagrams. However, the most convenient way toderive the kernel is to start from the conformal SD equations: − J ( G + 3 α G G ) ∗ G = δ ( u ) − J ( G + 3 α G G ) ∗ G = δ ( u ) (4.28)– 27 –here ∗ means convolution in the Matsubara time domain, and perturb them by G → G + v . The equations we obtain this way are Kv = v (4.29)where K is the kernel and v = ( v , v ) is the vector the kernel acts on. Explicitly wehave m Kv = (cid:18) J G ∗ (( G + α G ) v + 2 α G G v ) ∗ G J G ∗ (( G + α G ) v + 2 α G G v ) ∗ G (cid:19) (4.30)Correspondingly the kernel is 2 × K = K ab ( u , u ; u , u ) (4.31)Four-point function F ab is also 2 × F ab = (cid:18) (cid:104) ψ ψ ψ ψ (cid:105) (cid:104) ψ ψ ψ ψ (cid:105)(cid:104) ψ ψ ψ ψ (cid:105) (cid:104) ψ ψ ψ ψ (cid:105) (cid:19) (4.32)We can write down the expression identical to eq. (4.10), which says that F ∝ (1 − K ) − F . We will use the conformal solution everywhere except in 1 − k . Thisway the conformal kernel is proportional to SYK kernel: K conf,ab (12; 34) = − J G conf (13) G conf (24) G conf (34) (cid:18) α α α α (cid:19) (4.33)Eigenvalue 1 eigenvector Ψ k =1 corresponds to reparametrizations. Because of non-zero α , G and G have to be reparametrized the same way, so Ψ k =1 has equalcomponents: Ψ k =1 ∝ (cid:18) (cid:19) (4.34)It is indeed easy to see that this vector is an eigenvector n of the conformal kernelmatrix in eq. (4.33). Because of this, the leading answers for all four 4-point functionin eq. (4.32) are going to be the same. Notice that the kernel does not contain ξ explicitly, as it is determined by the ladder diagrams. The ξ is actually present innon-conformal correction to G and hence in the eigenvalue shift 1 − k (2 , n ). In thecoupled model this correction is: G exact, / = G conf + δG / (4.35) m Possible sign difference(overall plus instead of minus) is related the last G - it has two time-arguments exchanged compared to eq. (4.8). n The eigenvalue 1 + 3 α will conveniently cancel with the extra 1 / (1 + 3 α ) in the conformalsolution (2.7). – 28 –et us discuss which of the terms in δG contribute to the eigenvalue shift k (2 , n ) conf − k (2 , n ) exact = 1 − k (2 , n ) exact . Since the conformal G / do not de-pend on ξ , eq. (2.7), they and the conformal kernel have Z symmetry 11 ↔ (cid:104) Ψ k =1 | δK | Ψ k =1 (cid:105) .The leading ( βJ ) h − correction to G comes from computing the 3-pt function, eq.(4.22), where in our case the lightest operator is O h = − ξ (cid:88) i (cid:0) ψ i ∂ u ψ i − ψ i ∂ u ψ i (cid:1) (4.36)Crucially, it is linear in ξ . In other words, it contributes with different signs to δG and δG : δG G conf = − α Gh ( βJ ) h − f h (4.37) δG G conf = + α Gh ( βJ ) h − f h (4.38)Because of this asymmetry the leading correction from f h , (cid:104) Ψ k =1 | δK | Ψ k =1 (cid:105) will van-ish. Including the subleading correction (4.23) we have δG / G conf = ∓ α Gh ( βJ ) h − f h − α G h ( βJ ) h − f h + h − α GSch βJ f + . . . (4.39)Therefore, the leading δG correction f h can only contribute to δk starting at quadraticorder. However, it can mix with f h + h : it is even in ξ , and hence can contribute tothe eigenvalue shift in the leading order. This is why the analytic computation ofthe eigenvalue shift seems very difficult and we resort to numerics.Note that because of the original Z symmetry at ξ = 0, the one point function (cid:104)O h (cid:105) vanishes(in the first order of perturbation theory). So the absence of ( βJ ) h − term in the MSY computation and in the kernel computation has the same origin. We will diagonalize the kernel and find the eigenvalues closest to 1 following ref. [23].It can be done numerically by introducing a 2D grid. We will fix β = 2 π and studydifferent J . Index n arises by noticing that the kernel is invariant under translationsand so n is the momentum: K n,ab ( u, u (cid:48) ) = (cid:90) π ds K ab (cid:18) s + u , s − u u (cid:48) , − u (cid:48) (cid:19) e − ins (4.40)Since we are interested in the asymmetric kernel, it would be convenient to anti-symmetrize u, u (cid:48) explicitly: K An,ab ( u, u (cid:48) ) = 12 ( K n,ab ( u, u (cid:48) ) − K n,ab ( u (cid:48) , u )) , (4.41)– 29 – 20 40 60 80 1000.9750.9800.9850.9900.9951.000 sn , s = × − J = single SYK sn , s = × − J = single SYK Figure 11 . k (2 , n ) versus n plot for J = 2500 and J = 3300 single SYK. Red line is theliner fit to guide the eye. it improves the numerical results.Let us start from a single SYK as a benchmark - Figure 11. We see a perfectagreement with theoretical prediction 1 − k (2 , n ) ∝ n .Now we need to understand what kind of eigenvalue shift we expect from thenon-local action. From the SYK discussion in Section 4.1 and eq. (3.2) it followsthat the eigenvalue shift is determined by g h ( n ) / ( | n | ( n − − k (2 , n )) nonloc = α K h ( βJ ) h − g h ( n ) | n | ( n − 1) (4.42)Coefficients α K h and α S h are related by α K h = 4 m h π b α S h (4.43)where m h is given by eq. (3.4). Therefore, at large n we expect the followingbehavior: (1 − k (2 , n )) nonloc = α K h n h − ( βJ ) h − (4.44)In fact, for our range of h , g h ( n ) is almost indistinguishable from a power-law exceptfor the first few n . This motivates us to try to fit our results with a combination ofa linear piece n (Schwartzian) and n h − (non-local). For h in the range 1 < h < / n h − is less than 1. It means that for large n and h not too close to 3 / n . In order to check this we have plotted k (2 , n ) for n = 2 , . . . , 100 for various values of J and α .– 30 –et us consider α = 1 . α = 1 . α = 1 . n (left side inthe plots). Presumably for large n the Schwartzian piece starts to dominate. Naivelog-log plot is not very instructive for two reasons: non-local contribution is notexactly a power-law and also we have a mixed expression with the linear Schwartziancontribution and the non-linear piece (4.44). Let us describe in detail Figures 12, 13,14. We start from the left part, which is k (2 , n ): • Blue dots are numerically obtained k (2 , n ). • We could try to extract the linear term at large n by fitting k (2 , n ) with a line sn (red line), keeping s unknown. However, it turned out that for our range of βJ the non-linear piece is still not negligible even at large n ∼ • So in order to extract the linear piece properly we perform the fit with thenon-linear piece (4.44) as well:1 − (cid:101) sn − (cid:101) An h theor − , (4.45)with unknown (cid:101) s, (cid:101) A and where h theor is the theoretical value of the scalingdimension. This is the orange curve. As we see from the plots, slope (cid:101) s isnot close to naive slope s . This means that the nonlinear piece is indeed notnegligible. We will use this slope (cid:101) s to subtract it from k (2 , n ) and compare theresult with the full non-local prediction g h ( n ) / ( | n | ( n − • To double-check that we are not overfitting by introducing to many parameterswe perform a fit with unknown ˆ s, ˆ A and unknown power in the non-linear part:1 − ˆ sn − ˆ An h − (4.46)This is green curve. In most cases it is indistinguishable from the orange one.The best value of h = h best is close to the theoretical h theor . From using 2 − points for solving SD equation and 60 , − , 2D grid points for findingthe kernel eigenvalues, we can estimate the uncertainty in h best . We see that h theor and h best are within the uncertainty. As one can observe from the zero-temperature plots of the spectral function, Figures 8, 10, the numerics seemto overestimate h . We attribute the difference h best − h theor to this systematicoverestimation.The right side is less intricate: we subtract (cid:101) sn from k (2 , n ) and compare the resultwith the full non-local shift g h ( n ) / ( | n | ( n − | α | < 1. In this case weexpect that the Schwartzian does dominate for large βJ :1 − k (2 , n ) = α KSch | n | βJ + α K h ( βJ ) h − | n | h − , n (cid:29) 20 40 60 80 100 n k ( n ) sn , s = × − ˜ sn + ˜ An h theor − , ˜ s = − × − h best = n − k ( n ) + ˜ sn − k ( n ) + ˜ sn full non-local J = α = ξ = Figure 12 . Results for k (2 , n ). Details can be found in the main text. For α = 1 . h theor = 1 . 45, whereas h best = 1 . ± . n k ( n ) sn , s = × − ˜ sn + ˜ An h theor − , ˜ s = × − h best = n − k ( n ) + ˜ sn − k ( n ) + ˜ sn full non-local J = α = ξ = Figure 13 . Results for k (2 , n ). Details can be found in the main text. For α = 1 . h theor = 1 . 31, whereas h best = 1 . ± . where now h > / 2. It means that presumably at small n the Schwartzian dominatesand then for large n the non-local piece starts to win. Moreover, from the analytic– 32 – 20 40 60 80 100 n k ( n ) sn , s = × − ˜ sn + ˜ An h theor − , ˜ s = × − h best = n − k ( n ) + ˜ sn − k ( n ) + ˜ sn full non-local J = α = ξ = Figure 14 . Results for k (2 , n ). Details can be found in the main text. For α = 1 . h theor = 1 . 24, whereas h best = 1 . ± . expression (3.3) we expect that α K h is now negative , so 1 − k (2 , n ) will still curvedownwards. We considered α = 0 . k (2 , n ) with eq. (4.47), keeping h unknown - Figure 15. We again see a very good agreement with theoretical results. Finally, the non-local action predicts that the non-linear term n h − in the eigenvalueshift behaves as 1 / ( βJ ) h − , eq. (4.44). The fitting strategy outlined in the previousSection allowed us to extract this coefficient. We considered α = 1 . α = 1 . J . After that, we fitted the result with cJ h − (4.48)keeping c and h unknown. The results are presented in Figure 16. We see that h best is again very close to the theoretical value. One can also check that the resulting α K h agrees well with c h in Figure 4. This computation requires using the conversions(4.43) and (2.22). N One of the nice feature of SYK-like models in the opportunity to study finite- N effects using exact diagonalization of the Hamiltonian. In our case the dimension of– 33 – 20 40 60 80 n k ( n ) sn , s = × − ˜ sn + An h theor − , ˜ s = × − h best = s = × − J = α = ξ = Figure 15 . Results for k (2 , n ). Details can be found in the main text. For α = 0 . h theor = 1 . 79, whereas h best = 1 . 400 500 600 700 J α K h vs J , β = π , α = large N numerics J h − , h best = J α K h vs J , β = π , α = large N numerics J h − , h best = Figure 16 . Coefficient α K h (here for convenience we included βJ inside α K h compared toeq. (4.44)) as a function of J . Left: α = 1 . h theor = 1 . 45. Right: α = 1 . h theor = 1 . the Hilbert space is dim H = 2 N (5.1)– 34 –o we can easily consider N up to 16 without using any special techniques. Similarcomputations for the case of SYK has been done in the literature before [11], [38],[18], [29]. We have performed finite N exact diagonalization for four reasons: • Cross-check our infinite N solutions of SD equations. • Probe the density of states near the ground state and see if it differs from the1-loop result (3.48). • See how the 2-point function behaves at very late times τ (cid:29) N/J . • Check if the spectral correlators obey random matrix theory predictions. A de-viation from them would indicate possible spin–glass phase at low temperatures[18].As a starter we present the full spectrum binned with 300 bins for a single realiza-tion of disorder. Figure 17 shows the full spectrum of N = 32 original Majorana SYK.Figure 18 shows the same quantity but for our coupled model with α = 1 . , ξ = 0 . − − − E Full spectrum , N = J = single SYK Figure 17 . Full spectrum of J = 1 , N = 32 single SYK for a single disorder realizationbinned with 300 bins. We can also average over several samples to produce a more smooth density - Figure19 The main takeaway from these plots is that the coupled model does not have agap between the ground state and the rest of the spectrum. The presence of suchgap would immediately imply that the conformal solution (2.7) does not representthe dominant thermodynamic solution.– 35 – − − E Full spectrum , N = α = ξ = J = Figure 18 . Full spectrum of J = 1 , N = 16 , α = 1 . , ξ = 0 . − − − E Full spectrum N = α = ξ = J = Figure 19 . Full spectrum of J = 1 , N = 15 , α = 1 . , ξ = 0 . As we have mentioned in the Introduction, we are solving (Euclidean) SD equations(2.6) by the standard iteration procedure [11], when we start from a free solution– 36 – , G ∝ sgn( u ) and iterate the equations (2.6) until we converge(the norm betweensuccessive solutions becomes small). A natural question is: how do we know that weconverge to an actual physical solution?One way to check this is to compare the resulting ground state energy to theactual ground state energy computed from finite N exact diagonalization. Thisrequires two extrapolations. In SD we have to extrapolate finite-temperature energyall the way to T = 0. This can be done using the prediction (2.20). In ED we haveto extrapolate finite N results to N = ∞ . We can do this by assuming the following o N dependence in the ground state energy E ( N ) at finite N : E ( N ) = ( E /N ) N + c + c /N (5.2)and extract E /N, c , c from the fit. The quantity E /N ∼ O ( N ) is supposed tomatch the result from SD.As usual, we first present the result for ξ = 0, which is supposed to have “con-ventional” Schwartzian physics - Figure 20. In Figure 21 we present the results for ξ = 0 . α . In all cases we see a good agreement between SDand ED. 10 11 12 13 14 15 16 N E ( N ) α = ξ = J = E / N = ± ED data Figure 20 . Finite N exact diagonalization results for α = 1 . , ξ = 0 . , J = 1 . 0. Theuncertainty comes from including a subleading term c /N in the fit. Ground state energyfrom numerically solving large N SD equations is E /N = 0 . o Our N is not very large, this is why included the subsub-leading c /N term. In fact, we haveperformed the fit with and without it and this way estimate the uncertainty in E /N . Uncertaintyin each individual point can be made very small by averaging over many samples. – 37 – N E ( N ) α = ξ = J = E / N = ± ED data 10 11 12 13 14 15 16 N E ( N ) α = ξ = J = E / N = ± ED data Figure 21 . Finite N exact diagonalization results for α = 1 . α = 2 . J = 1 . , ξ = 0 . 5. The uncertainty comes from adding/removing a sub-subleading term c /N in the fit. Ground state energy from numerically solving large N SDequations is E /N = 0 . α = 1 . E /N = 0 . α = 2 . It is very interesting to check the prediction (3.48) for the density of states: ρ ( E ) non − loc, − loop ∼ E h − (5.3)Famous Schwartzian result predicts [37] square-root edge √ E density of states nearthe ground state: ρ ( E ) Sch,exact ∼ √ E (5.4)On ED side, working with the density of states directly is not good, because itdepends on bin size. In order to eliminate this dependency we can plot ”cumulativedistribution function”(CDF) which is just the number of states in a given energyinterval from the ground state:CDF( E ) = (cid:90) EE dE (cid:48) ρ ( E (cid:48) ) (5.5)The results for the original SYK and the coupled model are shown in Figure 22.We see a very good agreement with √ E for the case of original SYK. For α = 1 . h theor = 1 . 31 so for the right part of Figure 22 the 1-loop result (5.3) predicts p E − . which is definitely not the case. This indicates that the non-local action is not 1-loop exact. This numerical analysis suggests that the density of states keeps thesquare-root edge even when the Schwartzian is not dominant. p The negative power should not be a concern as the density ρ ( E ) is still normalizable. Forexample, for N = 1 SUSY Schwartzian ρ ( E ) SUSY,Sch ∝ / √ E [14] – 38 – .0 0.1 0.2 0.3 0.4 E N u m b er o f e i g e n v a l u es R dE ρ ( E ) , J = single SYK N = EDE p , p best = E N u m b er o f e i g e n v a l u es R dE ρ ( E ) , α = ξ = J = N = EDE p , p best = Figure 22 . CDF for original N = 32 SYK(Left) and coupled model with N = 16 , α =1 . , ξ = 0 . J = 1. Thepower p best was determined from a fit with AE p . p best obviously depends on the energyinterval where the fit is performed. Changing this interval introduces 0 . . 07 uncertainty for the coupled model. Quantization of the Schwartzian action can be reduced to Liouville quantum mechan-ics [38, 39]. At very late Euclidean times τ (cid:29) N/J it results in a universal behavior N/τ / in the 2-point function. In a single SYK, it is possible to see a power-lawdecay in ED even at moderate N . However one has to use large values of N to seeanything close to the power 3 / 2. We would like to see what happens in the coupledmodel. Unfortunately, in the coupled model we are limited to N = 15. Our resultsfor a single SYK(for comparison) and the coupled model for α = 1 . , ξ = 0 . (cid:104) | ψ i ( τ ) ψ i (0) | (cid:105) = (cid:88) E n |(cid:104) n | ψ i | (cid:105)| e − ( E n − E ) τ (5.6)Finally, we averaged over 100 samples. We can confirm qualitative 1 /τ p behavior,but we cannot reliable determine the power p . It seem to slowly increase with N .Our modest results suggest p > 1. Presumably these results can be easily improvedby studying larger N , but using low-lying states only. Another interesting quantity is the energy level statistics. A general expectation forchaotic models is that after making the energy density uniform, the energy gaps are– 39 – log ( τ ) − − − l o g ( G ( τ )) N = J = EDlinear , slope = − − log ( τ ) − − − − − l o g ( G ( τ )) N = J = α = ξ = EDlinear , slope = − Figure 23 . 2-point Green function at finite N and large times. Left: original SYK. Right:the coupled model and G . Almost exactly the same results hold for G . In both caseswe see a power-law behavior. distributed the same way as in a random matrix ensemble. A deviation from thisindicate possible spin-glass phase [18]. In this Section we are going to show thatin the coupled model the level statistics obey random matrix theory predictions,suggesting no spin-glass phase. Compared to the rest of the paper, in this Sectionparameter ξ is absorbed into J , J , C couplings, making the fermionic operatorssquare to one.First of all, instead of unfolding the spectrum we consider another quantity: theratio r n between the adjacent energy gaps: r n = E n +1 − E n E n − E n − (5.7)This quantity does not require unfolding. “Wigner-surmise”-like computation [40]predicts the following r distribution q : P β ( r ) = 1 Z β ( r + r ) β (1 + r + r ) β/ , (5.8)where as usual β = 1 correspond to Gaussian Orthogonal Ensemble(GOE), β = 2to Gaussian Unitary Ensemble(GUE) and β = 4 to Gaussian Symplectic Ensem-ble(GSE). For comparison, for Poisson distributed levels the distribution is P P oisson ( r ) = 1(1 + r ) (5.9) q Normalization factors are Z = 8 / , Z = 4 π/ (81 √ , Z = 4 π/ (729 √ – 40 –ow we need to understand what ensemble the coupled SYK Hamiltonian (2.1)corresponds to. Also we need to project out all global symmetries. The symmetry ψ i ↔ ψ i is broken by ξ term, so we should not worry about it. For even N wehave two independent and commuting symmetries: ψ i → − ψ i , ψ i → − ψ i . Thecorresponding operators are Γ = i N/ N (cid:89) i =1 ψ i (5.10)Γ = i N/ N (cid:89) i =1 ψ i (5.11)For odd N only Γ = Γ Γ is a symmetry. Having projected on eigenvalue subspaceof these operators, we need to ask if we have any anti-linear symmetries. It is alwayspossible to represent ψ i as real matrices and ψ i as purely imaginary matrices. Thenthere are three anti-linear symmetries: K s = C (5.12) K = (cid:32) N (cid:89) i =1 ψ i (cid:33) C , K = (cid:32) N (cid:89) i =1 ψ i (cid:33) C , (5.13)where C is complex-conjugation operator r . They obey the following commutationrelations for odd N : K s Γ = Γ K s , K , Γ = − Γ K , (5.14)Hence, for odd N we have two sectors, Γ = ± K s acts within them. Since K s = 1 we have GOE. Whereas for even N the commutation relations are: K s Γ , = ( − N/ Γ , K s (5.15) K Γ , = ( − N/ Γ , K , K Γ , = ( − N/ Γ , K (5.16) K s = 1 , ( K , ) = ( − N/ (5.17)and there are four sectors: Γ , = ± 1. For even N/ 2, operators K s, , act within thesectors and we have GOE. For odd N/ r For example, C i = − i C – 41 – r , consecutive level spacing N = α = ξ = samples RMT GOEPoissonED data r , consecutive level spacing N = α = ξ = samples RMT GUEPoissonED data Figure 24 . Distribution of r for various α, ξ, N . Random matrix prediction uses thesurmise (5.8). For comparison, we included the exact result (5.9) for Poisson-distributedgaps. In this paper we have presented a simple coupled SYK model. In the limit of large N and low energies this model, like SYK, has an approximate time-reparametrizationsymmetry. However, unlike any previously known SYK-type model, the actionfor reparametrizations is dominated by a non-local action rather than the (local)Schwartzian. To verify this claim studied numerically different physical quantities,such as thermodynamic energy, subleading correction to 2-point function and 4-pointfunction. Our approach was to solve large N equations numerically. We saw thatthe non-local action indeed dominates everywhere. We double-checked some of ourresults using finite N exact diagonalization.Also we discussed other physical features of the coupled model and the non-localaction. It turned out that the residual entropy and (maximal) chaos exponent areexactly the same as in SYK. However, the heat capacity and diffusion constant(inchain models) are very different from the models dominated by the Schwartzian. Alsocertain aspects of time-ordered 4-point function are different too. Also we presenteda limited discussion of 1 /N corrections. We computed the density of states near zeroand saw that it does not agree with 1-loop prediction of the non-local action. Thisshows that the partition function is not 1-loop exact.Let us comment on other models which can have an operator with dimension1 < h < / < h < / 2, [29]: O ,n = (cid:88) i (cid:0) ψ i ∂ n +1 u ψ i + ψ i ∂ n +1 u ψ i (cid:1) (6.1)– 42 –heir dimensions are determined by2( α + α )1 + 3 α g A ( h ) = 1 (6.2)Therefore the dimension of O , can be in the window (1 , / 2) for − < α < 0. How-ever the operator O , introduces non-diagonal(in 1 , L (cid:48) int = (cid:88) ijkl B i ; jkl ψ i ψ j ψ k ψ l + B ijk ; l ψ i ψ j ψ k ψ l (6.3)Compared to eq. (1.5) it couples 3 fermions from one side to 1 fermion from theother side. The resulting SD equations and the spectrum of conformal dimensionsare very similar to the ones we studied. We again expect that in a certain range ofparameters this model is dominated by the non-local action.Let us conclude by a list of open questions: • The most interesting question is to fully quantize the non-local theory (1.2). Isthe Schwartzian piece important for this? Could it be that it starts dominatingagain in the strong-coupling region βJ (cid:29) N ? • Can we learn anything about JT gravity with matter from studying this model? • Is there spin-glass phase? Our results about the level statistics suggest thatthere is no such phase. • The model we described has an obvious generalization to q interacting fermions. • Unfortunately, we could not obtain much analytic progress in the large q limit.Solving the model in this limit will give a partial analytical control over themodels without the Schwartzian dominance. • What is tensor-model counterpart? Some tensor models are different from SYKin 1 /N corrections and they are not captured by the Schwartzian. • What would be the physics of eternal traversable wormhole [13]? • What is the physics of the spectral form factor [15] ? • It would be instructive to incorporate complex fermions(or global symmetriesin general) and study the interplay between them and the non-local action [23].Models with complex fermions can have operators with dimensions 1 < h < / Schwartzian term gives rise to the famous linear-temperature dependence ofelectrical resistivity in certain models [35]. It would be very interesting togeneralize these models so that they are dominated by the non-local action.Presumably it will lead to a tunable temperature dependence in the resistivity. • It would be interesting to investigate the dynamics of entanglement [16] in thechain models. • Finally, it is worth mentioning that in our model the point | α | = 1 seems to bespecial. At this value of α there is a field with h = 3 / 2. However, because ofcos( πh ) in m h , the 2-point function of reparametrizations (3.2) blows up. Acknowledgment The author is forever indebted to I. Klebanov, G. Tarnopolsky and W. Zhao formany comments and discussion throughout this project. I am grateful to A. Gorsky,J. Turiaci and especially D. Stanford and Z. Yang for comments, and F. Popovfor discussions and very useful comments on the manuscript. I would like to thankC. King for help with the manuscript and moral support. This material is based uponwork supported by the Air Force Office of Scientific Research under award numberFA9550-19-1-0360. It was also supported in part by funds from the University ofCalifornia. Use was made of computational facilities purchased with funds fromthe National Science Foundation (CNS-1725797) and administered by the Center forScientific Computing (CSC). The CSC is supported by the California NanoSystemsInstitute and the Materials Research Science and Engineering Center (MRSEC; NSFDMR 1720256) at UC Santa Barbara. A Lorentzian Schwinger–Dyson equations Self-energies in Lorentzian signature are:Σ > = − J (cid:0) G > ) + 12 α G > ( G > ) (cid:1) Σ > = − J (cid:0) G > ) + 12 α G > ( G > ) (cid:1) (A.1)The relation between the self-energy and the retarded Green’s function is G Ra ( ω ) = 1(1 − ξ a ) ω − Σ Ra , ξ = ξ, ξ = − ξ (A.2)To close the system we need the fluctuation–dissipation theorem to relate G > to G R : G >a ( ω ) = 2 in F ( ω ) Im G Ra ( ω ) , n F ( ω ) = 1 e βω + 1 (A.3)– 44 –ote that we can easily put β = + ∞ . This equations can be solved by iterations,exactly like the Euclidean case. However one has to introduce a large interval in thetime domain. So there will be two cut-offs: the time step dt and the interval length L . References [1] S. Sachdev and J. 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