Coupled Sachdev-Ye-Kitaev models without Schwartzian dominance
CCoupled Sachdev–Ye–Kitaev models without Schwartzian dominance
Alexey Milekhin
Department of Physics, University of California at Santa Barbara, Santa Barbara, CA 93106, U.S.A. ∗ We argue that in certain class of coupled Sachdev–Ye–Kitaev(SYK) models the low energy physicsat large N is governed by a non-local action rather than the Schwartzian action. We present a partialanalytic and extensive numerical evidence for this. We find that these models are maximally chaoticand have the same residual entropy as Majorana SYK. However, thermodynamic quantities, suchas heat capacity and diffusion constant are different. INTRODUCTION
Recently Sachdev–Ye–Kitaev(SYK) model [1–5] andtensor models [6–9] have been the focus of much theo-retical research. Also there have been several proposalof possible experimental implementation [10–14]. Themost prominent features of this family of models arenon-Fermi liquid behavior, maximal [15] Lyapunov ex-ponent, non-zero residual entropy and an approximatetime reparametrization symmetry at large N and low en-ergies. Maldacena and Stanford [16] and Kitaev and Suh[17] derived the following (Euclidean) Schwartzian actionfor reparametrizations in the original SYK model: S Sch = − N α
SSch J (cid:90) du Sch ( τ [ u ] , u ) , (1)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) In all known SYK-like modes, the Schwartzian has beenidentified as the dominant [18] low-energy action and itsphysics has been explored in a variety of situations [19–30].
In this Letter we present a coupled SYK model whichis dominated by the following non-local action instead ofthe Schwartzian action: 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 (2)It was suggested by Maldacena, Stanford and Yang [31]that this action may appear when a theory contains alocal irrelevant operator with the dimension h within theinterval 1 < h < /
2. Holographically this correspondsto a light matter field in
AdS having a source term atthe boundary. In our model h is tunable and can beanywhere between 1 and 2.We would like to stress that the Schwartzian term isstill present in our model. The key point is that at large N it gives a subleading in T /J contribution, where T isthe temperature.Our approach is semi-analytical. We consider strictlarge N limit and obtain various predictions of the non-local action. Then we check them against numerical so-lutions of exact large N equations. Also we demonstratethat the Lyapunov exponent is still maximal and studythe transport in chain models. In the accompanying longer paper [32] we provide more details on numericsand theoretical computations, explore other aspects ofthese models and study 1 /N corrections. MICROSCOPIC FORMULATION
The model consists of two independent Majorana SYKmodels( L kin + L SY K ), a marginal interaction( L int ) be-tween them, and an innocuously-looking kinetic termtwist( L ξ ): L T = L kin + L SY K + L int + L ξ , (3)where L kin = 12 (cid:88) i (cid:0) ψ i ∂ u ψ i + ψ i ∂ u ψ i (cid:1) (4) L SY K = 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) (5) L int = 32 α (cid:88) ijkl C ij ; kl ψ i ψ j ψ k ψ l (6) L ξ = − ξ (cid:88) i (cid:0) ψ i ∂ u ψ i − ψ i ∂ u ψ i (cid:1) (7)There are 2 N Majorana fermions ψ ai , i = 1 , . . . , N, a =1 ,
2. This action with ξ = 0 has been studied in the lit-erature before [27, 33, 34] and it was argued that it isdominated by the Schwartzian. It is important that ten-sors J , , C are different. When they are the same theground state is actually gapped and the model is proneto Z symmetry breaking [35]. Low energy physics is notdescribed by the conformal solution in this case. We willgive a brief analytical argument below why this does nothappen in our model. Also we cross-checked this with ex-act diagonalization [32] at finite N . The non-local actionemerges only when both α (cid:54) = 0 , ξ (cid:54) = 0. Non-zero α ren-ders the two SYK models coupled and also it controls thedimension h in the non-local action. Specifically, we need a r X i v : . [ h e p - t h ] F e b | α | > ξ explicitly breaks ψ i ↔ ψ i Z symmetry and theneeded irrelevant operator appears in the conformal per-turbation theory(more on it below). Tensors J , J , arestandard SYK disorders(totally antisymmetric with i.i.d.Gaussian components). Tensor C ij ; kl has a Gaussian dis-tribution too, but it has a separate skew-symmetry in ij and kl indices: C ij ; kl = − C ji ; kl = − C ij ; lk (8)With 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 (9)Euclidean Schwinger–Dyson(SD) equations read(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 )(10)with ∗ denoting time convolution. G / are the time-ordered Green functions: G aa ( u ) = (cid:104) T ψ ai ( u ) ψ ai (0) (cid:105) , a = 1 , ξ can be reabsorbed into J , J variances.Mixed correlators (cid:104) ψ i ψ i (cid:105) do not appear up to 1 /N or-der. These correlators serve as order parameters for thegapped Z -symmetry broken phase. This suggests thatthis breaking does not happen in our model and thephysics can be described by the following conformal so-lution. At low energies we can neglect the kinetic termand ξ disappears. Assuming G = G , one obtains thefamiliar SYK conformal solution G , = G conf = b sgn( u )(1 + 3 α ) / π (cid:114) Jβ sin (cid:16) π | u | β (cid:17) (12)with b = 1 / (4 π ) / . The full solution can be obtainedby numerically solving SD equations. Our numerical ap-proach is a simple iteration procedure commonly usedin SYK literature [16]. We cross-checked our numericalsolution against the results of exact diagonalization atfinite N [32].It is important that the equations are coupled, sothere is only one reparametrization mode. Timereparametrizations act on Green functions by G aa ( u , u ) → ( τ ( u ) (cid:48) τ ( u ) (cid:48) ) / G aa ( τ ( u ) , τ ( u )) ,a = 1 , ψ have conformal dimension1 /
4. It is easy to compute the dimension h of the follow-ing bilinear operator [35]: O , = (cid:88) i (cid:0) ψ i ∂ u ψ i − ψ i ∂ u ψ i (cid:1) , (14) it is given by the smallest solution of1 − α α g A ( h ) = 1 , g A ( h ) = −
32 tan ( π ( h − / / h − / | α | >
1, the dimension h is in the range: 1 < h < /
2. This bilinear operator is exactly the L ξ term in theLagrangian. A PERTURBATIVE ARGUMENT
As in the standard SYK, we can go beyond the con-formal sector by perturbing [17, 36] the exact conformalLagrangian L conf = L SY K + L int (16)by a set of irrelevant operators L T = L conf + (cid:88) h α h O h (17)Unfortunately, there is no ab initio way to compute α h .The set of these irrelevant operators is constrained bythe symmetries of the model. For ξ = 0 there is 1 ↔ N and the operator (14) does notappear. This is why we need the L ξ term in the UVLagrangian. However, the corresponding IR parameter α h does not have to be linear in ξ [32]. Now we can seethe origin of the non-local action (2). In perturbationtheory, one can derive it by taking the 2-point function of O h and dressing it with reparametrizations [31]. Higher-order terms are suppressed by 1 /N . This agrees withholographic expectations of weakly interacting fields inthe bulk in the large N limit. Another consequence ofthis formalism is the leading non-conformal correction tothe 2-point function: δG ( u ) = α h (cid:90) du (cid:48) (cid:104)O h ( u (cid:48) ) ψ i ( u ) ψ i (0) (cid:105) ∝ Ju ) h − / (18)The last equality is valid for | u | (cid:28) β . We checked thisprediction by numerically computing the spectral densityat zero temperature and real frequency: ρ / ( ω ) = Im G R, / ( ω ) , (19)where G R is retarded Green function. Conformal behav-ior (12) results in ρ ∼ / √ ω for ω (cid:28) J , whereas thenon-conformal correction (18) gives ρ ∼ ω h − / . There-fore, we expect the following behavior at small ω : √ ωρ / = b ± b ω h − (20)Constant b can be extracted from the conformal solu-tion, but b is related to α h and has to be extracted fromthe numerics. The result is presented in Figure 1. Wesee a good agreement with the numerical results. 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 1. Results for G . The fit was performed with b + b ω h − . Uncertainty in h best results from changing the fittinginterval. Theoretical h is obtained from eq. (15). THERMODYNAMICS
Due to overall N factor, we can treat the non-localaction (2) classically as long as T (cid:29) J/N / (2 h − . Itscontribution to free energy is equal to the classical ac-tion evaluated on the thermal solution τ ( u ) = tan( πu/β ).This computation produces a divergence [31]: − ∆ F nonloc N = T h − α S h π h − / J h − Γ (1 / − h )Γ (1 − h ) (21)where we assumed a fixed cut-off u ∼ /J in Euclideantime. The divergent term E/N = E + c h T h − + 2 π α SSch
J T + . . . (22)where c h = (2 h − α S h π h − / J h − Γ (1 / − h )Γ (1 − h ) (23)For 1 < h < / G and G . The comparison between theprediction (22) and the numerics is shown in Figure 2,which demonstrates good agreement.Conformal solution (12) has the same form as in theoriginal SYK. Zero temperature residual entropy can beextracted from the conformal solution [2, 29]. Hence theresidual entropy should be twice Majorana SYK residualentropy. Our numerical results support this claim. T − − − − − − − E n er gy × − − T T + T h − , h best = α = ξ = Figure 2. Energy
E/N vs T for J = 2 π . Blue points are nu-merical data. There is a clear deviation from Schwartzian T prediction. For α = 1 . h theor = 1 .
31. Altering temperaturerange( β = 50 , . . . , − ) and removing c Sch T term results in h best = 1 . ± .
02. The fit was performed using eq. (22)with unknown E , c h , c Sch . KERNEL AND 4-POINT FUNCTION
As in the original SYK, connected 4-point function F is given by a sum of ladder diagrams: F ∝ / (1 − K ).”Flavor” indices 1 , K into 2 × v = ( v , v ): Kv J = (cid:18) G ∗ (cid:0)(cid:0) G + α G (cid:1) v + 2 α G G v (cid:1) ∗ G G ∗ (cid:0)(cid:0) G + α G (cid:1) v + 2 α G G v (cid:1) ∗ G (cid:19) (24)Using the translation symmetry, it is possible to separatethe Matsubara frequency n and re-write the kernel as afunction of two times only [29]: K n,ab ( u, u (cid:48) ) = (cid:90) β dsK ab (cid:18) s + u , s − u u (cid:48) , − u (cid:48) (cid:19) e − πins/β (25)Because of the reparametrizations, this kernel has eigen-value 1 in the conformal approximation for any integer n except 0 , ±
1. The eigenvalue shift 1 − k (2 , n ) can beread off from the reparametrization action:1 − k (2 , n ) = α K h ( βJ ) h − g h ( n ) | n | ( n −
1) + α KSch | n | βJ (26) g h ( n ) = n (cid:18) Γ( n + h )Γ(1 + n − h ) − Γ( h − − h ) (cid:19) (27)where the first term in 1 − k (2 , n ) comes from the non-local action, and the second linear in | n | term comes from n k ( n ) a n − a n − a n h − , h best = n − k ( n ) − a n full non-local J = α = ξ = Figure 3. Results for k (2 , n ). Blue points are numericallyobtained eigenvalues of K n in eq. (25). Top: since g h ( n ) isactually numerically close to n h +1 , we have performed thefit with 1 − a n − a n h − at large n to extract h best . For α = 1 . h theor = 1 .
24, whereas h best = 1 . ± .
04 Bottom:fit with eq. (26) with α KSch, h unknown: a = α KSch /βJ . the Schwartzian [37]. The relation between α K h and α S h is α K h = − (2 π ) h +1 π ( h − cos( πh )Γ(2 h ) 4 π b α S h (28)Using the exact numerical solutions G , and fixing β = 2 π , we build the kernel (25) for each n = 2 , . . . , k (2 , n ) closest to 1.For this we used a uniform 2D grid in ( u, u (cid:48) ) plane with60 , − , points. The comparison between theprediction (26) and the numerics is shown in Figure 3.Performing the same procedure for different J , we ver-ified J h − dependence in eq. (26) [32]. The eigenvalueshift is sensitive to exact form of reparametrization ac-tion. We take the high levels of agreement with thenumerics as the main evidence for the non-local actiondominance. CHAOS EXPONENT
Let us demonstrate that the Lyapunov exponent ismaximal in the out-of-time ordered correlation function(OTOC). We are interested in the connected 4-pointfunction F = (cid:104) ψ i ( u ) ψ i ( u ) ψ j ( u ) ψ j ( u ) (cid:105) conn (29)in the OTOC region: u = − β/ it, u = 0 , u = β/ it, u = β/ /βJ ) contribution comes from averagingthe disconnected 4-point function over the reparametriza-tions. Delegating details to the Supplementary Material,we write down the final answer for OTOC: F = − p h ( βJ ) h − N α S h m h exp (cid:18) πtβ (cid:19) + [non-increasing] (31)with coefficient p h = π − h Γ(2 − h )4Γ(1 + h )( ψ (1 + h ) − ψ (2 − h )) G conf ( β/ (32)and where ψ ( x ) = Γ (cid:48) ( x ) / Γ( x ). The Lyapunov expo-nent 2 πt/β is maximal. However, the prefactor ( βJ ) h − is smaller compared to Schwartzian-dominated originalSYK, where it is ( βJ ) . TRANSPORT IN CHAIN MODELS
We can arrange the coupled model into an array andstudy transport properties. In SYK chain models domi-nated by the Schwartzian [33, 38], the diffusion constantis temperature-independent and the heat conductivity islinear in the temperature. Here we find that the non-local action renders the diffusion constant temperature-dependent, but leaves the heat conductivity proportionalto the temperature. Specifically, we consider the follow-ing construction - Figure 4: L T,chain = (cid:88) x L T,x + L int,chain (33)where L T,x is an independent copy of the coupled model(3). The interaction between the sites should be care-fully chosen in order to avoid a possible spontaneous Z symmetry breaking. We study the following interactionterm between the sites: L int,chain = 12! (cid:88) x,ijkl (cid:16) 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:17) (34) Figure 4. Illustration of couplings in the chain model.
Each V / ,xij ; kl i.i.d. Gaussian and is skew symmetric in ij and kl : V / ,xij ; kl = − V / ,xji ; kl = − V / ,xij ; lk , (35) and has x − independent variance (cid:104) (cid:16) V / ,xij ; kl (cid:17) (cid:105) = V N . (36)Assuming the two-point functions to be x -independentwe arrive at single-site SD equations (10) with effective (cid:101) J and (cid:101) α : (cid:101) J = J + V , (cid:101) α = α J J + V (37)It is important to keep in mind that the conformal dimen-sion h in the non-local action is now determined by thisrenormalized (cid:101) α . In Supplementary Material we derivethe following low-energy action for reparametrizations: 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 (38)The action is valid in the hydrodynamic regime ω (cid:28) T, p (cid:28)
1, where p is the momentum conjugate to x . Inthis regime the reparametrization modes (cid:15) ω,p are propor-tional to stress-energy tensor [32, 38]. We clearly see adiffusion pole in (cid:15) propagator with the diffusion coeffi-cient: 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 ) (39)This result differs a lot from SYK chains wherethe Schwartzian dominance results in temperature-independent diffusion constant. Standard hydrodynamicexpectation is that thermal conductivity is given by dif-fusion constant times the heat capacity. This can beseen explicitly by the aforementioned mapping of (cid:15) ω,p to stress-energy tensor. In our case the heat capacity c v is proportional to N T h − , eq. (22). Therefore thethermal conductivity is linear in the temperature, as inSchwartzian-dominated theories: κ = c v D ∝ N T (40)
CONCLUSION
In this Letter we studied a coupled SYK model whichshares a lot properties with the original SYK: the sameconformal solution, residual entropy and maximal chaosexponent. However, in contrast to all known SYK-like models, the low energy physics at large N is dominatedby the non-local action for reparametrizations (2) insteadof the Schwartzian action. The reason behind this is thepresence of the light conformal field O , , eq. (14), withthe dimension h , 1 < h < /
2. This conformal field is ex-plicitly “brought to life” by the ξ term in the Lagrangian(3). We would like to emphasize that this situation isnot exotic: we studied the simplest coupled SYK and itis pretty easy to manipulate with conformal dimensionsin the coupled SYK-like models [35, 39]. Actually in ourmodel for | α | <
1, the key operator O , , eq. (14), hasthe dimension 3 / < h <
2. This does not result inthe dominance over the Schwartzian, but does result inthe leading non-conformal correction being different fromSYK [32].The non-local action, compared to the Schwartzian,produces a different heat capacity, eq. (22), OTOC pref-actor, eq. (31), and temperature-dependent diffusionconstant in chain models, eq. (39). Interestingly, thethermal conductivity is still linear in the temperature.This result suggests that this linearity is a more generalfeature which perhaps can be derived from the conformalsolution. It would be very interesting to consider modelswith U (1) symmetry and see if the electrical conductivityis linear in the temperature too as in strange metal.Obviously, there are a lot of other open questions.There exists huge literature on SYK and the Schwartzian.Essentially, all the questions asked there could be askedfor the non-local action. The most interesting of them isto study the strongly-coupled regime of this model, whenthe temperature T (cid:46) J/N / (2 h − and the non-local ac-tion is not semiclassical. Up to non-perturbative e − N correction is this equivalent to quantizing 2D Jackiw–Teitelboim gravity on a disk with light matter fields. ACKNOWLEDGMENT
The author is forever indebted to I. Klebanov,G. Tarnopolsky and W. Zhao for a lot of discussionsand comments throughout this project. It is pleasureto thank A. Gorsky, J. Turiaci and especially D. Stan-ford and Z. Yang for comments, and F. Popov for usefulinput and insightful suggestions on the paper. I wouldlike to thank C. King for moral support and help with themanuscript. This material is based upon work supportedby the Air Force Office of Scientific Research under awardnumber FA9550-19-1-0360. It was also supported in partby funds from the University of California. Use was madeof computational facilities purchased with funds from theNational Science Foundation (CNS-1725797) and admin-istered by the Center for Scientific Computing (CSC).The CSC is supported by the California NanoSystemsInstitute and the Materials Research Science and En-gineering Center (MRSEC; NSF DMR 1720256) at UCSanta Barbara. ∗ [email protected][1] S. Sachdev and J. Ye, Phys. Rev. Lett. , 3339 (1993),arXiv:cond-mat/9212030 [cond-mat].[2] O. Parcollet, A. Georges, G. Kotliar, and A. Sengupta,Physical Review B , 3794–3813 (1998), arXiv:cond-mat/9711192 [cond-mat].[3] A. Georges, O. Parcollet, and S. Sachdev, PhysicalReview B , 134406 (2001), arXiv:cond-mat/0009388[cond-mat.str-el].[4] A. Kitaev, “A simple model of quantum holography,” http://online.kitp.ucsb.edu/online/entangled15/kitaev/ , http://online.kitp.ucsb.edu/online/entangled15/kitaev2/ , talks at KITP, April 7, 2015and May 27, 2015.[5] D. J. Gross and V. Rosenhaus, (2016), arXiv:1610.01569[hep-th].[6] R. Gurau, Commun. Math. Phys. , 69 (2011),arXiv:0907.2582 [hep-th].[7] E. Witten, (2016), arXiv:1610.09758 [hep-th].[8] I. R. Klebanov and G. Tarnopolsky, Phys. Rev. D95 ,046004 (2017), arXiv:1611.08915 [hep-th].[9] I. R. Klebanov and G. Tarnopolsky, JHEP , 037(2017), arXiv:1706.00839 [hep-th].[10] A. Chew, A. Essin, and J. Alicea, Physical Review B (2017), 10.1103/physrevb.96.121119, arXiv:1703.06890[cond-mat.dis-nn].[11] I. Danshita, M. Hanada, and M. Tezuka, Progressof Theoretical and Experimental Physics (2017), 10.1093/ptep/ptx108, arXiv:1606.02454 [cond-mat.quant-gas].[12] A. Chen, R. Ilan, F. de Juan, D. Pikulin, andM. Franz, Physical review letters , 036403 (2018),arXiv:1802.00802 [cond-mat.str-el].[13] O. Can, E. M. Nica, and M. Franz, Physical Review B (2019), 10.1103/physrevb.99.045419, arXiv:1808.06584[cond-mat.str-el].[14] C. Wei and T. A. Sedrakyan, Physical Review A (2021), 10.1103/physreva.103.013323, arXiv:2005.07640[cond-mat.quant-gas].[15] J. Maldacena, S. H. Shenker, and D. Stanford, (2015),arXiv:1503.01409 [hep-th].[16] J. Maldacena and D. Stanford, Phys. Rev. D94 , 106002(2016), arXiv:1604.07818 [hep-th].[17] A. Kitaev and S. J. Suh, JHEP , 183 (2018),arXiv:1711.08467 [hep-th].[18] Here we talk about strict large N limit. In certain ten-sor models subleading 1 /N corrections are different fromSYK and are not captured by the Schwartzian.[19] J. Maldacena and X.-L. Qi, (2018), arXiv:1804.00491[hep-th].[20] W. Fu, D. Gaiotto, J. Maldacena, and S. Sachdev,(2016), arXiv:1610.08917 [hep-th].[21] J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski,P. Saad, S. H. Shenker, D. Stanford, A. Streicher, andM. Tezuka, JHEP , 118 (2017), [Erratum: JHEP 09,002 (2018)], arXiv:1611.04650 [hep-th].[22] Y. Gu, A. Lucas, and X.-L. Qi, Journal of High EnergyPhysics (2017), 10.1007/jhep09(2017)120.[23] J. Yoon, (2017), arXiv:1707.01740 [hep-th].[24] G. Gur-Ari, R. Mahajan, and A. Vaezi, Journal of HighEnergy Physics (2018), 10.1007/jhep11(2018)070.[25] J. Maldacena and A. Milekhin, (2019), arXiv:1912.03276[hep-th].[26] A. Almheiri, A. Milekhin, and B. Swingle, (2019),arXiv:1912.04912 [hep-th].[27] Y. Chen, H. Zhai, and P. Zhang, JHEP , 150 (2017),arXiv:1705.09818 [hep-th].[28] P. Zhang, (2019), arXiv:1909.10637 [cond-mat.str-el].[29] Y. Gu, A. Kitaev, S. Sachdev, and G. Tarnopolsky,(2019), arXiv:1910.14099 [hep-th].[30] K. Su, P. Zhang, and H. Zhai, (2021), arXiv:2101.11238[cond-mat.str-el].[31] J. Maldacena, D. Stanford, and Z. Yang, PTEP ,12C104 (2016), arXiv:1606.01857 [hep-th].[32] A. Milekhin, to appear in the same submission.[33] Y. Gu, X.-L. Qi, and D. Stanford, Journal of High En-ergy Physics , 125 (2017), arXiv:1609.07832 [hep-th].[34] A. Altland, D. Bagrets, and A. Kamenev, Phys.Rev. Lett. , 106601 (2019), arXiv:1903.09491 [cond-mat.str-el].[35] J. Kim, I. R. Klebanov, G. Tarnopolsky, and W. Zhao,Phys. Rev. X9 , 021043 (2019), arXiv:1902.02287 [hep-th].[36] M. Tikhanovskaya, H. Guo, S. Sachdev, andG. Tarnopolsky, (2020), arXiv:2010.09742 [cond-mat.str-el].[37] The author thanks D. Stanford and Z. Yang for usefuldiscussions about this computation and the Lyapunovexponent computation.[38] X.-Y. Song, C.-M. Jian, and L. Balents, Physical ReviewLetters (2017), 10.1103/physrevlett.119.216601, , 162 (2020), arXiv:2006.07317 [hep-th]. Supplementary materialOTOC computation
Averaging the product of two Green functions over thereparametrizations we get (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) (41) θ i are variables on the thermal circle, θ = 2 πu/β , and y, y (cid:48) , x, x (cid:48) are their combinations: x = θ − θ , x (cid:48) = θ − θ , y = θ + θ , y (cid:48) = θ + θ θ = − π − θ, θ = 0 , θ = π − θ, θ = π (43)Therefore we have the 4-point function (41) proportionalto (cid:88) | n |≥ e in ( π/ θ ) n cos πn g h ( n ) = (cid:73) C dn n e iπn − e in ( π/ θ ) g h ( n )(44)where the contour C encloses ± , ± , . . . . Pulling thecontour to infinity picks up zeros of e iπn − g h ( n ).We will be interested in the analytic continuation θ →− πit/β to OTOC region. The only exponentially grow-ing contribution comes from n = 1. Computing theresidue at n = 1 yields the expression (31) in the maintext. Hydrodynamic action in chain models
To derive the action for reparametrization we need tostudy the kernel. Now there are two types of ladderdiagrams: on-site and the ones jumping between sites: x → x ±
1. Taking a Fourier transform in x , we arrive atthe following result for the kernel: K chain = K ren + K p , (45)where K ren is exactly on-site kernel (24), with renormal-ized J, α . K p is proportional to cos( p ) − K p v = 2 V (cos( p ) − (cid:18) G ∗ ( G v ) ∗ G G ∗ ( G v ) ∗ G (cid:19) (46)Separately, K ren and K p has an eigenvalue correspondingto reparametrizations. One subtlety is that K p and K ren are not proportional as matrices, hence its hard to findthe eigenvalues of K chain precisely. We can ignore thisissue in the long-wavelength limit 1 − cos( p ) ≈ p / (cid:28)
1, and just use the conformal eigenvalue for K p . The eigen-value shift for K ren is controlled by reparametrizations,exactly as in a single coupled model. Putting everythingtogether, we have 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 (47)Analytically continuing from the upper-half plane andkeeping only n term in g h ( nn