Solvable Sachdev-Ye-Kitaev models in higher dimensions: from diffusion to many-body localization
SSolvable Sachdev-Ye-Kitaev models in higher dimensions: from diffusion tomany-body localization
Shao-Kai Jian and Hong Yao
1, 2, 3, ∗ Institute for Advanced Study, Tsinghua University, Beijing 100084, China State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China Collaborative Innovation Center of Quantum Matter, Beijing 100084, China (Dated: November 17, 2017)Many aspects of many-body localization (MBL) transitions remain elusive so far. Here, we proposea higher-dimensional generalization of the Sachdev-Ye-Kitaev (SYK) model and show that it exhibitsa MBL transition. The model on a bipartite lattice has N Majorana fermions with SYK interactionson each site of the A sublattice and M free Majorana fermions on each site the of B sublattice,where N and M are large and finite. For r ≡ M/N < r c =1, it describes a diffusive metal exhibitingmaximal chaos. Remarkably, its diffusive constant D vanishes [ D ∝ ( r c − r ) / ] as r → r c , implyinga dynamical transition to a MBL phase. It is further supported by numerical calculations of levelstatistics which changes from Wigner-Dyson ( r
Metal
Many-body Localization 𝑟 𝑐 =1 𝑟 (b) FIG. 1. ( a ) The 1D generalization of the SYK model con-sists of N SYK Majorana fermions ψ i on each site of the A sublattice and M free Majorana fermions η α on each site ofthe B sublattice. The hopping between two types of fermionsis represented by t iα,x and t (cid:48) iα,x . (b) The phase diagram ofthe 1D model in Eq. (1) as a function of r = M/N . tions and the other hosting M free Majorana fermions.Two sublattices are coupled via random hopping. Thefermion number ratio is denoted as r = M/N . Here, weconsider the case that both N and M are large but fi-nite while the ratio r is fixed. For r (cid:28)
1, SYK physicsdominates such that this phase exhibits a finite diffusiveconstant D and maximal chaos with the Luyapunov ex-ponent satisfying the upper bound λ L =2 π/β , where β isthe inverse temperature. It is a diffusive metal, similarto the one studied by Gu et al. [57]. For r (cid:29)
1, the “free”Majorana fermions on the 1D lattice dominate over theSYK fermions such that weak SYK interactions are irrel-evant around the Anderson-localization “fixed” point offree disordered Majorana fermions [73], leading to MBL.Consequently, we expect that there should be a dy-namic phase transition between a thermal (diffusive)phase and a MBL phase as the ratio r varies from small tolarge. Indeed, for r<
1, our analytical calculations showthat the diffusion constant vanishes as D ∝ (1 − r ) / when r →
1. This implies that a dynamical phase transition toa MBL phase should occur at r = r c =1. The MBL nature a r X i v : . [ c ond - m a t . s t r- e l ] N ov for r>r c is further supported by our numerical calcula-tions of the many-body level statistics, which qualita-tively changes around r = r c : it follows Poisson distribu-tion for r>r c but Wigner-Dyson for r
We first introduce thegeneralized model on 1D lattices, as shown in Fig. 1(a),and consider the cases of more than 1D later. The Hamil-tonian of the generalized SYK model in 1D reads: H = L (cid:88) x =1 (cid:104) (cid:88) ijkl J ijkl,x ψ i,x ψ j,x ψ k,x ψ l,x + (cid:88) iα (cid:0) t iα,x iψ i,x η α,x + t (cid:48) iα,x iη α,x ψ i,x +1 (cid:1)(cid:105) , (1)where ψ i,x and η α,x are SYK Majorana fermions andfree Majorana fermions residing on the A site and the B site of the unit cell x , respectively, with i =1, · · · , N and α =1, · · · , M . The number of unit cells in the chain is L , and the periodic boundary condition is assumed. TheSYK fermions on the A sublattice have on-site all-to-all random four-fermion interactions J ijkl,x with meanzero and variance (cid:104) J ijkl,x (cid:105) = J /N . Here, t and t (cid:48) are nearest neighbor random hopping of Majoranafermions within the same unit cell and between neighbor-ing unit cells, respectively, with mean zero and variance (cid:104) t iα,x (cid:105) = t / √ M N and (cid:104) t (cid:48) iα,x (cid:105) = t (cid:48) / √ M N . Hereafter,we assume t (cid:48) (cid:28) t . When we take the large- N limit, wekeep the ratio r ≡ MN fixed. Note that the time-reversalsymmetry ( ψ → ψ , η →− η and i →− i ) is assumed for thegeneralized model such that hopping between the sametype of fermions is forbidden.Like in the original SYK model, we use a replicatrick to get an effective disorderless model (see theSupplemental Material for details) and introduce bilo-cal variables: G mm (cid:48) ψ,x ( τ , τ )= N (cid:80) Ni =1 ψ mi,x ( τ ) ψ m (cid:48) i,x ( τ ) and G mm (cid:48) η,x ( τ , τ ) = M (cid:80) Mα =1 η mα,x ( τ ) η m (cid:48) α,x ( τ ), as well asΣ mm (cid:48) ψ,x ( τ , τ ), Σ mm (cid:48) η,x ( τ , τ ) as Legendre multipliers to im-plement the above identities, where m, m (cid:48) are replica in- dices. At the large- N limit, different replicas do not in-teract, so the bilocal fields are diagonal in replica indices,i.e., G mm (cid:48) = Gδ mm (cid:48) and Σ mm (cid:48) = Σ δ mm (cid:48) . We obtain thefollowing effective action: SN = L (cid:88) x =1 (cid:20) −
12 [tr log( ∂ τ − Σ ψ,x ) + r tr log( ∂ τ − Σ η,x )]+ 12 (cid:90) (cid:90) (cid:16) Σ ψ,x G ψ,x + r Σ η,x G η,x − J G ψ,x − √ r ( t G ψ,x G η,x + t (cid:48) G η,x G ψ,x +1 ) (cid:17)(cid:21) , (2)where G and Σ are collective bosonic modes and (cid:82)(cid:82) ≡ (cid:82) dτ dτ (integration over two times appears because thereplica trick couples fields at different times). The large- N structure is manifest in the effective action above. Thesaddle-point equations obtained by varying these collec-tive modes are G − ψ,x ( iω ) = − iω − Σ ψ,x ( iω ) , G − η,x ( iω ) = − iω − Σ η,x ( iω ) , (3)Σ ψ,x ( τ ) = J G ψ,x ( τ ) + √ r [ t G η,x ( τ ) + t (cid:48) G η,x − ( τ )] , (4)Σ η,x ( τ ) = [ t G ψ,x ( τ ) + t (cid:48) G ψ,x +1 ( τ )] / √ r, (5)where τ = τ − τ . These saddle-point equations are equiv-alent with Schwinger-Dyson equations obtained from di-agrammatic methods [48, 57]. Diffusive metals:
For r (cid:28)
1, it is expected that theSYK fermions dominate over the free Majorana fermionsin the infrared [61]. Similar to features of the originalSYK model, the time-derivative terms in Eq. (2) or the − iω terms in Eq. (3) are irrelevant in low energy. Re-markably, Eqs.(3-5) in the infrared limit of ω → τ → f ( τ ),˜ G a,x ( τ , τ ) = [ f (cid:48) ( τ ) f (cid:48) ( τ )] ∆ a G a,x ( f ( τ ) , f ( τ )) , (6)where f (cid:48) ( τ )= dfdτ , a = ψ or η , and the scaling dimensions∆ ψ = , ∆ η = . Like in the SYK model, this is an emer-gent time reparametrization symmetry at low energy thatis explicitly broken by high-energy degrees of freedom inthe microscopic model [or the time derivative-terms inthe effective action Eq. (2)].Helped by the emergent reparametrization symmetry,we obtain the following solutions of the Schwinger-Dysonequations in the infrared [Eqs.(3-5) in the limit of ω → G sψ,x ( τ ) = (cid:16) − r πJ (cid:17) / sgn( τ ) | τ | / , (7) G sη,x ( τ ) = 12( t + t (cid:48) ) (cid:104) r J π (1 − r ) (cid:105) / sgn( τ ) | τ | / . (8)The solutions above are spatially uniform while non-trivial in the time direction, exhibiting local criticality[57, 77, 78]. Note that the saddle-point solutions are validbelow a cutoff frequency ω c which scales as ω c ∼ (1 − r ) / when r → S = − r r S SYK , where S SYK = C + π log 28 π ≈ . C ≈ .
916 is Catalan constant. When r →
1, thezero temperature entropy vanishes which implies a hintthat there is a phase transition at r c =1.Note that the saddle-point solutions of Eqs.(7-8) spon-taneously break the continuous reparametrization sym-metry to SL(2, R ). Owing to the spontaneous and ex-plicit breaking pattern, site-dependent reparametrizationmodes (cid:15) x = f x ( τ ) − τ would contribute dominant low-energy fluctuations on top of the saddle-point one, whichdetermine the low-energy physics especially dynamicslike transport and the butterfly effect. Note that because t (cid:48) (cid:28) t the relative reparametrization fluctuation withineach unit cell (namely, f ψ,x − f η,x ) is at high energy anddoes not affect the physics in the low energy we considerhere.The effective action for the reparametrization modesis given by fluctuations around the saddle-point one,i.e., S eff [ f ] = S [ ˜ G ( f )] − S [ G ( τ )], where ˜ G a,x ( τ , τ ) = f (cid:48) ∆ a x ( τ ) f (cid:48) ∆ a x ( τ ) G a ( f x ( τ ) , f x ( τ )) is the Green’s func-tion of a = ψ, η fermions associated with the spatiallydependent time reparametrization f x ( τ )= τ + (cid:15) x . Notethat though the saddle-point solution of G a,x in Eqs.(7-8) is homogenous, its fluctuation associated with thereparametrization modes is generically inhomogeneous.By assuming weak reparametrization (cid:15) x as well as per-forming ε expansion and series summation (see the Sup-plemental Material for details), we obtain the effectiveaction up to the quadratic in (cid:15) , S eff N = πβ (cid:88) n,p (cid:16) α | ω n | + α p (cid:17) | ω n | (cid:104) ω n − (cid:0) πβ (cid:1) (cid:105) | (cid:15) ω n ,p | , (9)where ω n =2 πn/β is the Matsubara frequency, p ismomentum, and (cid:15) x ( τ )= √ Lβ (cid:80) n,p (cid:15) ω n ,p e − iω n τ + ipx .As shown in the Supplemental Material, α = π (cid:16) √ − rJ + Jt + t (cid:48) (cid:113) r − r (cid:17) and α = π rt (cid:48) t + t (cid:48) .Since J and t are both relevant at the UV Gaussianpoint, they increase as energy scales lower. Thus, α becomes extremely small due to the emergentreparametrization symmetry, while α is also small inthe homogenous limit, i.e., t (cid:48) (cid:28) t . These lead to strongfluctuations of reparametrization modes which dominatethe low-energy dynamics.Having obtained the effective action for thereparametrization modes, we are ready to calculatetheir contributions to energy transport and OTOC inthe limit of N (cid:29) βJ (cid:29)
1. The energy density for smallmomentum is given by T ω n ,p = iNα π ω n [ ω n − (cid:0) πβ (cid:1) ] (cid:15) ω n ,p .Using the effective action for reparametrization modes,the real-frequency correlator (see the SupplementalMaterial for details) (cid:104) T − ω, − p T ω,p (cid:105) = Nα πβ Dp − iω + Dp , where the diffusive constant D is D = π r √ − rJt (cid:48) (1 − r )( t + t (cid:48) ) + r J . (10)Some remarks come with this expression for diffusivetransport of energy. First, when t (cid:48) = 0, different unitcells decouple from each other, and the diffusive con-stant D vanishes as expected. On the other hand, when r →
0, the free Majorana fermions vanish, and the systembecomes decoupled islands of SYK Majorana fermionsand cannot conduct energy. A more interesting obser-vation is that when r →
1, the diffusive constant scalesas D ∝ (1 − r ) / , and we expect the system enters alocalized phase.We are now in a position to calculate the OTOC. Con-sider the following four-point correlation function F ψψ,xx (cid:48) ( τ τ τ τ ) = 1 N (cid:104) T τ (cid:88) ij ψ i,x ( τ ) ψ i,x ( τ ) ψ j,x (cid:48) ( τ ) ψ j,x (cid:48) ( τ ) (cid:105) = G sψ ( τ τ ) G sψ ( τ τ ) + 1 N F ψψ,xx (cid:48) ( τ τ τ τ ) , (11)where T τ denotes imaginary time ordering, G sψ isgiven by the saddle-point solutions in Eqs.(7-8), and F ψψ,xx (cid:48) is the connected part coming from the fluc-tuations around the saddle-point, F ψψ,xx (cid:48) ( τ τ τ τ ) ≡(cid:104) δG ψ,x ( τ τ ) δG ψ,x (cid:48) ( τ τ ) (cid:105) and is dominated by thereparametrization modes. Similar calculations apply tothe OTOC of other operators. In order to evaluate theOTOC, let τ = β + it , τ = β , τ = β + it , τ = β , and wearrive at (see the Supplemental Material for details) F ab,xy ( τ τ τ τ ) G sa ( β ) G sb ( − β ) ∝ − ∆ a ∆ b π √ α α (cid:114) β π e πβ (cid:0) t − | x − y | vB (cid:1) , (12)with v B = πβ D and a, b = ψ, η . We first note that the quan-tum analog of the Lyapunov exponent defined by OTOCin this phase still saturates the bound λ L = πβ [41]. Sec-ond, the butterfly velocity, Lyapunov exponent, and dif-fusive constant here satisfy a simple and elegant relation: D = v B λ L [79, 80]. Such a relation was previously obtainedin incoherent black holes [81, 82] and higher-dimensionalgeneralizations of the SYK model [57, 64, 83]. As thebutterfly velocity v B ∝ (1 − r ) is vanishing for r →
1, itfurther indicates that the system shall undergo a local-ization transition as r crosses the critical value r c =1. MBL phase:
For r (cid:29)
1, it is expected that the An-derson localization of “free” Majorana fermions for largebut finite N dominate in determining low-energy physicsand the SYK interaction J is irrelevant. Consequently,the system should fall into a localized phase [73]. Sim-ilar to the case of r (cid:28)
1, we also make a translationalinvariant ansatz for r (cid:29)
1, with which the saddle-pointequation can be approximated by G − ψ = − iω − Σ ψ , G − η = − iω − Σ η , (13)Σ ψ = √ r ˜ t G η , Σ η = ˜ t G ψ / √ r, (14)where ˜ t ≡ t + t (cid:48) . The exact solutions of the aboveSchwinger-Dyson equations are obtained in the Supple-mental Material. Here, let’s explicitly expand the inversepropagators G − a around small frequency: G − η = − rr − iω − r / ( r − ˜ t ( iω ) + O ( ω ) , (15) G − ψ = r − √ r iω − rr − iω − r / ( r − ˜ t ( iω ) + O ( ω ) . (16)Although in G − η the bare term ∝ − iω is renormalized bya factor rr − , its self-energy is subdominant at low energy,indicating the free Gaussian fixed point of η Majoranafermions is stable. (In the limit of r →∞ , G − η →− iω ,as expected from a free theory.) However, for the ψ fermions, the self-energy actually dominates the behaviorof G ψ in low energy, which generates a large anomalousdimension to ψ . For simplicity, we keep the leading termin Eq. (16) and make a Fourier transformation, G ψ ( τ ) = √ r − γ )( r −
1) sgn( τ ) | τ | , (17)where γ ≈ .
577 is the Euler-Gamma constant. From thepropagator of ψ fermions, one deduces its scaling dimen-sion [ ψ ]=1, as expected from the r (cid:29) J . In-cluding J terms leads to a correction to the self-energy, δ Σ ψ ( τ )= J G ψ ( τ ) = r / J − γ ) ( r − sgn( τ ) | τ | . By a Fouriertransformation, δ Σ ψ ( iω ) ∝ ω , which is subdominant inlow energy, compared with leading terms in Eq. (16). Thesame is true for δ Σ η ( iω ). Thus, we can conclude the freefixed point with [ η ]=0 and [ ψ ]=1 is stable against weakinteraction J , which self-justifies the assumption we havemade. One important consequence is that, as all levelsof the free Majorana fermions with random hopping arelocalized for large and finite N [73, 84], MBL emerges inthe presence of the weak but irrelevant SYK interaction J [85]. It is consistent with vanishing diffusive constant for r>
1. Note that the system has ( M − N ) L single-particlezero modes localized on the B sites for r >
1. However,these localized states do not change the MBL phase be-cause they are isolated from the rest of the many-bodystates and only cause macroscopic degeneracies. One wayof removing these extensive localized zero modes but pre-serving the low-energy physics is to add weak quadraticcouplings on B sites, as we show in the SupplementalMaterial.
Numerical evidences of MBL transitions:
Wenow show numerical evidence of such a phase transi-tion between the thermal and MBL phases. For a MBLphase, its level statistics satisfies the Poisson distribu-tion according to the Berry-Tabor conjecture [86] while athermal phase’s level statistics follows the Wigner-Dyson(WD) distribution. Suppose { E n } denotes many-bodyeigenstate energies in an ascending order and the levelspacings between adjacent eigenstates are ∆ n = E n +1 − E n log s - - - P ( log s ) Wigner-DysonPoisson (a) log s - - - P ( log s ) (b) log s - - - P ( log s ) (c) FIG. 2. The distribution of level-spacing ratios for the casesof ( N , M )=(6,4), (5,5) and (4,6) are shown in (a), (b) and (c),respectively. The results (red solid line) are obtained by ex-actly diagonalizing the generalized SYK model on the six-sitechain with N + M =10 Majorana fermions in each unit cell andwith J = t =1, t (cid:48) =0.5. The Wigner-Dyson distribution (dashedline) implies thermalization while Poisson distribution (dot-ted line) implies MBL. with ∆ n ≥
0. The ratio between two consecutive gaps s n = ∆ n +1 ∆ n can be employed to characterize the levelstatistics [9, 87]. The distribution of ratios in MBLphases follows Poisson level statistics p ( s )= s ) , whilein thermalized phases, it follows WD level statistics p ( s )= √ π ( s + s ) (1+ s + s ) (assuming Gaussian unitary ensem-ble).Following Ref. [46], we plot the distribution of log s ,i.e., P (log s ) = p ( s ) s , as shown in Fig. S1. Thedata are obtained from exactly diagonalizing the modelwith J = t =1, t (cid:48) =0.5 on a six-site chain with N + M =10Majorana fermions per unit cell. The distribution for( N, M )=(6,4), (5,5), (4,6) is shown in Fig. 2(a,b,c), re-spectively. When
N >M (namely, r< N
1, each many-body en-ergy level has an extra degeneracy due to the presenceof the single-particle zero modes localized on B sites. Inthe calculation of energy level statistics, we have ignoredthese trivial degeneracy. The degeneracy can be lifted byadding weak quadratic couplings on B sites, and for suchmodified case we also calculated the level statistics andobtained the qualitatively same results, as shown in theSupplemental Material.
SYK model on 2D lattices:
Our construction of theSYK models in 1D can be straightforwardly generalizedto more than 1D. For instance, we consider the general-ization to the square lattice as shown in Fig. 3. Each unitcell consists of two sites represented by a square and adisk, where N SYK Majorana fermions and M free Ma-jorana fermions reside, respectively. The model is given e e 𝜓 𝑖,𝑥 𝜂 𝛼,𝑥 𝑡 𝑡 𝑡 𝑡 (a) MBL (cid:1)
Diffusive metal (cid:1) D r (cid:1) (b) FIG. 3. (a) The generalized SYK model on the square lattice.Each unit cell consists of two sites represented by a squareand a disk, where N SYK Majorana fermions and M freeMajorana fermions reside, respectively. t denotes the vari-ance of random hopping within a unit cell, while t (cid:48) denotesthat between neighboring unit cells. (b) The energy diffusiveconstant D along the e or e direction as a function of r . Weuse the parameter J = t =1, t (cid:48) = t (cid:48) =0.1, t (cid:48) =0. by H = (cid:88) x (cid:104) (cid:88) ijkl J ijkl, x ψ i, x ψ j, x ψ k, x ψ l, x + (cid:88) iα (cid:16) t iα, x iψ i, x η α, x + (cid:88) a t (cid:48) iα, x a iη α, x ψ i, x + b a (cid:17)(cid:105) , where x represents unit cells, and b a label thevectors connecting neighboring unit cells with b = e =(1 , b = e =(0 , b =(1 , (cid:104) J ijkl, x (cid:105) =3! J /N , (cid:104) t iα, x (cid:105) = t / √ M N , and (cid:104) t (cid:48) iα, x a (cid:105) = t (cid:48) a / √ M N . (Note that the limit of t (cid:48) =0 correspondsto the honeycomb lattice). The analysis of the gen-eralized model on 2D and higher-dimensional latticesgoes like the 1D chain case. For r<
1, the generalizedmodels on 2D lattices possess similar features, includingdiffusive energy transport, zero-temperature entropy,and maximum quantum chaos, which are the same asthe model on a 1D chain. For instance, the diffusiveconstant in 2D as a function of r is also given byEq. (10), which is plotted in Fig. 3(b). For r → D ∝ r because in diffusive metal, the SYK Majorana fermionsdiffuse via free Majorana fermions; while for r → D ∝ (1 − r ) / , which indicates that the system couldundergo a dynamical transition into a MBL phase. Concluding remarks:
We have shown that the MBLtransition in the generalized SYK models is qualitativelydistinct from previously studied ones in other models likethe XXZ model in various ways. Intuitively, we thinkthat the qualitative differences are mainly due to thelarge- N degrees of freedom on each site in the gener-alized SYK models. In the large- N limit, due to the all-to-all interactions, we can define an effective dimensions d SYK →∞ such that the effective dimensions of the gener-alized model on the d -dimensional lattice is d ∗ = d SYK + d ,which approaches infinity. As a consequence, for the SYK model on the d =1 lattice, there is no subdiffusive phasearound the MBL transition because its effective space di-mension d ∗ is much larger than 1. Moreover, the Harriscriterion is not violated by ν =0 when d ∗ is considered asthe effective space dimension.Note that there are questions that remain open. Toinspire readers, we provide a few here. First, what is thecritical theory governing this MBL transition? Our anal-ysis cannot be applied directly at r =1, and the criticaltheory remains unknown. Second, is time-reversal sym-metry spontaneously broken in the MBL phase ( r (cid:38) J term is irrelevantwhen r (cid:29)
1, it is possible to be dangerously irrelevant for r (cid:38)
1. Third, how robust is the critical point when othertypes of interactions are included in the model?
Acknowledgements : We would like to thank Xin Dai,Yingfei Gu, David Huse, Xiaoliang Qi, Cenke Xu, andShixin Zhang for helpful discussions. This work is sup-ported in part by the MOST of China under Grant No.2016YFA0301001 (H. Y.) and by the NSFC under GrantNo. 11474175 (S.-K. J. and H. Y.).
Note added : After the completion of the present work,we became aware of an upcoming work [88] studying adifferent generalization of the SYK model with a zero-temperature insulating phase (but not MBL). ∗ [email protected][1] J. M. Deutsch, Phys. Rev. A , 2046 (1991).[2] M. Srednicki, Phys. Rev. E , 888 (1994).[3] M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) , 854 (2008).[4] L. Fleishman and P. W. Anderson, Phys. Rev. B , 2366(1980).[5] T. Giamarchi and H. Schulz, Europhys. Lett. , 1287(1987).[6] B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Levitov,Phys. Rev. Lett. , 2803 (1997).[7] I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Phys.Rev. Lett. , 206603 (2005).[8] D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Ann.Phys. (Amsterdam) , 1126 (2006).[9] V. Oganesyan and D. A. Huse, Phys. Rev. B , 155111(2007).[10] A. Pal and D. A. Huse, Phys. Rev. B , 174411 (2010).[11] B. Bauer and C. Nayak, J. Stat. Mech. (2013) P09005.[12] R. Nandkishore and D.A. Huse, Annu. Rev. Condens.Matter Phys. , 15 (2015).[13] E. Altman and R. Vosk, Annu. Rev. Condens. MatterPhys. , 383 (2015).[14] R. Vasseur and J. E. Moore, J. Stat. Mech. (2016) 064010.[15] J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys.Rev. Lett. , 017202 (2012).[16] D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal, andS. L. Sondhi, Phys. Rev. B , 014206 (2013).[17] M. Serbyn, Z. Papic, and D. A. Abanin, Phys. Rev. Lett. , 127201 (2013).[18] D. A. Huse, R. Nandkishore, and V. Oganesyan, Phys. Rev. B , 174202 (2014).[19] A. Chandran, V. Khemani, C. R. Laumann, and S. L.Sondhi, Phys. Rev. B , 144201 (2014).[20] D. Pekker, G. Refael, E. Altman, E. Demler, and V.Oganesyan, Phys. Rev. X , 011052 (2014).[21] R. Vosk and E. Altman, Phys. Rev. Lett. , 217204(2014).[22] S. A. Parameswaran and S. Gopalakrishnan,arXiv:1608.00981 [Phys. Rev. Lett. (to be published)].[23] Y.-Z. You, X.-L. Qi, and C. Xu, Phys. Rev. B , 104205(2016).[24] D.-L. Deng, X. Li, J. H. Pixley, Y.-L. Wu, and S. DasSarma, Phys. Rev. B , 024202 (2017).[25] R. Fan, P. Zhang, H. Shen, and H. Zhai, Sci. Bull. ,707 (2017).[26] Y. Huang, Y.-L. Zhang, and X. Chen, Ann. Phys.(Berlin) , 1600318 (2017).[27] B. Swingle and D. Chowdhury, Phys. Rev. B , 060201(2017).[28] R.-Q. He and Z.-Y. Lu, Phys. Rev. B , 054201 (2017).[29] Y. Chen, arXiv:1608.02765.[30] X. Chen, T. Zhou, D. A. Huse, and E. Fradkin, Ann.Phys. (Berlin) , 1600332 (2017).[31] R. Vosk, D. A. Huse, and E. Altman, Phys. Rev. X ,031032 (2015).[32] A. C. Potter, R. Vasseur, and S. A. Parameswaran, Phys.Rev. X , 031033 (2015).[33] Y. Bar Lev, G. Cohen, and D. R. Reichman, Phys. Rev.Lett. , 100601 (2015).[34] S. Gopalakrishnan, K. Agarwal, E. A. Demler, D. A.Huse, and M. Knap, Phys. Rev. B , 134206 (2016).[35] M. Znidaric, A. Scardicchio, and V. K. Varma, Phys.Rev. Lett. , 040601 (2016).[36] A. Kitaev, “A simple model of quantum holography”,KITP, April 7, 2015 and May 27, 2015 (unpublished).[37] S. Sachdev and J. Ye, Phys. Rev. Lett. , 3339 (1993).[38] A. I. Larkin, Y. N. Ovchinnikov, Sov Phys JETP, ,1200 (1969).[39] A. Kitaev, “Hidden correlations in the Hawking radia-tion and thermal noise”, in Proceedings of FundamentalPhysics Prize Symposium, November 10, 2014.[40] S. H. Shenker and D. Stanford, J. High Energy Phys. (2014) 67.[41] J. Maldacena, S. H. Shenker, and D. Stanford, JHEP, (2016) 106.[42] J. Maldacena, D. Stanford, and Z. Yang,arXiv:1606.01857.[43] K. Jensen, Phys. Rev. Lett. , 111601 (2016).[44] J. Engelsoy, T. G. Mertens, and H. Verlinde, JHEP, (2016) 139.[45] J. Polchinski and V. Rosenhaus, JHEP, (2016) 001.[46] Y.-Z. You, A. W. W. Ludwig, and C. Xu, Phys. Rev. B , 115150 (2017).[47] A. Jevicki, K. Suzuki, and J. Yoon, JHEP (2016) 007.[48] J. Maldacena and D. Stanford, Phys. Rev. D , 106002(2016).[49] A. M. Garcia-Garcia and J. J. M. Verbaarschot, Phys.Rev. D , 126010 (2016).[50] J. S. Cotler, G. G.-A. M. Hanada, J. Polchinski, P. Saad,S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka,J. High Energy Phys. (2017) 118.[51] A. M. Garcia-Garcia, J. J. M. Verbaarschot, Phys. Rev.D 96, 066012 (2017).[52] V. Bonzom, L. Lionni, and A. Tanasa, Journal of Math- ematical Physics , 052301 (2017).[53] D. J. Gross and V. Rosenhaus, J. High Energy Phys. (2017) 092.[54] D. Bagrets, A. Altland, and A. Kamenev, Nucl. Phys. B921 , 727 (2017).[55] D. I. Pikulin and M. Franz, Phys. Rev. X , 031006(2017).[56] W. Fu and S. Sachdev, Phys. Rev. B , 035135 (2016).[57] Y. Gu, X.-L. Qi, and D. Stanford, J. High Energy Phys. (2017) 125.[58] D. J. Gross and V. Rosenhaus, J. High Energy Phys. (2017) 093.[59] M. Berkooz, P. Narayan, M. Rozali, and J. Simon, J.High Energy Phys. (2017) 138.[60] W. Fu, D. Gaiotto, J. Maldacena, and S. Sachdev, Phys.Rev. D , 026009 (2017).[61] S. Banerjee and E. Altman, Phys. Rev. B , 134302(2017).[62] E. Witten, arXiv:1610.09758.[63] I. R. Klebanov and G. Tarnopolsky, Phys. Rev. D ,046004 (2017).[64] R. A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen,and S. Sachdev, Phys. Rev. B , 155131 (2017).[65] C. Peng, M. Spradlin, and A. Volovich, J. High EnergyPhys. (2017) 062.[66] C. Krishnan, S. Sanyal, and P. N. B. Subramanian, J.High Energy Phys. (2017) 056.[67] G. J. Turiaci and H. Verlinde, arXiv:1701.00528.[68] Z. Bi, C.-M. Jian, Y.-Z. You, K. A. Pawlak, and C. Xu,Phys. Rev. B , 205105 (2017).[69] T. Li, J. Liu, Y. Xin, and Y. Zhou, J. High Energy Phys. (2017) 111.[70] Y. Gu, A. Lucas, and X.-L. Qi, SciPost Phys. , 018(2017).[71] X. Chen, R. Fan, Y. Chen, H. Zhai, and P. Zhang, Phys.Rev. Lett. , 207603 (2017).[72] X.-Y. Song, C.-M. Jian, and L. Balents,arXiv:1705.00117.[73] For N = ∞ , the random-hopping Majorana fermion modelwith J =0 would have a finite diffusive constant, which isactually an artifact of N = ∞ . When N is large but not in-finity, the system with r> J =0 would be Andersonlocalized.[74] A. B. Harris, J. Phys. C , 1671 (1974).[75] J. T. Chayes, L. Chayes, D. S. Fisher, and T. Spencer,Phys. Rev. Lett. , 2999 (1986).[76] A. Chandran, C. R. Laumann, and V. Oganesyan,arXiv:1509.04285.[77] Q. Si, S. Rabello, K. Ingersent, and J. L. Smith, Nature(London) (2001).[78] T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, Phys.Rev. D, , 125002 (2011).[79] S. A. Hartnoll, Nature Physics , 54 (2015).[80] M. Blake, Phys. Rev. Lett. , 091601 (2016).[81] M. Blake, Phys. Rev. D. , 086014 (2016).[82] M. Blake and A. Donos, J. High Energy Phys. (2017)013.[83] Violation of such relation in inhomogeneous SYK chainsis studied in Ref. [70].[84] D. C. Herbert and R. Jones, J. Phys. C , 1145 (1971).[85] R. Vosk and E. Altman, Phys. Rev. Lett. , 067204(2013).[86] M.V. Berry and M. Tabor, Proc. R. Soc. A , 375(1977). [87] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux,Phys. Rev. Lett. , 084101 (2013). [88] C.-M. Jian, Z. Bi, and C. Xu, Phys. Rev. B , 115122(2017). SUPPLEMENTAL MATERIALA. Replica action
The Hamiltonian in main text is given by H = L (cid:88) x =1 (cid:104) (cid:88) ijkl J ijkl,x ψ i,x ψ j,x ψ k,x ψ l,x + (cid:88) iα (cid:16) t iα,x iψ i,x η α,x + t (cid:48) iα,x iη α,x ψ i,x +1 (cid:17)(cid:105) , (S1)where ψ and η are SYK Majorana fermions and free Majorana fermions, J ijkl,x , t iα,x and t (cid:48) iα,x are independent randomcouplings with zero mean and variance (cid:104) J ijkl,x (cid:105) = J /N , (cid:104) t iα (cid:105) = t / √ M N , and (cid:104) t (cid:48) iα (cid:105) = t (cid:48) / √ M N . Note that N , M are the numbers of SYK Majorana fermions and free Majorana fermions in each sub-lattice respectively, while L is the number of unit cells in the chain. Replica trick utilizes the identity log Z = lim n → e n log Z − n = lim n → Z n − n .Instead of disorder averaging the logarithm of partition function which is difficult to do, one averages over n copiesof the system, then take n → S = (cid:88) m,x (cid:90) ( 12 ψ mi,x ∂ τ ψ mi,x + 12 η mα,x ∂ τ ψ mα,x )+ (cid:88) m,m (cid:48) ,x (cid:90) (cid:90) (cid:104) − J N (cid:16) ψ mi,x ψ m (cid:48) i,x (cid:17) − (cid:16) t √ M N ψ mi,x ψ m (cid:48) i,x η mα,x η m (cid:48) α,x + t (cid:48) √ M N ψ mi,x +1 ψ m (cid:48) i,x +1 η mα,x η m (cid:48) α,x (cid:17)(cid:105) , (S2)where m, m (cid:48) is replica index. As explained in main text, here we consider the diagonal parts. Introduce two collec-tive modes, G ψ,x ( τ , τ ) = N (cid:80) i ψ i,x ( τ ) ψ i,x ( τ ), G η,x ( τ , τ ) = M (cid:80) α η α,x ( τ ) η α,x ( τ ), and corresponding Legendremultipliers Σ ψ , Σ η , one arrives at SN = 12 (cid:88) x [ − tr log( ∂ τ − Σ ψ,x ) − r tr log( ∂ τ − Σ η,x )]+ 12 (cid:90) dτ dτ (cid:88) x (cid:104) Σ ψ,x G ψ,x + r Σ η,x G η,x − J G ψ,x − √ rt G ψ,x G η,x − √ rt (cid:48) G η,x G ψ,x +1 (cid:105) , (S3)where Majorana fermions are integrated out. B. Two-point functions in diffusive metal
The solutions of saddle point function in frequency domain are given by Fourier transforming the time domainsolutions, i.e., G ψ ( iω ) = i [(1 − r ) π ] / sgn( ω ) (cid:112) J | ω | , G η ( iω ) = i √ r sgn( ω )( t + t (cid:48) )[(1 − r ) π ] / (cid:112) J | ω | . (S4)While finite temperature solutions are obtained by substitution τ → tan πτβ , i.e., G a ( τ ) = Λ a sgn( τ ) | βπ sin πτβ | ∆ a . HereΛ ψ ≡ (cid:16) − r πJ (cid:17) / , Λ η = t + t (cid:48) ) (cid:16) r J π (1 − r ) (cid:17) / , ∆ ψ = and ∆ η = . Following the same lines in [61], the cutofffrequency for validity of these conformal solutions are given by ω c = min( ω , ω ), where ω ≈ J √ π (1 − r ) / + J √ π r (1 − r ) / , ω ≈ (cid:16) √ π t + t (cid:48) ) J √ − rr (cid:17) / . (S5)When r approach the transition point r c = 1, ω c ∼ (1 − r ) / →
0, the saddle point solutions break down.
C. Zero temperature entropy
In this subsection, we use I to denote action while S denotes entropy to avoid confusion. The homogenous saddle-point action generalized to q -body interaction is given by IN L = 12 [ − tr log( − Σ ψ ) − r tr log( − Σ η )] + 12 (cid:90) dτ dτ (cid:104) Σ ψ G ψ + r Σ η G η − J q G qψ − √ r ˜ t G ψ G η (cid:105) . (S6)where time derivative terms are neglected since they are irrelevant in zero temperature. Equations of motion are G ψ ∗ Σ ψ = − , G η ∗ Σ η = − , Σ ψ = J G q − ψ + √ r ˜ t G η , Σ η = 1 √ r ˜ t G ψ , (S7)from which we have J G ψ ∗ G q − ψ = − (1 − r ) with the solutions G ψ ( τ ) = b sgn( τ ) | τ | , G η = √ rJ (1 − r )˜ t G q − ψ , Σ ψ = J − r G q − ψ , Σ η = ˜ t √ r G ψ (S8)where J b q π = (1 − r )( − ∆) tan π ∆, and ∆ = q . Free energy is given by F = T I , from which we can get entropythrough S = − ∂F∂T = − I − T (cid:88) α (cid:16) δIδG α ∂G α ∂T + δIδ Σ α ∂G α ∂T + ∂I∂T (cid:17) , (S9)The second term of Eq. (S9) vanishes when one plugs the solutions; the zero temperature entropy is given by S = − lim T → I . Plug the solutions into the second term of the action Eq. (S6), we have12 J (cid:104) − r − q (cid:105) (cid:90) dτ dτ G qψ ( τ , τ ) = 12 βb q J (cid:104) − r − q (cid:105) (cid:90) β dτ (cid:0) βπ sin πτβ (cid:1) = 0 . (S10)The vanishing of above integral can be seen by analytical continuing τ → τ + it . Thus only the first term in saddlepoint action is not vanishing, S = N L − Σ ψ ) + r tr log( − Σ η )] = N L (cid:88) n log( − Σ ψ ( iω n )) + r (cid:88) n log( − Σ η ( iω n ))] . (S11)Fourier transforming the self-energy leads toΣ ψ ( w ) = C ψ i sgn( ω ) | ω | − , Σ η ( w ) = C η i sgn( ω ) | ω | − (S12)where C ψ and C η are two constants independent of ∆. These constants are not important since we will take derivativewith respect to ∆. According to Ref. [48], the zero temperature entropy for SYK model satisfies ∂ S SYK (∆) ∂ ∆ = − π ( 12 − ∆) tan π ∆ . (S13)where S SYK = S SYK N . Here, analogous to SYK model, we have1 N L ∂S ψ (∆) ∂ ∆ = − π ( 12 − ∆) tan π ∆ , N L ∂S η (∆) ∂ ∆ = − rπ ( 12 − ∆) tan π (1 − ∆) , ∂ S (∆) ∂ ∆ = − − r r π ( 12 − ∆) tan π ∆(S14)and where S = N + M ) L ( S ψ + S η ). The boundary condition can be set by S ( ) = 0 because when q = 2, the systemhas only quadratic term and unique ground state. Then for our interest case, ∆ = , the ground state entropy isgiven by S = 1 − r r S SYK (S15)where S SYK = C + π log 28 π ≈ . C ≈ .
916 is Catalan constant.
D. Effective action for reparametrization modes in diffusive metal
Inspired by the reparametrization symmetry in infrared, we redefine Σ( τ , τ ) → Σ( τ , τ ) + δ ( τ − τ ) ∂ τ , and bringthe action to S UV N = (cid:88) x (cid:90) (cid:90) (cid:104) δ ( τ )( ∂ G ψ,x + r∂ G η,x ) − √ rt (cid:48) ( G η,x G ψ,x +1 − G η,x G ψ,x ) (cid:105) , (S16) S IR N = 12 (cid:88) x − [ tr log( − Σ ψ,x ) + tr log( − Σ η,x )] + (cid:90) (cid:90) (cid:104) Σ ψ,x G ψ,x + r Σ η,x G η,x − J G ψ,x − √ r ( t + t (cid:48) ) G ψ,x G η,x (cid:105) . (S17)The redefinitions effectively collect the time derivative term to S UV . Moreover, since each unit cell decouples fromothers in S IR , the saddle point equations given by S IR are exactly conformal invariant whose solutions are obtainedabove. Since S IR has reparametrization symmetry, it vanishes for reparametrization modes. However, this symmetryis explicitly broken by UV part, i.e., S UV will give a small action to reparametrization modes which dominants thefour-point correlator in the infrared. The effective action for reparametrization modes is given by fluctuations aroundsaddle point action, i.e., the effective action for reparametrization modes reads S [ f ] = S UV [ ˜ G ( f )] − S UV [ G ( t )]. Forthe first term in S UV , we obtain the effective action using ε -expansion [47, 67], where ε = − ∆ ψ : S (1) UV N = (cid:88) x − π ( √ − rJ + Jt + t (cid:48) (cid:114) r − r ) (cid:90) dτ { f x , τ } . (S18)where { f, τ } = f (cid:48)(cid:48)(cid:48) f (cid:48) − (cid:16) f (cid:48)(cid:48) f (cid:48) (cid:17) denotes Schwartz derivative. For small reparametrization f x = τ + (cid:15) x ( τ ), S (1) UV N = 1128 π ( √ − rJ + Jt + t (cid:48) (cid:114) r − r ) (cid:88) n,p n ( n − (cid:15) − n, − p (cid:15) n,p (S19)where (cid:15) x ( τ ) = π √ L (cid:80) n,p (cid:15) n,p e − inτ + ipx , and we have set β = 2 π for simplicity. For second term in S UV , thereparametrization modes are defined as ˜ G ψ,x = G ψ + δG ψ,x , ˜ G η,x = G η + δG η,x . Though the saddle point solutionsare uniform in spatial direction, their fluctuations are position-dependent [57]. Then the effective action reads S (2) UV N = 12 (cid:88) x (cid:90) (cid:90) √ rt (cid:48) ( δG η,x δG ψ,x − δG η,x δG ψ,x +1 ) , (S20)where (cid:82)(cid:82) = (cid:82) dτ dτ . Explicitly, for conformal solutions G ( τ ) = Λ sgn( τ ) | τ | , the reparametrization gives rise to δG x ( τ , τ ) = i ∆ π G ( τ ) (cid:88) n (cid:15) n,x h n ( τ ) e − in ¯ τ . (S21)where h n ( τ ) ≡ sin nτ cot τ − n cos nτ and τ = τ − τ , ¯ τ = ( τ + τ ). Then (cid:88) x (cid:90) (cid:90) ( δG η,x δG ψ,x − δG η,x δG ψ,x +1 ) = √ r π ( t + t (cid:48) ) (cid:88) n,x | n | ( n − (cid:15) − n,x (cid:15) n,x − (cid:15) − n,x (cid:15) n,x +1 ) (S22)= √ r π ( t + t (cid:48) ) (cid:88) n,p | n | ( n − − cos p ) (cid:15) − n, − p (cid:15) n,p (S23)Plug these results in to S (2) , we get S (2) UV N = (cid:88) n,p π rt (cid:48) t + t (cid:48) (1 − cos p ) | n | ( n − (cid:15) − n, − p (cid:15) n,p (S24)Finally, in terms of the infinitesimal modes, the effective action is given by S eff N = S (1) UV N + S (2) UV N = 12 (cid:88) n,p (cid:104) α n ( n −
1) + α − cos p ) | n | ( n − (cid:105) (cid:15) − n, − p (cid:15) n,p , (S25)0where α = π (cid:16) √ − rJ + Jt + t (cid:48) (cid:113) r − r (cid:17) , α = π rt (cid:48) t + t (cid:48) . To restore the dimension, note that dim[ (cid:15) x ( τ )] = −
1, anddim[ (cid:15) n ] = −
2, where dim[ ... ] denotes engineer dimension (not to confuse with scaling dimension), we obtain S eff N = πβ (cid:88) n,p (cid:104) α ω n (cid:16) ω n − (cid:0) πβ (cid:1) (cid:17) + α p | ω n | (cid:16) ω n − (cid:0) πβ (cid:1) (cid:17)(cid:105) (cid:15) − ω n , − p (cid:15) ω n ,p . (S26)where we have expanded for small momentum. E. Diffusive constant in diffusive metal
According to Noether’s theorem, the reparametrization modes in time direction couple to energy density, i.e., δS = (cid:82) dτ (cid:15)∂ τ T , where (cid:15) is reparametrization mode and T is energy density. At zero momentum limit, the effectiveaction is given by S eff N = α (cid:90) dτ (cid:2) ( (cid:15) (cid:48)(cid:48) ) − ( (cid:15) (cid:48) ) (cid:3) = α (cid:90) dτ (cid:15)∂ τ ( (cid:15) (cid:48)(cid:48)(cid:48) + (cid:15) (cid:48) ) , (S27)Then one finds that T = α N ( (cid:15) (cid:48)(cid:48)(cid:48) + (cid:15) (cid:48) ) or in frequency domain, T n = iα N π ( n − n ) (cid:15) n . The correlation of energydensity is given by (cid:104) T − n T n (cid:105) = N α π ( n − n ) (cid:104) (cid:15) − n (cid:15) n (cid:105) . (S28)For small momentum, one can directly generalize this function, (cid:104) T − n, − p T n,p (cid:105) = N α π ( n − n ) (cid:104) (cid:15) − n, − p (cid:15) n,p (cid:105) (S29)Plug the propagator from effective action into the correlation, we have (cid:104) T − n, − p T n,p (cid:105) = N α π n ( n − α n ( n −
1) + α p | n | ( n −
1) =
N α π | n | ( n − | n | + α α p . (S30)Note that dim[ T ( τ )] = 1, we can also restore temperature: (cid:104) T − ω n , − p T ω n ,p (cid:105) = N α π πβ | ω n | (cid:104)(cid:0) βω n π (cid:1) − (cid:105) | ω n | + α α p . (S31)Analytical continuation from upper half-plane, i.e. iω n → ω + iδ , the retarded correlation function is (cid:104) T − ω, − p T ω,p (cid:105) = N α πβ Dp − iω + Dp , (S32)where D ≡ α α and we have implicitly extracted a contact term [57, 64], i.e., (cid:104) T − ω, − p T ω,p (cid:105) − (cid:104) T − ω, T ω, (cid:105) . F. Chaos and butterfly velocity in diffusive metal
Four-point functions are defined as F ab,xy ( τ , τ , τ , τ ) = 1 N a N b (cid:88) i,j (cid:104) a i,x (1) a i,x (2) b j,y (3) b j,y (4) (cid:105) = G a ( τ , τ ) G b ( τ , τ ) + 1 N F ab,xy ( τ , τ , τ , τ ) , (S33)where F ab,xy ( τ , τ , τ , τ ) ≡ (cid:104) δG a,x ( τ , τ ) δG b,y ( τ , τ ) (cid:105) and a, b = ψ, η . Using the translational symmetry, F ab,xy ( τ , τ , τ , τ ) = 1 L (cid:88) p F ab,p ( τ , τ , τ , τ ) e ip ( x − y ) , (S34)with F ab,p ( τ , τ , τ , τ ) = L (cid:104) δG a, − p ( τ , τ ) δG b,p ( τ , τ ) (cid:105) . Thus F ab,p ( τ , τ , τ , τ ) G a ( τ , τ ) G b ( τ , τ ) = N ∆ a ∆ b π (cid:88) n (cid:104) (cid:15) − n, − p (cid:15) n,p (cid:105) h n ( x ) h n ( x ) e − in ( y − y ) . (S35)1For OTOC, let τ = β + it, τ = β , τ = β + it, τ = β , then x = β , x = − β , y = β + it, y = β , and set β = 2 π for simplicity, we have F ab,p ( τ , τ , τ , τ ) G a ( π ) G b ( − π ) = N ∆ a ∆ b π (cid:88) n (cid:104) (cid:15) − n, − p (cid:15) n,p (cid:105) h n ( π ) h n ( − π ) e nt − i nπ (S36)= N ∆ a ∆ b π (cid:88) n (cid:104) (cid:15) − n, − p (cid:15) n,p (cid:105) n (cos nπ e nt − i nπ (S37)= N ∆ a ∆ b π (cid:88) n ≥ ,even (cid:104) (cid:15) − n, − p (cid:15) n,p (cid:105) ( − n n ( e nt + e − nt ) . (S38)Plug the propagator obtained from effective action into above equation, we have F ab,p ( τ , τ , τ , τ ) G a ( π ) G b ( − π ) = 2∆ a ∆ b π (cid:88) n ≥ ,even ( − n n cosh ntα n ( n −
1) + 2 α (1 − cos p ) | n | ( n − . (S39)To evaluate above summation, consider the integral K = (cid:90) i ∞− i ∞ dω πi π πω ω cosh ωtα ω ( ω −
1) + α p | ω | ( ω − . (S40)Making a large semicircle contour to w → + ∞ of complex plane [57], one finds according to residue theorem K = − (cid:88) n ≥ ,even ( − n n cosh ntα n ( n −
1) + α p | n | ( n − − π tα + α p . (S41)or (cid:88) n ≥ ,even ( − n n cosh ntα n ( n −
1) + α p | n | ( n −
1) = − π tα + α p − K. (S42)There is no exponential term in K [57], thus the only exponential growth part is F ab,p ( τ , τ , τ , τ ) G a ( π ) G b ( − π ) (cid:51) − ∆ a ∆ b π cosh tα + α p . (S43)Fourier transform to real space, then we have F ab,xy ( τ , τ , τ , τ ) G a ( π ) G b ( − π ) (cid:51) − ∆ a ∆ b π L (cid:88) p cosh tα + α p e ip ( x − y ) = − ∆ a ∆ b π (cid:90) dp π cosh tα + α p e ip ( x − y ) (S44)= − ∆ a ∆ b π √ α α e − | x − y | vB coth t ≈ − ∆ a ∆ b π √ α α e t − | x − y | vB , (S45)The exponent growth of OTOC is now given by (we have restored dimensions) F ab,xy G a ( π ) G b ( − π ) ∼ − N ∆ a ∆ b π √ α α (cid:114) β π e πβ ( t − | x − y | vB ) . (S46)Since x, y is dimensionless here (we set the lattice constant to 1), [ v B ] = 1 and the butterfly velocity is given by v B = α α πβ = πr √ − rt (cid:48) − r ) t + t (cid:48) J + r J ] 2 πβ . (S47)When approaching transition point, r →
1, butterfly velocity vanishes indicating a MBL phase.2 log s
Wigner-DysonPoisson - - P ( log s ) (a) log s Wigner-DysonPoisson - - P ( log s ) (b) log s Wigner-DysonPoisson - - P ( log s ) (c) FIG. S1. The distribution of level-spacing ratios for the cases of ( N , M )=(6,4), (5,5) and (4,6) are shown in (a), (b) and(c), respectively. The results (red solid line) are obtained by exactly diagonalizing the generalized SYK model on the six-sitechain with N + M =10 Majorana fermions in each unit cell and with J = 0 . t = 1 . t (cid:48) = 0 . V = 0 .
2. The Wigner-Dysondistribution (dashed line) implies thermalization while Poisson distribution (dotted line) implies MBL.
G. Many-body localized phase
Similar to the case of r (cid:28)
1, we also make a translational invariant ansatz for r (cid:29)
1, with which the saddle pointequation can be approximated by G − ψ = − iω − Σ ψ , G − η = − iω − Σ η , (S48)Σ ψ = √ r ˜ t G η , Σ η = ˜ t G ψ / √ r, (S49)where ˜ t ≡ t + t (cid:48) . The exact solutions of the above Schwinger-Dyson equations can be obtained: G η = 2 − iω + i ( r − t √ rω − i sgn( ω ) (cid:113) ( r − ˜ t rω + r +1)˜ t √ r + ω , (S50) G ψ = 2 − iω − i ( r − t √ rω − i sgn( ω ) (cid:113) ( r − ˜ t rω + r +1)˜ t √ r + ω . (S51)The self-energy part Σ a seems to dominate the propagator G a at low frequency or energy due to the terms ∝ ω in thedenominators of Eqs. (S50-S51). It is true for Σ ψ ; but for Σ η the ω terms are cancelled in the limit of low frequency. G. A modified model