Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models

Journal of High Energy Physics, May 2017

Abstract The Sachdev-Ye-Kitaev model is a (0 + 1)-dimensional model describing Majorana fermions or complex fermions with random interactions. This model has various interesting properties such as approximate local criticality (power law correlation in time), zero temperature entropy, and quantum chaos. In this article, we propose a higher dimensional generalization of the Sachdev-Ye-Kitaev model, which is a lattice model with N Majorana fermions at each site and random interactions between them. Our model can be defined on arbitrary lattices in arbitrary spatial dimensions. In the large N limit, the higher dimensional model preserves many properties of the Sachdev-Ye-Kitaev model such as local criticality in two-point functions, zero temperature entropy and chaos measured by the out-of-time-ordered correlation functions. In addition, we obtain new properties unique to higher dimensions such as diffusive energy transport and a “butterfly velocity” describing the propagation of chaos in space. We mainly present results for a (1 + 1)-dimensional example, and discuss the general case near the end.

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Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models

Received: February Local criticality, di usion and chaos in generalized Sachdev-Ye-Kitaev models 0 Open Access , c The Authors 1 1 Einstein Dr , Princeton, NJ 08540 , U.S.A 2 382 Via Pueblo Mall , Stanford, CA 94305 , U.S.A 3 School of Natural Sciences, Institute for Advanced Study 4 Department of Physics, Stanford University The Sachdev-Ye-Kitaev model is a (0 + 1)-dimensional model describing Majorana fermions or complex fermions with random interactions. This model has various interesting properties such as approximate local criticality (power law correlation in time), zero temperature entropy, and quantum chaos. In this article, we propose a higher dimensional generalization of the Sachdev-Ye-Kitaev model, which is a lattice model with N Majorana fermions at each site and random interactions between them. Our model can be de ned on arbitrary lattices in arbitrary spatial dimensions. In the large N limit, the higher dimensional model preserves many properties of the Sachdev-Ye-Kitaev model such as local criticality in two-point functions, zero temperature entropy and chaos measured by the out-of-time-ordered correlation functions. In addition, we obtain new properties unique to higher dimensions such as di usive energy transport and a butter y velocity" describing the propagation of chaos in space. We mainly present results for a (1 + 1)-dimensional example, and discuss the general case near the end. - Sachdev-Ye-Kitaev 1 Introduction 2 Review of the SYK model 3 The generalized SYK model 3.1 3.2 3.3 De nition of the chain model The e ective action and the saddle point Fluctuations of the collective elds and four-point functions 3.4 Symmetry breaking and pseudo-Goldstone mode 4 The OPE region and the energy transport The OPE at nite p The h = 2 contribution for small p Energy transport 5 Chaos and the butter y velocity 6 General construction and higher dimensions Generalized SYK model on higher dimensional lattices Models with global symmetry General q 7 Conclusion and discussion A Diagrammatic derivation for four-point functions B Summation trick and the prefactor C Di usion and the butter y velocity in general dimensions Introduction Holographic duality, also known as the anti-de-Sitter space/conformal eld theory (AdS/CFT) correspondence, refers to the duality between quantum many-body systems in d spatial dimensions and gravitational theories in (d + 1)-dimensional asymptoticaly anti-de Sitter geometries [1{3]. Various pieces of evidence suggest that holographic duality is a generic phenomenon that applies beyond the super Yang-Mills theories where the original conjecture was proposed. However, it is di cult to nd concrete models for which the duality can be veri ed directly by comparing bulk and boundary calculations. A very interesting development in this direction is the Sachdev-Ye-Kitaev model [4], which is a (0 + 1)-dimensional model describing random interactions between N Majorana fermions. When N is large and the temperature is low, the model has an emergent approximate time reparameterization symmetry which is weakly broken. In this limit, two-point functions and four-point functions can be computed. The results suggest that the model has a weakly coupled holographic dual, which includes dilaton gravity in an approximate AdS2 geometry [5{10], weakly coupled to an in nite number of matter elds [11]. The SYK model is a modi cation of a quantum spin model proposed by Sachdev and Ye more than 20 years ago [12] (which was also related to holographic duality in ref. [13]). In the large N and low temperature limit N the inverse temperature and J the average coupling strength), the behavior of the model is controlled by a large the role of a semi-classical \order parameter" with small uctuations suppressed by N1 . At low temperature, G( ; 0) has a power law dependence on 0 in the infared, suggesting that the low energy dynamics of this model might be conformally invariant [14]. In fact, the (0 + 1)-d conformal symmetry is only approximate, and in fact the low-temperature dynamics are dominated by the speci c way in which the symmetry is broken [4, 11]. A particularly interesting type of four-point function is the out-of-time-order corremeasure of chaos in quantum systems [15{20] (see refs. [21{24] for experimental proposals for measuring OTOC). Physically, the decrease of this four-point function measures the increase of the size1 of the operator anticommutator f j (t); k(0)g, which indicates how sensitive the system is to an initial perturbation created by acting with the fermion operator j (0). The exponential time dependence of the connected part of the OTOC F (t)conn: / e Lt de nes an inverse time scale analog of Lyapunov exponent. Ref. [25] proved a general upper bound L which can be considered a quantum regularized form of OTOC with imaginary time evolution), which is saturated for theories with an Einstein gravity dual. Interestingly, the SYK model also saturates this upper bound [4, 11, 26]. Many other aspects of the SYK model (and a similar model for complex fermions) have been investigated recently [8, 10, 26{32]. Given the interesting properties of the (0 + 1)-dimensional SYK model, it is natural to look for higher dimensional cousins. In this paper, we propose a family of higher dimensional variants of the SYK model, which remain solvable in the large N limit. Our model can be de ned on an arbitrary discrete lattice in arbitrary spatial dimensions. There are N fermions on each site with an SYK Hamiltonian. Di erent sites have independent SYK couplings, and neighboring sites are coupled by random four-fermion terms. Using the same techniques as those applied to the SYK model, we can study two-point functions and four-point functions in space-time. The spatial locality of the model allows us to study transport properties and propagation of quantum chaos in space-time. We nd that the disorder-averaged two-point functions vanish between di erent lattice sites, and have the 1At in nite temperature, the \size" of the anticommutator simply means its 2-norm. At nite temperature, it is generalized to the thermal expectation value hf j(t); k(0)g2i . Note that we study the anticommutator instead of the commutator because the operators are fermionic. same local critical behavior as in the SYK model within each site. Our model also has the same zero temperature entropy at each site as the SYK model. Correlation between di erent sites begins at the level of four-point functions. Similar to the SYK model case, one can consider the fermion four-point function as being mediated by a series of collective elds, with the leading contribution coming from energy uctuations. In our generalized models, the four-point function allows us to study the dynamics of collective elds in space-time. In particular, we nd a di usive dynamics of energy density, which means this model describes a di usive strongly correlated metal phase. The OTOC can also be studied, which now has both spatial and temporal dependence. We show that at low temperature the Lyapunov exponent still saturates the chaos bound 2 . The propagation of quantum chaos in space-time yj=vB + (const). Interestingly, the di usion constant D and butter y velocity vB in our generalized SYK model satisfy metal and agrees with the holographic calculation on incoherent black hole [35{37]. The remainder of the paper is organized as follows. In section 2 we will brie y review the properties of the original SYK model. In section 3, we de ne the generalized SYK model in higher dimensions. For concreteness, we work on a (1 + 1)-dimensional example and study its correlation functions and thermodynamic properties in detail. In section 4 we study the dynamics of collective elds in this system based on an operator-product expansion of fermion four-point functions. In particular, we show that one of the collective elds describes the di usion of energy in this system. In section 5, we study the OTOC of the (1 + 1)-dimensional model and obtain the Lyapunov exponent and the butter y velocity. In section 6 we discuss the general form of the model in generic dimensions and graphs. In section 7 we end the paper with a summary and discussion of further topics. Review of the SYK model In this section, we brie y review some basic facts about the SYK model. The SYK interaction [4] H = 16j<k<l<m6N f j ; kg = jk mensionful constant that sets the scale of the Hamiltonian, and the factors 31! N 3 are for later convenience. The interaction is all-to-all, so that there is no spatial locality, and this model should be considered as a (0 + 1)-d quantum mechanical system. The model is solvable at large N , and exhibits holographic behavior at strong coupling N 1. In this limit, 2See refs. [33, 34] for other discussions on di usion and butter y e ect in solid state systems. has emergent PSL2(R) symmetry3 G ( 1; 2) = b b = = 1=4 where we denote 1 2 by 12 for simplicity here and below. One can also compute the four-point function in the holographic limit [4, 11]. The interesting piece of the four-point function is the connected part, which begins at order N1 N F ( 1; 2; 3; 4) := G( 1; 2)G( 3; 4): clidean) times with the ordering 1 > 2 > 3 > 4, then the correlator F factorizes: G( 12)G( 34) + O(1= J ) 2:852. The factorization indicates that the OPE of two fermion operators is dominated by the conserved quantity in this model | the Hamiltonian itself, and the uctuation of energy determine the four point function in this con guration. Quantum chaos can be diagnosed by the out-of-time-ordered correlator (OTOC). The OTOC is calculated by starting with the Euclidean correlator with a \j-k-j-k" con guration of times, e.g. 1 > 3 > 2 > 4, and then giving the times 1 and 2 large real-time parts. Making the particular choice that all four points are evenly spaced on the imaginary time circle, we have [11] + O(1= J ) which exhibits an exponential growth for real time t greater than the dissipation time . This growth was recognized as a signature chaos bound proposed in ref. [25]. The two-point function and four-point function shown above can be calculated via standard Feynman diagrams with large N simpli cation. Another approach is to use the disorder-averaged e ective action. After introducing a pair of auxiliary bi-local elds \Green's function" G( 1; 2) and \self-energy" ( 1; 2) and carrying out the disorder average of the partition function, the fermions can be integrated out, leaving [4, 38]: Z = exp ( N Se [G; ]) 1 Z Se [G; ] = 3This symmetry acts as f ! acff++db where f = tan , and The large N prefactor implies that the problem is essentially classical. The saddle point reproduces the Schwinger-Dyson equations that can also be derived via Feynman diagrams: G(i!) = ( ) = J 2G( )3 where we assumed time translation symmetry to simplify the equations. One can also obtain the connected part of the four point function by considering uctuations around the saddle point. Among all the quantum uctuations, there is a special class induced by the reparametrization of the time circle f 2 Di (S1), which contributes the leading piece at strong coupling 1. The e ective description of this part turns out to be well-captured by a local action proportional to the Schwarzian derivative [4, 11]. Remarkably, the same form of e ective action also appears in the AdS2 Einstein-dilaton theory [5{10]. One can also derive the thermodynamic properties from the large-N saddle point = U In the second line we write the free energy in a low temperature expansion,4 where 0:0406J is the ground state energy, S0 0:232 is the zero temperature entropy [4, 38], and T = cv = 16p2K J 0:3J96 is the speci c heat [11]. The entropy term can be derived by inserting the conformal saddle point solution (2.2) in the e ective action. The speci c saddle, but the energy requires the exact (numerical) nite J solution. The generalized SYK model In this section, we will present a simple way to generalize the SYK model to higher dimensions while keeping the solvable properties of the model in the large-N limit. For concreteness of the presentation, in this section we focus on a (1 + 1)-dimensional example, which describes a one-dimensional array of SYK models with coupling between neighboring sites. It should be clear how to generalize, and we will discuss more details of the generalization to arbitrary dimensions and generic graphs in section 6. De nition of the chain model In (1 + 1)-d, our model describes a coupled array of SYK model sites, as shown in gure 1. Each site containins N 1 Majorana fermions with SYK interactions drawn independently for each site. Each pair of neighboring sites are then further coupled via random four 4Starting at T 3:77, this expansion is expected to also involve non-integer powers given by the dimensions of irrelevant operators in the model. 1 fermion with SYK interaction. The coupling between nearest neighbor sites are four fermion interaction with two from each site. fermion interactions with two of the fermions from each site. The Hamiltonian has the following form: H = x=1 @16j<k<l<m6N are the Majorana fermion operators satisfying anti-commutation relations and periodic boundary condition: f j;x; k;yg = xy jk, j;0 j;M . N is the number of Majorana fermions on each site and M is the number of the sites, or equivalently, the length of the chain. In this expression, we restrict the range of indices in the sum such that each term only appears once. The rst term describes the on-site SYK interaction, while the second term is the nearest neighbor random four fermion coupling. The random couplings fJjklm;xg and fJj0klm;xg are drawn independently for each value of x, from a distribution with zero mean and variance de ned in the following way: Jjklm;x = Jj0klm;x = 0; The normalization, especially the factors of N , are chosen to make the large N limit uniform and ensures the dimension 1 coupling constants J0 and J1 represent the average strength of the thermal bath seen by each fermion eld. Comparing to the original SYK model, our model clearly has spatial locality. One can view our model as either M coupled SYK sites, or equivalently as a big SYK site with N M Majorana fermions but with inhomogeneous coupling strength | the non-local couplings (between sites jx yj > 1) are suppressed. As will be discussed in section 6, it is not essential to have the 2-2 coupling between neighboring sites. Introducing more generic couplings such as j;x k;x+1 l;x+2 m;x+3 is straightforward as long as the coe cients of these terms are all independent variables. However, in this section we will focus on the 2-2 coupling case for simplicity. A rst observation of the model is that the Hamiltonian doesn't contain any quadratic term, so that the free propagator is diagonal not only in avor indices but also in spatial coordinate hT nate under random average of disorder elds (dashed line). (a) Replicon diagonal (b) O -diagonal 1=N 2 connects same replica index. Di erent replica indices can be connected only by dashed lines, which to the diagonal diagram on the left. interaction vertices under random average only contribute to the diagonal part, see gure 2. and spatial coordinates.5 In addition, the diagonal Green's functions hT Therefore, the dressed Green's function hT j;x( 1) k;y( 2)i is also diagonal in both avor are independent of j due to an SO(N ) symmetry of the model after averaging over disorder. Therefore we have hT The e ective action and the saddle point In this section, we employ the large N e ective action approach to analyze the model. In principle, to study the quenched problem, one should introduce n replicas and analyze the disorder averaged partition function Zn. However, as in the original SYK model, our chain model self-averages at large N . One can verify this by checking the replicon o -diagonal contribution to the averaged partition function Zn. Figure 3 shows the leading replicon directly work with the disorder averaged partition function and correlators. In other words, quenched equals annealed at the order we work, so we can study annealed quantities. To derive the e ective action, we begin by integrating the partition function Z[fJjklm; Jj0klmg] over disorder fJjklmg and fJj0klmg with Gaussian distributions. This 5For the speci c 2-2 interaction we choose in the chain model, the two-point functions connecting di erent sites vanish even without averaging over disorder, because of a Z2 fermion parity conservation on each site. However, even in the more general models that we will discuss in section 6, the cross-site two-point functions still vanish after averaging over disorder, as a consequence of local SO(N ) symmetry on each site after random average. will produce eight-fermion interaction terms that are non-local in time: Z = S[ ] = X <Z x=1 : 0 Integrating out the fermion elds jx( ) one obtains the e ective action of the bilocal elds: The expression of the averaged partition function can be further simpli ed by introducing bilocal auxiliary elds Gx( 1; 2) and x( 1; 2). x( 1; 2) are Lagrange multipliers which impose the constraints Gx( 1; 2) = Z = Gx(i!) = x = J02 1 Z Se [G; ] = X 1 Gx( 1; 2)2Gx+1( 1; 2) All terms except the last term / J12 describe decoupled SYK models at each site, and the J12 term couples the bilocal elds Gx( 1; 2) on neighboring sites. In the large N limit, the saddle point of this action determines the leading order behavior of the two-point function Gx( 1; 2). The saddle point equation Se = 0 and Se = 0 produce the Schwinger-Dyson G equations (assuming time translation symmetry): with Gx( ) = Gx( + 1 ; 1) and G(i!) its fourier transformation, and similarly for x( ) and x(i!). This set of equations can be equivalently derived via Feynman diagrams: where the thick lines represent dressed Green's functions and the gray disk represents the b = = 1=4: 1 Z Here and below, we will denote the e ective coupling pJ02 + J12 in our model as J . In summary, the two-point function in our chain model is local in space and powerlaw decaying (at low temperature) in time, a behavior known as local quantum criticality [39, 40]. This saddle point solution (and the nite J corrections to it) is the starting point for studying other properties of the model. For example, inserting the solution into the action, we derive the saddle point approximation to the partition function. This gives the order N term in the free energy: Notice that the chain model under disorder average is translation invariant and also translation invariant solutions of the Schwinger-Dyson equations: Gx( ) = Gs( ) ; Gs(i!) = x( ) = s( ) = (J02 + J12)Gs( )3 : Comparing to equation (2.7), one sees the Schwinger-Dyson equations reduce to exactly the Therefore, we can directly apply the SYK model results in refs. [4, 11]. In particular, we immediately know that the solution in the conformal limit N 1 has the following familiar form: Here we have used space translation symmetry to simplify the notation. (M is the number of lattice sites, i.e. the spatial volume of the system.) Using the saddle point solution F one can see that the free energy density NM agrees exactly with that in the SYK model same zero temperature entropy per fermion S0 0:232, and speci c heat cv fermion. It should be noted that the zero temperature entropy is extensive. We should remark here that the exact agreement on thermodynamic properties with related to the discussion of the two-point function, where the agreement with the SYK model only holds at leading order. uctuations around the large N saddle point, are di erent in our chain model than in the SYK model. Fluctuations of the collective elds and four-point functions The fermion four-point function can be determined from the uctuations about the saddle point just discussed. At large N the saddle is sharp, and the connected four point function is small, of order N1 . This is determined by the Gaussian Gx( 1; 2) and x( 1; 2) about the saddle. uctuations of bilocal elds It is convenient to expand about the saddle using variables gx; x de ned by x( 1; 2) = where we have rescaled the uctuation elds gx; x by prefactors jGsj 1 and jGsj for convenience. It should be noticed that although the saddle point is uniform in space and translation invariant in time, the elds have generic space-time dependence. Using eq. (3.4), the connected, averaged four-point function of the fermions (de ned analogously to eq. (2.3)) can be written as the connected two-point function of gx( 1; 2): 1 Z that F is of order one at large N , so the connected correlator is of order N1 . More precisely, NF is the connected part of the fermion four point function. We will see To compute the hgxgyi correlator, we expand the e ective action to second order in the uctuation elds g; , which leads to Se [g; ] = x( 1; 2)Gs( 13) jGs( 34)j Gs( 42) jGs( 21)j x( 3; 4) The spatial kernel Sxy is a tight-binding hopping matrix Sxy = x;y + It is straightforward to integrate out x and obtain a quadratic action for gx alone. We de ne Ke as the (symmetrized) four-point function kernel of the SYK model [4, 11]: The e ective action of gx is Se [g] = which determines the fermion four-point function: N Fxy ( 1; 2; 3; 4) = jGs( 12)Gs( 34)j hgx( 1; 2)gy( 3; 4)i Here we have written the correlator in a compact matrix form. X gx( 1; 2) hKe 1 ( 1; 2; 3; 4) xy Comparing eq. (3.20) with the four-point function of the SYK model, the only di erence result, as expected. The behavior of the four-point function (3.20) can be analyzed by diagonalizing the S). First of all, due to translation symmetry it is straightforward to do a spatial Fourier transformation to (x y) and de ne the Fourier component N Fp( 1; 2; 3; 4) = with s(p) = 1 + Then one can diagonalize the temporal kernel Ke in the same way as in the SYK model. We write the antisymmetric eigenfunctions h;n ( 1; 2) where n labels the fourier mode for the sum of the two times, and h speci es the dependence on the di erence of the times. Writing the corresponding eigenvalues of Ke as k(h; n), the four-point function can be expressed as N Fp( 1; 2; 3; 4) = (where we have used the fact that the symmetrized kernel is Hermitian). The only di erence from the original SYK model is the factor of s(p) 6 1 in the denominator.6 The details of the eigenvectors and eigenvalues of the temporal kernel, k(h; n) and h;n have been worked out in ref. [11], and we will use them in the following sections. It should be noted that form of the e ective action. Symmetry breaking and pseudo-Goldstone mode Although the general discussion above is su cient for calculating four-point functions, it is helpful to gain more physical understanding by analyzing symmetry properties of the saddle point solution. As in the SYK model, our e ective action (3.5) admits an approximate reparametrization symmetry of time in the IR limit ! ! 0, where one can ignore the i! in the rst term. The reparameterization transformation is de ned by Gx(f ( 1); f ( 2)) : The symmetry Di (S1) is broken spontaneously to PSL2(R) by the solution in equation (3.12). If this symmetry were exact, the spontaneous symmetry breaking would have produced in nite number of Goldstone modes corresponding to the spatially dependent reparametrizations fx 2 Di (S1)= PSL2(R). The i! term in the e ective action (3.11) plays the role of a small explicit symmetry breaking eld, which selects the solution in eq. (3.12) and turns the Goldstone bosons to pseudo-Goldstone bosons. This is similar to the situation in a ferromagnetic spin chain with small external magnetic eld (see gure 4). 6It is interesting to note that the same type of modi cation was found in a di erent generalization of the SYK model in appendix G of [11]. (a) Green's function (b) Space-time picture for chain model (c) Analogy to ferromagnetic spin chain with pinning eld de ned on the imaginary time circle. It transforms covariantly under the reparametrization fx 2 Di (S1). (b) The space-time picture for the chain model. The Schwinger-Dyson equation at conformal limit has global reparametrization symmetry, but the conformal solution spontaneously breaks Di (S1) to PSL2(R). Moreover, the UV term i! in (3.11) breaks the emergent reparametrization and lifts the Goldstone modes to pseudo-Goldstone modes. (c) The situation in the SYK chain model is similar to a ferromagnetic spin chain with a small pinning eld, where the SU(2) symmetry is \almost spontaneously" broken to U(1), leading to a pseudo-Goldstone mode. The e ect of this symmetry breaking term is small at large J , which means that the pseudo-Goldstone bosons have large uctuations and make the most important contributions to the long-wavelength dynamics. In particular, they are responsible for the di usion of energy that we will analyze in section 4 and the chaos characteristics we will study in section 5. From the view point of the e ective action, the pseudo-Goldstone modes are those uctuations along the \nearly- at" direction around the saddle point in the potential of Se [G]. As we have discussed, these uctuations correspond to the spontaneously and explicitly broken reparametrization symmetry, and therefore can be parametrized by residue saddle point in the following way s f G ( 1; 2) ! Gx( 1; 2) := f x0( 1)f x0( 2) Gs(fx( 1); fx( 2)): For small deformations of time fx( ) = + x( ), the quadratic action for x( ) can be determined by diagonalizing the kernel Ke and using (3.19). Building on the results of [11], we will nd below that to quadratic order in the spatial momentum p, this leads to the action S = fx( ) = 2 K n2(n2 n;p = p M x=1 0 imaginary time imaginary time (a) j-j-k-k order (b) j-k-j-k order at left gives the propagating bilocal elds. The con guration at right can be continued to the OTOC which diagnoses chaos. The rst term is familiar from the SYK model. It is local in time, and can be interpreted as a quadratic approximation to the Schwarzian derivative action at each site. The coe cient J limit the reparameterization elds are soft modes, due to the approximate reparameterization symmetry. The second term describes a simple coupling of the reparameterization modes at di erent sites, but with a nonlocal form as a function of time.7 We will see that together, these two terms determine both the energy di usion dynamics and chaos behavior.8 In the following two sections, we will analyze two di erent regions of the four-point function with di erent ordering of the time variables 1; 2; 3; 4 as shown in The four-point functions with ordering 1 > 2 > 3 > 4, which we will sometimes refer to as j-j-k-k order, determines the dynamics of collective elds in our model, while the order 1 > 3 > 2 > 4 which we will refer to as j-k-j-k determines OTOC after analytic continuation, and characterizes chaos in our system. The OPE region and the energy transport As we have seen from the two-point functions and four-point functions, single fermion elds jx( ) do not propagate spatially in our model. The only elds that have nontrivial dynamics are the collective bilocal elds gx( 1; 2), which are singlet in the avor SO(N ) symmetry. In the four-point function Fxy ( 1; 2; 3; 4), if we denote = 12 ( 1 + 2 and take the limit j 2 expansion (OPE) of the fermion j j, we are e ectively taking an operator product j;x( 1) j;x( 2) and similarly for The four-point function becomes a sum over the propagators of di erent collective that appear in the OPE. Roughly speaking, di erent collective elds correspond to di erent families of eigenvalues in eq. (3.22). 7An interesting question is what is the full non-linear form of the e ective action which generalizes the Schwarzian action in (0 + 1)-d case, which we will leave for future work. 8Notice that this action has three zero modes, n = 1; 0; 1 for each point in space. These are set to zero by the quotient in Di (S1)= PSL2(R). The eigenvalues and eigenfunctions of the temporal kernel have been studied in refs. [4, 11]. In the strong coupling limit J 1, the two-point functions take the conformal form (3.12). This correlation function is covariant under Mobius transformations (PSL2(R)) of the time coordinate, which is used in diagonalizing the temporal kernel. It turns out [4, 11, 26] that the eigenvalues of the kernel are given by the simple formula k(h; n) ! k(h) = 2; 4; 6; : : : ; = I: The label set I consists of a discrete series and a continuum [11, 26]. For small momentum we study the case of nite p where the conformal large J limit is straightforward. JJ2 , we will see that 1J corrections to this formula are important. First, however, 1 The OPE at The fact that the eigenvalues (4.1) don't depend on n is a consequence of the conformal symmetry of the eigenvalue problem at large J . It makes it possible to do the sum over n in eq. (3.22), which gives particular hypergeometric functions of the cross ratio h=hm(p) The only di erence in (4.3) from the regular SYK case [11] is that we have the factor of s(p). Here 2F1(h; h; 2h; ) is the hypergeometric function, which is regular and approaches constant 1 at small . The values fhm(p)g that we sum over are the in nite set of positive solutions of the pole equation 1 spaced, approaching hm(p) ! 23 + 2m for large m. Taking the residues, we have Fp = c2m(p) = 3k0(h) tan h=2 (2h) s(p)2 h=hm(p) h=hm(p) If one considers the OPE limit J 1 = 13 ' 24, we have The sum and integral over the remaining eigenvector index h I can then be done following ref. [11] by deforming the contour of integration over the continuum portion of I. As the contour is pushed to in nity, one encounters two sets of poles: one set conveniently cancels the contribution from the discrete h I and the other gives the answer (in the region 1> 2> 3> 4 where Fp = tan h=2 1 s(p)k(h) (2h) Fp / G ( 12)Gs( 34) X c2m(p) hm(p) s time PSL2(R) representation of the operator m appearing in the OPE. It is also useful to consider the fourier transform of Fp back to position space.9 One can show that c2m(p) is analytic in a strip surrounding the real p axis, so the fourier transform will decay exponentially in x. More precisely, c2m(p) has both a pole and a branch cut for complex p. The pole is at the location where hm(p) becomes an even integer, and the m, one can show that both singularities are at a distance of order log m from the real p axis, so the contribution of the m mode will decay in space as e log(m)jxj for large x and large m. Here x is measured in lattice units, so the modes only propagate a few lattice sites. However, note that the decay factor in the exponential, log(m)jxj log(h)jxj, grows less rapidly with h than in a conformally invariant (1 + 1) dimensional theory, where we would have hjxj. The h = 2 contribution for small p for h = 2 have the form 2 ( 1; 2) = 2;n( 12) has the following explicit form: of the factor of tan h2m in the denominator of eq. (4.5), this leads to a large OPE coe cient c20(p) that diverges like p 2 as p ! 0. It will be important to understand how this is cut I eigenfunctions of Ke . In the non-conformal corrections to k(h; n). The correction can be derived by including the ref. [11]. In the following we will apply the SYK results to our generalized model. To start, we note that due to time translation symmetry, the eigenfunctions 2;n( 12)e i 2 n 1+2 2 . In the conformal limit J 2;n( 12) = fn( 12) := 3 In the above expression, n2 = 2jnj(n2 1) is the prefactor to normalize perturbation theory, one can use these zeroth order (conformally covariant) eigenfunctions 2;n. In the rst order to obtain the rst order corrections to the eigenvalues, which are given in ref. [11] by k(h = 2; n) ' 1 2 K jnj + : : : 9Here we consider only the modes m > 1 that have smooth behavior for p ! 0. We will consider the m = 0 (i.e. h 2) contribution in detail below. (a) OPE from the conformal part (b) OPE from the correction of h = 2 parts are locally critical and short-range correlated in space (see equation (4.6)). (b) The non-conformal corrections of the four-point function corresponds to the reparameterization eld, which has a di usive dynamics in space-time. 2:852. Summing over the eigenfunctions with this improved formula for the Fp;h=2( ; 12; 34) Gs( 12)Gs( 34) 2 K n jnj(n2 s(p) in the long wavelength limit s(p) ' 1 be shown below, is the energy di usion constant of the system: 3JJ122 p2, and de ned a constant D which, as will D = 2 J 2 Notice that eq. (4.9) has a smooth p ! 0 limit, which reduces to the corresponding formula of the SYK model in ref. [11]. Energy transport In subsections 4.1 and 4.2 we discussed contributions to the fermion four-point functions that are naturally organized in terms of the OPE. The operator product of two fermion 212 ) generates an in nite family of collective elds, including strong coupling limit J which has di usive behavior as can be seen from eq. (4.9) and (3.25), also see gure 6 for an since the time reparameterization eld is related to di eomorphisms. satis es the following equality [9]: Here cv is the speci c heat per fermion, such that N Fp=0;h=2 ( ; 12; 34) = N M cv = @hHi = hH2i is the thermal uctuation of the total energy. Here H is the Hamiltonian of the chain model. Thus eq. (4.11) can be rewritten as N Fp=0;h=2 ( ; 12; 34) = M (@Gs( 12)=@ ) (@Gs( 34)=@ ) (@hHi=@ ) = M @Gs( 12) @Gs( 34) scribes the thermal uctuation of the two-point function Gs( 12) induced by the uctuation of total energy. If we de ne the energy density of the chain model as T 00(x), and de ne its Fourier component as T 00(p) = M 1=2 P x T 00(x)e ipx, we have T 00(p = 0) = H= It is natural to generalize eq. (4.13) to nite (small) momentum and express the energy density correlation function as hT T 00( p; )T 00(p; 0)iconn: ' Fp;h=2( ; 12; 34) Eq. (4.14) is expected to hold in the limit 12; 34 ! 0 (which guarantees the j-j-k-k order Fp;h=2( ; 12; 34) Gs( 12)Gs( 34) Evaluating @ G( ) for small and using cv = 16p2K J and eq. (4.14), we nd hT T 00( p; )T 00(p; 0)iconn: = with !n = 2 n . This equation directly gives the Matsubara correlator C00(p; i!n). By analytically continuing10 i!n ! ! + i in C00(p; i!n) and subtracting a contact term [41], 10One may concern that the fourier coe cient in eq. (4.15) looks divergent at large j!j which makes the analytic continuation less rigorous. However, the expression in the denominator only keeps the leading terms where ci are coe cients that can be determined by numerical calculations, and hi are the negative roots of the rst negative root h1 = 1 corresponds to the j!nj term. For i > 2, the roots are generally non-integers with spacing roughly 2: h = 2:77354; 4:67946; 6:63197; : : :. These higher power terms regulate the large j!j behavior for the coe cients, but won't contribute the low energy physics discussed later. one obtains the retarded energy-density correlation function in the (p; !) space:11 C0R0(p; !) = i! + Dp2 i! + Dp2 Here we have taken the small frequency limit and omitted the higher order terms in !. Eq. (4.16) tells us that energy density perturbations in our system satisfy a di usion equation with di usion constant D given by eq. (4.10). From energy density correlations one can also derive the thermal conductance: 0(!; p) = Re In summary, in this section we analyzed the fermion four-point function in the limit 12; 34 ! 0, which determines the behavior of SO(N ) singlet collective elds in the chain model. The only collective eld with nontrivial spatial dynamics is the time reparameteri eld, the dynamics of which describes di usion of energy in this disordered system, with a temperature independent di usion constant. Although there are in nite number of other collective elds in this system, all of them are locally critical and decay exponentially in space with order 1 correlation length. Chaos and the butter y velocity Another interesting aspect of the model is the OTOC, which have been studied as a diagnostic of chaos. In addition to the exponent L that determines the exponential growth of (anti)commutators in the large N dynamics within a single site, the spatial locality in our generalized model allows us to study the dynamics of chaos in space. An important parameter is the speed vB [16, 42] de ned as the speed of growth of the \ lled light cone" that marks the region where operators have large (anti)commutators with an initial operator. It has the interpretation of the speed at which the butter y e ect spreads in space, and has also been related to the Lieb-Robinson velocity [43] recently [44, 45]. In this section we will evaluate vB. The OTOC can be computed by analytic continuation of the imaginary time ordered correlation function. A convenient special case to study is the \regularized OTOC," where the operators are equally spaced in imaginary time as shown in gure 7: F (x; t) = Tr (r j;x(t)r k;0(0)r j;x(t)r k;0(0)) ; havior of the OTOC. For xed x we expect the following behavior for F (x; t): at early 11We would like to thank Subir Sachdev for pointing out an error in our earlier version. t3 = i 2 t4 = 0 (Note that here we use the convention that the real/imaginary part of t is the real/imaginary time, which is di erent from our convention in previous section.) times it should be approximately equal to the disconnected product sign is because we have fermions), and at late times it should be small, indicating a large anticommutator between j;x(t) and k;0(0). The butter y velocity vB is de ned as the rate at which the region where F is small expands outwards as we increase t. Of course, to actually see that F becomes small, we would have to sum all orders in the N1 expansion. We can only compute the rst N1 term, but we assume that the exact F becomes small at around the time where this correction becomes comparable to the order one disconnected In the imaginary time con guration, the N1 part of the four point function is given by F , where F is the function studied in the previous section. More precisely, we studied N the spatial fourier transform Fp. To compute the N1 term in F (x; t), we will continue Fp to the con guration in eq. (5.1), and then nally fourier transform back to position space. We will see that there are some subtleties involved in getting an expression for Fp that is accurate for the small momenta that dominate this fourier transform. To begin, we will warm up by studying the case where p2 the conformal limit of the kernel. From eq. (4.2), we have that the cross ratio of the times in the con guration (5.1) is is greater than one. The four point function and this continuation are discussed in detail for the SYK model in ref. [11]. The only di erence in our case is that we have to insert a factor of s(p). After the contour manipulation and expansion of the relevant hypergeometric functions [11], one nds that the growing term at small is G( 12)G( 34) h) = k0(h) 4 (h 1=2) 3 tan( h=2) s(p)kR0(1 Using eq. (5.2), we see that this implies an exponential growth Fp This exponent is largest at small p, where we have h (p) more accurately by directly continuing eq. (4.9) to the con guration (5.1). In appendix B, we show that the growing term after this continuation is Fp;h=2(t) 2 K be modi ed to e exponent proportional to 1 . This expression has a smooth p ! 0 limit. However, notice that the exponential growth 2 t independent of p, whereas we have argued that at nite p the exact answer should J , we expect a modi cation of the growth can perturbatively compute both of them as follows. We have two small parameters, p2 and 1J , and we consider both to be rst order quantities, of order . If we expand in , we expect to nd B = b1 + b2 2 + : : : ; L = 1, we have b11 e 2 t. Comparison with (5.5) then determines b1. At order 0 we 2 t coming from the small shift in L. Since this is at order 0, we can compute it in the theory with p = 0 and in this limit, which for our case evaluates to 6 te J = 1. Indeed, ref. [11] did nd such a term . Matching to the expected term, we nd 1 = 3b1, giving the formula b(p) = result eq. (5.3) to the claimed order in p2. The exponent in eq. (5.7) is correct to order p2 and order 1J and the inverse of the prefactor We now have an expression that is su ciently accurate at small p, so we can do the nal step and transform eq. (5.7) to position space to compute F (x; t). Approximating the discrete fourier transform as an integral, we have = 1 1 Z 1 dp eipx e 2 [1 3b(p)]t + : : : x, leading to12 q 2 , which dominates the behavior for large B2 = This is the main result of this section, giving the butter y velocity vB. The order N1 term competes with the order one term, indicating that the anticommutator has become large, identi ed in holographic theories, see ref. [37]. It is remarkable to nd a simple relation between the di usion constant and butter y velocity of this type. In this model, it is a consequence of the fact that the same reparameterization degrees of freedom are responsible both for energy di usion dynamics and the OTOC chaos behavior. This is a property that the model shares with conventional holographic theories, where the gravitational eld in the bulk determines both of these higher orders in 1 . At least naively, because other modes besides reparameterizations will become important, one would not expect this equality to persist beyond in nite J . There is an interesting subtlety in the fourier transform that we glossed over above. In fact, the pole dominates only for x vBt . This means that for x . vJB log N , the pole J analysis will not be correct at the time when the anticommutator becomes large. For such x, we can approximate (5.8) another way by replacing b(p) in the denominator by b(0) and doing the Gaussian integral. This leads to which is accurate for x vBt . In this region, we nd that the \butter y cone" is rounded out, see gure 8. It is rather striking that these two di erent regions of behavior characterized by (5.10) and (5.9) were also identi ed in the analysis of stringy corrections to the holographic computation of F (x; t) in [46]. We will close this section with one further comment. A surprising feature of the large x behavior (5.9) is that the growth as a function of time is given by e L in eq. (5.7). These corrections, which are both negative for real momenta, cancel against each other when we evaluate at the pole at imaginary p. It would be interesting to know if this persists at higher orders in 1 . Note that in the small x region (5.10), the growth as a function of time is decreased by a 1J correction, with the same coe cient as in the original SYK model. 12Some constants that appear in the following equations are a1 = 2 K a2 = p a4 = 4 J12 F = 1 1 , and the blue/dashed line corresponds to F = 21 N . The blue/dashed curve is the butter y cone: operators above this location have large anticommutators with the operator at the origin. The gray solid line marks the transition between the behavior (5.9) to the right, and (5.10) to the left. An example of two-dimensional square lattice model. The Hamiltonian could contain all kinds of random four-fermion terms, for example, Jjklm;uuwz j;u k;u l;w m;z and Jjklm;xxyy j;x k;x l;y m;y as shown in the gure. General construction and higher dimensions In previous sections, we have focused on a (1 + 1)-dimensional chain example of the generalized SYK model, but our construction actually applies to generic dimensions and more general forms of interactions. In this section, we will discuss the general form of our model. We will start from the simple case of higher dimensional regular lattices and then discuss the even more general cases beyond that. Generalized SYK model on higher dimensional lattices In the chain model example, the di erent sites are only coupled by a 2-2 coupling with two fermions from each site. This restriction is chosen for simplicity, which is not necessary. In general, our model can be de ned on arbitrary graphs, including higher dimensional translation invariant lattices and non-translation invariant graphs (see gure 9 for an illustration on square lattice). We denote the set of sites in the graph as , and label the sites here in the sum over x; y; z; w, one can always de ne an order of the sites, and avoid duplication due to di erent order of x; y; z; w. For xed sites x; y; z; w, we can further restrict the range of indices j; k; l; m in the sum to avoid duplication due to the di erent order of j; k; l; m.13 This de nition makes sure that each 4-fermion term only appears once in the Hamiltonian. fJjklm;xyzwg are random couplings which are completely independent, with mean and variance Jjklm;xyzw = 0; Jj2klm;xyzw = where Jxyzw is xed in the large N limit. This model includes SYK model and the chain J uniform for all x; y; z; w, which gives a completely non-local Hamiltonian (or equivalently it can be treated as the case only contains a single site). The chain model is obtained by setting Jxyzw to zero except fJxxxxgx2 and fJxxyygx2 for y = x+1. Remarkably, this general model can still be solved in the same way as the chain model. From the Feynman diagrams shown in gure 10 one can directly see that the disorder averaged two-point function is still diagonal between di erent sites, even if there is no symmetry reason for it to vanish. In path integral approach, we can write down the e ective action of the same collective elds Gx( 1; 2) and coupled via random four-fermion interactions H = x;y;z;w2 j;l;k;m Se [G; ] = d 1d 2Gx( 1; 2)Gy( 1; 2)Gz( 1; 2)Gw( 1; 2) ; where Cxyzw are combinatorial factors which depend on how many sites in xyzw coin For example Cxxxx = 214! , Cxxyy = 2 2! 2! for x 6= y.14 1 In the large N limit, the corresponding saddle point equation always admits a translation invariant solution e ective coupling J 2 = Here we denote j j as the total number of sites. 1 6 j < k 6 N and 1 6 l < m 6 N , as we did in the chain model. multiplicity of site i. For example, xxxx corresponds to partition (4) and xxyy corresponds to the partition (a) A watermelon diagram (b) Ladder in generalized the (1 + 1) d chain model, in the general model the coupling Jxyzw is also diagonal, such that only fermions with the same avor and spatial coordinate are connected under random average of disorder elds (dashed line). (b) A typical ladder in the generalized model. One needs to sum over all possible z and w in the middle to get the ladder kernel Kxy. Similar to the chain model, we can expand Gx( 1; 2) around the saddle point by quadratic order of gx( 1; 2) leads to Se [g] = kernel Sxy: Sxy = The e ective action has the same form as equation (3.19) except for a di erent spatial (Cxyzw+Cxzyw + Cxzwy + Czxyw + Czxwy + Czwxy)Jx2yzw : The discussion so far does not rely on translation symmetry, and applies to general graphs. If the graph is a d-dimensional lattice and the coupling has translation symwhere the labels x; y should be considered as d-dimensional vectors. In this case the spatial kernel can be diagonalized by Fourier transformation. If Sxy is short-ranged, the Fourier transformation s(p~) is a smooth function of p~, which can be expanded at small p as s(p~) ' 1 di usion, and ai determines the di usion constant of energy along the i-th direction. Following the same approach as in the chain model case, we can also study the OTOC in the general model. Similar to the (1 + 1)-d case, one nds a Lyapunov exponent 2 saturating the chaos bound, and a butter y velocity vB (for translation invariant systems). More details of the higher-dimensional calculation is given in appendix C. Models with global symmetry As we learned from refs. [12, 27, 29], a complex fermion version of the SYK model can be de ned, which has similar properties such as local critical two-point functions. The Hamiltonian of the complex fermion at chemical potential = 0 is H = P j;k;l;m Jjklmcjycykclcm with 1 6 j < k 6 N , 1 6 l < m 6 N . Jjklm are also independent random variables. Since one can always write complex fermion operators in Majorana operators, the complex SYK The coe cients in this Majorana model are di erent from that of an SYK model because of the U(1) symmetry requirement. It is natural to generalize the single-site complex fermion SYK model to models dened on a general graph with a generic global symmetry. The most general form of the Hamiltonian is given by H = x;y;z;w2 j;l;k;m=1 a;b;c;d;P . The index a carries a representation of a global symmetry group. (For Majorana operators the representation has to be real. In other words, i;a;x is transformed under a subgroup of SO(L).) abcd for each P is an invariant rank-4 tensor in the symmetry group. For example, for symmetry group U(1) ' SO(2), there are two possible invariant abcd = ab cd, 1abcd = We assume the couplings JiPjkl are independent JiPjkl;xyzw = 0; Once the Hamiltonian is de ned one can try to study the e ective action in the same way as before. Now we have to introduce the matrix bilocal eld Gaxb( 1; 2) = and the corresponding Lagrange multiplier axb( 1; 2). The e ective action after integrating out fermions is Se [G; ] = 1 X d 1d 2 aPbcd a0b0c0d0 Gaxa0 ( 1; 2)Gbyb0 ( 1; 2)Gczc0 ( 1; 2)Gdwd0 ( 1; 2) P Although the e ective action is more complicated, the large N saddle point approximation still apply, at least if we keep L nite in the large N limit.16 The saddle point that maps the representation carried by ab to the representation P . However, it should be noted that the choice may be redundant. In other words, due to the anti-commutation of Majorana fermion operators, one may not need all representations P to expand an invariant tensor. 16The suppression of inter-replica coupling by large N still applies to in this limit, so that we expect the disorder averaged e ective action with a single copy is still meaningful. condition of this e ective action gives the Schwinger-Dyson equations of G and general there may be saddle points where Gab and ab have o -diagonal matrix elements, which we don't know how to solve analytically. However, there always exists a diagonal SYK model with a coupling J 2 = Therefore one can always take an expansion around this saddle point and study its stability using our knowledge about SYK model solution. If this saddle point is stable, these models have the same locally critical two-point functions as the SYK model, but have di erent uctuations since the uctuation gxab( 1; 2) is a matrix eld. This will lead to di erent four-point functions. We will leave more systematic investigation of these models to Another type of generalization one can consider is to include interactions that involve q fermions at a time, rather than just four. For q > 4, the detailed analysis of the chain The relation v B2 = 2 D= will remain correct at large coupling. One can also consider including terms with di erent values of q in the same model. In general, the terms with higher values of q are more RG-irrelevant, so the terms with the lowest values of q will dominate in the infrared. Nevertheless, the subleading terms can have interesting e ects. Suppose we have terms with q1; q2, with q1 < q2, and we use the same dimensionful coupling J for both interactions. Then a perturbative analysis of the Schwinger Dyson equations indicates that for 1 we will have (omitting coe cients) p = The rst term is the naive conformal limit in the pure q1 theory. The second term is the correction to the conformal limit, again in the pure q1 theory; this term leads to the correction to the kernel that eventually gives the action for the reparameterization modes. The third term is the new feature of the theory with both q1 and q2. It re ects the contribution of the irrelevant q2-fermion operator, which a ects the correlator at quadratic order. The dimension of this operator near the IR is = qq21 , and when so this third term will dominate over the second in (6.12). In such a case, the analysis of the reparameterization action would have to be redone, following e.g. appendix D of [9]. It would be interesting to compute the energy and chaos dynamics in the resulting model. Conclusion and discussion In this paper, we generalize the (0 + 1)-dimensional SYK model of Majorana fermions to higher spatial dimensions. The generalized model retains many interesting properties of the SYK model, such as local criticality, extensive zero temperature entropy and maximal chaos. On top of that, the spatial locality of our generalized model leads to many new physical properties. We nd that single Majorana fermions in our model do not propagate between di erent sites, but collective modes made by pairs of fermions have nontrivial spatial dynamics. In particular, the most important collective mode in the low-energylong-wavelength limit is the time reparameterization eld, the dynamics of which describes the di usion of energy in this system, with a temperature independent di usion constant D. This result tells us that our model describes a strongly correlated di usive metal. The dynamics of the same reparameterization eld also determines OTOC of fermion operators, from which we can also study the butter y e ect in this model. Our result shows that chaos spreads in space with a \butter y velocity" vB. Remarkably, at strong coupling, the consistency with the proposal in the literature about incoherent metals [35{37]. Our model pointed out a new class of solvable interacting lattice models in condensed matter physics. Usually, solvable models are mapped to weakly interacting theories such as mean eld theories, so that they are not \chaotic", while the interesting phenomena in chaotic systems cannot be studied in solvable models. The generalizations of SYK model is a rare example of solvable but still chaotic systems. Therefore this model provides an interesting platform for studying various properties of strongly correlated systems, such as thermalization, entanglement propagation, dissipative transport, etc. It is also natural to ask whether further generalizations of these models allow us to investigate the possibility of many-body localization and phase transition between localized and delocalized phases. From the perspective of holographic duality, the generalized SYK models might be considered as models that are dual to some kind of incoherent black hole (see [37] and references therein), but the details of this duality require further work. At strong coupling, the models do share a key property with conventional holographic systems, which is that a single set of degrees of freedom dominate and describe both energy di usion and the chaos behavior. In a holographic theory, the relevant degrees of freedom are the bulk gravitational eld. Here, it is a reparameterization of time that can vary from place to place. It would be interesting to derive the action (3.25) for these degrees of freedom from a subset of the metric degrees of freedom on some black hole background, in a similar way as the derivation of (0+1)-dimensional Schwarzian action from the Einstein-dilaton theory in approximately AdS2 background [7, 9]. It would also be interesting to understand the relationship of this action to recent work on hydrodynamic actions [47, 48]. Another natural question is whether there are higher-dimensional translation invariant generalizations of SYK models which are dual to weakly coupled gravity theories in the bulk. Acknowledgments We would like to thank Mike Blake, Richard Davison, Luca V. Delacretaz, Wenbo Fu, Tarun Grover, Sean Hartnoll, Alexei Kitaev, Juan Maldacena and Steve Shenker for helpful discussions. We especially acknowledge Subir Sachdev for helpful discussions and comments on the draft. This work is supported by the National Science Foundation through the grant No. DMR-1151786 (YG and XLQ). D.S. is supported by the Simons Foundation n = jx yj, one need at least n ladders. And the four-point function includes all possible such diagrams and the partner terms with ( 3 $ 4). Diagrammatic derivation for four-point functions In this section, we present a diagrammatic derivation of the fermion four-point functions. The four-point functions are the leading correlation functions which couple di erent sites. In the language of collective eld Gx( 1; 2), the four-point function comes from its quantum uctuations. The four-point functions are essential for understanding transport properties and also the OTOC measure of chaos, as discussed in section 4 and 5. The connected four-point function is de ned as Fxy( 1; 2; 3; 4) := We follow the main text to use Gs to denote the saddle point of bilocal eld, which is the Green's function here. Note that the four-point function is non-vanishing after disorder average only if the spatial and avor indices appear in pairs, i.e. SO(N ) singlet. Similar to the SYK model, in the large N limit, the leading contributions to the where the thick lines are dressed Green's functions solved in the previous sections. The interaction vertices are paired by random disorder elds Jx and/or J x0. Comparing to those in the SYK model [4, 11, 26], the ladder diagrams in the chain model have extra labels indicating the spatial coordinates ( gure 11). Each ladder can either couple fermions on the same site, or bring two fermions at site x to the neighboring site x 1, as is shown in gure 12. The on-site terms are contributed by both Jx and J x0 terms in the Hamiltonian, while the nearest neighbor terms are only from the J x0 term. In general, we can complete the summation of ladder diagrams (equation (A.2)) using the Schwinger-Dyson equation: (a) J-J contraction (b) J0-J0 contraction, (c) J0-J0 contraction, y = x 1 by the interactions between fermions at same sites; type (b) comes from the interactions between fermions at site x and nearest neighbor y = x 1, but the \rails" carry the same site indices, therefore, the e ect of interaction doesn't propagate to next site; type (c) also comes from the interactions between nearest neighbor sites, and the rails get shifted by where the gray box represents the dressed interaction vertex. In terms of algebraic formula, this equation is written as on functions of one spatial coordinate x and two time variables. There are two kinds of nonzero matrix elements of K, given by the diagrams in gure 12 (a) (b) and those in (c) F = F0 + KF =) F = (1 with F = K = F0 = Kxx( 1; 2; 3; 4) = Kx;x+1( 1; 2; 3; 4) = 3; y = x 4; y = x = (3J02 + J12)Gsx( 13)Gsx( 34)2Gsx( 42) =: K1 = J12Gsx( 13)Gsx( 34)Gsy( 34)Gsx( 42) =: K2 y = x 1 For the translation invariant saddle point solution, Gsx( ) is independent from x, so that K1 and K2 are only di erent by the coe cient in front. Therefore Kxy( 1; 2; 3; 4) has the separable form Kxy( 1; 2; 3; 4) = SxyK( 1; 2; 3; 4) Sxy = x;y + K( 1; 2; 3; 4) = 3J 2Gs( 13)Gs( 34)2Gs( 42) The spatial kernel Sxy is a simple tight-binding hopping matrix ( i.e. an identity matrix plus a lattice Laplacian), and the temperal kernel K( 1; 2; 3; 4) is identical to that of the (0 + 1)-d SYK model with coupling constant J . (One should be reminded that we directly apply the results in the SYK model [4, 11] to diagonalize the kernel and solve the four-point functions in our model. where j h;n;pi = h;n;p( 1; 2; x) is some antisymmetric eigenfunctions in time which n labels the fourier mode for the sum of the two times, and h speci es the dependence on the di erence of the times, and p labels the fourier mode for space. Technically, one can further simplify the calculation using the symmetrized kernel17 [11]: Kexy( 1; 2; 3; 4) := = SxyKe ( 1; 2; 3; 4) n=1 2 Gs( 42) jGs( 21)j. The simpli cation works by the following steps: (1) add two rungs (with absolute value for convenience) to the ladders in F (one on the left,18 one on the right), gure 13; (2) we get a multiple of \curved boxes", then express it as a power of the symmetrized kernel; (3) in the end, sum over all ladders, which is now a geometric series: = 2 X(Ke n)xy where the factor of 2 comes from the counting for extra term with 3 $ 4. This is the general formal expression for the connected Euclidean four-point function. We can proceed write the general expression above in momentum space: Fp( 1; 2; 3; 4) = s(p)k(h; n) h;n( 1; 2) h;n( 3; 4) where k(h; n) is the eigenvalue of temporal kernel. This agrees with equation (3.22) derived from e ective action. Summation trick and the prefactor In this appendix we determine the large t behavior of the contribution to the OTOC from 12 = 34 = j Re( )j < =2 18the extra factor 3J2Sxx0 is needed to construct the kernel. 17Roughly speaking, this trick is used to avoid the computation of the inner product h jF0i. poles on the positive real axis. 1+ 2 3 4 is the center of mass time separation, and the requirement Fp;h=2( ) = p 2 K n>2 even ( 1)n=2 cos(n 2 We would like to continue this we consider the following integral I = 2 sin( !=2) !2 Convergence is guaranteed by j Re( )j < 2; 4; 6: : : : on R+. When we deform the contour to the right (see gure 14), we nd =4. The integrand has poles at ! = 1 and I = ( 1)n=2 cos(n 2 The key point is that when we continue to large real time = it, the integral I remains convergent and non-growing in time, so we must have ( 1)n=2 cos(n 2 2 n + Dp2 = This directly gives (5.5). di usion constant and butter y velocity. sion in terms of momentum eigenvalue: Di usion and the butter y velocity in general dimensions In this section, we sketch the computation relevant to the di usion and the butter y velocity in general dimensional models with translation symmetry. We won't derive the exact formula for most general case, but will instead present key steps that determines the In the mode with transnational symmetry, the four-point function has simple expresFp~( 1; 2; 3; 4) = k(h; n)s(p~) h;n( 1; 2) h;n( 3; 4) Notice here the momentum p~ represents a general high dimensional vector. Further restrict the model to be local, we have a small p expansion for eigen-value s(p~) ' 1 As we have noticed in the SYK and SYK chain model, the contribution relevant to the that determines the di usion constant has the simple form: 2JK jnj + : : :. Therefore, the pole 2 Kjnj Here the spatial coordinate ~x represents a high dimensional vector. Analogous to the eigen-functions, which is the leading contribution: F (p~; t)h=2 / ( 1)n=2 cos(n 2 di usion pole: Dj = p 2 K Next is to compute the butter y velocity, which can be extract from the OTO correlation function. Again, we choose the time con guration which placed four time equally spacing around imaginary time circle, i.e., we are computing: Tr r j;~x(t)r k;0(0)r j;~x(t0)r k;0(0) Using the trick in appendix B and analytic to real time, we have a formula in momentum F (p~; t)h=2 / 2 + P argument in section 5, we have In general dimension, we need to compute the following fourier transformation to get the butter y velocity in xj direction: log N is a large parameter. Therefore, we can integral over all direction except Then this fourier transformation essentially goes back to the 1-dimensional case, for which we know that at large xj , the integral is dominated by the pole 2 determines the exponential decaying pro le for F (xj ; t) at real space, + Dj pj2 = 0. The pole with butter y velocity F (xj ; t) / exp jxj j=vB;j ) Open Access. 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Yingfei Gu, Xiao-Liang Qi, Douglas Stanford. Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, Journal of High Energy Physics, 2017, 125, DOI: 10.1007/JHEP05(2017)125