$\mathcal{N}=(0,2)$ SYK, Chaos and Higher-Spins

We study a 2-dimensional SYK model with $\mathcal{N}=(0,2)$ supersymmetry. The model describes $N$ chiral supermultiplets and $M$ Fermi supermultiplets with a $(q+1)$-field interaction. We solve the model analytically and numerically in the $N\gg 1$, $M\gg 1$ limit with $\mu\equiv \frac{M}{N}$ being a free parameter. Two distinct higher-spin symmetries emerge when the $\mu$ parameter approaches the two ends of its range. This is verified by the appearance of conserved higher-spin operators and the vanishing of chaotic behaviors in the two limits. Therefore this model provides a manifest realization of the widely believed connection between SYK-like models and higher-spin theories. In addition, as the parameter $\mu$ varies we find the largest Lyapunov exponent of this model to be slightly larger than that in models with non-chiral supersymmetry. A tensor model without random couplings that shares the same infrared physics is also introduced.

The finite anomalous dimensions suggest [6] that the SYK model could be thought of as a deformation of the vector models that have a tower of higher spin operators with small anomalous dimensions. Such deformation from a Gross-Neveu vector model to an SYK-like model is discussed explicitly in [40]. It is shown in [40] that there is a transition from the vector model to the SYK-like model, which is similar to other phase transitions observed in the SYK-like models [41][42][43][44][45][46][47]. Different bulk duals of the tower of operators are proposed in [10,[48][49][50][51], other discussions about the relations between the two sides can be found in [52][53][54][55][56][57][58][59]. Most of the analytic results of the SYK model are derived in a new type of large-N limit that is shared in particular by models without random couplings .
One could also consider models whose left-and right-moving sectors are not symmetric. In this paper we study some 2d SYK-like models of this kind. The models have an N = (0, 2) supersymmetry in the UV. In the infrared, these theories are dominated by the set of melonic diagrams in the large-N limit and can be solved as all other SYK-like models.
The N = (0, 2) supersymmetry plays an important role of this model. The N = 2 supersymmetry in the right-moving sector makes the IR solution reliable. On the other hand the absent of supersymmetry in the left-moving sector gives some room for interesting properties that are not observed in previous models. In particular, due to the smaller number of supersymmetry it is possible to study a one parameter family of such models. As a result, one could move on the moduli space of such models and understand their peculiar features, as well as their possible connections with other well studied models. In this paper we study two examples of such interesting consequences.
Firstly, the Lyapunov exponent of the supersymmetric model considered in [102], see also [128], is λ L = 0.5824, which does not saturate the chaotic bound [129]. It is then an interesting question to ask if there are other 2d SYK-like models that have larger or maximal Lyapunov exponent. In this paper, we show that in our N = (0, 2) setting, as we dial the free parameter, there is a continuous family of theories that have slightly larger Lyapunov exponent comparing to the supersymmetric models considered in [102]. This is discussed in detail in section 3.2.
Another interesting consequence is the existence of certain higher-spin limits. By continuously tune the parameter to some limiting values, we observe the emergence of higher-spin conserved currents explicitly. Besides, we observe the correlation of the emerging of the higher-spin symmetry and the fading of the chaotic behavior. This provides a manifestation of a connection between higher-spin like models and SYK-like models. The details of such higher-spin limits are analyzed in section 4. In this section we review some properties of 2-dimensional theories with N = (0, 2) supersymmetry. We work in Euclidean signature, where the two coordinates are x 0 , x 1 . We define and the derivatives become 2) The N = (0, 2) supersymmetry is generated by 2 supercharges. In the superspace formalism they read (2. 3) The super-derivatives are (

2.4)
It is easy to check that the supercharges anticommute with the super-derivatives. We consider models of two kinds of superfields. The chiral/anti-chiral superfields We also consider Fermi multiplets are (anti-)chiral superfields where the subscript a labels different chiral superfields. The Fermi supermultiplets satisfyD The supersymmetry transformation of the fields in the chiral supermultiplet are (2.14) The supersymmetry transformation of the fields in the Fermi supermultiplet are

15)
Q +λ = √ 2Ḡ ,Q +Ḡ = 0 , Q +λ = − √ 2Ē , Q +Ḡ = −2 √ 2∂ zλ + ∂Ē ∂φ aψ a . (2. 16) In the rest of the paper we consider special models with E = 0. Given these transformations, propagators of the different components of a chiral supermultiplet are related by The similar relation for the Fermi multiplet is For the case with E = 0, we simply get The D-terms of a chiral and a Fermi superfields are respectively In addition, we turn on holomorphic superpotentials that contribute F-term potentials. For the N = (0, 2) models, the holomorphic superpotential takes a general form where G(x, θ,θ) is some fermionic superfield that satisfiesD + G = 0. It is easy to check that the above results agree with the Euclidean continuation of the results [130] in Lorentzian signature.

The
We consider a special model of N chiral multiplets and M Fermi multiplets with the F-term potential where i, j, k, . . . label the Fermi multiplets and a, b, . . . label the chiral multiplets. The J ia 1 ...aq coupling has dimension ( 1 2 , 1 2 ) and is from a Gaussian distribution In component form, the above action reads When M = N the supersymmetry is enhanced to N = (2, 2) and the theory reduces to the models discussed in [113,131] whose action is recast in (A.3).
The self-energies of the fields are The calculation is done for q = 3, µ = 1.5.
It is easy to get the set of N = (0, 2) supersymmetric solutions of the form where n λ n q φ = − (q−1)q 2π 2 J 2 (µq 2 −1) and One can solve the Schwinger-Dyson equation numerically to confirm that the model indeed flows to this IR solution. The numerical solution is shown in figure 1.
We should comment on one subtlety in this computation. Since we look for supersymmetric solutions, we only need to solve the Schwinger-Dyson equation of one component of each multiplet; the equation of the other component is then automaticaly satisfied due to supersymmetry. When we solve the equations of the chiral multiplet, we notice that in the UV regime the Fourier transform involves an integral of the form rdrdθ r 2 µq−1 µq 2 −1 −3 e iθ e ir cos θ . (2.34) If we directly count the power of r, it seems that there is a divergence in this Fourier transformation. However, when we check the behavior near r = 0, we can expand the e ircosθ factor to get We observe that the first term vanishes due to the e iθ in the θ integral. 1 So the leading term at r ∼ 0 is drdθ ir Therefore as long as we focus on the models with µ > 1 q , this integral converges and the model does flow to the IR solution we found above.

Four-point functions
In this section, we consider 4-point functions of this model. Because there are two different types of multiplets in the model, there will be a few different 4-point correlation functions. As in the 1-dimensional cases [100,132,133], the correlation function can be computed either in terms of superfields or component fields. In the rest of the paper we work in the component formalism.

Operator spectrum
We are interested in the 4-point function There are in total 9 kernels that contribute to these 4-point functions.
The kernels take the following expressions where we use the short hand notation z ij = z i − z j . The following ansatz turns out to be the eigenfunctions of the above kernels where the * denotes a convolution in position space. Making use of the following integral formula [131] , (3.14) one finds the non-vanishing eigenvalues to be It is convenient to organize the eigenvalues into an 8 × 8 matrix 3.24) and the final eigenvalues come from diagonalizing this matrix.
Here the presence of the extra σ 1 = 0 1 1 0 matrix is due to the form of the interaction in (2.26). To illustrate it, we consider, for example, the action of the kernel K φψ on the eigenfunction Φ ψ . As shown in figure 2, after the action of the kernel, the upper leg becomes a conjugate field, namely the direction of the arrow is flipped. Therefore the actual eigenfunctions come in conjugate pairs. This means the actual action of the kernel should be like in figure (3). Further notice that due to the form of the IR propagators, the two entries  in the kernel matrix share the same expression. This leads to the extra tensor product with the σ 1 matrix in (3.24). Diagonalizing this matrix, there are 4 eigenvalues being the 4 roots of the following equation which we will call the symmetric eigenvalues. There are another 4 eigenvalues being the solution of the equation which we will call the antisymmetric eigenvalues. Their presence is a result of the complex fundamental fields in our model (2.26), similar to the 1-dimensional cases [100,128,133]. We have not succeeded in getting simple expressions of the eigenvalues. But the equation (3.25) and (3.26) pass a few consistency checks.
First, as we discussed above we expect the result to reduce to that of the N = (2, 2) model in the µ → 1 limit. Indeed, we can solve (3.25) and (3.26) at µ = 1 to find the following 8 eigenvalues 27) which are consistent with the N = (2, 2) result. At generic µ, the eigenfunctions can be considered as deformations of the eigenvalues (3.27). Another consistency check is the presence of the stress-energy tensor at any generic µ. To see this, recall that in this model the 4-point functions are sums of ladder diagrams. Therefore there is always a factor where i = 1, . . . , 8 are the 8 eigenvalues obtained from solving (3.25) and (3.26). Then an operators with dimension (h * ,h * ) running in each channel can be represented by a pole in the factor (3.28) at (h,h) = (h * ,h * ). For the special case of the stress-energy tensor, we simply expand the coefficients of (3.25) and (3.26) to the first order of h − 2 and then find the eigenvalues to the first order of h − 2. It turns out that there is always a solution behaves like which corresponds to the deformation of k F B (h − 1 2 ,h). Therefore (h,h) = (2, 0) is always a solution that sets some eigenvalue to 1. This can also be confirmed by explicitly verifying that E(1, 2, 0, µ, q) = 0.
To get the complete spectrum of the operators in the IR limit of the model, we need to find their corresponding (h,h) that makes some eigenvalues to 1. Therefore the dimension of the operators should solve where we use the subscript a, s to present operators in the symmetry and antisymmetric channels. For instance, we get the lowest dimensions of scalar operators running in the 4-point functions of the fundamental fields by solving for generic µ and q. The equations are easily solved numerically. The solutions to the equation E c (±1, h s , h s , µ, q) = 0 is shown in figure 4. As we can see the dimension of the scalar operators in the symmetric channel approach zero as the µ approaches 1 q . In addition, the dimension of the operators in the antisymmetric channel is larger than those in the symmetric channel. This is as expected since the operators in the antisymmetric channel involves one more derivative.   We also computed other 4-point functions where fermionic operators runs in the ladder. The details are elaborated in appendix . We only outline here that one can check explicitly that at the µ = 1 point, we do observe both (h,h) = ( 3 2 , 0) and (h,h) = (0, 3 2 ) operators that correspond to the supercharges in the holomorphic and antiholomorphic sectors. This again confirms that at µ = 1 our model has an enhanced N = (2, 2) supersymmetry. We further check that as long as µ = 1 the left-moving supercharges are lifted and stop generating supersymetry in the left-moving sector. This is as we expected since the model only has N = (0, 2) supersymmetry at generic µ.

Chaotic behavior
One can further go to the chaos region and study the out-of-time-ordered correlators. For this we simply consider retarded kernels. We start from the Euclidean propagators on a periodic τ direction and a noncompact spatial direction x The retarded propagators can be computed from analytic continuation of (3.32) We also need the set of ladder rung propagators between the rails. They can be obtained from a simple analytic continuation The set of retarded kernels can be obtained from (3.15)-(3.23) by replacing the propagators on the rails by the corresponding retarded ones (3.33) or (3.34) and replacing the ladder rung propagators by the ones in (3.35).
Following [102], we introduce the new variable for the retarded propagators on the upper rail and for the retarded propagators on the lower rail. We consider the following ansatz where h i ,h i labels the dimensions of the operators at t i , x i . In terms of the t and x coordinate, (3.38) becomes Our goal is to find eigenfunctions that grows exponentially with time but remain normalizable in the spatial direction. This requires h −h to be imaginary and we can follow [102] to reparametrize h andh as Finding the largest chaotic behavior then means to find the largest λ L that renders at least one eigenvalues to 1. As in the case discussed in [102], the convolution integral in the eigenequation factorizes into two 1-dimensional integrals in the u, v variables with the eigenfunction (3.38), each of which can be carried out straightforwardly. The resulting eigenvalues are In principle, we diagonalize the matrix of retarded kernels, which is the retarded version of (3.24), to get the equation of the eigenvalues Then we solve for the h andh that set the eigenvalue to 1. In our case we do not get a simple expression of the eigenfunctions. But we can still find the maximal values of λ L by a direct analysis of the eigenfunction equation: since we are interested in eigenvalue 1, we can simply set x = 1 of the equation (3.56) that determines λ L as an implicit function of µ and p. We can then find the largest value of λ L by tuning µ and p.
To proceed we start with a sanity check by focusing on the µ = 1 case and check if the result agrees with the N = (2, 2) result. There are two ways to do such a check. The first approach is noticing that at µ = 1 we can solve the retarded eigenequation (3.56) directly to find that there are 4 eigenvalues and indeed k µ=1 1 is identical to the k BB R function in [102]. Notice that the other k µ=1 i are due to the super-descendents of the eigenfunction corresponding to k µ=1 1 . Hence they are not expected to be related to the k BF R , k F B R , k F F R functions in [102] that are due to different primaries of the third operators in the eigenfunctions.
The second approach is to follow the method we discribed above, namely plug x = 1 into the equation (3.56), then setting µ = 1 and looking for maximal λ L by   tuning p. We indeed find a miximal λ L = 0.5824 at p = 0. This agrees with the result in [102] and confirms the validity of our procedure. We now move to general µ. Because the integral (3.41) again factorize into two 1-dimensional ones, by a similar monotonic argument as in [102] we expect the largest λ L is reached at p = 0. This is indeed true as one can check explicitly. We present the p dependence for some special values of µ in figure 5a. It is also straightforward to check the q dependence of λ L , a few examples of which are presented in figure 5b. We find the largest λ L appears close to the smallest q that leads to nontrivial interactions, namely q = 2.
Then we look for maximal λ L as a function of µ. The general dependence is shown in figure 6. One finds a maximum of λ L as we change µ. Interestingly, this maximal value does not appear at the special point µ = 1; rather it appears at µ 0.9802 where the maximal value is λ L (p = 0, µ = 0.9802) = 0.5825. This maximal value is only slightly larger than the value for the N = (1, 1) and N = (2, 2) model where λ L (p = 0, µ = 1) 0.5824. Notice that in determining this Lyapunov exponent we do all the computation analytically, except for the very last step where we find the solution to a given equation numerically. Since the error of this last step is very well controlled, our result is genuinely different from the exponent in the N = (1, 1) or N = (2, 2) model.
It is not clear what is the physical reason of why the maximal value of λ L in the class of models we consider here is only slightly higher than that found in the special case µ = 1, and why the correspnding µ is only slightly smaller than 1. The slightly larger Lyapunov exponent probably indicates that there should be a wider class of similar models that are continuously related. Our model, and the N = (1, 1), N = (2, 2) models are only examples that sit on a generic point on the moduli space.  It is conceivable that there are special theories on (some corners) the moduli space that have larger, or even maximal, Lyapunov exponent.

Two higher-spin limits
It is widely believed that the SYK-like models have close relations with higher-spin theories; higher-spin theories should be thought as a subsector of some tensionless limit of string theory, while the SYK model should be holographically dual to some string theory with finite tension [6]. Therefore it is tempting to find a direct relation between SYK-like models and models with higher-spin symmetry. In 1-dimension, one example of such relation is discussed in [132]. In this section we give another explicit example of such connection in 2 dimension. The basic idea is to tune the free parameter to some critical/singular value where the model develops some properties that is characteristic for models with higher-spin symmetry. In particular, we find two singular limits at the two ends of the range of µ, where the model (2.26) develops emergent higher-spin symmetries. In the following we discuss the two limits separately.

4.1
The µq → 1 + ("classical chiral") limit The Fourier transform (2.36) at the special value µ = 1 q has a logarithmic divergence. This means a proper renormalization analysis of the model at µ = 1 q is needed. We will not do it here and will postpone this in future work. Nevertheless, we consider the limit (4.1) Taking the limit in this manner, all our previous computation are valid since there is no divergence in the limiting process. The IR dimensions of the various fields in this limit are where the dimension of the φ and ψ fields take the values in a free chiral multiplet. This is a first hind that we should expect a larger higher spin type symmetry to emerge in this limit. This is consistent with the result from the chaos analysis in the previous section; as shown in figure 6 the Lyapunov exponents all vanish as µ → 1 q + for any q > 1. In the following we confirm the existence of a higher-spin symmetry from a few different aspects.
• A tower of conserved higher-spin operators. Given the above motivation, we look for a tower of higher-spin operators in the limit (4.1). Recall that in 2 dimensions, conserved higher-spin operators are represented by (anti-)holomorphic primary operators with vanishing (right) left conformal dimensions. Therefore we extend the computation in the previous subsection to find such (anti-)holomorphic primary operators in the limit (4.1). For this we go back to (3.30) and look for solutions of that correspond to the holomorphic higher-spin operators in the symmetric and antisymmetric channel. Because our model has a manifest N = (0, 2) supersymmetry, the operator spectrum of the holomorphic and anti-holomorphic operators could be different. Therefore we need to find the spectrum of the anti-holomorphic operators seperately, which corresponds to solving Furthermore, the operators running in the channel tha is detected by our ladder diagrams (3.1)-(3.10) are all bosonic, so we only look for solutions with integer s, at any q.
The equations are again easily solved numerically, and we get a functionh(µ) or h(µ) of any given s and q. We summarize the operators that become (anti)-holomorphic in the limit (4.1) in the following table From this result we indeed observe that in the µ → 1 q + limit the dimensions of some operators in 4-point functions approaches the dimensions of the (anti-)holomorphic higher-spin operators shown in the above table. This result indicates that a tower of conserved higher spin currents emerges in the limit (4.1).
The above holomorphic operators close among themselves under Operator Product Expansion (OPE) and hence generate a higher-spin type W-algebra. A similar result holds for the anti-holomorphic operators. Moreover, for each spin we find an operator in the symmetric channel and one in the antisymmetric channel, which is identical 2 to the structure of the generators of the bosonic subalgebra of the N = 2 supersymmetric SW 1+∞ algebra that governs the symmetry of many supersymmetric higher-spin theories in 2 dimension [134,[136][137][138][139][140]. This spectrum is as we expected since the model has a right-moving N = 2 supersymmetry.
It is slightly more surprising that the anti-holomorphic operators have a same spectrum as an N = 2 supersymmetric SW 1+∞ algebra. We have checked that in the limit (4.1) the left-moving sector does not have supersymmetry as well. Therefore the leftmoving section only has bosonic higher-spin symmetry emerging, which is consistent with the N = (0, 2) supersymmetry of our model. • Anomalous dimensions As we go away from the limit (4.1), namely as µ becomes larger, this emergent higherspin symmetry is broken, which is characterized by the anomalous dimensions acquired to those (anti-)holomorphic operators. This is another way to interpret figure 7, in particular the small figures inside figure 7. Before analyzing the results, we first remind the reader that in the following we call the operators that become the (anti-)holomorphic in the higher-spin limit µ → 1 q + "almost (anti-)holomorphic higher-spin operators", even after we move away from the higher-spin limit (4.1) at generic µ. We keep in mind that they are only strictly related to conserved currents in the limit (4.1).
Coming back to the results, in figure (7a) we plot the anomalous dimension of the holomorphic higher-spin operators in the symmetric channel (solid lines) and the anti- (a) Anomalous dimensions of the holomorphic higher-spin operators. Here we plot the left dimensionh of the operators and the right dimension is h =h + s.  shows that the anomalous dimensions of these operators approach zero in the limit The solid curves denotes operators in the symmetric channel, the discrete data points denote the operators in the antisymmetric channel. The plots are computed for q = 3. The N = (0, 2) supersymmetry is manifest in the plots. symmetric channel (discrete points). We observe that for the holomorphic higher-spin operators, the anomalous dimension of a spin-s operator in the antisymmetric channel is identical to the anomalous dimension of a spin-(s + 1) operator in the symmetric channel. This is consistent with the N = 2 supersymmetry in the right-moving section: the two operators are respectively the top and the bottom component of a single multiplet that consists of operators with spin (s, s + 1 2 , s + 1). 3 This confirms that the right-moving N = 2 supersymmetry is always perserved at generic value of µ.
On the other hand, from figure (7b) we observe that there is no such relation among the almost anti-holomorphic operators in the symmetric and antisymmetric channels away from the limit (4.1). This is again compatible with the fact that there is no supersymmetry in the left-moving sector.
It is also illustrative to study the dispersion relation, namely to understand how do the anomalous dimensions, which is the same ash for the anti-holomorphic higher-spin operators or h for the holomorphic higher-spin operators, depend on the spin once we  Figure 8: The anomalous dimension of the higher spin operators as a function of the spin of the operators. The yellow stars, red dots and the blue crosses are for q = 2, 3, 5 respectively and µq − 1 = = 0.01. The horizontal axis is log(s) where s is the spin of the operator. The vertical axis is the anomalous dimension. It is observed that when s is relatively large, the anomalous dimension is propotional to the log of the spin. This is the same behavior as the result from a perturbative higher-spin CFT computation [141], as well as the dispersion relation of a classical rotating string in AdS spacetime in the large-spin limit [142]. move away from the higher spin limit. One can find this by solving to determine the implicit functionh (s). And similarly the implicit function h (s) for the anti-holomorphic operators. This is again easily achieved numerically. The result for the anomalous dimension of the almost holomorphic higher-spin operators in the symmetric channel is plotted in figure 8. The results for the operators in the antisymmetric channel and the anti-holomorphic operators have similar structure. In figure 8 we observe that for each given q, the anomalous dimension γ (s) =h (s) behaves likeh for relative large spin s near the higher-spin limit (4.1). This result, which comes from diagonalizing the kernels of the SYK model and taking a higher-spin limit (4.1), agrees well with the result from a direct higher-spin CFT perturbation computation [141], as well as the dispersion relation of a classical rotating string in AdS spacetime in the large-spin regime [142]. This agreement means we have a consistent picture describing the properties of a model with an approximate higher-spin symmetry: we get the same results by either going away from a theory with exact higher-spin symmetry or going towards a higher-spin limit from a model that does not have a higher-spin symmetry. Since our computation is from a different approach, it serves as an independent evidence to support the general picture that the SYK-like models can be related to some finite tension string theory, while the higher-spin theories can be regarded as some (truncation of) tensionless string theory. In our model tuning the µ parameter away from the limit (4.1) drives the model from a higher-spin-like regime to SYK-like regime, which mimics the process of turning on the string tension.
• The Lyapunov exponent The higher-spin limit µ → 1 q + is also detected from the chaotic behavior. Indeed the Lyapunov exponent vanishes in the limit µ → 1 q + as can be seen from figure 6a. This vanishing Lyapunov exponent agrees with previous expectations that theories with an infinite dimensional higher-spin symmetry is not chaotic [6,143,144].
• The q-dependence A further comment is about the relative magnitude of the anomalous dimensions for different q. We notice that the anomalous dimensions become smaller as q becomes larger. This is what we expected for many previously studied SYK-like models; the larger the value of q the less relevant the interaction. Consequently, we expect the anomalous dimensions to be smaller for a larger q.
• This is a singular limit A last comment is that this µ → ( 1 q ) + limit is singular: one cannot naively take all the formula and plug in µ = 1 q since the Fourier transform (2.36) diverges. Instead one has to consider setting µ = 1 q + δ with 1 δ > 0 and extract the result by taking the limit → 0, which is how all the above results are computed. It is not very surprising that the higher-spin limit is singular: we have learned from many other cases that the higher-spin limit of different models are often singular [145][146][147] , see [148][149][150][151] for recent development. A different way to phase the singular nature of this limit is that the system undergoes a phase transition when going into/away from this limit. This is a first order phase transition because the free energy of the system, which can be expressed in terms of the IR propagators similar to [5][6][7], is discontinuous in this limit. We call this a "classical chiral" limit since the chiral mulitplet has classical dimension. But this is not quite a free field limit since the coupling remains large in this limit and the dimension of the Fermi multiplet does not take its classical/free value.

The µ → +∞ ("classical Fermi") limit
We can consider a different limit where the IR dimension of the various fields are In this limit another tower of higher-spin operators emerges. To illustrate this we carry out a set of computation that is in parallel with what we have done in the previous subsection.
• A tower of higher-spin operators As in the previous limit Therefore, we find a set of holomorphic higher-spin operators in the right-moving sector only at q = 2. This spectrum matches with the spectrum of generators of the bosonic subalgebra of the N = 2 SW 1+∞ algebra. This is again consistent with the N = 2 supersymmetry in the right-moving sector. In addition, we find a tower of higher-spin operators in the left-moving sector for any q > 1. Therefore we expect an N = (0, 2) supersymmetric higher-spin symmetric in the limit (4.8) at q = 2. Therefore at q = 2 (a) The q = 2 case. The anomalous dimension of the higher-spin operators vanish in the limit µ → ∞. They become conserved higherspin currents in this limit. The spin-2 operator, namely the stress-energy tensor, is always conserved for the whole range of µ where our SYK solution is reliable.  we find both the holomorphic and anti-holomorphic conserved higher spin operators at each integer spin s. But for other q > 2, we only find anti-holomorphic higherspin operators that generate an anti-chiral bosonic W 1+∞ algebra in the left-moving sector. Notice that this tower of chiral higher-spin operators is slightly unfamiliar since a general conserved higher-spin currents should have both the holomorphic and anti-holomorphic components. Therefore we believe the tower of higher-spin operators at q = 2 are related to the usual higher-spin currents that generate the higher-spin symmetry. Some examaples of the dimensions of the holomorphic higher-spin operators are shown in figure 9a, where we plot the left dimensionh of the spin-s operators both in the symmetric (solid lines) and the antisymmetric (discrete points) channels as a function of 1 µ for the whole range of 1 µ . From this result we indeed see that in the µ → +∞ limit the left dimensionh s (µ) all approach zero, which indicates that there is a tower of holomorphic operators with dimension (h,h) = (s, 0). We also plot the anomalous dimension of the anti-holomorphic operators in figure 10. We do not see any coincidence of the dimensions between the symmetric and antisymmetric channels. This indicates that in the infrared the model only has N = (0, 2) supersymmetry, namely there is no supersymmetry in the left-moving section, away from the higher-  spin limits (4.1) and (4.8).
• The q-dependence We notice that the tower of holomorphic higher-spin operators only emerges at q = 2. We have solved the equation for a few different q's and we find that the operators with higher spin have nonvanishing anomalous dimension for all q > 2. This is shown in figure 9b. We do not have a good physical explanation of why q = 2 is special in this sense. 4 For completeness we list the dimensions of the chiral and Fermi multiplet in the limit (4.8) at q = 2 lim µ→+∞,q=2 (4.14) We notice that in the limit (4.8) the chiral multplet gets its largest possible dimension at q = 2.  Figure 11: The anomalous dimension of the higher-spin operators as a function of their spin. The plot is around the higher-spin limit (4.8). We observe that for large enough spin the anomalous dimension is proportional to the log of the spin.
• Anomalous dimensions The anomalous dimensions again have a logarithmic dependence on the spin, as can be seen from figure 11. This is similar to the behavior in the other limit (4.1) and agrees with the result from pure higher-spin theory computation [141]. We only plot the holomorphic operators in the symmetric channel; the holomorphic operators in the anti-symmetric channel are in the same supermultplets and have identical anomalous dimensions.
• The Lyapunov exponent We can compute the Lyaponov exponent of our model for the whole range of µ. We find vanishing Lyaponov exponent at µ → ∞ only at q = 2. This is compatible with the fact that a normal higher-spin symmetry emerges in this limit only at q = 2, as we have discussed above.
• This is a singular limit The limit (4.8) is singular for a similar reason as the limit (4.1): the result depends on how the limit is taken. In the above analysis, we solve for the anomalous dimension at a given µ around µ → +∞ and track how does this anomalous dimension behave as µ approaches +∞. If we instead take an unphysical approach by simply plugging  Figure 12: The Lyapunov exponent as a function of µ for the whole range of µ. We observe that near the µ → +∞ limit only the model with q = 2 has a vanishing Lyapunov exponent. This is consistent with the fact that only at q = 2 the set of higher-spin operators have vanishing anomalous dimension and generate a higher-spin symmetry.
solve the set of Schwinger-Dyson equations, and then take the µ → ∞ limit of the solution. We do not expect to recover the same result if we first take a naive µ → +∞ limit of the Schwinger-Dyson equation and then solve it. As in the previous limit (4.1), we believe this subtlety reflects the common feature that a tensionless limit of string theory is usually singular and the model undergoes a first order phase transition in this limit.

Relations with higher-spin theories
Our model is defined in 2-dimensions where conformal field theories with higher-spin symmetries and their holographic dual have been extensively studied, see e.g. [152] for a review. In a supersymmetric context, the higher-spin conserved currents generate a supersymmetric SW ∞ -algebras. It is well known that the SW ∞ algebra allows a continuous deformation that preserves the higher-spin symmetry; there is a family of higher-spin SW ∞ [λ] algebra. In our model there are two parameters. As we have shown in previous sections, the µ parameter controls the breaking of the higher-spin symmetry. While in the higher-spin limit (4.1), there is no requrement of the q paramter: the higher-spin symmetry emerges at any given q > 2. It is thus natural to conjecture that the different higher-spin algebras emerging in the models with different q should be identified with the SW ∞ [λ(q)] algebras via an explict map λ = λ(q). At the moment we have not yet identified this map, but a natural conjecture is We plan to study this point in detail in future works. In figure 13 we draw a cartoon summarizing the properties of the model (2.26) on the moduli space spanned by the q and µ paramerters. On the graph we mark out the two limits where higher-spin symmetries emerge. Outside the shaded region the Fourier transform in (2.36) diverges. As suggested in [102] one probably has to turn on a negative mass counterterm to reach another fixed point. This is an indication that in this region the model might be gapped. It is interested to clarify this point further, and we will defer this into future works. The existence of the µ parameter and the appearance of a higher-spin limit as we vary µ has some resemblance to the ABJ triality [153] in one dimension higher. In our model there are indeed two global symmetry U (N ) × U (M ) corresponding to the rotations of the N chiral multiplets and the M Fermi multiplets respectively. Notice that this symmetry is only manifest once we average over the random coupling. Furthermore since we consider only singlets under these two global symmetry groups, we do not expect the results discussed in this paper to alter significantly once we gauge the U (N ) × U (M ) group [97]. As a result, tuning our parameter µ to reach limits with higher-spin symmetries has a qualitative similarity with, in the context of the ABJ triality, dialing the ratio M N to reach a conjectured dual of N = 6 matrix extended Vasiliev higher-spin theory [153]. But there are also qualitative differences between our model and the case of the ABJ triality. Firstly, the ABJ theory consists of (anti-)bifundamental fields, while our model (2.26) is built from (anti-)fundamental fields of the two global symmetry groups. Therefore we expect a different mechanism of the emergence of the higher-spin fields. Secondly, as the ratio M N increases there are two phase transitions separating three different phases with different thermodynamic properties [153]. But in our model, at least for q = 2, as we increase µ the model moves from a higher-spin behavior to a chaotic SYK-like behavior and then back to a higher-spin behavior (although a different one comparing to where the model started).
There could be other parameters that we can turn on while maintaining the properties of the models we have discussed in this paper. Notice that this possibility goes along with our previous analysis based on the chaotic behavior of this model, where the fact that the Lyapunov exponent does not saturate the chaos bound [129] could be a hint that there should be other directions on the moduli space of 2d SYK-like models along which the model ets more and more chaotic. Gapped phase ? Figure 13: An illustration of the moduli space of our model. In the shaded region the model has SYK-like behavior. In the two limits, which are on the boundary of the shaded region, one observe emergent higher-spin symmetries. At q = 1, the interaction is quadratic, which can be regarded as a free "mass" term and the model is essentially free. Outside the shaded region the Fourier transform in (2.36) diverges. As suggested in [102] one probably has to turn on a negative mass counterterm to reach another fixed point. This is an indication that in this region the model might be gapped.

Tensor models
As in 1 dimension, the SYK-like model we have discussed in this paper has tensor analogues that do not involve any random coupling. The construction of the tensor models is similar to the 1d case [101] where we promote each multiplet to carry tensor indices. In this section we construct a tensor model that reproduces the physics of the model (2.26), which is a direct generalization of [60,101]. The action of the tensor model is where we have omitted all the tensor indices that are contracted in the unique way so that each term in the action is a singlet under the global symmetry (5.1). This model is the q = 2 analogue of (2.26), and it is easy to generalize to models with higher q. In the large-N, M limit, the model is dominated by the same set of the melonic diagrams as in the model (2.26). This can be proved in essentially the same as that in [101]. The 2-point functions in the infrared can be determined again by solving the set of the SD equations with the self energies the same as ( A 2d SYK model with N = (2, 2) supersymmetry In 2 dimension, the N = (2, 2) superfields are constructed with the help of two complex fermionic coordinates θ + and θ − withθ A bosonic chiral superfield is defined as where y ± = x ± − iθ ±θ± . A simple 2d SYK-like model with N = (2, 2) supersymmetry is [102,113] In component form the interaction is It is straightforward to solve the Schwinger-Dyson equation of the two point functions in the IR limit where G I c (z 1 , z 2 ) are the propagators in the infrared and Σ I c (z 1 , z 2 ) are the corresponding self-energies of the various fields A supersymmetric solution of these equations reads 10) and the other propagators are related by supersymmetry The existence of such solutions is nontrivial. Indeed one can write down a similar set of Schwinger-Dyson equations for various models without supersymmetry [102], but it is not clear if the theory actually flows to the assumed SYK-like IR fixed points. The presence of supersymmetry improves the UV behavior and the potential divergences are avoided. In practice, the supersymmetry allows a natural regularization and renormalization of the possible divergences in a naive direct solution.
For later purpose it is useful to recast the N = (2, 2) model (A.3) as a model with a smaller number of supersymmetry. In terms of the N = (0, 2) superfields where D − = ∂ ∂θ − + 2θ − ∂ z and the action can be written as The perturbed theory has the same action as in section 2.2, the only difference is that now we take the ratio µ = M/N to be µ = 1 + , where 1 is treated as a small parameter, and we expand the propagators of the deformed model as In frequency domain, the perturbation can be denoted as where indicates a Fourier transform. The self energies of the deformed model are, to the leading order in where δG I (p) = G I (p) − G I c (p) and δΣ I (p) = Σ I (p) − Σ I c (p). In components, the equations are Now we look for a solution of the above equations with N = (0, 2) supersymmetry, which means These relations simplify (B.16) to Now we can Fourier transform back to get Further using Multiplying by (Σ λ c ) −1 G ψ c and using the known results for the N = (2, 2) conformal propagators and the self energies, we get Then using Plugging in the known results for the N = (2, 2) conformal propagators, we get a trivial identity. Similarly, the equations (B.15) turns out to be a trivial identity as well. This means that the N = (0, 2) SUSY is compatible with the 4 equations and it further reduces those to two equations. To solve the rest 2 equations we consider the following ansatz In the later computation, the following Fourier transform Strictly speaking, the integral on the LHS is only convergent when 1 4 < a < 1. However, we do not worry to much about the 1 4 end since it is an IR divergence and can be regulated by not going very deep in the IR. We only makes sure that a < 1 is satisfied in the following computation.
Using (B.27) we immediately compute Similarly, we have In addition, we get where we have used With these Fourier transformed quantities it is straightforward to solve (B.13), (B.17) and the solution is Notice that the form of the correction is compatible with the infinitesimal form of the correction to the conformal 2-point function (1 + (n + δ log(|z 1 − z 2 | 2 )) + O( 2 )) . (B.38) where we have takenδ I = δ I to retain locality. Therefore, we can rewrite the above perturbative result into an equivalent form This is compatible with the solution (2.32), (2.33) when expanded around µ = 1.

C Fermionic operators in the model
In this appendix we compute some other 4-point functions in which fermionic operators propagate. From this computation we get the fermionic spectrum of the model (2.26) and observe the emergence of fermionic higher-spin operators in the two limits (4.1) and (4.8).

C.1 Theφψ and φψ sector
We first consider the φ iψi ψ jφj and φ iψi G jλj correlators. There are 6 contributing kernels The functions that diagonalize the above kernels are of the form which satisfy One can explicitly evaluate the integral and get The final eigenvalues are obtained from diagonalizing the following matrix The expressions of the eigenvalues are not very eluminating. Here we only give the dimensions of the higher-spin operators that is of most interest to us. This is obtained in a similar manner as what we did to find the behaviors of the bosonic higher-spin operators. Namely we find the lowest dimensions, as a function of µ, that set some eigenvalue to 1 at a given spin, then we check the behavior of the dimensions as µ approaches the two limits (4.1) and (4.8). Instead of give the explicit expression of the dimensions of the operators, we simply present the results in figure 14. From the plot, we indeed observe that there is a tower of half-interger spin operators become conserved fermionic higher-spin operators in the limit (4.1) at any q. One the other hand, we only observe the emergence of the tower of conserved operators in the limit (4.8) at q = 2. These behavior are identical to those observed in the bosonic operators, as we expected due to supersymmetry. In adddition, the right-moving supercharges is present at any value of µ, in accord with the fact that the model has N = (0, 2) supersymmetry at any µ. We call the tower of higher-spin operator running in these 4-point function in theφψ sector since they are expect to have the form   where ". . ." represents other terms with different distributions of the derivatives on the two fields; the correct combination is determined by requiring the operator to be a primary field. There is a set of conjugate kernels that gives identical eigenvalues. Hence there is a second tower of fermionic higher-spin operators appearing in the two limits (4.1) and (4.8) with identical behaviors as the parameters change. They are referred as in the φψ sector. They complete the right-moving N = 2 multiplets at each spin.
manner as above and the result is present in figure 15. From the plot, we indeed observe that there is no half-integer higher-spin operators in the limit (4.1) at any q. These behavior is compatible with the results from the bosonic operator spectrum since we do not expect conserved anti-holomorphic fermionic operators due to the absence of supersymmetry in the left-moving sector. The only conserved operator appears at µ = 1, s = 3 2 , which correspond to the left-moving supercharges emerging as the model develops an enhanced N = (2, 2) supersymmetry at µ = 1, as we expected.